The most recent and complete health model documentation is available on Pardee's website. Although the text in this interactive system is, for some IFs models, often significantly out of date, you may still find the basic description useful to you.
The IFs health model allows users to forecast age, sex, and country specific health outcomes related to 15 cause categories (see table) out to the year 2100. Based on previous work done by the World Health Organization’s (WHO) Global Burden of Disease (GBD) project[1 ], formulations based on three distal drivers – income, education, and technology – comprise the core of the IFs health model. However, the IFs model goes beyond the distal drivers, including both richer structural formulations and proximate health drivers (e.g. nutrition and environmental variables). Integration into the IFs system also allows us to incorporate forward linkages from health to other systems, such as the economic and population modules. Importantly, IFs provides the user the ability to vary model assumptions and create customized scenarios; as such, IFs is a tool exploring how policy choices might result in alternative health futures.
This documentation supplements the third volume of the PPHP series, “Improving Global Health,” (Hughes et al, 2011) by providing technical details of health model integration into the IFs system. It includes the specific equations used to forecast outcomes and drivers, relative risk values for proximate drivers, and data manipulations related to model initialization and projection. We intend the IFs model to be fully transparent to all users, and invite comments and questions at http://www.ifs.du.edu/contact/index.aspx.
- 1 Cause groups in IFs
- 2 Structure and Agent System: Health
- 3 Dominant Relations: Health
- 4 Health Flow Charts
- 5 Distal Drivers and Basic Indicators
- 6 Smoking and Smoking Impact
- 7 Proximate Drivers and Risk-Specific Population Attributable Fractions
- 8 Child Undernutrition
- 9 Body Mass Index and Obesity
- 10 Indoor Use of Solid Fuels
- 11 Outdoor Urban Air Pollution
- 12 Water and Sanitation
- 13 Specialized Models: Deaths from AIDS and Vehicle Accidents
- 14 Forward Linkages from Health
- 15 Forward Linkages of Health to Population and Labor Supply
- 16 Forward Linkages of Health to Capital Stock
- 17 Forward Linkages of Health to Economic Productivity
- 18 Health Equations
- 19 International Futures Help System
- 19.1 Distal Driver Formulation
- 19.2 Forecasting Income and Education for the Distal Driver Formulation
- 19.3 Forecasting Technology for the Distal Driver Formulation of Mortality
- 19.4 Diabetes and Chronic Respiratory Diseases
- 19.5 Equations: Diabetes
- 19.6 Equations: Chronic Respiratory Disease
- 19.7 Cardiovascular Disease
- 19.8 Equations: Diarrhea, Malaria, and Respiratory Infections
- 19.9 Equations: HIV/AIDS
- 19.10 Equations: Road Traffic Accidents
- 19.11 Mental Health
- 19.12 Modifications to the Basic Health Model
- 19.13 Mortality Transition for Low-Income Countries
- 19.14 Maintaining Increase with Age in Non-Communicable Death Rates
- 19.15 Elasticity of Child Mortality with Health Spending
- 19.16 Data Initialization
- 19.17 Normalization and Scaling Factors
- 19.18 Cause-specific Mortality: Infants and the Elderly
- 19.19 Incorporating Proximate Drivers
- 19.20 Adjusting Mortality Due to Changes in a Single Risk Factor
Cause groups in IFs
Group I – Communicable, Maternal, Perinatal, and Nutritional Conditions
- Diarrheal diseases
- Respiratory infections
- Other Group I causes
Group II – Noncommunicable Diseases
- Malignant neoplasms
- Cardiovascular diseases
- Digestive diseases
- Chronic respiratory diseases
- Mental health
- Other Group II causes
Group III – Injuries
- Road traffic accidents
- Other unintentional injuries
- Intentional injuries
Structure and Agent System: Health
Hybrid structure using distal driver formulations supplemented by proximate drivers; integrated with larger IFs systems such as population and governance
Population by age-sex; stunted population; HIV prevalence
Births, mortality and morbidity
Key Aggregate Relationships
(illustrative, not comprehensive)
Distal driver formulations driven by income, education, and time as a proxy for technological advance
Proximate driver formulations driven by various social patterns and behaviors
Key Agent-Class Behavior Relationships
(illustrative, not comprehensive)
Behavior related to proximate drivers such as smoking, indoor solid fuel use, obesity
Dominant Relations: Health
Health forecasting systems typically can help us either (1) to understand better where patterns of human development appear to be taking us with respect to global health, giving attention to the distribution of disease burden and the patterns of change in it; or (2) to consider opportunities for intervention and achievement of alternative health futures, enhancing the foundation for decisions and actions that improve health.
Broad structural models (e.g., that of the Global Burden of Disease or GBD) assist in the first purpose by relating deep or distal development drivers to outcomes. More specialized structural formulations and the inclusion of proximate drivers open the door to the second, allowing for consideration of interventions in the pursuit of alternate health futures. A more hybrid and integrated model form like that of IFs can help with both purposes and provide a richer overall picture of alternative health futures.The figure shows the general structure. Formulations based on distal drivers (the GBD methodology) sit at its core. There is no inherent reason, however, that income, education and time (the distal drivers of the GBD approach) should be equally capable of helping us forecast disease in each of the major categories (let alone each of the specific diseases) that the GBD models examine. For example, distal driver formulations tend to produce forecasts of constantly decreasing death rates. Yet we know, for instance, that smoking, obesity, road traffic accidents, and their related toll on health tend to increase in developing societies among those who first obtain higher levels of income and education; with further societal spread of income and education, at least smoking and road traffic deaths (and perhaps also obesity) typically decline.[1 ] A hybrid model can therefore help us identify opportunities for interventions to improve health futures. These interventions might also occur in the form of super-distal drivers (for example, policy-driven human action with respect to health systems). The sociopolitical and environmental modules in IFs act in part as super-distal foundations for variables such as undernutrition and indoor air pollution which, in turn, facilitate analyses of proximate risk factors and human action around them.
The integrated nature of the IFs modeling system further allows us to think about feedback loops between health outcomes and larger development variables such as economic progress and population structure.
 It is partly for this reason that the creators of the GBD models added exogenous specification of smoking impact to the otherwise mostly monotonically (one-direction only) changing specifications.
Health Flow Charts
Mortality from most causes of death is a function of a small number of distal or deep drivers and a larger number of proximate or more immediate drivers. For two specific mortality types, however, specifically deaths from AIDS and vehicle accidents, there are more specialized representations that rely on a number of more cause-related drivers.
Distal Drivers and Basic IndicatorsTo forecast mortality related to most of the major cause clusters we use the regression models and associated beta coefficients prepared for the GBD project (Mathers and Loncar 2006). Age, sex, cause, and country-specific mortality rate is a function of income (using GDP per capita as a proxy), adult education, technological progress. For specific death causes, smoking impact (for malignant neoplasms, cardiovascul
A number of parameters control technology in the distal functions. In the default mode (hlmortmodsw = 1), IFs modifies the technology (time) coefficient in recognition of slower than expected historical progress in many countries, an approach developed in the Global Burden of Disease (GBD project). Those country differences are controlled by hltechbase ,hltechlinc, and hltechssa. Setting the switch value to 0 activates an alternative IFs project approach to the impact of those parameters.
