REDUCING TOBACCO USE THROUGH TAXATION IN TRINIDAD AND TOBAGO: MODELLING THE LONG TERM HEALTH AND ECONOMIC IMPACT REDUCING TOBACCO USE THROUGH TAXATION IN TRINIDAD AND TOBAGO: MODELLING THE LONG- TERM HEALTH AND ECONOMIC IMPACT I CONTENTS Acknowledgments VII Abstract IX Introduction 1 Summary of Methodology 5 Methodology 5 Assumptions 5 Limitations 6 Full Methodology 9 Data Collection 9 The Microsimulation Model 11 Development of Scenarios 12 Results 17 Smoking Prevalence (Percentage) 17 Summary 18 Cases of Disease per Year 20 Cumulative Cases of Disease 20 Mortality 21 Mortality cases avoided 22 Direct Cumulative Costs Avoided 22 Discussion 25 References 28 Appendix A: Microsimulation model 35 Module One: Preductions of Smoking Prevalence Over Time 35 Multinominal Logistic Regression for Smoking Prevalence 35 Bayesian Interpretation 37 Estimation of Confidence Intervals 38 Module Two: Microsimulation Initalization – Birth, Disease, and Death Models 38 Population Models 39 Reducing Tobacco Use Through Taxation in Trinidad and Tobago: Modelling the Long-Term Health and Economic Impact The Risk Factor Model 42 Relative Risks 45 Modelling Diseases 45 Methods for Approximating Missing Disease Statistics 46 Approximating attributable cases 52 Potential Years of Life Lost 53 Premature Mortality Rate 53 Costs Module 53 Premature Mortality Costs 54 Propagation of errors equation 54 Software architecture 54 Aim of the Model 54 Summary of the Architecure of the Existing Model 55 Main C++ classes used by the model 56 Tperson C++ class 56 Tdisease C++ class 57 Tscenario C++ class 60 Appendix B: Cigarette Tax Scenarios Output, 2015-2017, Ukraine (Results from TaXSim Modelling) 62 References 64 LIST OF FIGURES Figure 1: Illustration of the microsimulation model 12 Figure 2: Male and female smoking prevalence by year for each scenario 17 Figure A1: Population pyramid, 2015, Ukraine 40 Figure A3: Ex-smoker relative risks as a function of time after smoking cessation 44 Figure A2: The model structure 56 Figure A3: Multistage disease architecture 58 IV // Contents LIST OF TABLES Table 1: Summary of total disease cases (epidemiological) and costs (economic) by parameter, year, and scenario, total population (values in parentheses are uncertainty values) 3 Table 2: Never, ex-smoker, and smoker prevalence (%) by age group and sex, 2011 9 Table 3: References for disease data 10 Table 4: TaXSiM Model results 15 Table 5: Smoking prevalence by year, sex and scenario (percentage) 17 Table 6: Summary table of total disease cases (epidemiological) and costs (economic) by parameter, year, and scenario, total population 19 Table 7: Cases per year, total population 20 Table 8: Cumulative cases for each disease by year, total population 20 Table 9: Cumulative cases avoided relative to scenario 0 for the total Trinidad and Tobago population by 2025 and 2035 21 Table 10: Mortality cases in the total population per year 21 Table 11: Mortality Cases avoided in the total population per year 22 Table 12: Direct cumulative healthcare costs avoided (million, TT$) 22 Table A1: Summary of the parameters representing the distribution component 40 Table A2: Parameter estimates for γ0 and η related to each disease 43 Table A3: Survival percentage for lung cancer 47 Table A4: C++ Tperson class 57 Table A5: C++ Tdisease class 59 Table A6: The C++ Tscenario class 61 V ACKNOWLEDGMENTS This report was prepared by a team led by Patricio V. Marquez, Lead Public Health Specialist, Health, Nutrition and Population Global Practice, World Bank Group. Team members include: Lise Retat, Senior Economic and Mathematical Modeller, UK Health Forum; Abbygail Jaccard, Chief Technology Officer, UK Health Forum; Laura Webber, Deputy CEO, UK Health Forum; Karl Theodore, Director, HEU, Centre for Health Economics, The University of the West Indies, St. Augustine, Trinidad; Althea La Foucade, Coordinator, HEU, Centre for Health Economics, The University of the West Indies, St. Augustine, Trinidad; Samuel Gabriel, Researcher, HEU, Centre for Health Economics, The University of the West Indies, St. Augustine, Trinidad; Christine Laptiste, Research Fellow, HEU, Centre for Health Economics, The University of the West Indies, St. Augustine, Trinidad. The comments and advice provided by the following peer reviewers were incorporated in the final version of this assessment: Sheila Dutta, Senior Health Specialist, World Bank Group; Santiago Herrera, Lead Economist, World Bank Group; Alberto Gonima, Consultant, World Bank Group. Washington, DC August 31, 2018 VII ABSTRACT Background: Tobacco is a major contributor to the rise in Non-Communicable Diseases (NCDs) and is often linked to the increase in cardiovascular and respiratory diseases and various forms of cancer. Trinidad and Tobago’s existing prevention and control interventions are in urgent need of strengthening if the country is to reduce its tobacco use. Tobacco taxation has been shown to be very effective. This study quantifies the impact of increasing tobacco tax in Trinidad and Tobago on the future burden of smoking-related diseases. Methods: The UK Health Forum microsimulation model (McPherson and others 2007) was used to simulate a virtual ‘Trinidad and Tobago’ population and quantify the impact of different tobacco taxation scenarios on the future burden of smoking-related disease. Results and conclusions: The results showed that the higher tax increase scenario yielded the most significant results. If tobacco tax is increased by 100% in each of the next three years it is estimated that 2,537 new cases of smoking-related disease will be avoided by 2035, saving TT$254.7million to the health system. These findings support the “go big and go fast” approach outlined in the World Bank report Tobacco Tax Reform – A Multisectoral Perspective: At the Crossroads of Health And Development (World Bank 2018). IX TRINIDAD AND TOBAGO’S EXISTI PREVENTION AND CONTROL INTERVENTIONS IN URGENT NEED STRENGTHENING IF THE COUNTRY TO REDUCE ITS TOBACCO USE. INTRODUCTION The rising prevalence of noncommunicable diseases (NCDs) in the Caribbean represents one of the region’s biggest challenges. If not confronted head-on, the epidemic threatens to rapidly reverse the substantial health and economic gains that have been realized across the Caribbean Community (CARICOM)1 in the last three decades. It is estimated that in the Region of the Americas, which includes CARICOM, 80 percent of all deaths and 77 percent of premature deaths among persons ages 30 to 70 can be attributed to NCDs. Tobacco is a major contributor to this rise in NCDs, and is often linked to the increase in cardiovascular and respiratory diseases and various forms of cancer. In the Region of the Americas, 14 percent of deaths among adults ages 30 years and under are linked to tobacco consumption, as are 16 percent of deaths from cardiovascular diseases, 52 percent of deaths from chronic respiratory diseases (PAHO 2016), and 25 percent of deaths from cancer. Associations between cigarette smoking and the risk of developing diabetes (Śliwińska-Mossoń and Milnerowicz 2017; Will and others 2001; Rimm and others 1995) and cerebrovascular diseases (Indira, Muralidhar, and Munisekhar 2014; Molgaard and others 1986) have also been found. In 2011, the STEPS2 NCD Risk Factor Survey in Trinidad and Tobago indicated that approximately 21 percent of the population smoked cigarettes, with an average daily usage of 11.5 cigarettes (PAHO 2012). To curb tobacco use and institute penalties for breaches of laws and regulations, tobacco control measures including the Tobacco Control Act of 2009 and the Tobacco Control Regulations of 2013 have been implemented. Further, the country signed the World Health Organization’s Framework Convention on Tobacco Control (WHO FCTC) in August 2003, ratifying it one year later. Tobacco Use and its Costs to Health and the Economy in Trinidad and Tobago According to the 2006 United Nations Common Country Assessment report for Trinidad and Tobago (Government of Trinidad and Tobago, Ministry of Health 2017), tobacco was linked to 7 percent of all NCD deaths, 9 percent of ischemic heart disease and 61 percent of lung cancer in 2004. Furthermore, in 2016, ischemic heart disease, diabetes and cerebrovascular disease were the top three causes of premature deaths (in terms of years of life lost) in the country (IHME 2015). 1 CARICOM members are: Antigua and Barbuda, Bahamas, Barbados, Belize, Dominica, Grenada, Guyana, Haiti, Jamaica, Montserrat, Saint Lucia, St Kitts and Nevis, St Vincent and the Grenadines, Suriname, Trinidad and Tobago. 2 This is the World Health Organization’s STEPwise approach to Surveillance – a simple, standardized method for collecting, analysing and disseminating data in WHO member countries. 1 Reducing Tobacco Use Through Taxation in Trinidad and Tobago: Modelling the Long-Term Health and Economic Impact The economic costs of NCDs are also substantial. A 2016 RTI International study (Government of Trinidad and Tobago, Ministry of Health 2017) in Trinidad and Tobago estimated that the economic burden of diabetes, cancer and hypertension was roughly TT$8.7 billion annually, or approximately 5 percent of GDP. Of that total, cancer, which has tobacco use as its most important risk factor (WHO 2018) and diabetes, cost the Government of Trinidad and Tobago TT$2 billion and TT$3.5 billion, respectively, each year. Taxation is Key to Strengthening Prevention and Control Trinidad and Tobago’s existing prevention and control interventions are in urgent need of strengthening if the country is to reduce its tobacco use. Tobacco taxation has been shown to be most effective way to do this (Blecher and others 2014). For instance, WHO-recommended best practices for tobacco taxation suggest that to substantially reduce consumption, excise taxes as a percentage of the final consumer price of tobacco products should be no less than 70 percent (WHO 2014a). However, in 2016 Trinidad and Tobago’s excise taxes share was substantially lower, at 14.7 percent3 (WHO 2017). Evidently, there is much work to do to achieve this goal. This report uses selected scenarios for increasing excise taxes on tobacco products in Trinidad and Tobago, with the aim of reducing smoking prevalence across the population. These scenarios provide inputs for modeling the long-term health and cost benefits to the population of proposed excise tax increases. The report provides evidence from the modeling exercise. Impacts are calculated relative to the status quo before the tax hike and are modeled beginning in 2017 for 2025 and 2035. A microsimulation model was employed to simulate the long-term impact of tobacco taxations on the future burden of a range of NCDs. Specifically, the disease outcomes quantified were for coronary heart disease (CHD), stroke, chronic obstructive pulmonary disease (COPD), and lung cancer. The microsimulation model has been deemed by the Organisation for Economic Cooperation and Development (OECD) as the most relevant method for NCD modeling based on risk-factor data (Oderkirk and others 2012). This report complements modeling work done to estimate the fiscal-revenue impact and expected reduction in consumption that might stem from proposed additional tobacco excise tax increases in Trinidad and Tobago. This work has been carried out by the World Bank, using a model based on the Tobacco Tax Simulation Model (TaXSiM) developed by WHO. Table 1 presents a summary of total disease cases (epidemiological) and costs (economic) avoided by parameter, year, and scenario, for the Trinidad and Tobago population. 3 For the most-sold brand. 2 // Introduction The model estimated that by 2035, the specified tax increase would result in the avoidance of 1,633 and 2,537 new cases of smoking-related disease for the two scenarios modelled. These reductions in disease will result in TT$2.09 million and TT$19.18 million in healthcare costs avoided for the two scenarios respectively. Consequently, the results showed that scenario two (the higher tax increase) yielded the most significant results, supporting the “go big and go fast” approach outlined in the World Bank report Tobacco Tax Reform – A Multisectoral Perspective: At the Crossroads of Health And Development (World Bank 2018). This figure is conservative because: (a) only a subset of smoking- related diseases has been included (for instance, diabetes has not been included in the model in this project); (b) indirect, social care and productivity costs have not been estimated due to a lack of input data. Finally, it is important to note that a non-statistically significant impact on premature deaths avoided was derived in the study. However, these results are based on preliminary analysis using a subset of tobacco-related diseases and limited availability of country- specific data inputs. Table 1: Summary of total disease cases (epidemiological) and costs (economic) by parameter, year, and scenario, total population (values in parentheses are uncertainty values) EPIDEMIOLOGICAL OUTPUTS YEAR BASELINE SCENARIO 1 SCENARIO 2 2025 88,645 [±95] 88,366 [±95] 88,194 [95] Cumulative cases 2035 210,146 [±135] 208,512 [±135] 207,609 [135] 2025 NA 279 [±134] 452 [±134] Cumulative cases avoided 2035 NA 1,633 [±191] 2,537 [±191] 2025 10,943 [±33] 10,863 [±33] 10,823 [±33] Cases per year 2035 14,090 [±33] 13,891 [±33] 13,785 [±33] ECONOMIC OUTPUTS 2025 NA 15.46[±24.1] 27.81[±24.1] Cumulative direct costs avoided (millions, TT$) 2035 NA 155.46[±44.41] 254.73[±44.39] 3 TRINIDAD AND TOBAGO’S EXISTI PREVENTION AND CONTROL INTERVENTIONS IN URGENT NEED STRENGTHENING IF THE COUNTRY TO REDUCE ITS TOBACCO USE. ING SUMMARY OF METHODOLOGY Methodology • The model simulates a virtual population of Trinidad and Tobago, based on latest population statistics.4 • Data on initial smoking prevalence by age and sex are extracted from the Trinidad and Tobago Chronic Non-Communicable Disease Risk Factor Survey (Pan American STEPS) • Scenarios take account of two different tax increases on cigarette prices, and the impact of these tax increases on smoking prevalence and subsequent disease burden. 5 ARE • Individual smokers included in the model have a specified smoking status, and a probability of contracting, dying from, or surviving a disease. • Future prevalence of smoking is calculated based on the numbers of smokers and non-smokers who are still alive in a particular year. D OF • Data for disease incidence and mortality are extracted from the Institute for Health Metrics and Evaluation, Global Burden of Disease database. • Relative risks of contracting diseases in smokers compared to never-smokers are extracted from DYNAMO-HIA. • A five-module microsimulation model is used to predict the future health and eco- G nomic impacts of tobacco taxes by 2025 and 2035. • The model quantifies the future impact on health and related costs of different levels of tax increase relative to a “no change” scenario. Assumptions • Smoking prevalence follows a static trend from 2011 smoking prevalence data. Y IS • A specified percentage of smokers who are affected by the tax increase move to the “ex-smoker” category in 2018, 2019, and 2020 in order to account for reductions in uptake due to price increases. • If a smoker quits as a result of the intervention, he/she becomes an ex-smoker for the rest of the simulation. 4 Ministry of Planning and Sustainable Development. 2012. Trinidad and Tobago 2011 Population and Housing Census Demographic Report. Central Statistical Office. 5 World Bank Group. 2018. Tobacco Taxation and Impact of Policy Reforms: Trinidad and Tobago 5 Reducing Tobacco Use Through Taxation in Trinidad and Tobago: Modelling the Long-Term Health and Economic Impact • Time since cessation is included in the model to account for changes in disease risk for an ex-smoker. • Smokers react quickly to tax changes so immediate effects are modelled in the year following the year of implementation of the tax rise. Limitations • The model does not take account of future changes in policy or technology. • No change in secondhand smoke exposure is modeled. • The baseline is static over time. • The simulation only includes four smoking-related diseases, so results are likely to underestimate the true effects. • No data on non-healthcare costs, for example lost productivity due to disease, were available. • No data were available to explore differences by social groups. • The simulation did not model a possible relapse in smoking among smokers who gave up as a result of tax-induced price increases. • No uncertainty analysis was conducted. 6 // Summary of Methodology 7 TRINIDAD AND TOBAGO’S EXISTI PREVENTION AND CONTROL INTERVENTIONS ARE IN URGENT NEED OF STRENGTHENING THE COUNTRY IS REDUCE ITS TOBA ING FULL METHODOLOGY Data Collection Smoking Prevalence Data Table 2 sets out the baseline prevalence data used (never, ex-smoker and smoker). The data are from 2011. Table 2: Never, ex-smoker, and smoker prevalence (%) by age group and sex, 2011 AGE SEX YEAR SAMPLE SIZE NEVER EX-SMOKER SMOKER 20–24 M 2011 116 68.5 8.6 22.9 25–29 M 2011 121.5 39.8 17.0 43.2 30–34 M 2011 121.5 39.8 17.0 43.2 35–39 M 2011 118.5 52.0 14.6 33.4 40–44 M 2011 118.5 52.0 14.6 33.4 45–49 M 2011 105.5 27.4 35.8 36.8 50–54 M 2011 105.5 27.4 35.8 36.8 55–59 M 2011 95 9.8 55.4 34.8 60–64 M 2011 95 9.8 55.4 34.8 20–24 F 2011 134 84.9 6.2 8.9 25–29 F 2011 125.5 69.2 16.5 14.3 30–34 F 2011 125.5 69.2 16.5 14.3 35–39 F 2011 150 79.2 13.5 7.3 G IF 40–44 F 2011 150 79.2 13.5 7.3 45–49 F 2011 180 78.6 13.5 7.9 50–54 F 2011 180 78.6 13.5 7.9 55–59 F 2011 165.5 74.4 19.4 6.2 60–64 F 2011 165.5 74.4 19.4 6.2 S TO Note: Children were assumed not to smoke. Disease Data The following smoking-related NCDs were modeled for this study: coronary heart disease (CHD), stroke, lung cancer, and chronic obstructive pulmonary disease (COPD). Incidence ACCO 9 Reducing Tobacco Use Through Taxation in Trinidad and Tobago: Modelling the Long-Term Health and Economic Impact and mortality data by age and sex were extracted from the Institute for Health Metrics and Evaluation, Global Burden of Disease, and the International Agency for Research on Cancer databases. Lung cancer data were grouped with trachea and bronchus data in the database, so these may have been overestimated for lung cancer only (IARC 2012). No survival data were available for these diseases in Trinidad and Tobago, therefore survival data was calculated from incidence and mortality using WHO DISMOD II equations (WHO 2014b). Relative risks for smokers compared to non-smokers were extracted from Dynamo-HIA for CHD (Song and others 2008; Baba and others 2006; Tolstrup and others 2014; Burns 2003; Cronin and others 2012; U.S. Department of Health and Human Services 2014), COPD (U.S. Department of Health and Human Services 2014; Prescott and others 1997; Johannessen and others 2005; Terzikhan and others 2016; Thun and others 2013), lung cancer (U.S. Department of Health and Human Services 2014; Thun and others 2013; Freedman and others 2008; Bae and others 2007), and stroke (Mannami and others 2004; Shinton and others 1989; Wannamethee and others 1995). As various cohort studies usually observed participants of different age groups, their estimates were compared and combined to cover the modeled population: Thus, relative risks for various age groups may derive from different studies. Ex-smokers’ relative risk was assumed to decrease post-cessation and was computed using a decay function method developed by Hoogenveen and others (Hoogenveen and others 2008). This function uses current smoker relative risk for each disease as the starting point, and then models the decline in relative risk of disease for an ex-smoker over time (see appendix A). Table 3: References for disease data INCIDENCE MORTALITY DIRECT HEALTHCARE COSTS Calculations based on data from The Cost Institute for Health Metrics Institute for Health Metrics and of Health Services in Trinidad and Tobago. CHD and Evaluation, Global Burden Evaluation, Global Burden of Disease 2013 HEU, Centre for Health Economics of Disease (unpublished) Calculations based on data from The Cost Institute for Health Metrics Institute for Health Metrics and of Health Services in Trinidad and Tobago. Stroke and Evaluation, Global Burden Evaluation, Global Burden of Disease 2013 HEU, Centre for Health Economics of Disease (unpublished) Calculations based on data from The Cost Institute for Health Metrics Institute for Health Metrics and of Health Services in Trinidad and Tobago. COPD and Evaluation, Global Burden Evaluation, Global Burden of Disease 2013 HEU, Centre for Health Economics of Disease (unpublished) Calculations based on data from The Cost International Agency Lung International Agency for Research on of Health Services in Trinidad and Tobago. for Research on Cancer cancer Cancer databases 2013 HEU, Centre for Health Economics databases (unpublished) 10 // Full Methodology Health Economic Data Calculation of Direct Healthcare Costs Disease cost estimations were conducted using estimates of the cost of disease and health services provided in the Cost of Health Services in Trinidad and Tobago report carried out for the Ministry of Health in 2013 (HEU, Centre for Health Economics, 2013). This report was used in conjunction with consultations with medical professionals in respect to the various inputs required for treatment of the four health conditions studied. The cost per case per year includes costs of: • diagnostics; • pharmaceuticals and or medical supplies; • rehabilitation, where applicable; and • inpatient and outpatient care. Diagnostic costs include the average costs of laboratory tests and imaging. Pharmaceutical costs were estimated using the most probable drugs to be prescribed for each of the diseases, together with information on the average frequency of use, length of use and the average price per drug. The cost of medical supplies/equipment was included where appropriate. The cost of inpatient care was calculated based on the average number of inpatient days per disease coupled with the average cost per inpatient day. Outpatient costs were estimated based on the average number of visits per disease condition multiplied by the average cost per outpatient visit. Where applicable, the cost of rehabilitation includes physiotherapy sessions, as well as other general rehabilitation techniques. Population Data To simulate the population of Trinidad and Tobago, the population by age and sex, births by mother’s age, and total fertility rate statistics were taken from the 2011 population prospects database. Total mortality rates were taken from the 2017 population prospects database. The Microsimulation Model The UK Health Forum (UKHF) microsimulation model was originally developed for the UK government’s Foresight inquiry (McPherson and others 2007; Wang and others 2011) and has been developed over the past decade to incorporate a number of additional interacting risk factors, including smoking (methods are described in greater detail in (UK Health Forum and CRUK 2016; UK Health Forum) and in appendix A). The model simulates a virtual population that reproduces the characteristics and behavior of a large sample of individuals (50 million). These characteristics (age, sex, smoker status) can evolve over the 11 Reducing Tobacco Use Through Taxation in Trinidad and Tobago: Modelling the Long-Term Health and Economic Impact life course based on known population statistics and risk factor data. Individuals can be born and die in the model, which is modular in nature (see figure 1). • Module 1 uses cross-sectional data on the prevalence of the risk factor – cigarette smoking in this case. For the current study, 2011 smoking prevalence data for Trinidad and Tobago were extrapolated forward to 2035. It was assumed that the proportions of the population within each smoking category as calculated in 2011 remained constant until 2035. • Module 2 is a microsimulation model which uses the prevalence of the risk factor over time, along with the specified data on the risks of developing diseases, to make projections of future disease burden. A wide range of different outputs is produced, including cumulative incidence. To the authors’ knowledge, no other studies have used a microsimulation model to quantify the future costs and health impacts of tobacco taxation policy scenarios in Trinidad and Tobago. Figure 1: Illustration of the microsimulation model Risk data Population data Disease data Health economic Intervention data scenarios Distribution programme Risk UKHF Microsimulation© programme Input Output data Software Output Source: UK Health Forum 2017. Development of Scenarios An initial modeling study was carried out by the World Bank Group (Marquez and others 2018) using a version of WHO’s TaXSiM model.6 Within this model, a scenario that reflects tobacco excise tax changes in 2017 was simulated to calculate the revenue impact as a result of this tax increase. 6 WHO tobacco tax simulation model (TaXSiM) http://who.int/tobacco/economics/taxsim/en/. 12 // Full Methodology The modified TaXSiM also calculated the percentage reduction in total cigarette consumption due to the suggested tax changes. These taxation changes result in non- smokers (predominantly young people) not initiating smoking; smokers quitting, and smokers reducing the number of cigarettes smoked. There was one baseline and two intervention scenarios: • Baseline: A baseline “static” trend This assumed that smoking prevalence stays constant at 2011 rates. • Scenario 1: The 2017 specific excise tax rate on cigarettes is increased by 50 percent in 2018 to TT$6.57 per 20 cigarettes; by 100 percent in 2019 (TT$13.14 per 20 cigarettes), and 100 percent in 2020 (TT$26.28 per 20 cigarettes). The consumption as a result of the previously stated tax applied to cigarettes is estimated to reduce by (only cessation included) a relative reduction of: • 1.95 percent in 2018 • 5.15 percent in 2019 and • 7.00 percent in 2020. • Scenario 2: The 2017 specific excise tax rate on cigarettes is increased by 150 percent in 2018 to TT$10.95 per 20 cigarettes; by 100 percent in 2019 to TT$21.90 per 20 cigarettes; and by 100 percent in 2020 to TT$43.80 per 20 cigarettes. As a result of the previously stated tax applied to cigarettes, consumption is estimated to reduce by (only cessation included) a relative reduction of: • 5.65 percent in 2018 • 6.55 percent in 2019, and • 7.95 percent in 2020. Assumptions for implementation of the scenarios in the UKHF microsimulation are as follows: • Several studies suggest that around 50 percent of the effect of price increases on overall cigarette consumption results from participation changes (Farelly and others 2001; Centers for Disease Control 1998). Therefore, it is assumed that 50 percent of smokers would quit and 50 percent would cut down their tobacco intake. An estimated 50 percent reduction in cigarette consumption was used as an estimate of the reduction in the total prevalence of smoking. While taxation that raises the real prices of tobacco might reduce the intensity of smoking, research suggests that people who cut down may actually inhale more, as measured by serum cotinine 13 Reducing Tobacco Use Through Taxation in Trinidad and Tobago: Modelling the Long-Term Health and Economic Impact levels (Fidler and others 2011). Further, the WHO target is focused on a total reduc- tion in smoking prevalence. Therefore, modeling proceeded with a focus on current smoking prevalence, as opposed to the number of cigarettes smoked. • Future uptake of smoking was not included in the current scenarios. • While these average changes were not the same for each group, and usually people under 30 years of age initiate smoking, the model did not take age differences into account, and the relative decline in percentages of current smokers was applied to all age groups. • A baseline “static” trend was included. This assumed that smoking prevalence remains constant at 2011 rates. The tax increase scenario was compared to this baseline. • The tax increase scenarios represent the tax change adopted in 2018, 2019 and 2020. • The scenarios are based on Monte Carlo7 simulations (individuals were sampled from the population and simulated over time). • The specified percentage of smokers who are affected by the tax increase move to the ex-smoker category in 2018, 2019 and 2020. • If an individual’s smoking status is changed by the scenario, their smoking status will remain fixed for the entire simulation. • An immediate reduction in smoking prevalence due to the tax increases in 2018, 2019 and 2020 was assumed. We learned via personal communication with Pro- fessor Joy Townsend that there are different views on the temporal impact of a tax: econometricians follow Becker’s model, assuming that, as tobacco is very addictive, the reaction to price increases is slow and greater in the long run. Becker, therefore, uses a lagged variable of y (t-1) (Becker and Murphy 1998). Townsend and Atkinson take the opposite view (Atkinson and Skegg 1973) – that smokers tend to react quickly to a price change. A model similar to theirs was used, with an immediate effect and then a linear trend, and in line with the modified TaXSiM model outputs (table 4). 7 A Monte Carlo simulation is a mathematical technique which uses stochastic processes to accurately reproduce a system – in this instance, a realistic population. 14 // Full Methodology Table 4: TaXSiM Model results SPECIFIC EXCISE TAX RATE 2018 INCREASED BY 100% IN 2020 TO SPECIFIC EXCISE TAX RATE 2019 SPECIFIC EXCISE TAX RATE 2017 INCREASED BY 100% IN 2019 TO INCREASED BY 50% TO TT$6.57 SPECIFIC TAX INCREASED TO TT$43.80 PER 20 CIGARETTES TT$26.28 PER 20 CIGARETTES 2018 INCREASED BY 100% TO TT$10.95 PER 20 CIGARETTES 2019 INCREASED BY 100% TO TT$21.90 PER 20 CIGARETTES 2017 INCREASED BY 150% TO TT$13.14 PER 20 CIGARETTES TT$4.38 PER 20 CIGARETTES SPECIFIC EXCISE TAX RATE SPECIFIC EXCISE TAX RATE SPECIFIC EXCISE TAX RATE BASELINE 2017: SIMPLE PER 20 CIGARETTES (SCENARIO 2) (SCENARIO 2) (SCENARIO 2) (SCENARIO 1) (SCENARIO 1) (SCENARIO 1) Total cigarettes taxed 1.16 1.12 1.00 0.86 1.03 0.90 0.76 (billion pieces) Average cigarette price 23.45 26.48 36.33 56.49 32.82 49.73 83.37 per pack (TT$) Average excise tax burden (excise tax as 18.7 24.8 36.2 46.5 33.4 44 52.5 percentage of price)a Average excise tax (per 219 328.5 657 1,314 547.5 1,095 2,190 1,000 pieces) (TT$) Average tax burden (total tax – import 29.9 36.0 47.4 51.7 44.6 55.2 63.7 excise and VAT as percentage of price Percentage change in -1.9 -3.9 -10.3 -14.0 -11.3 -13.1 -15.9 total cigarette taxed Source: WBG Staff estimates. Note: a Based on assumptions for elasticity price and elasticity income for high-income countries (Marquez and others 2018). 15 TRINIDAD AND TOBAGO’S EXISTI PREVENTION AND CONTROL INTERVENTIONS ARE IN URGENT NEED OF STRENGTHENING IF THE COUNTRY IS TO REDUCE ITS ING RESULTS Smoking Prevalence (Percentage) Table 5 shows smoking prevalence for males, females, and both males and females combined for the baseline scenario, and scenarios 1 and 2. By 2035, smoking prevalence decreases to 12.74 and 10.64 for scenario 1 and 2 respectively. More specifically, smoking prevalence in men reduces to 20.80 and 17.33 for scenarios 1 and 2 respectively, with the smoking prevalence among women falling to 4.80 and 4.05 respectively. Table 5: Smoking prevalence by year, sex and scenario (percentage) SCENARIO 0 (BASELINE) SCENARIO 1 SCENARIO 2 YEAR MALE FEMALE TOTAL MALE FEMALE TOTAL MALE FEMALE TOTAL 2017 25.81 6.48 16.18 24.39 6.12 15.28 23.44 5.87 14.69 2020 26.26 6.52 16.41 23.94 5.95 14.96 22.52 5.59 14.07 2025 26.58 6.44 16.51 22.83 5.54 14.18 20.67 5.03 12.85 2030 26.90 6.33 16.57 21.71 5.13 13.39 18.88 4.50 11.66 2035 27.46 6.27 16.78 20.80 4.80 12.74 17.33 4.05 10.64 Figure 2: Male and female smoking prevalence by year for each scenario FIGURE 2A. MALE 30.00 Male smoking prevalence (%) 25.00 G 20.00 15.00 10.00 5.00 Y 0.00 2020 2025 2030 2035 2015 2040 Scenario 0 (baseline) Scenario 1 Scenario 2 FIGURE 2B. FEMALE 7.00 Female smoking prevalence (%) S 6.00 5.00 17 4.00 3.00 2.00 FIGURE 2A. MALE 30.00 Male smoking prevalence (%) 25.00 20.00 Reducing Tobacco Use Through Taxation in Trinidad and Tobago: Modelling the Long-Term Health and Economic Impact 15.00 10.00 5.00 0.00 2020 2025 2030 2035 2015 2040 Scenario 0 (baseline) Figure 2: Male and female smoking prevalence Scenario 2 Scenario 1 by year for each scenario, Cont. FIGURE 2B. FEMALE 7.00 Female smoking prevalence (%) 6.00 5.00 4.00 3.00 2.00 1.00 0.00 2020 2025 2030 2035 2015 2040 Scenario 0 (baseline) Scenario 1 Scenario 2 Summary There are a number of outputs from the microsimulation. Epidemiological Indicators Results from the microsimulation are presented as rates per the Trinidad and Tobago population, 2011. Incidence The total number of new cases of disease, divided by the total number of susceptible people in a given year presented as a rate per population. Cumulative incidence rate per year, per Trinidad and Tobago population To calculate the cumulative incidence rate per year, the total number of new cases of disease was divided by the total number of susceptible people in a given year and accumulated over a specified period of the simulation from the year 2016. Therefore, the cumulative number of cases represents a sum of all of the cases from the start of the simulation. Cumulative incidence avoided per Trinidad and Tobago population over the simulation period The total number of cases of disease avoided or gained as compared to baseline (i.e., scenario 0) was estimated. A positive value represents the number of cases avoided, whereas a negative value represents the number of cases gained. Mortality per Trinidad and Tobago population over the simulation period The number of deaths from a disease was estimated. 18 // Results Mortality cases avoided per Trinidad and Tobago population over the simulation period The number of deaths from a disease avoided or gained as compared to baseline (i.e., scenario 0) was estimated. Economic outputs Direct costs avoided These are cumulative direct costs across the period of the simulation. The result for 2020 represents the cumulative costs avoided for the period 2016 to 2020. These costs are scaled to the total population of Trinidad and Tobago. Table 6 presents a summary table of total disease cases (epidemiological) and costs (economic) by parameter, year, and scenario as rates per Trinidad and Tobago population. The model estimated that by 2035 the specified tax increase would result in the avoidance of 1,633 and 2,537 new cases of smoking-related diseases respectively for the two scenarios modelled. These reductions in disease will result in the TT$2.09 million and TT$19.18 million in healthcare costs respectively avoided for the two scenarios. Table 6: Summary table of total disease cases (epidemiological) and costs (economic) by parameter, year, and scenario, total population SCENARIO 0– EPIDEMIOLOGICAL OUTPUTS YEAR SCENARIO 1 SCENARIO 2 BASELINE 2025 88,645 [±95] 88,366 [±95] 88,194 [95] Cumulative cases 2035 210,146 [±135] 208,512 [±135] 207,609 [135] 2025 NA 279 [±134] 452 [±134] Cumulative cases avoided 2035 NA 1,633 [±191] 2,537 [±191] 2025 10,943 [±33] 10,863 [±33] 10,823 [±33] Cases per year 2035 14,090 [±33] 13,891 [±33] 13,785 [±33] ECONOMIC OUTPUTS 2025 NA 15.46[±24.1] 27.81[±24.1] Cumulative direct costs avoided (millions, TT$) 2035 NA 155.46[±44.41] 254.73[±44.39] 19 Reducing Tobacco Use Through Taxation in Trinidad and Tobago: Modelling the Long-Term Health and Economic Impact Cases of Disease per Year Table 7 presents the annual cases for each disease by year. Table 7: Cases per year, total population LUNG YEAR CHD COPD STROKE TOTAL CANCER 2025 Scenario 0 5,312 [±27] 2,829 [±13] 332 [±0] 2,470 [±13] 10,943 [±33] Scenario 1 5,299 [±27] 2,802 [±13] 332 [±0] 2,430 [±13] 10,863 [±33] Scenario 2 5,299 [27] 2,789 [13] 332 [0] 2,404 [13] 10,823 [33] 2035 Scenario 0 7,092 [±27] 3,240 [±13] 345 [±0] 3,413 [±13] 14,090 [±33] Scenario 1 7,065 [±27] 3,187 [±13] 332 [±0] 3,307 [±13] 13,891 [±33] Scenario 2 7,052 [27] 3,147 [13] 332 [0] 3,254 [13] 13,785 [33] Cumulative Cases of Disease Table 8 presents the cumulative cases for each disease by year, and table 9 presents the cumulative cases avoided. By 2035, the cumulative cases avoided for scenario 1 compared to scenario 0 are 239, 452, 120,823 respectively for CHD, COPD, lung cancer and stroke. Similarly, by 2035, the cumulative cases avoided for scenario 1 compared to scenario 0 are 359, 691, 199, and 1,288 respectively for CHD, COPD, lung cancer and stroke. Table 8: Cumulative cases for each disease by year, total population LUNG YEAR CHD COPD STROKE TOTAL CANCER 2025 Scenario 0 43,214 [±66] 23,200 [±53] 2,789 [±13] 19,442 [±40] 88,645 [±95] Scenario 1 43,174 [±66] 23,134 [±53] 2,762 [±13] 19,296 [±40] 88,366 [±95] Scenario 2 43,134 [66] 23,094 [53] 2,749 [13] 19,216 [40] 88,194 [95] 2035 Scenario 0 103,253 [±93] 52,709 [±66] 6,109 [±27] 48,074 [±66] 210,146 [±135] Scenario 1 103,014 [±93] 52,258 [±66] 5,989 [±27] 47,251 [±66] 208,512 [±135] Scenario 2 102,895 [±93] 52,019 [±66] 5,910 [±27] 46,786 [±66] 207,609 [±135] 20 // Results Table 9: Cumulative cases avoided relative to scenario 0 for the total Trinidad and Tobago population by 2025 and 2035 LUNG SCENARIO 1 CHD COPD STROKE TOTAL CANCER 2025 40 [±93] 66 [±80] 27 [±13] 146 [±53] 279 [±134] 2035 239 [±133] 452 [±93] 120 [±40] 823 [±93] 1,634 [±191] LUNG SCENARIO 2 CHD COPD STROKE TOTAL CANCER 2025 80 [±93] 106 [±80] 40 [±13] 226 [±53] 452 [±134] 2035 359 [±133] 691 [±93] 199 [±40] 1,288 [±93] 2,537 [±191] Mortality Table 10 and table 11 present the mortality cases for scenarios 1 and 2, relative to scenario 0. Note that it was not possible to derive premature mortality costs due to the non- significance of the results (this is explained in the discussion section). Relative to scenario 0, by 2035, 53 and 93 mortality cases are avoided for scenarios 1 and 2 respectively. Table 10: Mortality cases in the total population per year YEARS SCENARIOS TOTALS 2025 Scenario 0 9,894 [±53] Scenario 1 9,867 [±53] Scenario 2 9,880 [±53] 2035 Scenario 0 11,872 [±66] Scenario 1 11,819 [±66] Scenario 2 11,780 [±53] 21 Reducing Tobacco Use Through Taxation in Trinidad and Tobago: Modelling the Long-Term Health and Economic Impact Mortality cases avoided Table 11: Mortality Cases avoided in the total population per year YEARS SCENARIOS TOTALS 2025 Scenario 1 – Scenario 0 27 [±53.12] 2025 Scenario 2 – Scenario 0 13 [±53.12] 2035 Scenario 1 – Scenario 0 53 [±66.4] 2035 Scenario 2 – Scenario 0 93 [±66.4] Direct Cumulative Costs Avoided Table 12 presents the cumulative direct healthcare costs avoided for scenarios 1 and 2, relative to scenario 0. By 2035, scenario 1 results in costs avoided of TT$155 million compared to scenario 0; similarly, by 2035, scenario 2 results in TT$255 million in direct costs avoided compared to scenario 0. Table 12: Direct cumulative healthcare costs avoided (million, TT$) LUNG YEAR CHD COPD STROKE TOTAL CANCER 2025 Scenario 1 relative to Scenario 0 -0.97 [±21.15] 0.78 [±7.29] 0.82 [±1.6] 14.82 [±8.83] 15.46 [±24.1] 2025 Scenario 2 relative to Scenario 0 0.2 [±21.15] 3.52 [±7.28] 2.71 [±1.59] 21.38 [±8.83] 27.81 [±24.1] 2035 Scenario 1 relative to Scenario 0 24.01 [±38.75] 25.09 [±13.52] 4.3 [±2.93] 102.07 [±16.69] 155.46 [±44.41] 8.86 254.73 2035 Scenario 2 relative to Scenario 0 49.05 [±38.75] 40.17 [±13.51] 156.65 [±16.67] [±2.92] [±44.39] 22 // Results 23 TRINIDAD AND TOBAGO’S EXISTI PREVENTION AND CONTROL INTERVENTIONS IN URGENT NEED STRENGTHENING THE COUNTRY IS REDUCE ITS TOBA USE. TOBACCO ING DISCUSSION This study explored the impact of two tobacco tax increase scenarios in Trinidad and Tobago on the future burden of four smoking-related diseases up to 2035. The results showed that small changes in smoking prevalence in one year can have relatively large impacts in terms of disease into the future. The results showed that scenario 2 (the higher tax increase) yielded more significant results, supporting the “go big and go fast” approach outlined in World Bank report Tobacco Tax Reform – A Multisectoral Perspective: At the Crossroads of Health And Development (World Bank 2018). This publication suggested that “tax strategies should focus on health gains first, then on fiscal benefits. This means going for big tobacco excise tax rate increases starting early in the process.” ARE The study included just four smoking-related diseases (CHD, COPD, stroke, lung cancer). However, we know that smoking is responsible for many more diseases, and harms almost every organ in the body (Centers for Disease Control 2016). Therefore, we are likely to see much wider epidemiological benefits than those observed here. Future work could update this study by including additional smoking-related diseases that would also increase D OF the impact of the scenarios on epidemiologic outputs such as cumulative incidence, incidence, and mortality. The relatively small number of premature deaths avoided can be explained by a Danish study (with 15 years follow-up of a large cohort of smokers) that found no evidence that heavy smokers who reduced their number of cigarettes had a lower risk of death from CHD or from all causes (Gotfredsen and others 2003). G IF It was not possible to derive premature mortality costs due to the non-significance of the results. There are three possible explanations for this: first, the threshold age of premature mortality was assumed to be 65 years, which is very close to life expectancy in Trinidad and Tobago. Second, the number of mortality cases avoided (by diseases studied) is small (see table 10). Consequently, given that premature mortality is a subset of mortality, S TO premature mortality costs would be expected to be small and not significantly different. Third, the relatively small change in smoking prevalence due to the intervention and the relatively short time span of the simulation will play a part. While the microsimulation method is advantageous in NCD modeling, a key disadvantage is that the model is data intensive and required detailed and up-to-date data. ACCO Unfortunately, it was not possible to include data on indirect costs (such as productivity losses) because the data by disease were not available (as is the case in many countries). However, we know from studies of total costs (Rezaei and others 2016) in Trinidad and Tobago that the relative non-health cost is colossal due to premature deaths and lost productivity. For example, the American Cancer Society’s Tobacco Atlas indicates that the 25 Reducing Tobacco Use Through Taxation in Trinidad and Tobago: Modelling the Long-Term Health and Economic Impact total direct and indirect cost of smoking in Trinidad and Tobago totals TT$1,858 million (US$275million) (WHO 2002). One systematic review estimated the direct costs of smoking to equal around 1.5–6.8 percent of national health system expenditures and 0.22–0.88 percent of GDP in the country studied (Rezaei and others 2016). In a study in Ukraine, using the same model, the premature mortality costs avoided of increasing tobacco tax was estimated at Hrv 16.5 billion (US$695 million) (Webber and others 2017), and in the UK, increasing the tobacco duty escalator to 5 percent (from an annual tobacco tax increase of 2%) would avoid £192 million in indirect costs by 2035 alone (Knuchel-Takano and others 2017). Therefore, wider societal costs such as losses in productivity are likely to be higher than direct costs, making a stronger case for the implementation of regular tax hikes for tobacco control (Action on Smoking and Health 2015). In fact, whereas the indirect cost of tobacco use has not been assessed in Trinidad and Tobago, in 2016 RTI International (RTI 2016) estimated annual indirect costs (in terms of productivity losses) for smoking-related cancers and diabetes to be TT$1.18 billion and TT$2.32 billion respectively. According to the study, the estimated costs represent 90 percent and 58 percent, respectively of the total economic burden of cancer and diabetes. Furthermore, it was projected that sustained prevention and control efforts can result in significant savings in terms of productivity losses. These costs are hardly comparable to most other studies, since they only include productivity losses while excluding other aspects such as morbidity costs and costs of premature retirement. Nevertheless, as noted by Rezaei et al. (Rezaei and others 2016), indirect costs exceeding direct costs is not an uncommon occurrence. Of the 14 studies reviewed Rezaei et al., seven reported substantially higher indirect costs, ranging from 53.3 percent to 81 percent of the total costs of smoking. If indirect cost data by disease becomes available, then the model can once again easily be updated in the future. Another data limitation was the lack of trend data on smoking prevalence. Therefore, only a static trend could be included. This may overestimate the impacts if smoking prevalence is actually falling, or underestimate the impact if smoking prevalence is actually increasing. We know that social groups react differently to tax increases (Krasovsky 2013). Due to small sample sizes, it was not possible to model the long-term health impacts on different social groups within the microsimulation. One specific limitation of any predictive model is that it does not take account of major future changes in circumstances, such as the behavior of the tobacco industry, or the introduction of new drugs or technologies. In theory, their effects can be estimated by altering parameters in the model, but these will significantly increase the degrees of 26 // Discussion uncertainty. However, they could be simulated as different scenarios in the future relative to a “no change” scenario. At present, the model does not take account of multi-morbidity and the joint effect of several risk factors on disease occurrence and related mortality. However, individuals can get more than one smoking-related disease in their lifetime. Future work could expand the scope of the model to take account of technological and economic changes and their potential effects, and also to model the clustering of risk factors and diseases in the same individuals. The model did not take account of passive smoking/secondhand smoke. Understanding the combined risk of smoking and passive smoking on later disease outcomes will enable us to model the combined impact of these risk factors on later disease outcomes. It was beyond the scope of this study, given the time constraints, to carry out an in-depth uncertainty and sensitivity analysis. We are aware that this is good practice; however, there is a lack of validated datasets with which to compare our outputs. Furthermore, the microsimulation is complex, relative to spreadsheet models, for example. It involves many thousands of calculations which are completed during the simulation of 50 million individuals. 