The user can also affect the mortality patterns directly with several parameters, including mortm , which allows simultaneous manipulation of all causes of death and hlmortm , which facilitates manipulation of each cause of death separately. Hlmortcdchldm changes the rates of all communicable diseases for children aged 5 and younger, while hlmortcdadltm affects rates of death from communicable diseases for adults aged 15-49.
Based on the mortality level, it is possible to compute the years of life lost to each cause of death (HLYLL). Using WHO-based estimates, IFs links mortality also to years of living with disability (HLYLD). The sum of the two is disability-adjusted life years lost (HLDALYS).
The forecast of mortality in this figure, dependent almost entirely on distal factors, is not actually the final calculation in the model. See the discussion of the entry of proximate drivers into the discussion of population attributable mortality fractions (PAFs), in interaction with distal-based mortality, for the rest of the story.
Because of the importance of smoking impact in the distal driver formulation, it is important that we elaborate that term. Body mass index is, at this point, only linked to diabetes and we discuss that in the context of the PAFs.
Smoking and Smoking ImpactOf the various specific health risks that the model treats, smoking has a special place because its impact is in the distal driver formulation of the IFs health model. The figure shows that the impact is driven by the rate of smoking (differentiated by males and females) 25 years earlier, with the relationship controlled by an impact elasticity (hlsmkel ). The user can also posit as a nearer term (in the model immediate) impact by setting a switch for that (hlsmkimeff ) at some fractional value of the full delayed impact–the value in the base case is 0.1 or 10 percent. For analysis purposes, another switch (hlsmimpsw ) can turn off the endogenous computation of smoking impact and leave it constant at the initial year value.
Smoking rate itself is computed in two different ways. The basic formulation uses only the initial condition and a function linked to the simple and squared values of GDP per capita at PPP. The more extended formulation is an algorithmic one based on the same general concept of a pattern that initially rises with GDP per capita, peaks, and then falls, but with a series of parameters that allow much more control over the stages.[1 ] This staged algorithmic approach (see Lopez et al. 1994; Shibuya et al. 2005; Ploeg et al. 2009) is turned on with a switch (hlsmokingstsw ).
Because control of tobacco is a major policy objective in many countries, there is also a representation of a tobacco control score on a 100-point scale (hlsmokingtcs ) with an associated parameter to control the elasticity of smoking with that score (hlsmokingtcsel ), as well as a multiplier on the score (hlsmokingtcsm ).
Finally, there is a multiplier that allows direct manipulation of the smoking rate, again by sex (hlsmokingm ). Cecilia Peterson developed this approach for IFs.
Proximate Drivers and Risk-Specific Population Attributable Fractions
OverviewAlthough mortality can be calculated solely from distal drivers such as income and education, it is better to calculate it from proximate or more immediate factors, such as undernutrition or exposure to pollutants. But IFs, and perhaps any model, will never be able to represent all such
The figure below shows such a combination. Each proximate driver (and there are many different ones in IFs, in spite of the generalized representation in the figure) can be associated with a fraction of the mortality of a society. That population attributable fraction or PAF (derivative from the risk exposure level relative to a theoretical risk minimum) can be used to adjust the mortality associated with any cause that would have an implicit risk-related mortality built into the distal driver formulation. IFs makes those implicit distal-driver associated risk levels explicit by using the distal drivers to identify a risk level that would be expected based on cross-sectional analysis using the distal drivers. That allows the computation of a distal-driver based PAF. In similar fashion a PAF can be calculated that relates an exposure level to the risk, calculated mostly elsewhere in IFs (such as in the food and agriculture model for undernutrition of children) to a PAF.
The complication mathematically lies in the interaction of (1) the distal-driver and proximate risk-based PAFs and (2) the multiple specific-risk PAFs, because avoidance of death from one will generally increase the risk of death from others. See the equations associated with PAFs for details.
Among the specific risk factors treated in IFs are overly high body mass indices and associated obesity, undernutrition of children, access to unsafe water and sanitation, indoor use of solid fuels, and levels or urban air particulates.
Child UndernutritionAlthough obesity is a growing problem and killer around the world, the most important risk factor for children in particular has traditionally been undernutrition (often simply referred to as malnutrition). The percentage of children undernourished (MALNCHP) affects mortality rates frdistal driver formulation by actual risk level in a country. The core of that approach is to compare the risk-specific population attributable fraction (PAF) of total morality as calculated from the distal drivers with the PAF calculated from the actual level of the risk in the country.
The figure below shows the approach for childhood undernutrition. The two key variables in the distal driver formulation at any point in time (ignoring the technology factor that adds dynamics over time) are GDP per capita at purchasing power parity and years of adult education. They are used in a cross-sectionally estimated function to calculate an implicit body mass index that then produces the associated implicit PAF. IFs uses an alternative and more risk-factor specific formulation to forecast values of child undernutrition over time. The PAF associated with this explicit representation of MALNCHP is compared with the PAF from the implicit calculation and the comparison alters the actual mortality pattern.
To calculate MALNCHP the explicit formulation also uses GDP per capita, as in the distal formulation, but augments it with calories per capita and with access to safe water and sanitation (unsafe water can cause diarrheal disease and undernutrition even with caloric intake would be adequate). A multiplicative parameter (malnchpm ) can be used to change child undernutrition in scenario analysis. Another parameter (malnchpsw ) can be used to hold the level of undernutrition at the level of the first year, an approach useful for counterfactual scenario analysis.
Although not used in the health model, IFs contains two other measure of undernutrition. The first is an alternative measure of child undernutrition developed by Smith and Haddad (2000); MALNCHPSH is computed as a function of the ratio of female and male life expectancy, of female secondary school gross enrolment rate, and of access to safe water. The second is a measure of rate of undernutrition for the entire population (MALNPOPP), computed as a function only of calories per capita.
Body Mass Index and ObesityThe distal driver formulation used for forecasting mortality in IFs contains a country’s average body mass index (HLBMI) for diabetes. HLBMI also affects mortality from cardiovascular disease in IFs via the mechanism that the model uses to modify cause-specific mortality from the distal driver formulation by actual risk level in a country. The core of that approach is to compare the risk-specific population attributable fraction (PAF) of total morality as calculated from the distal drivers with the PAF calculated from the actual level of the risk in the country.
The figure below shows the approach for body mass index. The two key variables in the distal driver formulation at any point in time (ignoring the technology factor that adds dynamics over time) are GDP per capita at purchasing power parity and years of adult education. They are used in a cross-sectionally estimated function to calculate an implicit body mass index that then produces the associated implicit PAF. IFs uses an alternative and more risk-factor specific formulation to forecast values of body mass index over time. The PAF associated with this explicit representation of HLBMI is compared with the PAF from the implicit calculation and the comparison alters the actual mortality pattern.
To calculate HLBMI the explicit formulation uses calories per capita as the sole driving variable. A multiplicative parameter (hlbmim ) can be used to change HLBMI in scenario analysis. A forecast of the obese population as a percent of the total population (HLOBESITY) is driven by the body mass index. A separate multiplicative parameter can modify it (hlobesitym ).