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The Appendix A: Microsimulation model Appendix A: Microsimulation model second module performs the microsimulation of a virtual population, generated with model The microsimulation consists of two modules. The first module calculates the predictions of risk demographic characteristics matching those of the observed data. The health trajectory factor trends over time based on data from microsimulation The The rolling consists cross-sectional two ofthe modules. The studies. The second first module module calculates the p ofThe microsimulation each consists the of individual from two modules. population first module is simulated over calculates time allowing predictions them to of risk contract, performs the microsimulation factor trends of a virtual population, over time studies. based on generated data with from module demographic characteristics rolling cross-sectional studies. The s factor ules. The first module calculates the predictions trends over time based on data from of risk rolling cross-sectional The second survive or die from a set matching those of of diseases orthe observed injuries performs data. related the The to health the trajectory analysed microsimulation of a risk factors. virtual of each individual Thegenerated population, from thewithpopulation demograp performs m rolling cross-sectional studies. The second module the microsimulation of a virtual population, generated with demographic characteristics ARE detailed description simulated of the two over time modules allowing matching is them those presented to of contract, the below. survive observed data.or die The from health a set of diseases trajectory of or each injuries individual fr matching population, generated with demographic characteristics those of the observed data. The health trajectory of each individual from the population is related to the analysed risk factors. simulated over detailed them The allowing time description to of the survive contract, two modules or die is presented from a set of belo dise simulated health trajectory of each individual from the population isover time allowing them to contract, survive or die from a set of diseases or injuries related related to the analysed Module One: Predictions of Smoking Prevalence Over Time risk factors. The detailed of therisk to the analysed description two factors. modules The detailed below. of the two module description Module tract, survive or die from a set of diseases or injuries One: Predictions Appendix A: Microsimulation model of Smoking Prevalence OverisTime presented etailed description of the two modules is presented below. For the risk factor (RF), let N be the number of categories for a given risk factor, e.g. N = 3 for Module One: Predictions of Smoking Prevalence Over Time Module One: Predictions of Smoking Prevalence Over Time For the risk factor (RF), let N be smoking. the Let 1, 2, …, N of ! =number categories number for a given these categories andrisk factor, #$ (&) e.g. denote the prevalence of the RF tha oking Prevalence Over Time For the risk factor (RF), let N be the number of the For risk factor categories for (RF), N be let risk a given the number factor, e.g. N =of3categories for for a given risk factor, The microsimulation Nsmoking. = 3 for smoking. of two consists Letcorresponds k= modules. to 1, 2, …,The the N first module category number smoking. ! at these Letcalculates time the t. We categories predictions estimate and # of using risk $ (&) denotemultinomial logistic regression Let ! = 1, 2, …, N number these categories er of categories for a given risk factor, e.g. N = 3 for and! 1, 2, #= (&) …, denoteN number the prevalence $ these categories of the RF and that#$ (&) denote the preva factor trends over time based on with rolling data from cross-sectional studies. the second asThe module themodel prevalence of RF category ! outcome, and time #$ (&) explanatory t as a single variabl D OF categories and #$ (&) denote the prevalencethe of prevalence corresponds the of category RF that to the ! at corresponds RF that time t. We corresponds to the estimate #$to the (&) category category using k at at time !time multinomial t. We t. We logistic estimate estimate regression using multinomial lo performs the microsimulation ofFor a virtual !<) population, , we have generated model with with demographic prevalence of RF characteristics category ! as the outcome, and time t as a single e model We estimate #$ (&) using multinomial logisticusing with regression prevalence multinomial of RF logistic category regression! as the outcome, model with and time prevalence t as a single of RF explanatory category k isvariable. as the matching those of the observed data. The health trajectory For ! < of ) ,each we individual from have the population For ! < s the outcome, and time t as a single explanatory variable.) , we have simulated over time outcome, allowing and t as a time them to contract, single survive or die explanatory from a set variable. For of ) ö =or p (,twe æk 0},by 1, …extrapolated extrapolated both is to usedto to cross-sectional, sex forecast manufacture forecast S= the{male, distribution the RFtime-dependent, female} distribution trendsand of eachage offor each group individual RF category RF discrete A= category {0-9, members distributions in the 10-19, in the of future. 6 = {# ..., For the future. 70-79, population. each For $ (&) 80+}. |! = The sex-and-age- fitted trends are group completion 1, … ); stratum, & > of 0} the the group set simulation. ,extrapolated is used to cross-sectional, of stratum,manufacture to the set forecast ofRF the time-dependent, cross-sectional, trends for individual distribution of discrete time-dependent, each members RF distributions category ofdiscrete the in the 6 = each {#$ ( distributions population. future. sex-and-age- & For )|! = each 6 = {#$ (&)|! = group stratum, the setsetof of cross-sectional, time-dependent, discrete distributions 6 6= {# (& )|! sex-and-age- = 1, … );group & > 0} stratum, ,group is … 1, used ); the to & > stratum, manufacture 0} , iscross-sectional, used toRF trends manufacture the set of cross-sectional, time-dependent, for individual RF trends members time-dependent,fordiscrete individual ofdistributions the discrete population. members of distributions = $ the{# $ (&)|! = population. 6 = {#$ (&)|! = 1, …1, ); … ); & > & 0} > ,0}is, used is used to tomanufacture manufacture RF RFtrends trends forfor individual individual members members of the of the population. population. 1, … ); & > 0}, is used to manufacture RF trends for individual members of the population. 8 The probability of the sex of a child can be made time dependent. 8 The probability of the sex of a child can be made time dependent. 8 The probability 8of the The sex of a child probability cansex of the be of made time a child dependent. can be made time dependent. The 8 8 probability The probabilityof the sexsex of the of a ofchild can a child be be can made time made dependent. time dependent. 33 8 The probability of the sex of a child can be made time dependent. 33 33 33 33 33 33 10 The probability of the sex of a child can be made time dependent. 41 he probability pmale=1-pfemale. In the baseline model this is taken to be the probability Nm/(Nm+Nf). The Population editor’ menu item Population Editor\Tools\Births\show random birthList creates an ditor\Tools\Births\show random birthList creates an instance of the TPopulation class and uses it to generate and list a (selectable) sample of mothers generate and list a (selectable) sample of mothersand the years in which they give birth. Reducing Tobacco Use Through Taxation in Trinidad and Tobago: Modelling the Long-Term Health and Economic Impact Deaths from Modelled Diseases The simulation models any number of specified diseases – some of which may be fatal. In the start d diseases – some of which may be fatal. In the start year the simulation’s death model uses the diseases’ own mortality statistics to adjust the eases’ own mortality statistics to adjust the probabilities of death by age and gender. In the start year the net effect is to maintain the same e start year the net effect is to maintain the same probability of death by age and gender as before; in subsequent years, however, the rates at which re; in subsequent years, however, the rates at which people die from modelled diseases will change as modelled risk factors change. The population as modelled risk factors change. The population dynamics sketched above will be only an approximation to the simulated population’s dynamics. The The Risk ximation to the simulated Factor Model population’s dynamics. The latter will be known only on completion of the simulation. simulation. The distribution of risk factors (RF) in the population is estimated using regression analysis The Risk Factor Model stratified by both sex S = {male, The distribution female} and of risk agefactors group (RF) A= the population in{0-9, is estimated 10-19, ..., 70-79, 80+}. using The regression analysis stratified ation is estimated using regression analysis stratified by both sex S = {male, female} and age group A = {0-9, 10-19, ..., 70-79, 80+}. The fitted trends are fitted trends are extrapolated to forecast the distribution of each RF category in the future. = {0-9, 10-19, ..., 70-79, 80+}. The fitted trends areextrapolated to forecast the distribution of each RF category in the future. For each sex-and-age- h RF category in the For each future. sex-and-age-group For stratum, each sex-and-age- group thethe stratum, setset of of cross-sectional, cross-sectional, time-dependent, time-dependent, discretediscrete distributions 6 = {#$ (&)|! = dependent, discrete distributions D={p_k distributions 6 = {#$ ((t)|k=1,…N; &)|! = 1, … ); t>0}, isused & > 0}, is used toto manufacture manufacture RF trends RF trends for individual for individual members of the population. ds for individual members members ofpopulation. of the the population. We model We model different risk factors, different risk some of factors, some which are of which continuous (such (such as are continuous BMI) and as BMI) some are and some are model different Wecategorical categorical risk (smoking (smoking factors, status). status). some of which are continuous (such as BMI) and some are 8 The probability of the sex of a child can be made time dependent. me dependent. categorical (smoking status). Categorical Risk Factors Categorical Risk Factors We model different risk factors, some of which are continuous (such as BMI) and some are Smoking Categorical Risk Factors Smoking is is the the categorical categorical categorical factor. risk factor. risk (smoking Each individual Each status). in individual in the the population population may belong to may belong one of to one of the the 33 three possible smoking categories {never smoked, ex-smoker, smoker} with their probabilities { three possible smoking categories 33 { never smoked , ex-smoker , smoker } with their probabilities {p0, p0 ,p p11, , Smoking is the categorical risk factor. Each individual in the population may belong to Categorical Risk Factors p p2}. These states are updated on receipt of the information that the person is either a smoker or a 2 }. These states are updated on receipt of the information that the person is either a smoker or a one of the three Smoking possible is the categorical risk factor. {never Each individual smoked, in the ex-smoker, population smoker may belong to one of the non-smoker. non-smoker. They They will be will be aasmoking never-smoker never-smoker categories or an ex-smoker or an ex-smoker depending depending on their on their original original } with state state their (an (an ex- ex- three possible smoking categories {never smoked, ex-smoker, smoker} with their probabilities {p0, p1, smoker can smoker probabilitiescan { p0, p1become never never a never-smoker). a , p2}. These become states are updated on receipt of the information that the never-smoker). p2}. These states are updated on receipt of the information that the person is either a smoker or a person is eitherset complete The complete The a smoker set of or a non-smoker. non-smoker. of longitudinal longitudinal They will smoking smoking beThey trajectories trajectories will and theaprobabilities never-smoker andbe a never-smoker the or an ex-smoker probabilities or depending of their their of an ex-smoker happening happening onare their original state (an ex- are generatedon depending generated for their for the smoker the simulation simulation can years original years never state byby (an become allowing ex-smoker allowing a all never-smoker). all possible possible transitions can never become transitions between between smoking categories: categories: a never-smoker). smoking The { complete {never never set} smoked smoked }of ®longitudinal ® {never {never smokedsmoking smoked , smoker , smokertrajectories } } and the probabilities of their happening are generated for {ex-smoker ex-smoker} the }®®{ simulation ex-smoker, {ex-smoker years , smokerby smoker} } allowing all possible transitions between smoking categories: { {smoker { smoker}}® ®{ ex-smoker, {ex-smoker , smoker smoker} } {never smoked} ® {never smoked, smoker} When the probability of {ex-smoker When the probability of being a smoker is p the } ® {ex-smoker allowed transitions , smoker} in the state are summarized When the probability ofbeing beinga smoker is pis a smoker the p allowed transitions the allowed are summarized transitions in the state are summarized in the update equation: update equation: { smoker } ® { ex-smoker, smoker} state update equation: When the probability '' of being a smoker is p the allowed transitions are summarized in the state 0ù é p0 update equation: é1 - p 0 0 ù é p0 0ù ê úê '' ú ê ú ê ú ê p1 1ú = ê 0 1 - p 1 - p p1 (0.14) (0.0) (0.0) úê 1ú ê p2 '' ú ë p p é p0 p p ë 2û ê ù û ú ê ú ' é1ë -2 p 2û 0 0 ù é p0 ù ê 'ú ê ê p 1ú = 0 1 - p 1 - p ú ê p1 ú (0.0) After After thethe final final simulation simulation year the the yearthe smoking smoking trajectories are trajectories ê completed until the ú ê person’s person’s ú maximum After the final simulation year smoking trajectoriesêare p2 ú are ê completed ' completed ë p Thep until theuntil the person’s p ú û ê ë maximum p2 calculation ú û possible age possible age of 110 by of 110 age supposing by supposing that that their smoking their smoking ë state û stays fixed. life expectancy maximum possible of 110 by supposing thatstate their stays fixed. The smoking life state fixed. calculation expectancy stays The life is equal is equal to the sum to the of the sum of probabilities of the probabilities of being alive in in each being alive possible year each possible of life. year of life. After expectancy calculation to simulation the final is equal the sum of year thethe smoking trajectories probabilities of being are completed alive until the person’s maximum in each In the In initial the initial year year of possible age of 110 by supposing that their smoking state stays fixed. The life expectancy calculation the simulation, of the simulation, a person may a person be in may be one of in one of the three smoking the three categories; after smoking categories; N after N possible year of life. updates there updates there will will be equal is´ be 3 3 ´ 2N 2 to the sum N possible possible of the probabilities trajectories. trajectories. of being trajectories These trajectories These alive will will in each each each have possible have a calculated a year of life. calculated probability probability In the occurring; of occurring; of initial year of the sum of the sum the of these these probabilities probabilities is in is 1. one of the three smoking 1. Insimulation, the initial a person year of the may be simulation, a person may be in one of the three smoking categories; after N categories; In each In each year year theN after the updates updates probability probability there of of will there being being a be will be 3 smoker a smoker3x 2Na ´or or possible apossible non-smoker non-smokertrajectories. trajectories. These depend will depend will These the trajectories ontrajectories on the forecast forecastwill willhave a calculated each smoking smoking scenario each have which scenario which a calculatedprobability provides provides exactly of occurring; exactly that that probability the Note information. information. of occurring; sum Note theof these that that sum these these ofprobabilities states statesprobabilities these are is 1. are two-dimensional is 1. and two-dimensional and cross-sectional {non-smoking, cross-sectional { smoking}, non-smoking, smoking }, and and they they are are turned into three-dimensional turned into states { three-dimensional states {never never In each year the probability of being a smoker or a non-smoker will depend on the forecast smoking smoked, In each year smoked, ex-smoker, ex-smoker, smoker the probability } smoker} ofas described being as describeda above. smoker The or above. The time a time evolution non-smoker of evolution of the will three the depend dimensional on the three dimensional states states scenario which provides exactly that information. Note that these states are two-dimensional and are are the the smoking smoking trajectories trajectories necessary necessary for for the the computation computation of of disease disease table table disease disease and and death death forecast smoking scenario which provides cross-sectional exactly {non-smoking, that information. smoking }, and they areNote turnedthat intothese states three-dimensional states {never probabilities. probabilities. smoked, ex-smoker, smoker } as described above. The are two-dimensional and cross-sectional {non-smoking, smoking}, and they are turned intotime evolution of the three dimensional states Smoking Smoking are the smoking trajectories necessary for the computation of disease table disease and death three-dimensional states {never smoked, ex-smoker, smoker} as described above. The time The microsimulation The probabilities. model microsimulation model applied to applied to smoking smoking enables enables usus to measure the to measure future health the future impact of health impact of changes in changes in rates of tobacco rates of consumption. This tobacco consumption. includes the This includes impact of of giving the impact up smoking giving up on the smoking on the Smoking following diseases: following diseases: a) COPD; b) a) COPD; CHD (or b) CHD acute myocardial (or acute infarction (AMI) (AMI) if myocardial infarction CHD data if CHD are not data are not The microsimulation model applied to smoking enables us to measure the future health impact of available); c) available); c) stroke; and d) stroke; and lung cancer. d) lung In the cancer. In the simulation each person person is simulation each categorized into is categorized one of into one the of the 42 three // Appendix changes in rates of tobacco consumption. This includes the impact of giving up smoking on the three smoking groups: smoking groups: smokers, ex-smokers and smokers, ex-smokers people who and people have never never smoked. who have Their initial smoked. Their initial following diseases: a) COPD; b) CHD (or acute myocardial infarction (AMI) if CHD data are not distribution is based distribution is based on the distribution on the distribution of of smokers, smokers, ex-smokers and never ex-smokers and smokers from never smokers from published published available); c) stroke; and d) lung cancer. In the simulation each person is categorized into one of the data. data. three smoking groups: smokers, ex-smokers and people who have never smoked. Their initial distribution is based on the distribution of smokers, ex-smokers and never smokers 34 from published evolution of the three dimensional states are the smoking trajectories necessary for the computation of disease table disease and death probabilities. Smoking The microsimulation model applied to smoking enables us to measure the future health impact of changes in rates of tobacco consumption. This includes the impact of giving up smoking on the following diseases: a) COPD; b) CHD (or acute myocardial infarction (AMI) if CHD data are not available); c) stroke; and d) lung cancer. In the simulation each person is categorized into one of the three smoking groups: smokers, ex-smokers and people who have never smoked. Their initial distribution is based on the distribution of smokers, ex-smokers and never smokers from published data. During the simulation a person may change smoking states and their relative risk will change accordingly. Relative risks associated with smokers and people who have never Duringhave smoked a personfrom been collected the simulation published may change data. smoking Theand states relative risks (RR) their relative associated risk will change with accordingly. ex-smokers (RRRelative risks ) areassociated related to thesmokers with riskpeople relativeand who have of smokers never (RRsmoker).smoked have been The ex-smoker ex-smoker collected from published data. The relative risks (RR) associated with ex-smokers (RR ex-smoker ) are relative risks are assumed to decrease over time with the number of years since smoking related to the relative risk of smokers (RRsmoker). The ex-smoker relative risks are assumed to cessation decrease (T over time cessation ). These relative with the numberrisks are computed of years the model in cessation since smoking using (Tcessation equations ). These 1.19 relative and risks are 1.20 computed in the (Hoogenveen model and using others equations 1.19 and 1.20 (Hoogenveen and others 2008). 