Indoor Use of Solid FuelsOne of the most important health risk factors in the developing world, especially for women and children under 5 is the use of solid fuels for cooking (and heating) indoors (ENSOLFUEL). It is a major cause of respiratory diseases. In IFs it affects mortality rates via the mechanism that the model uses to modify cause-specific mortality from the distal driver formulation by using information concerning actual risk level in a country. The core of that approach is to compare the risk-specific population attributable fraction (PAF) of total morality as calculated from the distal drivers with the PAF calculated from
The figure below shows the approach for indoor air pollution from the use of solid fuels. The two key variables in the distal driver formulation at any point in time (ignoring the technology factor that adds dynamics over time) are GDP per capita at purchasing power parity and years of adult education. They are used in a cross-sectionally estimated function to calculate indoor air pollution (linked to solid fuel use) that then produces the associated implicit PAF. IFs uses an alternative and more risk-factor specific formulation to forecast values of solid fuel use over time. The PAF associated with this explicit representation of ENSOLFUEL is compared with the PAF from the implicit calculation and the comparison alters the actual mortality pattern.
To calculate ENSOLFUEL the explicit formulation also uses GDP per capita, as in the distal formulation, but augments it with access to electricity. For the actual equation, see the topic on equations for solid fuel use in the infrastructure documentation. A multiplicative parameter (ensolfuelm ) can be used to change solid fuel use in scenario analysis. Another parameter (ensolhldsw ) can be used to hold the rate of solid fuel use at the level of the first year, an approach useful for counterfactual scenario analysis.
Major factors affecting the health impact of indoor solid fuel use are the efficiency and ventilation of the stoves. The model provides a coefficient (ensfvent ) for scenario analysis concerning those factors.
Much analysis on this health issue will want to use control of solid fuel use, partly through the use of a multiplier (ensolfuelm ). There is also targeting of solid fuel use and the model provides two different kinds of targeting parameters, absolute and relative. The absolute (or universal) targeting allows the setting of a year (ensolfueltrgtyr ) by which solid fuel use would be eliminated; it is available country by country. The relative targeting approach, available only globally across all countries, allows the setting of a value based on the typical rate of solid fuel use at different levels of GDP per capita (estimated cross-sectionally). A target rate (ensolfuelsetar ) would normally be no higher than the typical rate at the country’s level of GDP per capita and could be, for instance, one standard error lower than the typical rate. An associated parameter (ensolfuelseyrtar ) identifies the number of years over which a country would move to the target level. If a country already meets or exceeds a relative target, it will not move (adversely) toward it. Moreover, only the absolute or relative target should be used in analysis, not both together–an attempt to use both together will result in neither being used.
Outdoor Urban Air PollutionOne of the more important health risk factors in the developing and developed world alike, especially for middle-income industrializing countries, is the concentration of particulate matter of diameter 2.5 micrometers or less per cubic centimeter in urban air (ENVPM2PT5).[1 ] It is a major cause of respiratory infections, respiratory diseases, and cardiovascular disease in adults 30 and older. In IFs it affects mortality rates via the mechanism that the model uses to modify cause-specific mortality from the distal driver formulation by using information concerning actual risk level in a country. The core of that approach is to compare the risk-specific population attributable fraction (PAF) of total morality as calculated from the distal drivers with the PAF calculated from the actual level of the risk in the country.
The figure below shows the approach for outdoor urban air pollution, focusing on the measure of ENVPM2PT5. The two key variables in the distal driver formulation at any point in time (ignoring the technology factor that adds dynamics over time) are GDP per capita at purchasing power parity and years of adult education. They are used in a cross-sectionally estimated function to calculate outdoor air pollution that then produces the associated implicit PAF. IFs uses an alternative and more risk-factor specific formulation to forecast values of outdoor urban air pollution use over time. The PAF associated with this explicit representation of ENVPM2PT5 is compared with the PAF from the implicit calculation and the comparison alters the actual mortality pattern.
To calculate ENVPM2PT5 the explicit formulation also uses GDP per capita, as in the distal formulation, but augments the spending of a country on health as a portion of GDP (which appears to serve reasonably well as a proxy for more general attention to the environment). For the actual equation, see the topic on outdoor urban air pollution equations in the health documentation . A multiplicative parameter (envpm2pt5m ) can be used to change urban air pollution in scenario analysis. Another parameter (envpm2hldsw ) can be used to hold the level of urban air pollution at the level of the first year, an approach useful for counterfactual scenario analysis.
 Initialized in IFs by converting World Bank data on PM10 concentrations.
Water and Sanitation
Although unsafe water and sanitation is a killer via its contribution to undernutrition of children in particular, it creates its own mortality risk especially via diarrheal disease. The variables of importance in IFs are access to safe water (WATSAFE) and safe sanitation (SANITATION). In IFs they affect mortality rates via the mechanism that the model uses to modify cause-specific mortality from the distal driver formulation by using information concerning actual risk level in a country. The core of that approach is to compare the risk-specific population attributable fraction (PAF) of total morality as calculated from the distal drivers with the PAF calculated from the actual level of the risk in the country.The figure below shows the approach for safe water and sanitation. The two key variables in the distal driver formulation at any point in time (ignoring the technology factor that adds dynamics over time) are GDP per capita at purchasing power parity and years of
To calculate WATSAFE and SANITATION (separately) the explicit formulation also uses both average years of adult education and GDP per capita, as in the distal formulation, but augments those with the spending of a country on health as a portion of GDP (which appears to serve reasonably well as a proxy for more general attention to the environment) and portion of the citizenry living on less than $1.25 per day. For the actual equations, see the topic on outdoor urban air pollution equations in the infrastructure documentation.
Both access to safe water and to safe sanitation have ladders of access quality ranging from none to household connections. Parameters affecting them must thus take into account those ladders and the specific level(s) the parameter affects. Multiplicative parameters (watsafem and sanitationm ) can be used to change access at any level on the two ladders (the model normalizes access across levels to assure summation to 100 percent. Another parameter pair (watsafehldsw andsanithldsw ) can be used to hold the level of access at that of the first year, an approach useful for counterfactual scenario analysis.
Other parameters control targeting, both universal and relative. With respect to absolute targeting, watsafetrgtval and watersafetrgtyr control those with no access to safe water (the proportion and the number of years to reach the target, respectively). Similarly, sanitationtrgtval and sanitationtrgtyr control those with access to household connections. The relative targeting approach, available only globally across all countries, allows the setting of a value based on the typical rate of access at different levels of GDP per capita (estimated cross-sectionally). A target level (watsafenoconsetar, sanithhconsetar, sanitnoconsetar ) would normally be no better (which could mean no higher or no lower) than the typical level at the country’s level of GDP per capita and could be, for instance, one standard error better (higher or lower depending on the variable being targeted) than the typical level. An associated parameter (watsafenoconseyrtar, sanithhconseyrtar, sanitnoconseyrtar ) identifies the number of years over which a country would move to the target level. If a country already meets or exceeds a relative target, it will not move (adversely) toward it. Only the absolute or relative target should be used in analysis, not both together–an attempt to use both together will result in neither being used.
Specialized Models: Deaths from AIDS and Vehicle Accidents
AIDS deaths depend very directly on the number (or stock) of HIV-infected individuals, which depends in turn on the HIV-infection rate (HIVRATE). Data on HIV infection rates were used to compute a basic, country/region-specific rate of HIV infection increase (hivincrate), which the user can alter, as they can exogenous assumptions about the peak year of the epidemic (hivpeakyr) and the infection rate in that year (hivpeakr). If a country is beyond the peak year of the epidemic, control will be bringing the rate down over time (HIVTECCNTL). The user may also rely upon a country/region-specific multiplier to move rates up or down (hivm).
There is both a policy and medical effort underway to reduce the growth in infections. An HIV technical advance rate (hivtadvr) represents the success of that in rate of reduction in annual infection growth, and a variable (HIVTECCNTL) shows the cumulative impact of changes past a peak rate and year. Although highly speculative, the user will recognize the long-term importance of such assumptions.
Turning from the infection rate to the death rate, the user can make changes in the initial AIDS death rate (aidsdrate) to reflect possible progress or lack of it in reducing the deaths from HIV (using aidsdrtadvr). When the deaths from AIDS are computed, they are used to compute an incremental number of deaths (since some are already in the mortality of the base year); and an exogenous vector spreads them by age and sex.
The computation of deaths from vehicle accidents starts with computing the number of vehicles per capita (VEHICFLPC) as a function of population density and GDP per capita. That allows computation of the total number of vehicles (VEHICLESTOT). The function used by IFs to compute the total number of road deaths uses th
Forward Linkages from Health
Chapter 7 of Hughes, Kuhn, Peterson, Rothman, and Solorzano (2011) elaborated the forward linkages of the health model to other parts of the IFs system at the time of that volume's completion. It begins by discussing a controversy in the literature about whether the effects on economic well-being (as indicated by GDP per capita) of improvements in life expectancy are positive or negative. It goes on to devote much attention to three major and general pathways of impact between health and GDP, each of which corresponds to an element in standard production functions and that in IFs. The diagram below shows the major pathways between health/demography and GDP, each of which requires elaboration by showing the variables and logic of the IFs system; those three are labor, capital, and multifactor productivity.
Forward Linkages of Health to Population and Labor Supply
The IFs demographic model captures the mechanical or accounting effects of mortality on population (see the solid paths in the figure below). A key pathway passes from mortality through adult age population to labor supply (including aging-related lags). Similarly, IFs captures the mechanical effect of mortality on fertility through the death of women of childbearing age.
The most important non-mechanical linkage is almost certainly the relationship between child mortality and fertility. IFs forecasts fertility as a relationship with infant mortality, the log of educational level of those aged 15 and older (neither the education of women alone nor the education of those 15-24 work as well), and the percentage use of modern contraception.
 IFs also includes income-based formulations for changing the female participation rate.
Forward Linkages of Health to Capital StockThe figure below sketches the primary paths between health (morbidity and mortality) and capital stock. Most capital stock consists of buildings and machinery for producing goods and services; some representations may include
Turning our gaze to the paths by which health affects investment, the three major ones run though health spending, which can crowd out savings and investment, through the age-structure of societies, which affects the savings rate, and through investment from abroad, which can augment that generated domestically.
With respect to health spending, to which we return later, the IFs model uses a social accounting matrix (SAM) structure. Thus the flow of funds into health spending automatically competes with other consumption uses and with savings and investment. The major current weakness of the model with respect to this path is that there is no linkage from morbidity (associated in IFs with mortality) and health expenditures. (There is a linkage in IFs back from health spending to mortality –all categories except for AIDS).
The paths in IFs that link age structure most directly to domestic savings have two important elements. The most fundamental one represents the understanding of life-cycle dynamics in income, consumption and savings. The cycle for income is fairly clear-cut with a peak in the middle to latter periods of the working years. Workers set aside some portion of income as savings and that portion, too, tends to peak in the middle and late period of working years. Society-wide savings themselves become negative after retirement age (65 in the Base Case scenario but variable in scenarios) even though some portion of the population will continue to work. The second fundamental element is that both the horizon of life expectancy and the average income level of a society can have an impact on the portion set aside for savings and the degree to which it rises and then falls. Thus, for example, the life-cycle “bulge” of savings may be earlier and flatter in developing countries.
We implemented the representation of savings and investment in accord with that understanding. Relying upon analyses of selected countries that Fernández-Villaverde and Kruegger (2004 and 2005) and Deaton and Paxson (2000) undertook, we extracted general stylized patterns of the savings life cycle to represent more and less developed (and lower life expectancy) countries. In forecasting we use the pattern for less developed countries when life expectancy falls below 40 years, use that for more developed countries when life expectancy exceeds 80 years, and interpolate in between for all other countries. The result of this largely algorithmic approach[1 ] is an adjustment factor (SavingsAgeAdj) that augments or reduces investment.
In addition, investment is somewhat augmented or reduced as a direct result of changing life expectancy. Life expectancy is compared over time with an expected value (tied to cross-sectional estimation with income). That difference is compared to the difference in the initial year and, if it rises, augments investment.
Although conceptually tied to savings rates, neither the life-cycle analysis nor the life-expectancy term directly affect savings in IFs. Instead, they affect investment directly and savings indirectly via the dynamics in IFs that balance savings and investment over time.
The path linking health to foreign direct investment is potentially quite important. Alsan, Bloom and Canning (2006: 613) reported that one additional year of life expectancy boosts FDI inflows by 9 percent, controlling for other variables. We have implemented that relationship in IFs. The representation of FDI in IFs captures the accumulation over time of FDI inflows in stocks of FDI, as well as the accumulation of FDI outflows in stocks. In addition, the stocks set up their own dynamics, including the tendency for stocks to reinforce flows. For that reason, we have set the base case parameter for the impact of each year of life expectancy on FDI flows to 0.05 (5 percent), lower than the estimate of Alsan, Bloom and Canning (2006).
 See the subroutine SavingsDemogAdj in routine Populat.bas, which draws upon table IncConSav in IFs.mdb with different patterns of income, consumption, and savings for more developed countries (MDCs) and less developed countries (LDCs) across age categories; in general, peaks of income, consumption, savings occur the in late 40s and savings turn negative at 65.
Forward Linkages of Health to Economic ProductivityHealth outcomes impact productivity through a variety of pathways (see the figure below). Overall the function for multifactor productivity from human capital (MFPHC) is a sum of terms linked to educational expenditures (GDS(EDUC)) as a portion of GDP and to educational attainment of
In the IFs health module, the prevalence of adult stunting (HLSTUNT) relates negatively to overall productivity via an elasticity (mfpstunt ). In extreme cases, stunting could cost as much as 1 percent of economic growth.
We compute HLSTUNT in the health model itself. We initialize adult stunting in a long-term lagged relationship (using a moving average of 25 years) with child malnutrition (MALNCPH) and forecast it as a function of both malnutrition and child mortality as a proxy for morbidity.
Turning to disability (which is driven by mortality rates), childhood malnutrition and morbidity do not give rise to all of disability in working years; much also comes from disabilities arising during the working years. IFs therefore also calculates millions of years of living with disability related to mortality rates specific to the working aged-population (HLYLDWORK).
Turning to the forecasting relationship between disability and productivity, the IFs approach drives changes in the growth of productivity from the changing difference between computed and expected values of disability. We used the world average disability rate as an “expected” value. Because we have replicated the practice of the GBD project and kept mental health disability rates constant over time, and because mental health generally dominates disability, forecasts of this disability term are relatively stable over time. Thus analysis with respect to this variable will depend on scenarios that increase or decrease those disability rates. Changes in disability levels (relative to expected ones) relate to change in productivity via a parameter,mfphlyld . <header><hgroup>
The hybrid IFs model for forecasting health (using distal and proximate drivers) provides forecasts of age, sex and country-specific mortality rates for most of the 15 cause clusters it represents. The model within IFs for mortality in those cause clusters builds on the Global Burden of Disease (GBD) methodology, which uses mainly distal (more distant) drivers to project mortality. IFs then extends that methodology in many cases by adding attention to a selected set of proximate drivers.