2008). RRex-smoker ( A, S , Tcessation ) = 1 + ( RRsmoker ( A, S ) - 1)exp( -g ( A)Tcessation ) (0.15) (0.0) g ( A) = g 0 exp( -h A) (0.16) (0.0) where where γ is γ is regression coefficient theregression the coefficientof of time dependency. time The constants dependency. γ0 and ηγ0 The constants are intercept η are and and regression coefficient of age dependency, respectively, which are related to the specified disease intercept and regression coefficient of age dependency, respectively, which are related to (see Table 13). the specified disease (see table A2). Table 13: Parameter estimates for γ0 and η related to each disease Disease γ0 η AMI A2: Parameter estimates Table for γ0 and η 0.24228 related to each disease 0.05822 Stroke 0.31947 0.01648 DISEASE γ0 η COPD 0.20333 0.03087 AMI 0.24228 0.05822 Lung cancer 0.15637 0.02065 Stroke 0.31947 0.01648 Source: Hoogenveen and others 2008. COPD 0.20333 0.03087 However, a minimum exists when the cessation time is equal to η-1. The minimum value was Lung cancer 0.15637 0.02065 calculated by the method detailed in equations (0.0), (0.0) and (0.0). Where time, t is equal to the age, Source: A of an individual. Hoogenveen and others 2008. r Exsmk ( t ) = 1 + ( r smk - 1) f ( t ) (0.0) f ( t ) = exp ( -g 0 ( t - t0 ) exp ( -ht ) ) Þ (0.0) f ¢ ( t ) = -g 0 f ( t ) e -ht ( -h ( t - t ) + 1) 0 43 The function f(t) has the following properties: (see Table 13 regression ). coefficient of age dependency, respectively, which are related to the specified disease (see Table Table Table13: 13 ). 13:Parameter estimatesfor Parameterestimates forγγ andη 00and relatedto ηrelated eachdisease toeach disease Table 13: Parameter estimates for Disease Disease γγ γ and η related to each 000 ηηdisease Disease AMI AMI γ 0.24228 0.24228 0 η 0.05822 0.05822 Reducing Tobacco Use Through Taxation in Trinidad and Tobago: Modelling the Long-Term Health and Economic Impact AMI Stroke Stroke 0.24228 0.31947 0.31947 0.05822 0.01648 0.01648 Stroke COPD COPD 0.31947 0.20333 0.20333 0.01648 0.03087 0.03087 COPD Lung Lung cancer cancer 0.20333 0.15637 0.15637 0.03087 0.02065 0.02065 Lung cancer Source: Hoogenveenand 0.15637 andothers others 2008. 0.02065 Source: Hoogenveen 2008. minimumand Source: Hoogenveen However, aminimum others exists when2008. cessationtime thecessation timeis equalto isequal toη η Theminimum ..The minimumvalue was valuewas -1 However, a exists when the -1 However, calculated calculateda minimum by the exists method when detailed in the cessation equations (0.0),time (0.0) is andequal (0.0). to η -1 Where . The minimum time, t is equal value to the However, by the method a minimum detailed exists when in theequations cessation(0.0), time (0.0) andto is equal (0.0). Where η-1. The time, t value minimum is equal wasto the age, was calculated age, A ofan Aof calculatedan by individual. bymethod the method individual. the detailed detailed in equations in equations (0.0), (0.0)(1.17), (1.18) and (0.0). and (1.19). Where time, tWhere to thet is is equaltime, equal age,to the A of anage, A of an individual. individual. rExsmk r Exsmk((tt))= =1 +((r 1+ rsmk -1 smk - )) ff ((tt)) 1 (0.0) (0.0) r Exsmk ( t ) = 1 + ( r smk - 1) f ( t ) (0.0) (0.17) ff ((tt))= exp((- =exp -gg00((tt- 00) exp ( -ht ) )) -tt ) exp ( -ht ) f (t ) Þ= exp ( -g 0 ( t - t0 ) exp ( -ht ) ) Þ (0.0) (0.0) ff¢¢((tt))Þ =- = -gg00ff ((tt))e e-- hhtt ((- -hh((tt- 00) + 1)) -tt ) + 1 (0.0) (0.18) f ¢ ( t ) = -g 0 f ( t ) e -ht ( -h ( t - t ) + 1) 0 The The The functionf(t) function function hasthe f(t)has has the the following following properties: properties: following properties: The function f(t) has the following properties: ff ((tt ) =1 00) = 1 (0.19) ff¢¢((tt 00))= h gg00e t0 0 =1- - e-- h t0 (0.0) (0.0) t( ) =a + h-- -h t0 ff(¢( t)) has t0has ag - minimum 0e minimum attt = at =tt 00 + h 11 (0.0) ff ((¥t¥) )) =A = has A a minimum at t = t0 + h -1 f (¥) = A order In In to order toavoid avoid the RR the RR ex-smoker ex-smoker from from increasing, increasing, the the cessation cessation time time was was set set equal toequal to η η-1 when -1 the when the cessation cessation time was greater time was thangreater than η-1 (see η-1 (see equation equation (1.20)). (0.0)). 35 35 35 ì 1 + ( RRsmoker ( A, S ) - 1) exp( -g ( A)Tcessation ) Tcessation < h -1 RRex-smoker ( A, S , Tcessation ) = í î1 + ( RRsmoker ( A, S ) - 1) exp( -g ( A)h ) Tcessation ³ h -1 -1 (0.0) (0.20) g ( A) = g 0 exp( -h A) (0.21) (0.0) The ex-smoker relative risks as a function of time after smoking cessation were plotted in Figure A3 The ex-smoker relative risks as a function of time after smoking cessation were plotted in for AMI, stroke, COPD, and lung cancer. Figure A3 for AMI, stroke, COPD, and lung cancer. Figure A3: Ex-smoker relative risks as a function of time after smoking cessation 1.6 Figure A3: Ex-smoker relative risks as a function of time after smoking cessation 1.6 1.4 Ex-smoker Relative risks 1.4 1.2 1.2 Ex-smoker Relative risks 1 1 0.8 AMI 0.6 0.8 Stroke 0.4 AMI 0.6 COPD 0.2 Stroke COPD Lung cancer 0 0.4 Lung cancer 0 5 10 15 20 0.2Time since cessation 0 0 5 10 15 20 Time since cessation // Appendix 44Relative Risks The reported incidence risks for any disease make no reference to any underlying risk factor. The microsimulation requires this dependence to be made clear. The risk factor dependence of disease incidence has to be inferred from the distribution of the risk factor in the population (here denoted 1.2 1.2 risk risk Ex-smoker Relative risk Ex-smoker Relative ris risks risks 1 1.2 1 1 1 Ex-smoker Relative risks Relative 1 Relative 1 Relative Relative AMI AMI AMI 0.8 1 AMI AMI 0.8 0.8 0.8 0.8 0.8 Stroke Stroke Stroke Ex-smoker Stroke Stroke Ex-smoker Ex-smoker 0.6 0.8 Ex-smoker 0.6 0.6 0.6 COPD COPD COPD 0.6 0.6 COPD COPD LungLungLung cancer cancer cancer 0.4 0.6 Lung cancer Lung cancer 0.4 0.4 0.4 0.4 0.4 0.2 0.4 0.2 0.2 0.2 0.2 0.2 0 0.2 0 0 0 0 0 0 5 10 15 20 0 0 0 5 5 5 10 10 10 15 15 15 20 20 20 0 0 5 5 100 10 15 15 20 20 cessation Time Time since Time since since cessation cessation cessation Time since Time since Time cessation 0 since cessation 5 10 15 20 Relative Risks Time since cessation The reported Relative Risks incidence risks for any Relative Risks Relative Risks Relative Risks disease make no reference to any underlying risk Relative Risks Relative Risks The The reported incidence risks for any The reported incidence risks for any disease make no reference to any underlying risk disease make no no reference to any underlying risk factor. The reported TheThe factor. incidence reported The incidence Themicrosimulation reported incidence reported risksrisks risks incidence for for for any requires any risks any for disease make disease this make no dependence Relative Risks disease any make disease noreference make reference noto reference to be to any toany made any reference underlying underlying to underlying clear. any risk The risk underlying factor. risk factor. risk factor factor. risk TheThe The factor. The microsimulation microsimulation microsimulation requires requires requires this this microsimulation dependence to be requires made this clear. dependence The risk to factor be made dependence clear. ofThe risk factor dependen disease microsimulation dependence microsimulation requires of disease thisthis dependence requires dependence incidence dependence hasThe thisincidence dependence to toto be be tomade reported be be made to made be inferred clear. incidence clear. clear. madefrom The The The risk risks clear.risk the factor risk for The factor any factor risk dependence disease distribution dependence dependence factor make dependence of the ofno of disease risk of disease reference disease of to any underlying disease factor incidence incidenceincidence has to has has be to be toinferred be inferred inferred from fromfrom the the the distribution has to distribution distribution of be the inferred of the of requires the risk riskrisk factorfrom the factor in factor the distribution in the population in the population population of (here the (here risk (here denoted factor denoted denoted in the population incidence has to be inferred incidence has to be inferred from from the microsimulation distribution the of distribution the risk of factor the this risk independence the factor population in the to be(here population made clear. denoted (here The denoted risk factor depen inas the population as p ita is );disaggregation (here denoted a disaggregation as p as process: ); it is ); is aadisaggregation disaggregation process: process: asppas );it ); p it is ); isit aa is disaggregation disaggregation process: process: as p); it is a disaggregation process: process: incidence has to be inferred from the distribution of the risk factor in the populatio as Suppose p ); it thatis a disaggregation a is process: Suppose Suppose Suppose Suppose Suppose that that that ais that a that is a isaa aa risk arisk is risk is a factor risk factor risk a isfactor factor factorstate state state ofstate state of of risk of of risk risk risk factor risk factorfactor factor factor AAand aand A and Arisk and denote andfactor denote denote denote denote by byp state p by A(d| by by pA A(d| of a p (d| risk A(d| pA¬(d|,a ,a,s) a,a,s) ,a,s) factor a the,a,s) the A the the incidence incidence and denote by pA(d|a,a,s) the inci incidence incidence Suppose that a risk factor state of risk factor A and denote by ap A(d|the a,a,s)incidence the incidence probability probability probability for for the for the the disease disease diseased given probability given dthe d given the riskthestate, risk risk for state, state, the ,,the disease a , is ,athe the person’s d person’sgiven person’s age, a, age,the age, a, of and risk a, gender, and state, and gender,a gender, s. , the person’s s. The age, a, and gender, s relative probability probability forforthe probability thefor disease disease dd the disease given given d the the given Suppose risk risk thestate, risk a that state, a a a state,, the the person’s a risk , the factor person’s age, person’s state a,age, and age, a,risk gender, a, and and s.The factor gender, The gender, A The s. and s. s. relative relative denote The relative The by pA(d|a,a,s) the relative risk r is defined by equation (0.0). risk r A risk riskis Ar rdefined is defined isAdefined by by by equation equation equation(0.0). (0.0). (0.0). A risk r A is risk defined r is by defined equation relative risk A defined by equation (1.22).by (0.0). equation probability (0.0). for the disease d given the risk state, a , the person’s age, a, and gende risk rA is defined by equation (0.0). A (( dAa( p (a a , ,s))= ds a,,= a),r r =|) s dr ((a = r (Aa s( s) ap a ),(s )Ap ( p (a 0 ,( dd,,a,s) a ,a ,,sas , )s ) ), = r A|d (a a, s ) pA ( d a 0 , a, s ) p p ,,a d , A (d a s A) ra |, a |d (a),ps A,( pd )a d a ( s a )0 s ) A p A d a a a sd a a d, p A ,a s ,A |d = a pA 0, d0 a 0 , a, A (0.0) A |d A (0.0) (0.0) ( ) A r |dr (ar (a a |d,, (s) aa0º a s1 ),0º )1,º , s )1 º 1 r p ( a d aa ,,sa , º s )1 = r (a a , s ) (0.22)( (0.0) p d a0 , a, s ) (0.0) |d (a 0(s s) º 1 A |d 0 rAA r 0A A|d 0Aa|d a a A A|d A r A|d (a0 a, s ) º 1 Where Where a isthe 0 is is the zero zero risk risk state state (for (for example, example, the moderate state for alcohol consumption). Where Where 0a a0 is the the zero Where zero risk risk a0state is the state (for (for zero example, risk state example, the moderate the theexample, (for moderate moderate state state thefor for alcohol state moderate alcohol consumption). for alcohol state for consumption). consumption). alcohol consumption). 36 36 36 36 36 0 The incidence incidence probabilities, The incidence probabilities, probabilities, as The incidence reported, as reported, reported, probabilities,can can be be beas expressed expressed reported, in in terms terms can of be expressedthe ofterms equation, the equation, equation, in terms of the equation, The The incidence as probabilities, as reported, can can expressed be expressed in terms of in the of the equation, p( p (d daa,,ss) =åp )= p( dda ,,a a, s,,)s ) )p (a pa s) s (,,,d A (( påA (aAa )a , a, s )p A (a a, s ) , pA d a s= a p d a s = å a p A d a a , a , s A A s (0.0) a a (0.0) (0.23) (0.0) (0.0) = =p = p pA (dd a0 , 0 , a, s ) å A ( d a0 A ,a ,s a, s) å=rpA r A A| A |( d (a 0 dd |d (a aaa a ,, , ,as,) s s p p)A )s A Aå( (a ara a a A, ,|ds , s( s ) )a a, s )p A (a a, s ) a a a a Combining Combining these equations allows the conditional incidence probabilities to to in written terms beprobabilities of Combining these Combining these these equations Combining equations allows equations these allows the allows equations the conditional conditional incidence the conditional allows probabilities incidence the conditional incidence probabilities incidence probabilities to be written to be written in be in terms of written to be terms of in in terms of written known quantities known quantities known quantities known terms quantities of known quantities p( p (dda a,,ss)) p ( d a, s ) p( s) ( ) p d a , s p (d da ,a a, a,,s =r )= (a a , s ) ) ( ) p d , a , s = a , s (0.0) (0.24) rA p ad a a ,,sa , s = r a a , s ( ) (0.0) A| |d d A|d å åb r r A A| |d d A|d |d | a , s ) A( ( b b | | a a , , s s ) p p å A A A (a ar a a a A|d , , , s ( s s ) b | a , s ) p A (a a , s ) (0.0) (0.0) b b b Previous Previous to any to anyseries seriesof Monte Carlo trials, the microsimulation program pre-processes the set of Previous Previous to to any any series series ofof of Previous Monte Monte to any Monte Carlo Carlo Carlo trials, trials, series of the Monte trials, the microsimulation the microsimulation Carlo trials, the microsimulation programprogram pre-processes microsimulation program pre-processes pre-processes the program set set of the pre-processes of the set of diseases diseases the set of diseases and and stores diseases and stores stores the the and the calibrated calibrated diseases and stores stores calibrated incidence incidence the statistics statistics calibrated the calibrated incidence statistics p p incidence A ( (d incidence pA |a d| A(d|a0 a0 0 , ,a , a a , ,s , s). s). statistics statistics ). pA(d|a0, a pA(d|0, , s). a, . Modelling Diseases Modelling Diseases Modelling Diseases Modelling Diseases Disease modelling Disease modelling Disease heavily relies heavily relies the on the on setsheavily sets on the mortality, of incidence, survival, relative risk and Modelling Disease Diseases modelling modelling relies heavily on the of incidence, sets of relies incidence, mortality, sets of incidence, mortality, survival, relative risk relative survival,mortality, and survival, and relative risk and risk statistics. prevalence statistics. prevalence prevalence statistics. prevalence statistics. Disease modelling In the simulation, relies heavily individuals on the are assigned sets a risk of incidence, factor trajectory mortality, giving their survival, relative personal risk risk factor In the In simulation, individuals the simulation, are assigned are In the simulation, individuals a a risk individuals riskare assigned factor factor trajectory assigned trajectory giving a risk their factor giving their personal trajectory risk risk their giving personal factor factorpersonal risk factor and prevalence history history for each for statistics. each year year of for lives. of their their history lives. Their Their each year ofprobability probability of of their lives. getting getting Their a particular a particular probability risk factor-related risk factor-related of getting disease disease a particular risk factor-related disease history for each year of their lives. Their probability of getting a particular risk factor-related disease in a particular in a in a particular year particular year in year will will will depend on depend a particular depend on year on their their will their risk risk factor factoron depend risk factor state state state in that thatfactor in risk their in year. year. state in that year. that year. In the simulation, individuals are assigned a risk factor trajectory giving their personal risk Once a person has a fatal disease (or diseases), their probability of survival will be controlled by a Once a person has a person a fatal fatal hasOnce diseasehas (or a diseases), probability their(or of survival will be Once factor history for a each ayear person disease (or of their fatal disease diseases), lives. their Their diseases), probability probability oftheir ofsurvival getting a be probability will controlled of survival controlled particular by will by risk a be controlled by a a factor- combination of combination of the the disease-survival disease-survival combination of thestatistics statistics and the and the probabilities disease-survival probabilities statistics andof of dying dying the from from other other probabilities ofcauses. causes. dying from other causes. combination of the disease-survival statistics and the probabilities of dying from other causes. related Disease Diseasedisease survival survival in a particular statistics statistics are are year modelled modelled will as as depend age- age- and and on their risk gender-dependent gender-dependent factor state in exponential exponential that year. distributions. distributions. Disease survival statistics are modelled as age- and gender-dependent Disease survival statistics are modelled as age- and gender-dependent exponential distributions. exponential distributions. Methods for Approximating Missing Disease Statistics Once a person has a fatal disease (or diseases), their probability of survival will be Methods for Approximating Missing Disease Statistics Methods for Approximating Missing Disease Statistics Methods for Approximating Missing Disease Statistics A large amount data are required for modelling these diseases. Where possible these datasets have A large amount A large amount controlled by adata data are required are A large amount required combination offor data for themodelling are modelling these required diseases. for these diseases. disease-survival Where modelling possible these Where statistics possible and these diseases. datasets Where these possible datasets the probabilities ofhave these datasets have have dying been collected been collected from from published published been collectedsources sources or or analysed analysed from published from from sources either either cross-sectional cross-sectional or analysed or or from either longitudinal longitudinal cross-sectional or longitudinal collected beenother from from causes. published Disease sources survival or analysed statistics from are either modelled cross-sectional as age- and or longitudinal gender-dependent datasets. Another datasets. Another limitation limitation is is that that often often this this data data needs needs to to be be in in a a specific specific format. to be in For format. For example, example, the For example, the the limitationAnother datasets. Another datasets. limitation is that often is that this data often needs to this be indata needs format. a specific a specific For format. example, the exponential model updates model updatesdistributions. individuals’ individuals’ model updates model updates individuals’ disease disease disease status status every every individuals’ status every year year disease so so year so the the status relative relative risks risks every year the relative so risks used used used in in the the model model the relative need risks used in the model need tothe model need to to needin to annual relative be annual be risks. relative be risks. annual relative risks. be annual relative risks. This section contains the methods used in this project in cases where in for data a particular disease This section contains This section the This contains methods section the used contains used the methods in this in methods this project used project in in cases where in this cases project where data data for a particular a cases for where data particular disease for a particular disease disease were unavailable. were unavailable. were unavailable. were unavailable. 