For several other causes or cause clusters (*), most of those related to communicable disease, the IFs system uses a variant of the distal driver approach, one that looks to a forecast of all communicable disease except HIV/AIDS and then subdivides that total by more specific disease type.
For still other causes of mortality (**), it uses more specialized models totally unrelated to the distal driver approach.
Cause Clusters in IFs 
- Other Group I diseases (excludes AIDS, diarrhea, malaria and respiratory infections)*
- Malignant neoplasms
- Cardiovascular diseases
- Digestive diseases
- Chronic respiratory diseases
- Other Group II diseases (excludes malignant neoplasms, cardiovascular diseases, digestive diseases, diabetes, chronic respiratory diseases, and mental health)
- Road traffic accidents**
- Other unintentional injuries (excludes road traffic accidents)
- Intentional injuries
- Respiratory Infection *
- Mental Health**
For help understanding the equations see Notation .
- Help During Use of IFs
- Understanding the Modeling Approach
- Understanding the Equations
- Demography / Population
- Structure and Agent System
- Dominant Relations
- Flow Charts
- Distal Driver Formulation
- Forecasting Income and Education
- Forecasting Technology
- Diabetes and CRD
- Cardiovascular Disease
- Diarrhea, Malaria, and Respiratory Infections
- Road Traffic Accidents
- Mental Health
- Modifications to the Basic Health Model
- Data Initialization
- Incorporating Proximate Drivers
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International Futures Help System
Distal Driver Formulation
For the basic forecast of mortality related to most of the major cause clusters (exceptions are deaths from HIV/AIDS and traffic accidents) we use the regression models and associated beta coefficients prepared for the Global Burden of Disease project (Mathers and Loncar 2006). Age, sex, cause, and country-specific mortality rate is a function of income, adult education, technological progress, and (in specific cases) smoking impact:
M is mortality rate in deaths per 100,000 for a given age category c, sex p, cause of death d and country or region r.
Y is GDP per capita at PPP
HC (human capital) is Years of Adult Education over 25
T is time
SI is Smoking Impact
Income and education (IFs variables GDPPCP and EDYRSAG25, respectively) are forecast endogenously in IFs. Time, a proxy for technological progress, is calculated as calendar year minus 1900 (for example, T for the year 2001 equals 101). Smoking impact, a variable meant to capture historical smoking patterns, is included only in the forecasts of mortality related to malignant neoplasms, cardiovascular disease, and respiratory disease.[1 ] As described in another section of this document, IFs uses both historical smoking rate estimates and SI projections to 2030 (as provided by GBD authors) to forecast the SI variable.
Using an historical database representing mortality data from 106 countries for the years 1950-2002, the GBD calculated sex-specific regression coefficients for seven age groups (<5, 5-14, 15-29, 30-44, 45-59, 60-69, and 70+) and ten major cause clusters–the first ten in the list above (Protocol S1, 1-3).[2 ] GBD estimations using the data from the 106 countries created separate low- and high-income regression models (not coefficients for each country separately), with low income defined as GDPPCP < $3,000 in the initial year. Both sets of coefficients are publicly available online.[3 ] In IFs we spread the coefficients for the seven age groups across 5-year subcategories; that is, we use the same coefficients for each subcategory within the larger GBD ones–normalization of mortality within each 5-year subcategory across causes and to total mortality rates for each subcategory (taken from UN Population Division data) does, however, create differences in mortality rates across those 5-year groupings.
We generally use the beta coefficients provided by GBD authors to forecast mortality related to six cause groups: Group I excluding detailed communicable causes, malignant neoplasms, digestive diseases, Group II excluding diabetes and mental health, other intentional injuries, and intentional injuries. However, for a few age and Group III cause groups where regression models provided low predictive value, we also follow the GBD in keeping mortality rates constant over time instead of using the regression equations. Affected groups include: unintentional injuries for males older than 70; unintentional injuries for females older than 60; intentional injuries for males and females under 5; intentional injuries for males older than 60; and intentional injuries for females older than 45.
Although we forecast mortality by age, sex, cause, and country as in the general GBD equation above (and the details can be seen in the specialized displays of the model on mortality by age, sex, and cause and the mortality J-curve), the major model variable for display is DEATHCAT, which is total deaths by country/region, cause, and sex. The equation for it, using the IFs variables for GDP per capita at PPP (GDPPCP), average years of education for adults aged 25 and older (EDYRSAG25), time (IY%), and smoking impact (HLSMOKINGIMP) is
The betas in the equation, as indicated earlier, are from the GBD work and are dimensioned also by country/region r (only as high income or low income), cause of death d, sex p, and age category c. The entire equation for mortality is adjusted in an algorithmic process so that the total across all causes of death equal the mortality rates from the UN Population Division’s data (using a normalization factor), while the relative weights for each disease match WHO data (using a scaling factor). The normalization and scaling factors are multiplicative, affecting everything in the equation. In the Base Case scenario we keep those factors constant, but we can control convergence of them (see the Normalization and Scaling Factors section).
The equation allows scenario modification with multiplicative parameters that change mortality overall (mortm ) or by cause of death (hlmortm ). Not shown in the equation, hlmortcdchldm changes the rates of all communicable diseases for children aged 5 and younger, while hlmortcdadltm affects rates of death from communicable diseases for adults aged 15-49.
Forecasting Income and Education for the Distal Driver Formulation
</hgroup></header> For the basic distal driver formulation, GDP per capita at PPP (GDPPCP) and years of adult education (EdYrsAg25) are forecast endogenously within IFs. GDP per capita is computed in the Economic Module as an annual flow variable (that is, it is generated anew each year), driven in part by underlying stocks such as capital supply and, in fact, years of adult education (both of which accrete or deplete very slowly over time). Adult education is computed in the Education Module using government spending on education as the main driver. For a more complete description of driver forecasting in IFs, please see www.ifs.du.edu. <header><hgroup>
Forecasting Technology for the Distal Driver Formulation of Mortality
</hgroup></header> Although two IFs variables, namely GDP per capita at purchasing power parity (GDPPCP) and years of adult education (EDYRSAG25) drive the distal formulation in most of our forecasting and scenario analysis, the technology parameter (the beta on time) in the distal equation is very powerful. We therefore want some control over it, ideally with ability to differentiate that control with respect to level of income of countries and with respect to the age structure of mortality. For a basic approach to providing such control, we follow the Global Burden of Disease (GBD) project in modifying the regression models for child mortality low-income countries. But we have extended that GBD approach to allow some additional parametric control.
The control system in IFs uses a switching parameter (hlmortmodsw ) in interaction with three other parameters (with their default values those are hltechbase =1,hltechlinc =0.25, and hltechssa =0); see the table below for a summary of the application of those parameters.
In the default mode (hlmortmodsw = 1), IFs uses the GBD approach to modifying the technology (time) coefficients for children under 5 in recognition of slower than expected historical progress in many countries.[1 ] Specifically, for children under 5 in low-income countries in four regions (Africa, Europe, SE Asia and West Pacific) the time variable is held constant (zero, or no technological advance, using hltechssa ); in low-income countries in the Middle East and North Africa the coefficient on time is reduced to 25 percent of its original value using the parameter hltechlinc .