45 Terminal and non-terminal single state disease incidence from prevalence Terminal and non-terminal single state disease incidence from prevalence Terminal and non-terminal single state disease incidence from prevalence Terminal and non-terminal single state disease incidence from prevalence For terminal diseases, For terminal For terminal diseases, to estimate to estimate For terminal diseases, to estimate incidence incidence diseases, to estimate incidence (knowing (knowing (knowing prevalence prevalence incidence (knowing prevalence and and mortality and mortality rates) rates) prevalence mortality and rates) one one one can can mortality can rates) one can proceed by by finding proceed by finding those findingproceed incidence incidence those incidence probabilities probabilities by finding that minimize minimize that minimize those incidence the the that probabilities distance distance between between minimize the known known the known the distance between the known proceed those probabilities that the distance between the K K å å A|d A|d A ( A () ) b b Previous Previous series to any to of Monte any series Carlo trials, of Monte Carlo the trials, microsimulation the microsimulation program pre-processes program the set pre-processes of set of the diseases diseases and stores the calibrated and stores incidence the calibrated incidence statistics pA(d|a statistics p 0A d,|s , (a a).0, a, s). Reducing Tobacco Through Taxation in Trinidad and Tobago: Modelling the Long-Term Health and Economic Impact UseModelling Diseases Modelling Diseases Disease Disease modelling modelling relies heavily relies heavily sets on theon theofsets incidence, of incidence, mortality, mortality, survival, survival, relative relative risk and risk and prevalence prevalence statistics. statistics. In the simulation, In the simulation, individuals individuals are assigned are assigned a risk factor a risk factor trajectory giving giving trajectory their personal their personal risk factor risk factor history for each history foryear eachof their year oflives. theirTheir lives.probability of getting Their probability of getting a particular a particular risk factor-related disease risk factor-related disease in a particular in a particular year year will will depend depend on theironrisk their risk factor factor state state in year. thatin that year. Once a person Once has a fatal a person has a fatal disease disease (or diseases), (or diseases), their probability of survival their probability will bewill of survival be controlled controlled by a by a Methods for Approximating Missing Disease Statistics combination combination disease-survival of the of statistics the disease-survival and the statistics and probabilities of dying the probabilities dyingother of from other causes. from causes. Disease A large amount data survival Disease survival statistics are required aremodelling statistics for modelled are modelled as age- theseasand age-gender-dependent and gender-dependent diseases. exponential Where possible exponential thesedistributions. distributions. datasets have been collected from published sources or analysed from either cross- Methods for Approximating Missing Disease Statistics Methods for Approximating Missing Disease Statistics Aamount A large large amount sectional or longitudinal data are data datasets. required are required Another for for modelling modelling limitation these is oftendiseases. thatdiseases. these Where this dataWhere possible needs these to bedatasets possible these in datasets have have been collected been collected from published from published sources sources or analysed or analysed from either cross-sectional from either or longitudinal cross-sectional or longitudinal a specific format. For example, the model updates individuals’ disease status every year so datasets. datasets. Another Another limitation limitation is that is that this often often data this needs to be in data needs specific toabe in a specific format. For example, format. the the For example, the relative risks modelused inupdates updates model the model individuals’need individuals’ diseaseto be annual status disease status relative every year every so risks. the year so relative the relative risksin risks used the model used need to in the model need to be annual be annual relative relative risks. risks. This section contains the methods used in this project in cases where data for a particular disease wereThis This section section unavailable.contains the methods contains the methods used in this in used project in cases this project cases where in where data data for for a particular a particular diseasedisease were unavailable. were unavailable. Terminal and non-terminal single state disease incidence from prevalence Terminal and non-terminal single state disease incidence from prevalence Terminal and non-terminal single state disease incidence from prevalence diseases, For terminal For For terminal terminal estimate todiseases, toincidence diseases, to estimate estimate (knowing incidence prevalence (knowing incidence (knowing prevalence mortality andand prevalence rates) mortality and mortalityone rates) rates) one canone can can proceedproceed by finding proceed by finding by finding those those those incidence incidence incidence probabilities probabilities probabilities thatminimize that that minimize minimize the thedistance the distance distance between between the known between the known K K K K the known p pre and p pre and prevalence computed and computed computed p pre prevalence prevalence p pre Non-terminal Non-terminal diseases diseases Non-terminal diseases are treated intreated are aare in away treated similar similar in – way – way although, a similar although, – although, obviously, obviously, the the mortality obviously, the mortality mortality probabilities probabilities arezero. probabilitiesare are zero. zero. Mortality statistics Mortality statistics Mortality statistics Mortality statistics Mortality statistics Mortality statistics Mortality statistics Mortality statistics Mortality statistics Mortality statistics 37 37 In In In any InIn any any year, anyIn In year, year, any anyInyear, In year, in in in a year, any in aasample sample sample sample in in year, a a sample of asample of of N sample in aof N people sample NN people people of of N who Nof people N who who people people who have have have who who people have the the the have have who the disease, disease, disease, the the have the disease, a a subset asubset disease, disease, a subset asubset a disease, subset NaNN subset N willwill w will N subsetNdie Ndie Nfrom die will will from from die diethe will the the from from die disease. disease. disease. from disease. thedisease. the the disease. any year, In in any any a year, year, in in a a of N sample sample people of of NN who people people have who who the have have disease, the the disease, disease, a aa w will subset subset subset w die from will the die die w will die disease. from from from the thethedisease. disease. w w w w w Mortality Mortality Mortality statistics statistics Mortality Mortality statistics Mortality record record record statistics statistics statisticsstatistics the the cross-sectional the record record cross-sectional cross-sectional the the record probabilities probabilities cross-sectional cross-sectional the probabilities cross-sectional of of of probabilities probabilities death death death of of probabilities as death death deathof as a as aa result asdeath result result as as a a of as of the of result result a the disease theof of disease disease the the – –– disease disease –– Mortality disease. Mortality Mortality Mortality record the statistics statistics statistics cross-sectional record record record the the the probabilities cross-sectional cross-sectional cross-sectional of probabilities probabilities of of probabilities a result death death of as as of aa result deaththe result resultas a the of disease of of the– the disease disease disease result of – – – possibly possibly possiblystratifying stratifying possibly possiblystratifying possibly by by age by stratifying stratifying age age by stratifyingbyage age by possibly stratifying possibly possibly by age stratifying stratifying the disease – possibly stratifying by age by age by age age NwN N Nww N Nw wN Nw p pwp p= = w= ppw= = p = w = (0.0) (0.0) (0.0) (0.0) (0.0) w = w w Nw N N pw N w (0.25) (0.0) (0.0) (0.0) (0.0) N N Nw w N of N people who have the Within Within Within Within disease, such such Within Within such such a such a aa such subset subset subset subset subset a N asubsetN subsetNN of of will of people people wof NN die people people of from thatthat that people the that die die die that disease. in indie in thatthat that die in year in that year year thatfrom year from from yearthe from the disease, the from thethe disease, the disease, disease, the distribution the disease, the distribution distribution distribution byby distribution thedistribution by year-of- year-of- year-of- year-of- byyear-of- Within Within such a such subset aN subset wof NN w of people people of people that that die die that in in thatdiethat in year year that from from yearthe the from disease, the disease, the disease, the the distribution distribution by by by by year-of- year-of- w Within Within such such aa w subset subset w w w w of w of people people that that die die in in that that year year from from the the disease, disease, the the distribution distribution by year-of- year-of- d the cross-sectional by probabilities disease disease disease is disease is not is disease not not year-of-disease of usually usually is is death usually not not as a recorded. recorded. is recorded. not usually usually result This usually recorded. recorded. of This the disease distribution distribution This distribution recorded. This This – would would would This distribution distribution be be be distribution would would most most most be be useful. useful. useful. most most Consider would Consider Consider useful.be useful. two most Considertwo two Consider important important important useful. two two Consider important important disease is disease not usually disease disease is is not is not not usually usually recorded. recorded. usually This distribution recorded. recorded. This This distribution This would be distribution distribution most be would would would be most useful. be most most useful. Consider useful. Consider useful. Consider Consider two two important two important two important important e idealized, idealized, idealized, idealized, special special special idealized, cases. cases. cases. special special cases. cases. idealized, two importantidealized, special idealized, idealized, special cases. special idealized, special cases. cases. special cases. cases. NSuppose Suppose the Suppose the true the true probabilities probabilities ofdying of of dying dyingofin inthe the the years years afterafter somesome age a age aa are { are { { { } } } } Suppose true Suppose pw Suppose Suppose = Suppose the Suppose w the the true Suppose the true probabilities true true the the the probabilities probabilities true probabilities true true of probabilities of dying probabilities probabilities probabilities of of the in dying dying of in of dying inyears in dying dying dying the the in inyears in in the after years years the after the the some after after years (0.0) years after some years yearsafter age some some age after after age 0age a some some some some 00 are 0 are a age a00p age age age are { w p are a a p 0 p, w 0 w 0 w { p 00,, are are ,w 0 0are arep p p p ,ww 1w w p { 0{ 1 0 1 , 1,, p ,w pp p p pp 2 w w w w , w w 0 w w1 0 p , , 2 1 2 , , 2,, p ,w pp p p p3 w w p w w , w 1 w w 2 1 p , , 3 2 3 , ,, ,p 3p w, p pp p4w w w p w w 2 w w 3 2 , 4 3 4 } , ,4 ,, p pp p pw w w w w34 3 ,} ,, 4 p p pw ww4 } 4} N 0 1 2 3 4 The The probability probability The probability The The The being ofbeing of of being probability probability ofalive of probability alive being being ofalive after after after alive aliveN NNyears years years after after NNis simply issimply simply is years years is thatthat that issimply simply you you you thatdo that do not do you you not die not do do die in die not not in each in each each die die inyear in year year each each year year TheThe probability The The probability of ofbeing probability probability being of being alive of being being alive alive after N alive alive after after years after after is N yearsN Nsimply N years years years is is simply that is is simply you simply simply that do that that that you not you you you do die do do do not in each not not not die die die die in ineach year in in each each eachyear year year year w of people that die in that year from the disease, the distribution by year-of- would be most p corded. This distribution p useful. pp survive ( pp survive survivea ( (0(a a + Consider ap p 00N p survive survive (+ + +(a a) N NN 0 ) = two ) 0+( + ()a ( = = a a = 1 NN( - ( +( ) 1)1p --(( pp)( important 1 + + == N - NN w ) 1 p 01 )w- = = = w- 0 survive 1 00 )( )( (( )( p 1- p 1 11 ( 1 w w 1 - - -0 p - - 0 )( )( - 0p p p wp p 1 p 1 1)( w- w 1- 1)( 1 )( )()( )( p 1- p )w0 ( 1 11 1 w w 1 - - - 1 )( 1)( p - --p p p wp pp 1 21)w w w - 2) .. )( - 2 0 )( )() p )(1 .. p 1 1 1 .. .. ww ( - ( - - - 2 ( 1 12) 1) p - p - pp- .. .. ww 1 (( p 1 )( p N p 1)- )w w - - .. .. 1 N )( .. N- p ( 1 - 1p 1 11 w )) )w - - - N p)) Nw-p - 1p 1 ) ( w N )) (0.0) (0.0) (0.0) (0.26) (0.0) (0.0) (0.0) (0.0) (0.0) (0.0) survive survive survive 00 w ww001 w ww12 1 w2 w 2 w N -1 2 w w N- N - -1 11 Survival models Survival models Survival models Survival models Survival models Survival models Survival models Survival models Survival models Survival models There There bilities of dying in the are years There There There Thereareare three after are There are three some three three are are There three There There in in use in in three three are in are areage use use ina use (they use in three three three (they (they (they use use (they 0 are are in in in { are easily are are (they (they use are use use p easily w easily 0, are are (they (they (they extended extended p easilyw extended easily are , easily easilypextended wextended easily extended are are 1 easily easily 2 , pif extended w ,if the if extended 3 ifp extended the the extendeddata the w4 }data if if data merit): the the the data ifdata if if merit): merit): data data the merit): the the merit): merit): merit): data data data merit): merit): merit): Survival model 0 alive after N years Survival model 0 Survival model 0 : a: Survival model 0 Survival model 0 :aasingle single single Survival model 0 aprobability ::a ::probability single single : probability a single ofdying dying of of probability probability dying of of probability{ p{p { dying dying { of p} {{} w0} w dying } p { pw0{}} } 0} is simply that you do Survival model 0 Survival model 0::not a single Survival model 0 Survival model 0 a die single in aeach a year probability single single of dying probability probability probability of of of dying w p 0dying dying ww00 0 p p p w w0 w0 ( a0 + N ) = (1 - pp p pp )( is is valid is 1p p 00 is valid -valid p 00 p valid for is is p )( for all for valid valid is for 1 all -all for all p for valid )years. years. years. years. all all for years... ( years. 1- all pwN -1 ) years. (0.0) wis0 valid for isall years. valid for all years. 0 is valid for all years. vive ww w 00 w ww w 1 w 2 w w0 w 0 Survival model 1 Survival model 1 Survival model 1 Survival model 1 Survival model 1 : two 1: : Survival model 1 two :two different different different two ::two : probabilities probabilities different different :: two probabilities differentprobabilities dying ofdying dying of of probabilities of of probabilities{ p{ { of{p dying dyingp , p{ 0dying{,p ,w1} pp { {} } ,,} pp },} } 1} Survival model 1 Survival model Survival model 1 Survival model 1 two :two different two different two probabilities different probabilities different of dying probabilities of of dying probabilities of w pdying w0 , p dying w w00 www 1 w0 101 p p p w w w w w1 0 0 , 0, 1 pp pw ww1 1 hey are easily extended if the data merit): pw p ppis is valid is pw p 00 is valid valid is 00 p valid for is p for the for valid valid is for the first the for for valid the first year; first the the for year; year; first first the year; p wp year; year; the first pp thereafter. thereafter. thereafter. pp year; thereafter. thereafter. 11 ppw thereafter. thereafter. 0 valid wis for isthe valid first for year; 0 is valid for the first first thereafter. year; wyear; thereafter. 1thereafter. 0w w 1ww11 w ngle probability of dying { pw w0} w w0 w 0 1 w w w11 Survival model 2 Survival model 2 ::three three : three Survival model 2 : Survival model 2 Survival model 2 :: Survival model 2 Survival model 2 different three Survival model 2 Survival model 2 different different three three : different different :: three different three three probabilities probabilities different of of probabilities probabilities different different dying ofdying dying probabilities probabilities of{ of probabilities of dying probabilities probabilities p{p { dying dying { of of of w p 0 p, ww{ p 0{ dying dying dying 0,, p ,wp p1 p w , w w { w 0 p { 1 0, 1, p ,, p p wp p p } p w 5ww 1,,5 ,, 5 1 w0 } } ,p p} p ppw w , 5 w1, , 5} {} p pp ww0 w1 w 5 } } w w 005 w w 1 1 w w 55 pw p ppis is valid is p p 00 is valid valid validis is 00 ppfor for the for valid valid is for the the thefirst for for is valid valid first year; first the the for first for year; first first the year; p year; wp year; the firstpp for year; first for the for pp year; for year; the second the for 1for p thepw second second the theforsecond for second to to to the second the the to the fifth the to second the fifth year; tofifth second the the to fifth to thep year; year; fifth fifth the year; pp thereafter. year; year; fifth p fifth thereafter. thereafter. pp year; thereafter. thereafter. 55 ppw thereafter. year; thereafter. thereafter. {000pis valid for the firstwyear; 1for the second to the fifth w 5year; 5thereafter. 0w w w 1ww11 w w 5ww55 w 0 , pw1} w w1 ww different probabilities 46 w of //0 dying Appendix w w w 1 w w11 w5 w 5 Remember Remember Remember that that Remember Remember Remember that different Remember that different different that that that different probabilities probabilities probabilities different different will will apply will probabilities probabilities different apply willto apply will probabilities probabilities will apply toapply different to apply apply will to different different todifferent to age age and different to different age age and and and gender gender age age different gender and and age groups. gender gender and gender groups. groups. Typically Typically Typically groups. groups. gender groups. groups.the Typically the the the Typically Typically the the Typically the Remember Remember that that different different probabilities probabilities will will apply apply to to different different age age and and gender gender groups. groups. Typically Typically the the year; pw1 thereafter. data data might data might databe might be might divided divided be divided into divided bedivided into 10-year 10-year into 10-yearage intointo age groups. groups. age 10-year groups. groups. agegroups. datadata might data might data data be bemight divided might might be beinto be into divided 10-year 10-year divided divided into into 10-year age age 10-year 10-year age groups. age age groups. groups. groups. Calculating survival from incidence and mortality dying { p , p , p } Calculating survival from incidence and mortality Calculating survival from incidence and mortality e different probabilities of Calculating survival from incidence and mortality Calculating survival from incidence and mortality psurvive (p a psurvive p 0(survive a0N + survive (a + ) (Na=)(+ 0 + 0 1 N- = )- ()1= N pw(01 = p)( (w 1 - 1 0-)( p-1p pw w0 )( - w 01)( p 1)( 1 - w11)( - p p 1 -w1 p - w1 )( )( w p 1 2 )w 1 -.. 2-( )1 p p ..- w2 (w 1)2p -)w.. .. (N1 p( -1 - w 1 )- N p -1w)N pw- 1) N - ) 1 (0.0) (0.0) (0.0) (0.0) Survival models Survival models Survival models Survival models There There There are three are There in three are are in use use three three (they inin are (they use use (they are are easily easily (they are extended easily extended easily extended if the if the extended if if data the data merit): the merit): data data merit): merit): Survival model 0 Survival model 0 Survival model 0: a single : a single Survival model 0 :a: probability a single single of dying probability probability probability { of dying ofofp }w{ {0p dying dying w 0}{ pwp0 w 0} } pw 0pis is p valid w 0valid pw is 0w 0 for is all all for valid valid for years. years. for all all years. years. Survival model 1 Survival model 1 Survival model 1 : two : two Survival model 1 : two different different : two probabilities different different probabilities of dying probabilities probabilities { of dying ofofp {0p dying dying w ,wp{ 0w { 1p ,p} w p ww 01,},w 0p p1}1} w pw 0pis isp valid w 0 pw valid is 0w 0 is for valid thethe for valid first for first for the year; year; the first first p 1pw1 pw wyear; year; p thereafter. thereafter. thereafter. thereafter. 1w1 Survival model 2 Survival model 2 Survival model 2 Survival model : three Survival model 2 2: : three three different :different : different three three different probabilities different probabilities probabilities of dying of of dying probabilities probabilities of{ ofp {0p dying dying dying w { ,wp 0w,{ p 1pp0 , ww p w,w 10 , ,p 5p} p ww 1w,1 5 } p p5 ,w w}5} pw 0pis is is pw p valid valid w 0 valid is for is for thethe valid valid first forfor first the year; year; the first first p p year; forfor year; 0w 0 for the first year;w1 w1 for p the pw the second for the second 1 the for 1wthe to secondto the the second second fifth to to to fifth the the the year; fifth year; fifth fifthpyear; w p5wthereafter. 5pw 5 pw thereafter. year; year; thereafter. thereafter. 