In the IFs implementation of the GBD approach to treatment of technology, we wished to change the patterns not just for children under 5, but also for older children and adults. We decided, however, to regularize the somewhat ad hoc assignment of countries by the GBD to the low-income category by defining low-income as being less than $3,000 per capita and high-income as being above that level. We use the parameter hltechlinc to control technological change for older children and adults in low-income countries regardless of geographical region; at its default setting that parameter reduces the coefficient on time to 25 percent of its original value.
|Age and Geographic Impact of the Parameters|
|Base or default values||For children under 5 (GBD Geographic Classification)||For older children and adults (IFs Geographic Classification)|
|hltechssa =0||Low Income Countries in mostly 4 regions (Africa, Europe, SE Asia and West Pac; also selected countries such as Haiti)||Not used|
| hltechlinc =0.25
||Low Income Countries in the Middle East and North Africa||Low Income Countries (GDPPCP < $3k in 2010)|
|hltechbase =1||All other countries, mostly High Income and including most of Latin America||High Income Countries (GDPPCP >= $3k in 2010)|
For children older than 5 and adults in what IFs classifies as high-income countries (countries with GDP per capita at PPP in 2010 above $3,000), IFs uses the parameter hltechbase . Thus in the default situation, technological change is unchanged from the basic value.
These GBD technology factor modifications (and their extensions by IFs to adults and older children in our definition of low- and high-income countries) can be turned off in the model (hlmortmodsw = 0).[2 ] When the switch is turned on, adjustments can also be made to hltechbase , hltechlinc , and hltechssa to build new scenarios.[3 ] It is important for the user to know, however, that regardless of age or income level of countries, the model uses hltechbase for mortality from cardiovascular causes (which uses a different regression model for forecasting).
Changes to hltechbase can also be adjusted by using a shift parameter called hltechshift (0 by default), which adjusts the technology factor depending on the level of initial GDP per capita at PPP.
This adjustment increases the technology factor for high income countries more quickly than for middle or low-income countries.
 The Global Burden of Disease (GBD) project made low-income modifications after recognizing that historical child mortality data did not match back projections of the model (Mathers and Loncar 2006b: 9). Note that the GBD approach to these modifications changed from the 2002 revision to the 2004 revision of the project. In the 2002 revision, the human capital (education) beta was reduced to half of its magnitude for sub-Saharan countries, and to 75 percent of its original magnitude for other low-income countries. This was done only if the beta on the human capital (education) term in the distal model formulation was negative (reducing mortality with increases in education). Technological advance factor (time) was left constant (no advance) for sub-Saharan Africa and reduced to 25 percent for other low income countries. The 2004 revision dropped the human capital modifications, but continued to reduce the coefficient on time.  Although not recommended, IFs also allows the user to use the original GBD 2002 modifications (as described in a previous footnote) by specifying hlmortmodsw = 2.
 Given the results found for Intentional Injuries, where mortality was reaching unrealistic levels, we have limited the changes to hltechbase and hltechlinc to at most 1.5 for this particular cause of death in the 2004 revision (hlmortmodsw = 1).
 See Protocol S1, Mathers and Loncar 2006 for more detail on the use of smoking impact in GBD projections.
 See Table 1, Protocol S1, for the cause clusters used in the GBD 2002 and 2004 projections. IFs does not use GBD coefficients for HIV/AIDS, relying instead on a structural model; mental health mortality is kept at a constant rate and coefficients for the other three communicable diseases (diarrhea, malaria, and respiratory infection) come from other sources.
 For regression results, see Tables S3 and S4 at http://www.plosmedicine.org/article/info%3Adoi%2F10.1371%2Fjournal.pmed.0030442#s5. <header><hgroup>
Diabetes and Chronic Respiratory Diseases
Two chronic cause groups, diabetes and respiratory, are so strongly influenced by specific risk factors that estimates based on distal drivers alone fail to accurately represent expected mortality rate trajectories. In the case of diabetes, rising population levels of overweight and obesity contradict suggestions that diabetes-related mortality will fall over time in line with other Group II causes. Conversely, declining smoking rates in many high income countries may temper projections of increasing chronic respiratory-related mortality (Protocol S1, 5-6). Therefore, IFs follows the GBD methods in modifying the distal driver formulation by adding proximate risk factors (BMI and SI, respectively) to forecast base diabetes- and chronic respiratory-related mortality rates. <header><hgroup>
</hgroup></header> To forecast diabetes, IFs uses the following formula:
Mr,c,d=Diabetes,p,r is diabetes-related mortality by country/region r, age category c, and sex p.
ONCDr,c,p is other Group II (non-communicable disease) mortality, derived using the basic distal driver equation . HLDIABETESRRr,c,p is a “Diabetes Relative Risk” factor, explained below.
In a population at the “theoretical minimum” level of body mass index (BMI), where BMI is 21, diabetes-related mortality is expected to fall at 75 percent of other Group II mortality.[1 ] The diabetes relative risk factor (HLDIABETESRR) captures the increased risk represented by a population above the theoretical BMI minimum level. For example, the factor is about 1 for young females in Vietnam (where BMI is close to the theoretical minimum level of 21). Comparatively, the RR is approximately 28 for middle-aged women in the United Kingdom where population BMI is much higher.[2 ]
The GBD project projected the RR variable for diabetes out to 2030 using fairly involved estimates of age and sex-specific levels (plus standard deviations) of population BMI. Our estimates in IFs of future BMI (HLBMI) are less sophisticated, and we only forecast country/region (r) and sex-specific (p) mean BMI (see this section for a description of our forecasts of BMI). As such, while we endogenize the RR variable by tying it to our forecasts of BMI, we also adjust our forecast by initializing RR using the GBD estimates for the year 2010 and computing an age-category specific shift factor (HLDIABSHIFT) in order to tie our forecast of expected RR with GBD estimates.
The RR for diabetes forecast in IFs (HLDIABETESRR) assumes that country-specific BMI is distributed normally, and also assumes a standard deviation of 10% of the mean:[3 ]LogRR is the change in log of RR per 1 unit change in BMI. [4 ] These values are age category (c) and sex (p) specific; the absolute relative risk of diabetes-related mortality in relation to a unit increase in BMI varies from between 1.47 (females under 45) and 1.2 (females over 80).[5 ] P(HLBMI) is a normal distribution function with mean of avgBMI; StdDev is a fixed 10% of avgBMI.[6 ]
 The slower decrease in diabetes-related mortality reflects assumptions that risk factors for diabetes will improve more slowly that risk factors for other Group II diseases (Protocol S1, 6).
 All RRs available in the IFs system, variable name HLDIABETESRR.
 We recognize, of course, that BMI is most likely not distributed normally in a population. However, we follow CRA authors in assuming normality in order to compare a given population with an ideal counterfactual population (James et al 2004).