5 thereafter. Remember Remember Remember Remember that that Remember that different different that that different differentprobabilities probabilities probabilities different will probabilities apply apply will probabilities will will will to apply different to different apply apply toto to ageage and different different differentand gender age gender andgroups. age and age andgender groups. gender groups. Typically groups. gender thethe the Typically Typically Typically groups. the data data might might data be be divided divided might be into divided into 10-year into age 10-year 10-year age groups. groups. age data might be divided into 10-year age groups.groups. Typically the data might be divided into 10-year age groups. Calculating survival from incidence and mortality Calculating survival from incidence and mortality Calculating survival from incidence and mortality Calculating survival from incidence and mortality Calculating When Whena person When a person When asurvival a (of person from (of person (of a given a given (of a incidence given gender) agender) given dies gender) dies gender) and fromfrom dies dies mortality disease, a from from a aa disease, they disease, must they disease, must they theyhave must have contracted contracted must have contracted have it at contracted atit atat it some it some earlier some earlier some earlier earlier age. age. When For survival age. For a For survival person modelmodel survival 2, (of a 2, this model given is this 2, expressed is expressed this is expressed gender) dies age. For survival model 2, this is expressed from a disease, they must have contracted it at some earlier age. For survival model 2, this is expressed ˆ p ( a() a = p ( (a -1 - ) )1p 0 + )( ) ( -) ˆ mortality p ˆ p ˆ p ()p a )a = =)+pinc p pinc a( a a - 1 1 p p)wp ++ ˆp pˆ (( aa))==pp ( (mortality aa --11 p = 0w+ 00+ mortality mortality inc inc w w 0 0 ( )( 0) mortality mortality mortality inc inc inc w ww0 p a inc ( a - 2 )(1 - p ++pp ((aa--22 )()( 11 -- + + +p p p + p p) inc inc ( )p 00inc a( p - a -+ 2 2 )( - 11 + 1 2 )( 1- - 1- p pw w wp0w 0 )0p ) p )w p w wp1+ 1w 1 + +1+ inc ( )( 0 )( 1) inc inc inc ww w 0 www 1 p a p p ++pp inc inc inc ((aa--33 )()( 11 -- + + +p+p p p ww inc inc w p)( 00 0 ( (1 )( inc a a( 1 - - a - --- 3 3 3 pp )( )( w 3)( ww1 11 1 1 1 ))p - - -1p w - ww p pw 11 1 w w +p +0w 0 )( )( 1 1 0 )( 1- - - 1- p pww wp1w 1 )12p ) p p)ww wp1+ 1w 1 + +1+ (0.27) + ++ p pinc ( ( ( a - a( a -- 4 )( 4 )( )( 1 12 4 )( - - p )( 1 p 2 pw 0 )(1 - pw1 ) pw1 + )( - ) )3 2 p ++ 0 )( 1 )pw 2 2 ++pp ((aa--44 )()( 11 --+p p p p)( inc )(a 11 - - -4pp 1 ))- 2 1p-ppw wp+ 0 + 1- 1- pww p 1 w p 1 + (0.1) inc inc inc wwinc w 000inc ww w 1 11 www 111 0 w 1w 1w 1 (0.1) (0.1) (0.1) inc ( )( 0 )( 1) + p a - 5 1 -3 p 1 - p p(0.1) (0.1) 1+ ( )( )( ) 3 3 + + p p + p ( a a( - a - 5 5 - )( 5 1 )( 1 - -1 -p p p )( 1 )( 1 - - 1 -p p p ) 3 )p p p + + + (( )()( )()( )) pp 33 w w w ++pp aa--55 11 --pp inc inc inc11 --pp w+ w 0 0w+0 w 1 w1w1 w1w1 w 1 inc inc inc www0 00 w ww1 11 www1 11 38 38 38 38 inc ( )( 0 )( 1) 4 + p p ( a a - 6 )( 1 -4 p p )( 1 - p p ) 4 p 4 4 5 + ++pp inc inc inc ( ( a a-- 6 6)()( 1 1- + +p - p + p w p)( inc inc ww00 0 ( )( inca 1(1 - a - - - 6 6 - p p)( w 6 )( ww1 11 1 1 ) ) 4 4- -1p - ppw w ww w w 5 p 55+ 0w 0 )( + 0 1 )( 1- - 1- pww wp1w 1 ) 1 )p pw w wp5w 5 + + 5 + inc ( )( )( ) 4( 5) 4 + + p p ( a a - - 7 7 )( 1 1 - - p p )( 1 1 - - p p ) 4 1 (1(- 1 - - p p )5p p 5 + + ( (1- )( 7 )( )( )( ) 1) ( ) )w 4 (( )()( +p p + p)( )(a a - 7 -- 1 ))- 1 (- (1 p p 1)- 1- pw p 1- pw p p p +5 + 55 )p 4 4 4 w 0 w 1 ++pp inc inc inc aa--77 11 -- p ww inc inc 00inc w 0 1 pp ww w 11 1 1 -w w-0 0wpp0 w ww5 pw ww55+ w 5 1 1w+ w w 5 5w w w 5 5w ... +... + ++... ... +... +... The The three three probabilities probabilities { { { p p {w 0,,p p 1,,p p 5}} } are are estimated estimated by minimizing byby minimizing 5} The The Thethree three three {{ probabilities probabilities probabilities p p , p,wp 1, p,wp are estimated are are by estimated estimated minimizing minimizing by minimizing 5}5} Thethree The probabilities threeprobabilities pp w ww00 0 ,,pp w ww1, 1 1,pp w ww5 are arew westimated estimated 0 0w 0 w w1w1 w w by 5 5wby minimizing minimizing ( )a) 2 (p( p ((aa)( a() ()a )a- ˆ p ( (aa() 2 (0.28) ˆp )) ) 2 2 p ( ) - )-p ( ) ((ppS å p a) ˆ p ˆ a 22 2 - = =(( ))--p mortality mortality SS= a a ˆp ˆ mortality mortality (0.2) å å (0.2) mortality mortality mortality mortality S = å å (0.2) (0.2) mortality mortality mortality mortality mortality mortality SS== s s s (0.2) (0.2) 2 2 2 aÎ a aÎ ÎAgeGroup AgeGroup AgeGroup ss aÎAgeGroup a Î 22 2 AgeGroup aÎAgeGroup s aÎAgeGroup 2 Whenthe When the longitudinal probability longitudinal probability ofdisease of the the disease incidence incidence at age the at age a satisfies a satisfies recursionthe relation When When the the longitudinal longitudinal probability probability ofof of the the disease disease incidence incidence atatage a satisfies satisfies the recursion relation When Whenthe thelongitudinalWhen longitudinal recursion the longitudinal probability probability relation of the ofthe probability disease disease incidence incidence the atdisease agea atage satisfiesat incidence asatisfies the age the age a recursion recursion the a satisfies the relation relation recursion relation recursion relation p pinc ( (a ( a() ) ) = = (1 (1(1 - --p (0))(1 pii p (0))(1 - -- p (1))..(1 pii p (1))..(1 - -- p pi ((aa - 1)) - 1)) p ( (a pii p a() ) (0.3) (0.3) (0.29) )pp i (ia ) a) pp (( ))==(1 aa (1 -pp p pinc a a= (1 = - p i (0))(1 (0))(1 -pp -aap (1))..(1 i (1))..(1 pp (( a- a))pii p ( a i (a - 1)) - (0.3)p 1)) (0.3) (0.3) (0.3) i (0))(1 ii (0))(1 i (1))..(1 ii (1))..(1 i( ii ( - - - i- -- 1)) i 1)) inc inc inc inc inc ii i Table A3 is taken from the Cancer Research UK website as an example. It gives 1, 5 and Table Table 14 is 14 taken from is taken the Cancer from the Research UK Cancer Research website as UK website an example. as an example. It gives 1, It gives 1, 5 and 10 5 and year 10 year Table14 Table 14isistaken takenTable Table from from 14 the the 14is taken is Cancer Cancer taken from fromthe Research Research theCancer UK Cancer UK Research website Research website as an asan UKUKwebsite website example. example. It Itas an as gives gives example. an 1,example. 1,55and and10It It 10 gives gives year year 1,1,5 and 5 and10 year 10 year 10 year survival survival survival percentages percentages percentages lung for lung for for lung cancer. cancer. cancer. survival survival survivalpercentages survival percentagesfor percentages forlung percentages cancer. lungcancer. for lung for lung cancer. cancer. Table 14: Table 14: Survival Survival percentage percentage for for lung lung cancer cancer Table14: Table 14:Survival Table Survival Table14: percentage14: percentage Survival forSurvival for lung lungpercentage percentage cancer cancer for lung for cancer lung cancer Table A3: Survival percentage Survival Survival Survival Survival for lung cancer Survival Survival percentage percentage percentage percentage percentageSURVIVAL percentage PERCENTAGE Cancer year 1 year year 5 year 10 year year 1 p 1 p 1 p Cancer Cancer Cancer55year 1 1 year 1 year10 5 year 5year 5 year 10 1010year year 11 1- 1 - pw p p 1- 1 - pw p p 1-- pw p p - 1- -1- 1w1 1 - 1- w00 w11 w55 Cancer Cancer 11 year year Cancer year 10year 1 year 5 year 10 year --pp w ww00 1 0 1--ppw ww11 1 1 1 w-- 0 0p wpwww 555 w w 5w 5 Lung Lung 32 32 10 5 0.32 0.75 0.71 Lung Lung 32 Lung 32Lung Lung 10 10 32 32 3255 10 10 10 10 5 55 5 0.32 0.32 0.32 0.75 0.32 0.75 0.75 0.32 0.32 0.71 0.75 0.71 0.71 0.750.75 0.71 0.71 0.71 Source Source: Source : Cancer : Cancer Research Cancer Research UK, data UK, Research UK, data at: www.cancerresearchuk.org/health-professional/cancer- at: www.cancerresearchuk.org/health-professional/cancer-statistics/statistics-by-cancer-type/lung-can- data at: www.cancerresearchuk.org/health-professional/cancer- Source Source Cancer : :Cancer Source Source Research Research cer/survival : Cancer Cancer :UK, ). UK, Research data data Research at: UK, UK,data at: data www.cancerresearchuk.org/health-professional/cancer- at: www.cancerresearchuk.org/health-professional/cancer- www.cancerresearchuk.org/health-professional/cancer- at:www.cancerresearchuk.org/health-professional/cancer- statistics/statistics-by-cancer-type/lung-cancer/survival). statistics/statistics-by-cancer-type/lung-cancer/survival). statistics/statistics-by-cancer-type/lung-cancer/survival). statistics/statistics-by-cancer-type/lung-cancer/survival). statistics/statistics-by-cancer-type/lung-cancer/survival). statistics/statistics-by-cancer-type/lung-cancer/survival). The The probabilities probabilities of being ofof being alive alive after after 1, 5 and 5 1,1, and 10 10 years years are are Theprobabilities The probabilities The ofThe of probabilities probabilities being being alive of aliveafter after being 1, 1,being and 55andalive alive 10 after 1, after years 10years 5 are are and 5 and10 years 10 years are are p psurvival p p ( ( ( a a(0 a +1 + +) 1 ) )1= =)= ( (1 ( 1(- - p pw p 0)) )0 ) pp survival survival survival(( aa000++ ))== 11 (survival ( 11--pp survival survival ww w 000 0a )0)+01 = 1 - 1- w wp0 0w survival ( ) ( )( 1) 4 p p ( a a + + 5 5 ) = = ( 1 1 - - p p )( 1 1 - - p p ) 4 (0.3) ( ( ) ) ( ( )( )) w 0w 0 w1w14 ) )( ) 4 4 pp (( aa +55 p p (1 ))==(survival 1 --pp a )( 0 a )(+ 115 + 5= --pp = 14 4 - 14 -pw p0 1 - 1 -pw p (0.3) (0.3) (0.3) (0.3) 00+ survival 0 w 0 w 1 survival survival survival 0 survival ww w 0000 0 w ww1 11 (0.3) survival ( 0 + 10 ) ( 4 pw 0 )( 55 pw1 ) ( 5) 5 p p ( a a + 10 ) = = ( 1 1 - - p )( 1 1 - - p ) ( 4 4 1 1 - - p p ) 5 5 (1--( a(0 )p )p ( )( )( )( 1) ) (1(- )5 ) 4 5 survival( ( ))p p (1 0a + )()(10 + 10 = = 1)- 1(1- (p p 1)- 1- pwp 55 ) w1w1 1-pwp 4 4 5 w pp survival survival aa00+ 0 +10 10 ==survival survival pp survival w ww000 0 1 1-- ww w 1 11 1- w w-0w 0 pp 0ww w 5 w 5w 5 47 Table 14: Survival percentage for lung cancer Survival percentage Reducing Tobacco Use Through Taxation in Trinidad and Tobago: Modelling the Long-Term Health and Economic Impact Cancer 1 year 5 year 10 year 1 - pw 0 1 - pw1 1 - pw 5 Lung 32 10 5 0.32 0.75 0.71 Source: Cancer Research UK, data at: www.cancerresearchuk.org/health-professional/cancer- statistics/statistics-by-cancer-type/lung-cancer/survival). The probabilities of being alive after 1, 5 and 10 years are The probabilities of being alive after 1, 5 and 10 years are psurvival ( a0 + 1) = (1 - pw 0 ) psurvival ( a0 + 5) = (1 - pw 0 )(1 - pw1 ) 4 (0.30) (0.3) Survival rates Survival rates Survival rates Survival rates to ( 10 )in (in )( 1) (1 )supposing 4 5 It isItcommon Survival rates It is is common It is common common practice practice practice p to describe to practice to describe a0 survival + survival describe survival describe 1 terms =terms in - pof terms survival in aof1 of terms - pwa survival aa survival survival of rate -p R, rate survival rate supposing R, rate R, 5supposing an exponential anexponential exponential an R, supposing an exponential Survival rates Survival rates survival w0 w Survival rates death-distribution. It isdeath-distribution. common In this practice to to formulation describe the survival probability in terms of aof surviving a survival survival ratet years R, from supposing some an time t0 is exponential given Survival It is rates death-distribution. In death-distribution. In common Survival rates practice this this Informulation describeformulation this formulation survival the the in probability probability the terms probability of of of surviving of survivingsurviving rate R,t t years years supposing It is common practice to describe survival in terms of a survival rate R, supposing an exponential t from from years ansome some from time time some exponentialt time t given isgiven 00is t0 is given It as is common death-distribution. as as as practice to In this thisdescribe formulationsurvival the inin terms of probability a survival of surviving rate R, years t years supposing from some an some exponential time t t0 is given death-distribution. It is common death-distribution. is common It death-distribution. In practice practice to formulation describe this Into formulation describe the survival probability terms of the probability survival in termsaof surviving survival rate of surviving a survivalt R, t years ratefrom supposing from R, an time exponential some time supposing is given 0 t0 is given as In this formulation the probability of surviving t years from some time t0 anis given as as death-distribution. In this formulation the probability tof surviving t years from some time t0 is given as exponential death-distribution. In this formulation the tt probability t of surviving t years from as psurvival ( t ) = 1 - R -1 - ((tt))= (1 )- ò due - Ru - Ru - Rt - Rt =Rue - Ru (0.3) t ò - Ruò ò 11 - pp psurvival survival survival = 1t- R =R t1 - R t due due due -1 - ==e e =e - Rt - Rt (0.3) (0.3) (0.3) some time t0 is given as ( ) -1 ò - Rt - - p psurvival ( t t ) = = 1 1 - - R R 10 due 1 00 - Ru - due Ru e 0= = e Rt (0.3) psurvival ( t ) = 1 - t- R ò due = e - Rt (0.3) (0.3) 39 survival psurvival ( ( t ) = 1 - R- ) -1 ò 1 t due - Ru - Rt - Ru = e - Rt (0.3) ò 0 ForFor a time For aa period time For time timeof period a period 1 ofyear of period1 year 1year p of 1 yearsurvival t = 1 - R 0 due 0 = e (0.31) (0.3) 0 0 For a a time ForFortime period period of of 1 1 year year a time period of 1 year For For For a time aatime period time period of of period of 1 11 yearpsurvival year year pp (1p)survival survival survival (= ( 11)e )= -R -R = (e 1e) = e - R - R p psurvival survival (1) (1 psurvival)(= Þ 1)e = -R eÞ = Þ -R -R - Re Þ (0.3) (0.3) psurvival ( ) (0.3) (0.3) psurvival 1 = e (1) = Þ e -R (ln (1 ) )survival ()ln -)p RÞ= ÞR- Þ R= ln = -- R p ln = ((- ppln survival ( p survival survival (( 1= 1 )= )))- ln (=1)- - 1 - ln =(( 1 -p- 1ln w ww)) w ) (p 1- p (0.3) (0.3) (0.32) (0.3) (0.3) R R=Þ = - R- ln =ln ( p ( psurvival psurvival - ln (1 ) ) (1)(1= = - ) )- ln =ln ( -(ln 1 1-- p (1pw) - ) pw ) (0.3) ( (1) ) = survival a ForFortime period of, for example, R4= 4 years,- ln (p - ln (1 - pw) w survival (1) ) = - ln (1 - pw ) For aa time For time period time of,for of, period a period for of,example, for example, example, 4 R =years, -4 years, lnyears, psurvival For a time period of, for example, 4 years, For a time period of, for example, 4 years, For For a a time time period of, for example, 4 years, For a time period of, period of, for for example, example, years, 4 years, 4 44 4 ( tp4 )== 4 R(1 - -)p 4 a time period of, for example, For ((years, ò òò ò -1 -1 - Ru - Ru -4 R -4 R psurvival = t4 ( 1 ) - )= R )- due =Rue - Ru = ((p- ( ) pw ) 44 4 (0.3) p t= 4 =11- - R due =e e = 1 p w) -1 -1 - - -4 R p survival survival survival 4 t 4 R =4 4 1 -R due due= == e 1 = w 1 w - (0.3) (0.3) (0.3) ( ) ( ) 4 4 ò ò 1 - Ru -4 R p psurvival ( t t = = 4 4 ) = = 1 1 - - R R -1 0 due 0= 1 00 - Ru - due =Rue e -4 R -= 4 R( = 1 1 - - p p ) 4 4 (0.33) (0.3) psurvival ( t = 4 ) = 1 - = (1 - 4 ) 4- (0.3) R due =-4e wp w (0.3) survival psurvival ( ( t = 4) = 1 - R- ) ò 1 4 due - - Ru = e -4 R = (1 - pw ) 4 Ru R ( ) w (0.3) òlog -1 0 In short, the survival ratep Inshort, In short, In short, the is survival thesurvival minusisminus rateis the survival rate t the minus rate = 4 thenatural the is minus = 1 - natural natural Rlog logof the natural 0 due of ofthe thethe 0 1-year 1-year = e = survival survival 1 - probability.p probability. (0.3) log log 1-year of survival the 1-year probability. survival probability. survival 0 w In short, the survival rate is minus the natural of the0 1-year survival probability. In short, In short, the survival survival thethe rate rate is is minus minus the natural natural thethe log loglogof of the the 1-year 1-year survival survival probability. probability. In short, survival Survival models 0, 1 and 2 Survival models 0, 1 and 2 rate is minus natural of the 1-year survival probability. Survival models 0, 1 and 2 Survival models 0, 1 and 2 In short, the survival rate is minus the natural log of the 1-year survival probability. Survival In For short, any models Survival models 0, 1 and 2 any Forany For the potentially For 0, survival potentially potentially 1rate terminal any potentially and is2minus disease, terminal terminal thethethe disease, terminal disease, disease, natural model themodel log model the canof the useuse can can model 1-year any use can of any anyuse survival three of any of three three of probability. survival three models, survival survival numbered models, survival models, models, numbered{0, {0, numbered {0, {0, numbered Survival models 0, 1 and 2 Survival models 0, 1 and 2 Survival models 0, 1 and 2 1, For For 2}. any 1, 1, The any potentially potentially 2}. 2}. parameters The The 1, 2}. parameters parameters The parameters describing terminal terminal disease, disease, describing describing describingthesethemodels thesethe thesemodel model models these models are can given are are models can use below. use any given given are of three below. givenany below. ofsurvival below.three survival models, models, numbered {0, For any potentially Survival models 0, 1 and 2 For any potentially terminal terminal disease, disease,thethe model model can use can any use of three any of threesurvival models, survival numbered models, numbered{0, {0, For 1, any 2}. Thepotentially parameters terminal disease, describing thesethe modelare models can use any given of three survival models, numbered {0, below. 1, For 2}. any numbered 1, The 2}. parameters potentially The{0, Survival model 0 1, 2}. parameters describing terminal The disease, parameters describingthesethemodels describing these model models are can given use these are below. any given of models three below. survival are givenmodels, below. numbered {0, Survival model 0 Survival model 0 Survival model 0 1, 2}. The parameters describing these models are given below. 1, 2}. The parameters describing these models are given below. Given the 1-year Survival model 0 Given Given Survival the1-year Given the Survival model 0 model survival 1-year 0 1-year the Survival model 0 probability survival survival probability survival probability psurvival probability pp (1p )survival survival survival (( 11)) (1) Survival model 0 Given the 1-year survival probability p (1 ) Given thethe Survival model 0 Given 1-year survival 1-year survivalprobability probability psurvival (1 ) psurvival (1) survival (1) Given Given The the the model The The 1-year Theuses model model 1survival uses model uses survival parameter 11parameter uses probability 1probability parameter {R} {R} parameter{R} p survival {R} survival (1) Given the 1-year survival probability p The The model model uses uses 1 parameter 1 parameter {R} {R}{R} The model uses 1 parameter The The The model uses model uses 11 parameter parameter model uses 1 parameter {R} {R}{R} R= RR- =ln = -R( (( (( )()( )))) ( ) ) -lnp ln p =survival -pln 1 1 1 1 psurvival (0.3) (0.3) (0.3) (0.3) -(ln 1) )1 survival survival R= R = R-- ln =ln psurvival p (1 ( psurvival ) ) () (0.3) (0.34) (0.3) R = - ln ( psurvival survival (1 ) R = - ln ( psurvival (1) ) (0.3) (0.3) (0.3) Survival model 1 Survival model 1 Survival model 1 Survival model 1 Survival model 1 The Survival model 1 The model model The The uses uses model model The uses model uses Survival model 1 two two parameters twoparameters two uses {p1,{p parameters parameters R} {p two parameters R} , ,R} {p1, R} Survival model 1 11 Survival model 1 The model uses two parameters {p , R} The model uses two Survival model 1 The model uses parameters two parameters{p11, {pR} 1, R} Given The the Given model Given 1-year the the 1-year uses Given survival two 1-year the probability survival parameters 1-year survival survival probability {p p 1, R} probability probability (1p)survival psurvival and (( the ))and 1 (1)the and 5-year the and 5-year 5-year the survival 5-year probability survival survival psurvival probability survival probability probability p (p5)(( 5)) (5) The model uses two parameters {p1, R} psurvival Given the 1-year survival probability 1 and the 5-year survival p probability 5 survival survival survivalsurvival Given Given the 1-year 1-year thethe survival survival probability probability p psurvival ( (1 1 ) ) psurvival ( and and the the 5-year 5-year survival survival probability probability 1) the 5-year survival probability survival p p ( survival ( 5 5 ) ) (5)(5) Given 1-year survival probability and the 5-year survival probability p 1( 1)p survival Given the 1-year survival probability pp and -(1 )survival p survival 1 = Given the 1-year survival probability psurvivalpp11( survival =- 1=) 11 pand - 1 -= ppthe 1 survival ((1 5-year p survival survival 1 )) survival (1) probability psurvival survival ( 5) p1 = p1 = 1- p11 11p - psurvival æpp ( (1) 1 )(æ1 (5)p)survival p1 = R = -1=- - p ln11 æ survivalæ 1(pp 1 survival survival ) (ö (55))öö( 5) ö (0.35) p1 =RR1== - 1-- R 4 1p ç = survival æ ln ln- p ç ç (1 ) survival survival ln ç( ( 5 1 ) ÷ ö ÷÷ ÷ (0.3) (0.3) (0.3) (0.3) R R= 1 - 1 ln 44æ è p survival æ 4p è p p survival 5 ( )((ø ö ))ö 1 psurvival 51 øø(1) ø survival ( 1 = R- =1 -ln ç æln ç pçèsurvival survival è5 survival survival survival ÷ ö ÷ (0.3) R= - 4 4 ln4è æ è ç p p psurvival (ø survival ( ( 5 1 ) ) ö 1) ø ÷ ÷ (0.3) (0.3) (0.3) R = - 4 ln survival (1) ø ç pè survival ÷ ø (0.3) 4 è è psurvival (1) ø 40 40 40 40 40 40 40 48 // Appendix 40 40 Survival model 2 Survival model 2 The Survival model 2 Survival model 2 Survival model 2 model uses three parameters {p1, R, R>5} Survival model 2 The model Themodel model The uses The uses model uses Survival model 2 three three three parameters parameters uses three {p11,, R, {p parameters parameters R,{p R>5 R >5 1 ,}}R, R>5} Survival model 2 The model uses Given The model uses three the 1-year Survival model 2 survival three parameters {p probability parameters {p1 ,pR, 1, R, R R>5>5} }(1) and the 5-year survival probability p survival ( 5) The model uses three parameters {p 1, R, R >5} ))and ((survival (5 ) (5) survival Given Given the Given the 1-year 1-year the Survival model 2 survival 1-year survival survival probability probability p probability 1 and (1 the ) the theand5-year the 5-year survival 5-year survival probability survival probability p( psurvival probability 5) Given the 1-year The model survival uses three parameters {p1, R, p probability Rsurvival survival >5 } p 1 and 5-year survival p probability survival survival Given the The model Given 1-year uses three the 1-year survival survival probability parameters {p , R, probability p (1 ) and the 5-year survival probability R>5}(1) and the 5-year survival probability psurvival ( 5) p ( 5 ) Given the 1-year survival probability 1 p survival (1) survival survival 1 = 1( 1)p pp survival - and the 5-year survival probability psurvival ( 5) Given the 1-year survival probability psurvivalpp 1) (= 1and 11 = 1 p survival -- 1p= the psurvivalp(( survival 1)) survival 1 - 5-year 1 (1) probability psurvival ( 5) Given the 1-year survival probability p p = 1 (1 - ) 1 survival ( ) ( ) p and æ pthe 1 survival 5-year 5 ösurvival probability psurvival ( 5) p =1 ---p æ( 1 ) survival (1 æ p(( )survival 5)) 1 survival pR1 = 1 ln p 11ç æ survival ppsurvival 1 ÷5 öö ( 5) ö (0.3) p 1 = R 1 R =11 = -4 -p -R ln survival æ èln = p p-çç ( 1 )ln ( ( ç 1 5 ) ) survival ö ø ÷÷ ÷ (0.36) (0.3) (0.3) (0.3) pR R = 1 - -1 1 p ln ln4 æ 4ç æ p survival p è survival è (p 41 ) survival survival ( psurvival ( è 5 5 ) p ) (( ö ö ÷11)) ø (1) ø ø (0.3) R= =- survival ç 5) )÷ 1 survival survival survival ( -1 4 ln æ p survival 1 öö (0.3) 4 ç è p psurvival (10 1 ) ÷ ø (0.3) R>R5 = -1 4 ln1 1è æ pæ ç è æ p survival 1p ( æ 1 5p)((ø ÷ ö 10 ø10÷ ) ) öö(10 ) ö (0.3) RR R = = 55 = 4 5 - 1R - -ln >5èæ çlnp ln =pçp -ç survival survival ln ( (10 survival survival ç 15)))÷ survival øø ö ÷ ÷ ÷ (0.3) R R >5 = = >> - - 1 4 ln 1 ln 5 5çæ æ è p p pèè survival survival 5 p psurvival survival survival survival ( (è 1) 10 10 survival ) p( )( ø 5ö)) ø ö 5 ÷ ø ( 5) ø survival(( )mortality R> 5 = -1 5 ln çç æp psurvival 5 )÷ ÷ survival ö Approximating single-state disease R> 5survival >5 = - 1 5 è 5 ln data è ç æp p p from survival(( survival survival (10 Approximating single-state disease survival data from mortality and prevalence 5) 5 10 )) ø ø ÷ ö and prevalence è p ( ) Approximating single-state disease survival data from mortality and prevalence ø Approximating single-state disease survival data from mortality and prevalence An Approximating single-state disease survival data from mortality and prevalence example R>5 = - 5 is provided here with a standard ln ç life-table survival analysis èlife-table analysis survival 5 for ÷ ø a disease d. An example Approximating single-state disease survival data from mortality and prevalence An is provided Anexample example isprovided An example is providedhere herewith here is providedwith with here a a a standard standard with standard psurvival 5 life-table life-table a standard ( 5) øanalysis analysis life-table analysis Approximating