 WHO Comparative Risk Assessment Methodology, Kelly et al, 2009
 See associated data table, Kelly et al 2009.
 avgBMI is our forecast of BMI, while BMI are the values from -3 standard deviations to +3 standard deviations away from that avgBMI. Cecilia Peterson determined the fixed 10 percent rate for StdDev from the literature. <header><hgroup>
Equations: Chronic Respiratory Disease
</hgroup></header> Again following GBD authors, IFs separately computes the two components of the chronic respiratory disease category–chronic obstructive pulmonary disease (COPD) (where smoking is the overwhelming related risk factor) and “other” respiratory disease (where smoking is somewhat less determinative). Both elements follow the same formulation:
SIR is the “smoking impact ratio, ” that is smoking impact in IFs (HLSMOKINGIMP) divided by an adjustment factor that is specific to three big regions (1. China, 2. World, and 3. SearD (Bangladesh, Bhutan, India, North Korea, Maldives, Myanmar, Nepal, Afghanistan, Pakistan)), age category c, gender p. CRDRR is the relative risk for chronic respiratory disease specific to age category c, gender p, and disease/death d type (COPD or other respiratory disease). Chronic respiratory disease is assumed to be declining at 75 percent of ONCD_Mort, which is other Group II related mortality.
 GBD authors provided the adjustment factor for SIR, and it is constant over the length of the IFs forecast. The computation is done with the hard-code value in a procedure called UpdateRespDisease. The name for SmokingImpactAdj in the model is sird. CRDRR is hard-code in the same procedure.
 RR ranges from approximately 10 for COPD to about 2 for other chronic causes. Again, GBD authors provided the relative risk estimates used in IFs. <header><hgroup>
The regression models used in the GBD project did not differentiate between-subject from within-subject variation. Particularly for cardiovascular-related outcomes in some age/sex groups, this model produced a perverse finding: a negative relationship between cardiovascular-related mortality and smoking impact (HLSMOKINGIMP). However our further statistical investigation showed, as expected, a positive relationship between cardiovascular-related mortality and HLSMOKINGIMP within a given country over time.
As such, we completed a more sophisticated mixed model regression analysis (using SAS, version 9.1) to capture both within and between-subject effects. We used the GBD mortality database described here, supplemented by our historical series of income per capita. All distal drivers were included as fixed effects, with random effects included for subject (country) and time (T). The revised coefficients (see Appendix Table 1) were used to forecast cardiovascular disease-related mortality. We created only one model for all countries (no separate low-income model) due to lack of data. Comparison with the original GBD models reveals fairly similar forecast outcomes overall. However, the positive change in the smoking/cardiovascular mortality relationship allows us to better examine how smoking intervention scenarios might impact cardiovascular-related mortality.
 Note that we did use historical estimates of education provided by the GBD project, instead of using the less complete historical series available through IFs. Future distal driver analysis may explore using alternate sets of education data, including those included in the IFs system. <header><hgroup>
Equations: Diarrhea, Malaria, and Respiratory Infections
</hgroup></header> IFs added three additional communicable diseases, namely diarrhea, malaria, and respiratory infection, after it had developed the modelling approach for distal drivers discussed above. The model uses the more general Group I (communicable disease and maternal mortality excluding AIDS) forecast to project mortality related to all three additions:
M is mortality rate in deaths per 100,000 for a given region r, age category c, sex p, general (Group 1) cause dg=1 and specific disease d within general cause group dg=1. Here dg=1 refers to all communicable diseases (other than HIV/AIDS because Mathers and Loncar 2006 did not use the same approach to that particular communicable disease) and d refers to diarrhea, malaria or respiratory infections.
The constants and beta coefficients for the above equation for the three diseases come from Mathers and Loncar (2005 and 2006c). For diarrhea and malaria we used their coefficients for infectious and parasitic diseases (Mathers and Loncar 2005: Table A-6 on page 115); for respiratory infections we used their coefficients for respiratory infections specifically (Mathers and Loncar 2006c, Table S5).
The results for these three subtypes are then subtracted from the mortality for the total Group I category (except HIV/AIDS). The reason for this is to make sure that the sum of all them does not exceed the total of Group 1 (excluding HIV/AIDS), which is a result the equation could theoretically produce. In fact, we want to be sure that there is room for the Other Group 1 cause of death, so IFs limits the sum for diarrhea, malaria, and respiratory infection to 95 percent of the total of Group 1 (excluding HIV/AIDS). If necessary, all three subcategories are reduced proportionally by a factor of 0.95/(SUM(3 subtypes)/Tot(big type)). Note that, if this restraint needs to be imposed, the denominator will always be higher than 0.95 and then the multiplicative adjustment factor will always be lower than 1.
 The extended process for using those is described in 2 working notes for the IFs project by Dale Rothman, Dealing with Diarrhoeal Diseases Including the Effects of Unsafe Water & Sanitation and Undernutrition (March 25, 2009) and Dealing with Effects of Indoor Air Pollution (October 8 2009). Titles of the files are Incorporating Diarrhoea 25 March 2009 and Incorporating Indoor Air Pollution 9 October 2009, respectively. <header><hgroup>
</hgroup></header> The ultimate objective of the calculations around HIV infections and AIDS is to forecast annual deaths from AIDS (AIDSDTHS) by age category and sex. We did not look to the forecast methodology of Mathers and Loncar (2006) for their approach on this particular communicable disease; in fact, they also used an approach that did not rely upon the general distal driver formulation.
The IFs approach begins by forecasting country-specific values for the HIV prevalence rate (HIVRATE). For the period from 1990-2007 we have reasonably good data and estimates from UNAIDS (2008) on prevalence rates and have used values from 2004 and 2006 to calculate an initial rate of increase (hivincr) in the prevalence rate across the population (which for most countries is now negative).[1 ]
There will be an ultimate peak to the epidemic in all countries, so we need to deal with multiple phases of changing prevalence: continued rise where rates are still growing steadily, slowing rise as rates peak, decline (accelerating) as rates pass the peak, and slowing rates of decline as prevalence approaches zero in the longer term. In general, we need to represent something of a bell-shaped pattern, but one with a long tail because prevalence will persist for the increasingly long lifetimes of those infected and if pockets of transmission linger in selected population sub-groups.[2 ] As a first level of user-control over the pattern, we add scenario specification via an exogenous multiplier on the prevalence rate (hivm ).
The movement up to the peak involves annual compounding of the initial growth rate in prevalence (hivincr ), dampened as a country approaches the peak year. Thus we can further control the growth pattern via specification of peak years (hivpeakyr ) and prevalence rate in those peak years (hivpeakr ), with an algorithmic logic that gradually dampens growth rate to the peak year:[3 ]
t is time, r is country or region. Names in bold are exogenously specified parameters.
As countries pass the peak, we posit that advances are being made against the epidemic, both in terms of social policy and technologies of control, at a speed that reduces the total prevalence rate a certain percent annually (hivtadvr ). To do this, we apply to the prevalence rate an accumulation of the advances (or lack of them) in a technology/social control factor (HIVTECCNTL). In addition, if decline is already underway in the data for recent years, we add a term based on the initial rate of that decline (hivincr ), in order to match the historical pattern; that initial rate of decline decays over time and shifts the dominance of the decline rate to the exogenously specified rate (hivtadvr ). This algorithmic formulation generates the slowly accelerating decline and then slowing decline of a reverse S-shaped pattern with a long tail:
Finally, calculation of country and region-specific numbers of HIV prevalence is simply a matter of applying the rates to the size of the population number.