single-state disease survival data from mortality and prevalence fora for for a a disease disease disease d d.. for a disease d. d. è Approximating single-state disease survival data from mortality and prevalence An theis An example Consider example four is provided here following provided here with standard life-table a standard with a states: analysis for life-table analysis a disease for a disease d Approximating single-state disease survival data from mortality and prevalence d. . example An Consider Consider Consider the is theprovided the four Consider four the four here with following following following four states: life-table a standard states: states: following states: analysis for a disease d. Approximating single-state disease survival data from mortality and prevalence An example is provided here with a standard life-table analysis for a disease d. Consider State Consider An example the theis four four following provided Description following here states: states: with a standard life-table analysis for a disease d. Consider the four State State State following states: Description Description Description STATE Consider the four following states: DESCRIPTION State 0 State Consider the four following Description states: disease d alive without Description State 00 0 Description alive alive without without alive disease without disease dd disease d 0State Description alive without disease d 0 1 State 0 alive without with disease Description alive without disease disease d d d 10 0 1 1 1 alive alive without alive alive alive with alive withdisease with without disease disease with disease disease disease d dd d d d 1 2 0 1 alive alive with dead without with disease disease fromdisease disease d dd d 21 2 2 2 alive dead deadwith dead disease from from dead from disease disease disease from d disease dd 1 alive with disease d d d 2 3 1 2 dead dead alive from from with disease another disease disease dddisease d disease 32 3 dead dead from from disease another danother 3 3 dead dead from from dead another another from disease 2 dead from disease d disease disease 3 p 2ik 3 is the probability dead of from disease dead from another anotherd disease d disease diseaseaged k incidence, 3 p pik pik isthe theprobability is dead probability is the from probabilityofdisease of another disease of disease d disease d incidence, incidence, d incidence, agedk aged k aged k 3 ik dead from another disease p3ik p ik wk is the the probability probability is the probability dead of disease of offrom dying disease disease from anotherd incidence, d the disease d incidence, incidence, disease aged aged k d, aged kaged kk pik p pw w k ispthe isthe is probability the probability is the probability of disease probability dying ofdying of of incidence, dfrom from dying the thefrom aged disease disease the kd aged d,, aged disease d,k k aged k pik k is the probability of disease d incidence, aged k w k pw # is the is probability of dying from other the than disease from d d,, aged disease k d, aged kk p ? ik w pwk# k @k$ is the the is the probability probability probability of ofdisease dying of dying dying from from d incidence, the from the than disease the disease aged disease k d, aged aged d, disease k aged pwk ? #@ ? @$ $ #? is theis @ is the $the probability is probability the probability probability ofdying of of dying dying other dying of other from the other thanfrom disease from than fromk disease d, aged k d, aged disease d, aged d,k k aged k # The p # ?@ $ state is is the the probability transition matrix probability of of isdying constructed dying other other from than as than the from follows from disease disease disease d , aged d, d, k aged aged k k #w ? ? @k$ @$ Thestate The isThe state the the probability probability transition transition state of of matrix matrix transition dying isdying is matrix is other other constructed constructed thanas constructed than from as followsfrom disease follows as follows disease d, aged d, k aged k #? @$ is the probability of dying other than from disease d, aged k The The state transition matrix is constructed as follows @ $ state is transition é p0probability the (k + 1 matrix ) ù of é(is constructed dying pwother as ) than follows from ( 1 - pwk d, disease pwk ) k - aged 0 0ù é p0 ( k ) ù k )(1 - p #? The state transition matrix 1 is -constructed constructed as pa k ik follows TheThe statetransition state ê é p00( ké+ é transition p ( k matrix + pú11 matrix ) () ù is kê + é ù is 1( é )( 11 - constructed ù- p p é (1 k)()( -1 1 p- - as )( pp as follows 1 ))follows - pik ( ( ) 1 - 1 - pw p ( 1k- -p p k) ) - p pp kk ) pa k 00úê 00ù ù0éép p00 0 ((k ú ù é))p k ù0 ( k ) ù ù The state transition é p1 p êê( ( k k + + 1 1 ) ) ù 0 matrix ê úúé ( is 1 ê( 1 p-p constructed - ú w )( w k w k1 ) -pik p w as k ) ik ik follows( 1- (1 p- p -p - wk wkp )( 1) - wwk p pa k ) 0 a a w k 0 0 0ù é p p ú êê( ( k k )) ù ê ú (0.3) ú ê p ê( k +(k1 ) ú ()ú ê ( ê (1 ê)( )p ) (( ))( )( ) p0 ú ê ú 0p 0( k ) ((ùú ))p ( k )ú wk 1 é ù 1 - ) úp -1 - p 11- -p --p p 0 0 ù)é 11( 1) p k = é p( ( kk) w k ) pik (wkk1 (p )+p11 k (11 1 )( p p ) ( p p kk) 1 - )( p kk) 0 ê(pp ) kk ê+ + p p p p pkwk p 0 w w w a 0 é p0 k + 1 ù úé - k- k 1 -1 - p ik 1 - k - k - p1 - k 1 - 0 0kù 0 é ú0 p k 0 ù ú1 é ê ê p p ê 2( 0 ( k k +1 + 1 ) ) ú ú ù ê ê é== (1êêê -( 1 p- w wk ê = p 0 )( w k1 )0 w w -p ik pik ) ( ik ik ik 1 -(1p - p- w ww wk p kkp-wkp w )( w w wk wk1 k )- pap a ak ka k ) aa 1 0 a0 0 ú ú ê ê ú p p é0p ê 0 ( (pk k 11 )) ú ú ê ùk ) úú ú(0.3) (0.3) ê êp p1ê( 0ê ( k p + ((k1 kê) )+ú ú11 ) ()úúê ê ()1úp- pêwk ) 0p 0 ( 1 - p wk - p w k )( p 1 - p akk ) 0 1 10ùúê ú ê ú1 ú ê( 2 0ê ( k p ) ) ((ú ú ê)p ú 2 (k ) ú (0.3) p +p k ê(+ 1 ( ) ( wkp )( p ) k wk 1 ê é k22+ 1 ú ù 2= 1 - wk p )( p ik ) 1 -( p - p k)- 1 p 0p ê é k 0 ú ù ê p ë p1 1 1 0 ê (k ê( k + + 1 1 )ú ê) û1 = = úúé ê ê ë ê 1 - ê()1ú- w kp pê 1 - w k ) p ik w 0 k p ik ik 1 (1 - p - - w p k wk p - pww p k )( k ww wkk p 1 - ap w ak ) 1 a k 0 0 1 0 0 0 ù ú ûê ú p ú1 ë1p ú 1 1 ê 0 (k ê( k22 ))ú ú ûk ê)ú (0.3) ú (0.3) (0.3) ú (p ) ( ) k êp ê 3 2 p1ë(p 2ë k k + 33( +(k1 k 1ë) ++ pú ú13=()û ) kûê ê+ 1 ëë( 1û - pë 0 wk p 0 p) wpkk ik pwk (1 - pwk - p pw wk kpp )(w k -p 1 p k ) 1 1 0 000 0úúê1p ê û0 p û 3 2 2ë ë (pk p k 1 33)((û ú k ú ë )pû3 ( k û )û (0.37) (0.3) 3( 1) 3( k) ê p2 k + ú =ê p 0 w w w pwkw w k w k wa 0 1 0ú 1 ê p 2 1 k ú 3( 1) 3( )ú k ik k k k ê ë p2 k ú û ê ë ú û ê ë p2 û +1 + p correctly sum p 0 1 0 (0.3) It is worth noting ë ê ë p2 ( k that ) û ú û the ë ê ë separate p0columns wk wk ptowk unity. 1 1 0úû û ë ê ë p23 (k )ú û û It is worth It is worthIt êis ë p ( 3noting worth 3 k noting + 1 )that noting that ú û the the ë ê that separate the separate p w k columns correctly sum columns separate columns correctly sum correctly p w w k to unity. ktosumunity. to unity. 0 1 ûê ú ë p 3 ( k ) ú û + 1) û p3 ( k that wk wk It is It The is worth disease worth noting ë mortality noting that the equation the ë separate separate pcolumns is that w k for state columns correctly 2, correctly sum sum p to towk unity. unity. 0 1û ë p3 ( k ) û It is worth The The noting disease The disease disease that mortality mortality the separate equation columns is that for correctly state 2, sum to unity. It is worth noting thatmortality the equation separate equation is thatis columns for thatstate for2, correctly state sum 2, to unity. It The is is disease worth mortality noting that equation the is that is separate ( for state columns ) 2, correctly( )+ sum to unity. 2 (k ) The It disease worth mortality noting that equation the separate that p for columnsk + state 1 = 2, correctly p p sum k topunity. (0.3) The disease mortality equation is that 2 for state 2,wk 1 The disease mortality equation is that p p for 2 ( 2(state k k+ + p1 2) 1 () 2,k= =+ p p1ww) kkp=11( p p( kk wk ) )p ++1( p p k2 (k 2)( +k)) p ( k ) 2 (0.3) (0.3) (0.3) The The disease disease mortality mortality equation equation is p that isp 2 ( that k for + 1 state for ) = p 2, state wk p 2,1 ( k ( k + 1) = pwk p1 ( k ) + p2 ( k ) ) + p 2 ( k ) (0.3) (0.3) p2 2 ( k + 1) = pwk p1 ( k ) + p2 ( k ) (0.3) p2 ( k + 1) = pwk p1 ( k ) + p2 ( k ) (0.3) p2 ( k + 1) = pwk p1 ( k ) + p2 ( k ) (0.38) (0.3) 41 41 41 41 41 41 41 41 41 49 Reducing Tobacco Use Through Taxation in Trinidad and Tobago: Modelling the Long-Term Health and Economic Impact The probability of dying from the disease in the age interval [k, k+1] is The The probabilityprobability ofdying of ofof from dying from thethe diseasethe in disease the age in the age interval interval [interval k+1] k,the [is p [ p k kp ,+1] +1] k pis (k kis p) ,pwhich pkp k() (is k is p,) , which not (not to be( ), which is not to be is )is toto not be k pis k The The probability probability The dying dying probability from from the of dying disease from disease inthe indisease the the ageage in interval age [k k ,kk interval 1is [k ,wk +1] which is not , which be is not to be k,+1]+1] p(k )p ( 1) which w k w 1 The probability The probability confused with of dying of dying from the fromdisease the (cross-sectional) the disease in the age in the disease mortality, interval interval age [k, k+1][ k,is p w p1 is ,k k which is not , which w 1to be to be confused confused with the (cross-sectional) (cross-sectional) with the with disease disease called mortality, pmor( mortality, pmor k). p wk (k1 called calledmortality, (k). pmor ).wk 1 not beconfused to confused confused confused with the with thethe confused with the (cross-sectional) with the (cross-sectional) (cross-sectional) disease disease (cross-sectional) disease mortality, (cross-sectional) disease disease mortality, mortality, called mortality, pmor(kp calledcalled (k). pmor(k). mor called called ).mor(k). p p1 (k) is known pp 1(k) is known (k as 1 (k ) as ) is is the known the known disease disease p 1 (k isThe ) as as prevalence, the the known disease prevalence, probability disease as the p prevalence, ppre of dying prevalence, disease pre (k ( k). Hence ).prevalence, Henceppre from ppre (k the ( the ).k). the relation Hence relation disease Hence the ppre(k the ).in relation the Hence theinterval age relation relation[k, k+1] is pwk p1 (k ), which is not to be p1(k) is p1(k) is (k)1 is known p1known known asas the as the the disease disease prevalence, disease prevalence, ppre(k). prevalence, ppre pp Hence (k). Hence the relation Hence the therelation relation 1 (k ) erval [k, k+1] is pwk pThe , which is not probability to be from of dying confused with the the disease the age interval disease in(cross-sectional) [k, k+1] is p p (k mortality, ), which called pmoris (k). not to be p pmor (wk )=p p (k ) ( k ) p mor ( k ) p (k ) wk 1 wk = morp mor alled pmor(k). confused with the (cross-sectional) disease pwk = pp (kk p pwcalled mortality, ( = ) ) ( mor k mor pp k p ) ( k= ( ). ) p(k). Hence (0.3) (0.39) (0.3) (0.3) (0.3) (0.3) p1(k) is known as the pk ( k ) p ( k disease prevalence, )k ppre the relation mor p = p = mor pre ( ) pre k wk pre k (0.3) (0.3) the relation wk w p pre pre (k p)pre ( k ) pre 1( )k p 1( )probability is known as kexponential the disease prevalence, For The Forexponential ofsurvival dying survival For exponential For exponential survival thep probabilities, from the disease probabilities, survival probabilities, the inpre(k the probabilities, ).age the probability Henceofthe probability theinterval probability of dying relation of dying from ,dying [kfrom probability k +1] of isfrom the dying disease thefrom pdisease kppthe from k( in thedisease )which , the in is the not to age-interval disease in the age- be age-interval For exponential survival probabilities, For exponential survival the probability probabilities, of dying the probability mor in the of dying w the age-interval disease from indisease the the age-interval in the age-interval For [ For exponential exponential survival survival probabilities, probabilities, the probability the of dying probability of dying p disease from from the =the disease in thein age-interval the age-interval (0.3) ( k ) k k, k+1] ,k interval+1][is confused [ k[,k is denoted k [,k k,+1] +1] with k+1] denotedis as is p Wk and denoted isdenoted the (cross-sectional) as Wk and p+1] is as as is given pW given by and the is is and k disease by the formula given is by the given formulaby mortality, formula the calledformula pmor (k). wk p pre ( k ) mor ( k ) [denoted k, k isas pW k andas denoted given p Wk and by isthe formula given by the formula [k, k+1][k,is k+1] is denoted denoted as pWkas (0.3) and pWis given k and by the is given pformula by the pformula wk = (0.3) k) p k) - Rk ( the p 1(k) is known as the disease prevalence, p k ( k). Hence - Rk(R relation ) ( ) - Rk p = 1 - ep = Þ 1 - e R = Þ- ln (1 - =p - ln ) 1 - p (0.3) (0.40) in (0.3) pre (1 ) (1 - pwk ) pre For exponential p wk = 1 - esurvival pw- - k R= w R k 1- Þ e pprobabilities, - R k wk = kR k1 k = Þ --- ln eln 1the Rk =Þ - k probability p -w ln w k R k -=p - lnof w k dying from the disease (0.3) the (0.3) age-interval (0.3) f dying from the disease in the age-interval [ k , k +1] pw k wk is = p 1 - wk denoted e = 1 k - as e p -R Þ and RÞk is = R given k = (1- by - ln the p( 1 wk ) - kp formula wk )w k (0.3) (0.3) k ) probabilities (survival Wk For When,exponential as is the survival case for probabilities, most cancers, the theseprobability survivalpmor of dying probabilities from the disease in the age-interval When, When,as as isWhen, is When,the the as as is is case case the for for the When, case most most case as isfor for cancers, cancers, the most most cancers, for these these casecancers, mostsurvival pthese these survival these cancers,probabilities survival these are known, are probabilities probabilities survival known,are are the are known, the probabilities microsimulation known, known, are the the microsimulation microsimulation the microsimulation known, the microsimulation k = survival (0.3) When, When, [k, kuse +1] as is is as the denoted is the case case for pWk astheymost for not and most cancers, is given these cancers, by the survival formula probabilities probabilities are known, are the known, the microsimulation microsimulation will them; will use them; will will use when use when microsimulation them; them; they will are when are when useare them; notthey they when known known areare they not are or not orknown not w are known are known too too p or old or old orofare are pre (to arek to p too ) too be too be old = of of 1 old old -any any toe k to be of any to - be use, Rbe use, k wk use, of the of Þthe any will use them; when they are not known or are too old to be of any use, the microsimulation anymicrosimulation microsimulation R use, use, =thethe - ln (1 - p uses microsimulation ) uses microsimulation uses uses uses (0.3) will will use them; survival when use them; when they not they are not known or areor known are too too old to be to old wk of be any use, anythe the microsimulation microsimulation use, uses uses = - ln (1 - pwk ) survival statistics survival statistics survival inferred from statistics inferred statistics inferred from the prevalence from thefrom prevalence (0.3) inferred survival statistics the prevalence the - Rk prevalence inferred and and mortality from the and mortality and statistics mortality statistics mortality prevalence and (equation statistics (equation statistics mortality (0.3)). (equation (0.3)). (equation (0.3)). (0.3)). statistics (equation (0.3)). survival the microsimulation For survival statistics exponentialstatistics inferred survivalinferred from uses pthe wk = from survival prevalence probabilities, When, 1 - the as is the eprevalence statistics the probability case Þ and inferred for R mortality and mostk = of from dying cancers, (1 - lnstatistics mortality- the from pthe thesewk ) statistics (equation (equation prevalence disease survival in (0.3)). (0.3)). and mortality the age-interval probabilities (0.3) are known, the microsimulation An alternative An derivation alternative equation derivation (0.3) equationis as follows. (0.3) is Let N k be the number of people in the obabilities are known, An [ k,the kalternative statistics+1] is Andenoted (equationderivation alternative microsimulation as An pW equation and derivation alternative (0.39)). kwill is given use (0.3)by equation derivation them; is the when as follows. formula (0.3) equation they asas is are Letfollows. Nknown follows. (0.3) not k be is asLet Let the Nnumber follows. or Nbek be the k are the Let too of Nold number k people number be to the be of of in of people the people number any inin of use, thethe people in the the microsimulation uses An When, alternative An population asalternative is the agedderivation case k population andderivation for most let aged equation n k be equation cancers, and the (0.3) letnumber n these beis (0.3) as the survival of is as follows. people number follows. Let probabilities in of N the Let people k be Nthe population in k be number are thethe of number known, aged population kpeople the withof in people microsimulation the aged the disease. k in with the the disease. population to be of any use, the population aged population microsimulation k and population uses let n agedsurvival k k and be k aged the let nnumber be the k kk and statistics let of number nk be the inferredpeople of from in the people number the population in people the population of prevalence aged in the and k with population mortality the aged statisticsdisease. k with the aged kdisease. with the(0.3)). (equation disease. will Then, population use thethem; aged number k when and aged of k let they and deaths nk from let are benot n the benumber kknown the the number or disease of are people of tooofold people people intothe be aged in population of k theany can population use, be aged the given in aged k with kthe microsimulation two with ways: the disease. as p disease. uses kn An alternative Then, rtality statistics (equation the (0.3)). Then, number Then, thethe of number derivationdeaths number Then, the equation of of frompdeaths wk the deaths number =1 of -from e - Rk disease (0.39) from deaths the the isofÞ asdisease people from R follows. disease the kof=of- aged people ln Let people disease k(1 can N- aged of aged pbe be wk k k people )the k given can can inbe number be aged two given given k can ways:of in be in two two as ways: pw given people ways: wk nk in asas (0.3) in ktwo p pnwknk k k as p wknk ways: Then, survival and, the Then, the number statistics equivalently, and, number of inferred as pdeaths equivalently,of( An deaths from k ) Nfrom as. the p alternativethe from prevalence Observing ( k )disease N the . disease that derivation of Observing and thepeopleof disease that equationpeople aged mortality the aged k prevalencecan statistics disease (0.3) is be k as cangiven is n be (equation prevalence follows./N in giventwo leads Let in (0.3)). is Nnways: totwo / N be the theas p ways: equation leads number wkas tonkthe nw p wkof equation k people in the and, equivalently, as pmor and, equivalently, mor (k )asNkp k Observing . mor (k)beN k. p that Observing the disease thatof the prevalence disease is nk/Nk prevalence k k leads is nk/N to k the is equation leads nkto kthe equation mor k k k and, kequivalently, as (k )N k. Observing that the disease prevalence k /N leads to the equation theand, When,population equivalently, and, equivalently, is as in aged the case pmor as for and as(kp most )N kletk)n .(Observing Nk cancers, the mor k. Observing thesethatnumber letdisease that the survival nthe people prevalence be disease in prevalence the nkpopulation ispeople/Nisknleads k/N k leads to aged the to the kwith equationequation population aged k and k probabilities the number are ofknown, the in microsimulation the population aged k with the disease. mor et Nk be the number of people the An alternative derivation equation (0.3) is as follows. Let Nk be the number of people in the the kdisease. will use them; Then, when the they number are not of deaths known orp n k from are =p = k too pfrom the old ( (n tokk) ) disease be=N p disease of any (N (= of )p kuse, N people k the aged microsimulationk can be given uses k ) (k ) n the population aged with theageddisease. nk bethe Then, thenumber pW kn of deaths p kn morWk k= N the pp k kWmor n k N of people Nwith aged k can be given in two ways: as p wknk k =( k ) ( kk) population k and let number pW of people in the population aged k the disease. Wk nkandp n Wk p k = Wmor kp N mor kmor k survival statistics inferred from the prevalence nmortality statistics (equation (0.3)). nkn mor mor k k aged k can be given in two inThen, ways: twoways: as p wk as nkdeaths and,and, equivalently,as equivalently, as pmor(k Nk. Observing )k Observing that thatthe thedisease prevalence is n /Nk leads to the equation the number of from the disease p pre ( kof) n people k (k )kp aged k kcan be ngiven in disease two ways: prevalence as p wknk k is p pre (k )= = p p pre N k( k ) n npre = = ( k) = k )pre (disease as(follows. N) ase prevalence is nkn /N k leads to the equation /Nk and, equivalently, as pmor(equation k)Nk. Observing p that k pthe= k = N pre prevalence is nk/Nk leads to the equation An leads to the alternative equation derivation (0.3) ispre k Let N kN bek the number N k of people in the k k k Nk Nk pWk n k = pmor ( k ) Nk (0.3) (0.3) (0.3) Þ (0.3) (0.3) population aged k and let nk be the number of people Þ in Þ theÞpopulation Þ aged k with the disease. (0.3) (0.3) (k ) Nk p n Þ = p Þ ( k ) N nk Then, the number of deaths from the disease of people W p k k = p pmor mor morp ( k k )= aged kp k mor can pmor (pk(pre k )be)(pk )= given ( k in two ways: as p wknk ) pWthe and, equivalently, as pmor (k)Nk. Observing that W k p = disease np pW( (Wk kk p= )prevalence ) k mor (pp) k Wk ( = k ) mor is n N equation k/Nk kleads to the p (k ) mor p k =p k = pre p pre ( k )= prep ( k ) p pre ( k ) (0.3) (0.3) Wk p pre N pre k (k Wk p)pre (pre k) Þ Approximating multi-state disease survival k )survival pmor (from data incidence and mortality, (0.3) assuming (0.41) Approximating Approximating multi-state Approximating multi-state Approximating disease multi-state pWk nk Þ disease survival disease multi-state = survival data survival disease from Ndata from k incidence data from incidence data pWk = incidence and pmor from ( ) and kand mortality, incidence mortality, assuming mortality, and assuming assuming mortality, assuming Approximating Approximating no remission multi-state no remission multi-state disease survival disease data data survival from from incidence incidence and mortality, and mortality, assuming assuming (k ) no remission no remission no remission no remission no remission p pre (k )that = nk ( k ) p p pre ( k ) Disease Disease mortality Disease statistics mortality give the the probability statistics give the a ak person probability that will die a person from the will diedisease from in a the given disease year in a given year mor (k ) mortality Disease statistics mortality give statistics Disease mortality probability give the statistics p = the kthat N probability give person that a probabilitywill die person that from will the die a person disease willthe from diein a disease fromgiveninyear the a given disease year in a given year Disease of Disease life. mortality They mortality make statistics no give statistics reference give the to the probability probability when the W that disease pafrom( that k person) a person will which die will the die from person the from disease dies the was disease in a in given contracted. a year given year ofof life. They make no reference to when the disease from which the person dies was contracted. Approximating multi-state disease survival data from incidence (0.3) was mortality, diesand contracted. assuming pre of life. They make life. They reference nolife. of make no They to when reference make the to disease when no reference Þ to from the disease when which thefrom the person which disease the from dies person which wasthecontracted. dies was person contracted. of of life. life. make They no reference They make to when no reference to when the disease the disease from which from which the person the person dies wasdies was contracted. contracted. from incidence and mortality, assuming no remission Approximating Approximating Disease Disease survival Disease survival Disease multi-state multi-state statistics survival statistics survival give give disease disease the statistics the statistics probability give probability give survival survival the the that that probability data data probabilitya apperson personfrom (from that ) awill that k aincidence willincidence person person die die from from willwill and and the the die die disease disease from mortality, mortality, from the in the in a a disease assuming assuming given disease given in year in yeara given a given year year Disease survival Disease statistics mortality give statistics the probability give the that probability a person that a will die person from will the die disease from in a givenin year yearthe disease a given year mor Disease Disease no remission survival statistics survival statistics give the give probability the p that probability = a person that a will person diewill from die from the disease the disease in a in given a given year noof of remission life life given given of ofin that life that life they given they givenofyear contracted that contracted that life given of they they life. the disease contracted the contracted that They theydisease make the in theW in contracted no k an disease an disease reference earlier p earlier the in (in k an year. an ) year. disease to earlier earlier when inthe year. year. an earlier disease year. from which the person dies was contracted. erson will die from the disease of lifeof Disease given life a mortality given that they given that contracted they give contracted statistics the the the disease disease probability in that in pre a an an earlierpersonearlier year. willyear. die from the disease in a given year m which the person Disease dies The was mortality contracted. connectionThe statistics between connection the give two between isthe provided the probability two isby the provided that equationby athe person of the equation willof form die the fromform the disease in a of The life. They connection Approximating make no between The connection reference the two between The connection multi-state Disease to iswhen theprovided between disease survival two the disease isthe provided survival bytwo statistics the from equation is data give by the which the provided from of equation by the incidence probabilitythe person form of that the equation adies and form person was of thecontracted. mortality, willform dieassuming from the disease in a given year The given connection The year connection of between life. They the between make twothe no istwo provided referenceis by provided to the by whenequation the equation the of the disease offormthe from form which the person dies no remission diseasesurvival son will die from theDisease in a given of life given that they contracted the disease year give the probability that a person will die from the disease in a given year in an earlier year. statistics p ( (aa) å ( ( ( på) ( ( å) ) ( )pa a p a( a 0) (the )0 )p ( å () = = pmor a (a a pinc a er year. contracted. was Disease mortality statistics of life given that they contracted give the pmor pmor the probability (a disease )p ) =mor pbetween(aa in0< å ( å å )a a that = an p p)w= wa ) ) = a a earlierp mor a0 person a aw ( 0 p aw aa )w p p iK 0 the then 0the )(,K, K p0w> then the (a (0.3) )), =p0 00the w ( a0 ) = 0 . . disease. (This analysis focusses only on the identified disease =1 and 0 gets the not disease allow age 0 for the possibility has the disease they can possibly from that that they the they die disease. die from from other (This other causes.) analysis causes.) Suppose focusses Suppose they theyonly acquire on acquire a =0 the è the the k disease identified disease at at ø age disease age a 0 00 in inandstage stage doesK, K, then thennot the theallow Once change a person disease hasinitial the stage state disease or they At vector initial they can subsequent state canis die At from subsequent determined vector possibly ages, isthe the from determined change state ages, the disease probabilities the initial fromstate stage probabilities conditions the or are given initial they can p i( conditions (a a 0 i) by = are die )(= the given fromd ) p recursion (the aby( ( ) 0 Kthe =0( d > )> equation recursion 0 , (0 p K )w ,((> ) )(= equation a 0 = ,a000p)w=. (a )=0 . 0) , K, w) a in state initial that K initial state given they state die vector as from vectoris other is determineddetermined causes.) from Suppose from the the initial they initial conditions acquire conditions the p disease p a at d age= 0 d a K 0 > K 0 p ap0 . 00 . i( 0) ( 0 in stage ) then the w( 0) initial state vector is determined a from the initial conditions d . 0 ) focusses iKi 0 iK 0 a |a nalysis The probability The only probability on of not the ofdied identified having dying 0from disease from for thethe andthe initialdisease 0 diseasedoespossibility state not and in stage allow vector being that is K for in at they age the determined stage die given possibility K at from from age that other the the initial causes.) conditions Suppose p p i (a a = 0 ) they = 0 d iK0acquire iK0 ( iK0K K 0 > > 0 0the 0 ) , , p disease p w ( a a 0 ) = =at 0 0 . disease. (This analysis At Disease survival focusses subsequent At only ages, subsequenton the the stateidentified ages, the disease probabilities state and are probabilities does not given by are allow the i for recursion given by 0 the possibility the equation recursion iK0 0 equation w 0 a = a0 -1 K At subsequent Atstate subsequent ages, ages, the the state state probabilities probabilities are are given given by thebyp the recursion recursion equation equation æ i ( a0 )from ( they 0)conditions ( a0 )the disease wasthey contracted at age initial a0 and that vector the person is determined was alive from at age the a-1 initial conditions 0 (.a , K ) = om other a , given causes.) that Suppose the disease was that acquire they diethe contracted from disease age At at At subsequent a other 0 in Once at in subsequent age state stage a person causes.) aK 00 ages, in has K, ages, Supposestage then the thethe state K, the theydisease statethen probabilities the initial theystate probabilities acquire the a can = a0 -1 are are given vector possibly disease æ given by the by determined is change recursion the disease recursion K at age a0 in stage K, then the ö recursion = d stage equation iK0the equation or K 0 >can initial , diepwfrom =pinc 0 0Õ a =0 è å ç1 - or is determined from the initial The probability of being initial dead conditions state (from vector the piAt (a is disease) subsequent disease. (This 0 ) = d iK0 determined at age (K a ages, from , analysis 0 > the given 0the )p,inc that (w state focusses p a( initialthe 0,a probabilities K0) 0) =only conditions= 0 . on Õpç è are the 1 ( a å given -identified ) = pd0k ( a ) by the disease ( ÷ K ø p0 K 0 > 0) , pw ( a0 ) = 0 0 and does equation not allow for the . (0.3) possibility k ase incidence that they die from other causes.) Suppose they acquire the disease a = 0 i 0 k =1 iK0 at age a0 in stage K, then the ages, the state probabilities At are given by the subsequent recursion ages, the state equation probabilities are the given by the recursion equation Disease survival disease robability was that a contracted person, who atat a agein state 0 does K0 not have the disease, first gets disease at age 0 Disease survival initial state vector is determined from the initial conditions piOnce ( a0 ) = diK0 ( K a person has 0 > 0) , pw ( a0 ) = 0 the disease they can . possibly change state K0 given as disease. (This analysis focusses only on the identified dis The probability of first getting the disease in a Once a person At in has subsequentthe state K0 givendisease ages, they no the can state probabilities disease possibly at change disease stage are given by the recursion equation or they can die from the 0 disease. (This analysis focusses only on the identified disease and does not that allow they for die the from other causes.) Suppose they acquire t possibility age 0 a = a0 -1 51 æ dieKfrom other ö causes.) Suppose they acquire the disease at age a in stage K, then the 43 43 pinc ( a0 , K0 ) that= they a =0 è Õ ç 1 - p0 k ( a ) ÷ p0 K0 å ø a givenfrom (0.3) initial 0 state vector is determined from the initial conditio The probability of dying from the disease in stage k =1 K at age that the the initial conditions pi ( a0 ) = diK At(subsequent 43 43 43 initial state vector is determined K0 > 0) , pages, w ( a0 ) =0 the . probabilities state 43 43 are given by t 0 43 disease was contracted at age a0 and that the person was alive at age a-1 ase survival At subsequent ages, the state probabilities are given by the recursion equation 43 Reducing Tobacco Use Through Taxation in Trinidad and Tobago: Modelling the Long-Term Health and Economic Impact At subsequent (a p æ p æ +ages, (a 1 the | a+,1K K ) probabilities | a) state ,ö ö æ p æ | a(given ( aare p aK , by)the | a) , ö K ö recursion equation æ p0 (a +0æ1 |p a0 (0+ (,,1 K+ 0) 0 0 |ö 0 ÷0 )ö ö æ p0 (a |a 00 0 ,p K0 () 0 |ö 0 ÷0 )ö ö 0) (0| a 0) ç 0ç a 1 ÷aK ,K ç 0ç æp a ÷ aK ,K 0 (a 0) 0 (a 0) æ ç p p aæ +1 p + 1 1pa ( ||0aa 0a 0, K K+| 1 a | ,a ö K ÷0 ,0 ) æ ç pp aæ |a p | a ( 0, a,0K K 0 a | ,a ö K ÷ 0 ,0 ) ç ç æ ç p p 1 ( ç 0 (a a a+ ç + +1ç1 1 | p||1p a a( 0 a , (,,a K K+ 0 0) 1 ) 0 1 |a ÷ ÷ öa |÷ 0 K,K ÷ )0) ÷ ÷ ÷ ç ç æ p ç p 1 0 (a ( ç a ç| a || aa 1ç a0 0 p, ,,p KK(1a 0 () ) 0 a|a ÷ ÷ öa |÷ 0 K,K ÷ ÷÷ )0 )÷ ÷ p a + K 0, ( () p ) 1K 0 ,0 1 0 0 1 0 ç ç 1 ç ç ç .. 10 0 .. 0 ÷ =0 T 0÷ a=÷, ÷ = T ( a , a0 ) ç T a aç , ç 0 a 1 ç ç...ç 0 0 ...0 0 ÷ ÷ 0÷ ç ( .. ) ÷ ( (, 1a0()p ... ) ÷ a) ç p ç a + ç1 ç +.. | a , K .. ÷ ÷ ÷ = Tç(p = çT pa , çaa | aç , K ÷ ... ÷ ÷ ç ç( (a a) , K0 ) ÷ ÷ 0.. ÷ = T ( a ,a ÷0 ) ç a 0ç ... (,,aK 0 ... a) , K0 ) ÷ ÷ ÷ ÷ 0 ç çp 1 p a Nç-11 |a 0 + 0 ,1 K |0 0÷÷ ç ç N| ç -a 0 |0 0÷÷ ç p N -1 ( aç + p.. 1 | a (10,a , (,KK ) ÷ = T (a ÷, a ) ç pN -1 0 ) ÷0 ç ç p N -1 ( aç | ... a10 K ( ) ÷ ÷)÷ 0) (1|a 0) ç (aa p a +0 1 )| ÷ a |0 ,K ( p a|÷ |0a0K ,K ç ç pp ç ( ç ( 1 1a |a + K 1 ) a ÷ ,0K 0) ø ÷ ÷ ç ( aa ) |p |ç a ( ,,1)K a)0 ÷a K,0 ÷ N -1 p ç++ w1 N a |-N + - 1 |0 a 0 ,ø K ÷ ç p p ç aN0 0 a N - K 0 ÷, 0÷ ç è p Nw è ( -1 a a 00 K ) 0 ÷ ç è p Nw -1 è ( a ) a - K 0 ÷ ø ø ((aè+è1 |)a (( ))|)ø +ç 1 | ( , ÷ w | ç 0 , 0 ÷ 0) ø (0a p aç + 1 | a (a , K ÷ p aç | a , K ÷ |pa ,a 0) è ç p ++ 10)1 |0 ø ÷ a ,K è p p(K |0a ,K pw ,0K ø ç )è |p a ÷a ,0K w w è N -1 w 0 0 K 0 N -1 aè a0 w ,w 0 0 0 0ø ø ç pw ( a + 1 | a0 , K 0 ) ø ÷ è ç pw ( a ) | a0 , K 0 ø ÷ è w 0 0 ø è w 0 0 ø ææ ææ ö ö ö ö æ ç æè ç æ ç 1 æ1 - å - ç åç 1 æ æ p - æ kå æ pkk0 ö ö-å p 0 ÷ k0 ÷ ö ö 0 0 0 ... ... ... 0 0 0 0ö 0ö ÷ 0÷ öö 00 0 ÷ öå ç è 1 ÷ >ø pø p 0÷ 00 ...... 0 ÷0 0 æ 1 -çå >0 ç1 - ç ç çè æ ç çkk >ç0ç pç k0 ø ÷ 0 k 0k÷ 0 ... ÷ ÷ ö ÷ ÷÷ è ø ç ç ç è 1 - å k >ç0 p ø ç çp ç p ÷ æ 1 - 1 - èç k 0 k >k0>0 æ ø å æ p å ö ö ( p 0 1 - ö p( 1 1 ( - a p | 1 a ( ) a ) | a ... ) ... ) ... 0 0 0 0 0 ÷ ÷ ÷ 0 ÷ ÷÷ ç ( ÷ ÷ ÷÷ p) è ç ø ç 1- å ç æ ÷ ÷ ö 1 (a ) ç 0÷ è>1 æ ö 1 ( a)0)) ) ...... 0 1 p0,1 k > 0 p1,k1k 1 -ø k pw w |aw 1 ... å 0,1 0,1 1, 1, 0 0 T ( a ,T a ()aº ,a ç 0 ) º çp ç ç pæç è >ö÷ øå - ( p ( - )( (a |0 00 0 ÷ ÷0 0 T ( a , a0 )Tº ç ç ç,a ( ) ç ... æ p0,1è ç 1 0,1 - å k çp1ç è ö - k 1 ø ÷ 1 -p1, pk1,÷ k( ÷1 a -| a p01 ) w w a |a ... 0 ÷ ÷ ÷ ÷÷ ( ,0 a)0ç º ( øa | a ) ... ... ) 0,1 k 1, k w 0 ÷÷ > 1 T ( a , a0 ) ç ÷ ÷ ç å 1,k ÷ ... w T º aç a ºp...ç è 1 - èp ø k ... >k 1> 1 -1 p ø ... 1 ( ... 0 ... ... 0 ÷ ... 0 ç ... ç k > 1 ... ... ... ÷ ç ç ÷ ... ÷ ÷ ÷ 0,1 0 T ( a , a0 ) º ç ç ç ... çp ... p p1 -... ... è ø ... ... ... ... N -... N -1 ... ... ÷ ... 1, N -1 (1, N -1p ( ) | a0 ) ) ... ( ( ) ) k > 1 1( ( ) ... 1 ... ÷ ç ç0, 1( ( ) ) ÷ 0÷ ÷÷ 1 1 1 p ç -ap |w a0 a ...- p 1 - paw | a0 a | a0 0 p p0, N -1 1, N -1 (1 p) ç ç p ... N -1 0, N -1 p 1 -p pw ... 1 (a 1 w ( |a ) ...) ) (1 - p N- w 1( ...(Na- |a 0-)1) a 0|÷ ( ... ÷ ÷ )) ÷ ç ç ç pç0, ç0, 0N -1 0 0, N -1 p1, N -1 ( Nç -ç ç 1 pw 1p- 1 ( ap p 1, N1,- N |w1- 1 a ((10(1 a ) a 1 - - | |aa p01 0w ) ))w 1 (a (a |a |0a)0) ... ... (1...-... p p N ... - N- w 1 (1 (pa(1 -a |-1| a pa ( p N ) w0w a N ) | )a - 1 (a )|0 1 a÷0) ÷ 0) a ÷0 (0.3) 1 0 ÷ ÷ (0.3) ÷ (0.3) 0 0 1, N -1 ( 1a ) ) (1 - 1) w (0.3) ç è è ç (a (p ) (a )10-() ÷ ø ø ÷ ç 1 (a N a÷ 1w 1- 1 p0, 0 ç -ç p p 1 1- p|wa a p|w (0a 0 a)0 ) ... pw wN- p w |a | 0a 0 (a 0 0 |1 ÷ (0.3) 0) N ÷ è ç ç è 0N è è1 p w w ( a | a 0 0 1 w) ( a |a |0 ... ... pw N... -w ( a pa | p N w0w - ) a |a 0) 1 ø ÷ ÷ ø 11 (0.3) ø ø Where, Where, for survival ç model for survival 0 model 2, 2, p 1 ( a | a ) ... p N -1 ( a | a ) 1 ÷ (0.3) Where, for survival è model 2, w 0 w 0 ø Where, Where, Where, for for for survival survival survival model 2, model model 2,2, (0.44) Where,for Where, forsurvival survival model model 2,2, ì pw ì pw K K0 ì p ( ( K a a = (0 0 ) a a = a ) ) p0wa = w0 K 0 (a ) ) ) ï ì p K0ï ì ( ì apwK = a0 (a = =a a0 ) pwK ( ( ap|K a( ) a w 0 K í ) | = a ï ) p K = w 1í ï 0 p( ( K a wï <( a 0 £ < aa +£ 0a 4 ) + 4 (0.3) (0.3) pw K K a | pw ( a | a a p K p0w ( ) ï = =a( p 0 w ï ì í a | ï a p| K K a K w )0ï) = =((a a 10 K 0 a p í = p K< < aa a 0( £ £a( a a a 0 < + < +a 4 a 4 ) £ 0 £ ) 0 ) a a + +4 ) 4 ) (0.3) (0.3) (wa w1 (4 0 < a) 0 1 0 0 w í p0 w K 5î p 0 í K 0 + w 1 a0 <+ a4 0 0 0 (0.45) (0.3) (0.3) î ) w K1 50 pw ( a | a0 ) = ï ( ï K K p w ï a + < 4 a < £ (a + 4 ) w 5 ( a0 0 K í î pw1 ( a î ï pw K w K5 5 îî p0 p K 0w 5 4 < a 0 + a 0 4< + )a ) < a (0.3) Disease mortality Disease mortality Disease mortality î w 5 ( a 0 + 4 < a ) Disease mortality Disease mortality Disease mortality K (p (a ) is ) is given Disease mortality The probability The of dying of dying probability from from the disease the disease in stage K at age in stage K at aage, is a , is denoted denoted K pmor K amor given by the by the The probability The of probability Disease mortality dying The probability from ofof the dying dying disease from from the in stage disease the disease K in at in age stage stage K a K , at is atage denoted agea a , is p denoted denoted ( a )pis KK p given ( ( a ) by ) is is the given given byby the the mor ( ) mor The Theprobability probability equation equation of of dying dying from from the the disease disease in in stage stage K at ageK at a , age is denoted, is denoted p mor K a is mor a given by is given the equation The probability equation of dying from the disease in stage K at age a , is denoted pmor ( a ) is given by the equation K by the equation equation equation K0 = N -1Ka0 =N 0= a -1 a0 = a -1 K pmor K (a pmor ( p) a K ) =(K = mor Ka0å ) å = N -1 a0 = a -1 K =N=-1 aå å K0- 0 =a K =1 0 å N 0 p = p -å N 1 w -1 Ka0 ( ( a =a 0 a =0 åp ap -| = a 1 |w -1( , K a a0 aK ,K | a) K 0) 0p, K0 p|K ( ( ) p- a a K1 - 1 (a| a- | a0 ) 1p ) |a p- 0) inc ( ( a a p0inc ,K 0 ,pK (a0)0 , K0 ) 0)a (0.3) (0.3) (0.46) (0.3) K p p K pmor ( a ) =mor K0 K0å K( mor0a( =1a ) K0 =1 0 ) = N -1 a å = a0 =å a K== a -1 pw ( 0 =1 0 awK 0 å a | a0w 0Kp ( w a( 0 a , K 0 ) pK | a a 0 ( a 0- 1K K ,0K, K)0p) p ( | a0 ) pinc ( a K a 0( a- 1 1 | inc a| a ) 0 0, K 0 ) p ( 0) 0 incinc ( a , K ,0K 0) ) 0 (0.3) (0.3) (0.3) Estimating survival Estimating survival Estimating survival K pmor K =1 a =0 (a ) = K å å w 0 =1 a0 =K K10= 0 0 =p K (0a0| a , K ) p 1 a = a0 =0 0 0 K ( a - 1 | a 0 ) p inc ( a 0 , K 0 ) (0.3) Estimating survival Estimating survival 0 0 Estimating survival Estimating survival When When When the the the disease disease disease mortality mortality mortality is is known, known, is known, here heredenoted here denoted K (p denoted K pmor K (Ka )sets ) ,, the amor ,, the the sets of survival setsof survival of parameters (3 for(3 for parameters survival parameters When the disease When When the Estimating survival mortality disease the disease is known, mortality mortality here is known, is known, denoted here p here denoted( denoteda )p the p ( sets ( a ) ,) ,of the survival the sets sets of parameters of survival survival (3 for (3 for (3(3 parameters parameters for for ( by) mor K When the disease mortality is known, here denoted pmor K a , the a sets of survival parameters (3each for each state, state, each state, and and possibly and stratified possibly possibly stratified by age) stratified age) by can by can be estimated age) be can estimated morbe by minimizing minimizing mor estimated by minimizing each When state, the disease each each each state, and possibly state, state, and mortality and and possibly stratified is known, possibly possibly stratified by age) stratifiedhere stratified by age) by by can can denoted age) age) be be estimated by can can pbe be estimated mor K (a ) minimizing estimated estimated by minimizing , the byby sets of survival parameters (3 for minimizing minimizing æ can each state, and possibly stratified by age) æ be estimated ( K ( K by K a ) ) (ö minimizing 2ö ö )) ÷2 2 ö 2 2 pmorK (p )- amor (p a ) -(p K a S= å S =ç æ æ å å ç ææ å( ( K ( a )( pmor ( ()p -pmor K pK mor ( amor K K (a) - ( a ) ( - )p ) ) ) p÷ 2 K öK ( (a a )) )ö )÷ (0.47) (0.3) (0.3) å ç å p a - p a ÷ mor S = Kç å ç ç mor å s 2 (a 2 s)mor( a ) 2 ÷ 2 mor ÷ ÷ (0.3) S = å å çAgeGroup mor mor åK S = stage stage S ç æ=aÎ stage ç K å aÎ AgeGroup è ç è aÎAgeGroup ( çÎ pmor ( a )2- K s s (a ( p)mor a ) K s (2a s ( 2 ) a()) a ÷ ö ÷ ) ø ø ÷ ÷ (0.3) (0.3) (0.3) S = å è å èè stage K ç è aÎ stage stage K AgeGroup Ka aÎAgeGroup AgeGroup ÷ ø ø ø ø (0.3) stage K ç aÎAgeGroup è s 2 ( ) a ÷ ø Approximating attributable cases Approximating attributable cases Approximating attributable cases Approximating attributable cases The number of smoking-attributable cases (I ) for a disease (d) is calculated by dividing Approximating attributable cases The Approximating attributable cases The number number of smoking-attributable of smoking-attributable casescases Approximating attributable cases (IA) for a (IA A ) disease calculated for a disease (d) is (d) is calculated by dividing by dividing the number the number The the number numberTheof of smoking-attributable number new of smoking-attributable cases a individuals of among disease (IA) cases casesamong disease for a individuals (I ) for (d) a calculated iswho disease (d) are is by dividing calculated either bythe smokers number dividing or the number Approximating attributable cases of Thenew The new number number ofcases of cases of ofof a smoking-attributable a disease disease among smoking-attributable individuals cases cases who (IA a(Idisease who ) forare A ) Afor either are disease aeither smokers (d) (d) or is smokersis calculated calculated ex-smokers or ex-smokers by by (n) dividing dividing by(n) thetheby the total number number total of new cases of a disease among individuals who are either smokers or ex-smokers (n) by the total of new cases The number of number of new number new of cases ex-smokers of of of (n) ofof a smoking-attributable cases people in people a by the disease the a in disease the population among totaldiseaseamong population individuals numberamong cases in individuals individuals in a given (Ia )given who a disease forare year. who who year. are eitherare (d) either is calculated either smokers smokers smokers or byor or ex-smokersex-smokers dividing the ex-smokers (n) by (n) number (n) the byby the total the total total in aof people year.in the population in a given year. A number of people number in of the population people given of new cases number number of of a of people disease people in inin among the the the population population individuals population in in a given aa in who given are given year. year. either year. smokers or ex-smokers (n) by the total number of people in the population in a given year. n( y) I Ad ( y) = (0.48) (0.48) N ( y) 44 44 44 44 44 44 Potential Years of Life Lost 44 The PYLL (Gardner and others 1990; Health and Social Care Information Centre 2015) for an individual (PYLL(i)) who dies in a given year will be calculated from the following equation (0.49). 52 // Appendix ì Ageref - Agedeath if Agedeath < Ageref PYLL(i) = í (0.49) î0 if Agedeath ³ Ageref For each individual the difference between the reference age (life expectancy) and the age of death n( y) Potential Years of Life Lost I Ad ( y) = (0.48) N ( y) n( y) I Ad ( y) = (0 The PYLL (Gardner and others 1990; Health and Social Care Information N ( y)Centre 2015) Potential Years of Life Lost n for an individual (PYLL(i)) who dies in aIgiven (y )== (y n( year ) ywill ) be calculated from the following The PYLL (Gardner and others 1990; HealthI d (y Potential Years of Life Lost and ) Social Care (0.48) ) Information Centre 2015) for an A (0.48) Ad N( N (yy) equation individual (0.49). The PYLL (Gardner (PYLL(i)) who dies in a given yearIwill and others n( y be calculated) 1990; Health and Social Care Information Centre 2015) for an Ad ( y ) = from the following equation (0.49). (0.48) N ( y ) individual (PYLL(i)) who dies in a given year will be calculated from the following equation (0.49). Potential Years of Life Lost Potential Years of Life Lost The PYLL The (Gardner and PYLL (Gardner and others others 1990;ì Age 1990; AgeSocial ref - and Health Careif Agedeath