The rate of death to those with HIV would benefit from a complex model in itself, because it varies by the medical technology available, such as antiretroviral therapy (ART) and the age structure of prevalence. We have simplified such complexities because of data constraints, while maintaining basic representation of the various elements. Because the manifestation of AIDS and deaths from it both lag considerably behind the incidence of HIV, we link the death rate of AIDS (HIVAIDSR) to a 10-year moving average of the HIV prevalence (HIVRateMAvg). We also posit an exogenously specified technological advance factor (aidsdrtadvr ) that gradually reduces the death rate of infected individuals (or inversely increases their life span), as ART is doing. And we allow the user to apply an exogenous multiplier (aidsratem ) for further scenario analysis:
We spread this death rate across sex and age categories. We apply a user-changeable table function to determine the male portion as a function of GDP per capita (at PPP), estimating that the male portion rises to 0.9 with higher GDP per capita.[4 ] To specify the age structure of deaths, we examined data from large numbers of studies on infections by cohort in Brazil and Botswana (in a U.S. Census Bureau database) and extracted a rough cohort pattern (aidsdeathsbyage ) from those data.
 The HIV/AIDS data were being update in October, 2013. The IFs pre-processor calculates initial rates of HIV prevalence and annual changes in it using the middle estimates of the UNAIDS 2008 data. When middle estimates do not exist, as in the case of the Democratic Republic of Congo, it uses an average of high and low estimates. The system uses data for total population prevalence, but also includes HIV prevalence for those 15-49.
 A more satisfactory approach would use stocks and flows and have a more strongly systems dynamics’ character. It would track infected individuals, presumably by age cohorts, but at least in the aggregate. It would compute new infections (incidence) annually, adding those to existing prevalence numbers, transitioning those already infected into some combination of those manifesting AIDS, those dying, and those advancing in age with HIV. But the data do not seem widely available to parameterize such transition rates, especially at the age-category level.
 Table 17 (pp 77-78) of the Annex to World Population Prospects: the 2002 Revision (UNPD 2003) provided such estimates for 38 African countries and selected others outside of Africa; the IFs project has revised and calibrated many of the estimates over time as more data have become available. By 2004-2006, however, quite a number of countries had begun to experience reductions, and this logic has become less important except in scenario analysis for countries where prevalence is still rising.
 Early epidemic data from sub-Saharan Africa and the United States supported this assumption. <header><hgroup>
Equations: Road Traffic Accidents
</hgroup></header> In forecasting mortality related to road traffic accidents, IFs replaces the GBD regression model with a structural formulation designed to better capture relevant drivers for this cause group. Specifically, IFs projects deaths due to traffic accidents (DEATHCAT, Traffic) as a function of deaths in traffic per vehicle (DEATHTRPV) and vehicle numbers (VEHICLESTOT), both computed in the automobile module of IFs. We first need to compute the total size of the vehicle fleet.
Total vehicles per capita (VEHICFLPC) is based on a formula proposed in a paper by Dargay et al (2007) in which fleet size per capita is a function of GDP per capita at PPP (GDPPCP). Translating the Dargay et al (2007) equation into one using IFs variable names yields:[1 ]
The parameter vehicfpcm allows scenario intervention. RF is an adjustment factor that compensates for different land densities, that is the ratio of population (POP) to land area (LANDAREA), taking the U.S. as the base:
The computation was only used when country R had higher density than the US. The paper also describes another adjustment factor related to urbanization as percentage of total population, but we did not use this additional adjustment factor in our model.
Given fleet size per capita and the population, we compute the total size of the fleet.
The number of deaths per vehicle is based on Smeed’s Law[2 ] , an empirical rule originally proposed by R.J. Smeed, which relates deaths to vehicle ownership. In the original conceptual form Smeed’s Law is:
D is annual road deaths
n is number of vehicles
p is population
In terms of IFs variable names this would translate literally (ignoring some unit issues) as:
The actual representation in IFs involves two steps. First we calculate the death rates per vehicle, adding a division by a multiplicative term that is equivalent to total vehicle numbers VEHICLESTOT. One of the virtues of this first step is that we can add an exogenous multiplier for death rates per vehicle, deathtrpvm .
The second step is to use the death rate per vehicle, the vehicle fleet size per capita, and information on the age and sex distribution of deaths from vehicles to compute the mortality rate from vehicle accidents by age and sex, putting the results into a variable internal to model named modmordstdet. In a third step that variable is used with population by age and sex to compute the total deaths from vehicle accidents (DEATHCAT). These second and third steps stylistically yield
After initialization in the base year (using GBD estimates of road traffic-related mortality and total vehicles from the automobile module in IFs), IFs calculates a multiplicative shift factor that is kept constant for the entire forecast horizon. If this initialization value is greater than 40 deaths per 1000 vehicles, we adjust the number of vehicles per capita to set 40 as our initialization value. We started using this limit after finding inconsistencies between estimates derived from Smeed’s Law and those from initial estimates.[3 ]
IFs also computes a ratio (in a variable internal to the model) of traffic accident mortality for males compared to females. The model converges that ratio to 1.5 over 100 years by preserving the total mortality for each age category but adjusting the distribution between males and females.
 Dargay, Gately, and Sommer 2007. “Vehicle Ownership and Income Growth, Worldwide: 1960-2030”. Joyce Dargay, Dermot Gately and Martin Sommer, January 2007.
Smeed, RJ 1949. "Some statistical aspects of road safety research". Royal Statistical Society , Journal (A) CXII (Part I, series 4). 1-24.
Adams 1987. "Smeed's Law: some further thoughts." Traffic Engineering and Control (Feb) 70-73
 The case of Bangladesh is illustrative, where the forecast calculation of 141 deaths/thousand vehicles contrasts with an expectation of 30 deaths/thousand vehicles using Smeed’s Law. We concluded that our mortality figures were consistent with WHO estimates, but sometimes the total number of vehicles was too low. For example, for Bangladesh our data showed 1 vehicle per thousand people, which meant about 141,000 vehicles, when several reports indicate the real number is much higher (850,000) (http://www.brta.gov.bd). <header><hgroup>
</hgroup></header> The IFs model assumes that the initial rate of mortality related to mental health remains constant across our forecast horizon. That rate is subtracted from the other Group II category. The scenario parameter hlmortm allows the user easy control over mortality from the cause <header><hgroup>
Modifications to the Basic Health Model
After examining in IFs the long-term behavior of the regression model forecasts using the Global Burden of Disease (GBD) distal driver approach and coefficients, we made a limited number of modifications. One set of modifications was made to the treatment of the technological change term in the distal formulation (see this section). We also allow countries to transition from low-income status, given expected improvements in development status over the long-term. An adjustment for monotonicity ensures that the forecast population comports with well-known patterns of rising chronic-cause mortality rates with age. Finally, we include health spending in our model in order to better forecast potential outcomes. <header><hgroup>
Mortality Transition for Low-Income Countries
</hgroup></header> As described in the discussion of distal driver coefficients for low-income countries we use Global Burden of Disease (GBD) regression coefficients developed separately for low and high income countries. However, given the long forecast horizon of IFs, we recognize that many low-income countries eventually will reach high levels of income and thus should follow a similar pattern of mortality. Therefore, we allow low-income countries to transition gradually by computing two mortality rates for low-income countries˗one using the low-income beta coefficients and the other using the high-income model. We start the transition when countries reach GDPPCP of $3,000, and finish the transition when countries reach $15,000. The transition is computed finding target mortality in between the two, interpolating depending on the current level of GDPPCP.
Given the target mortality, we compute how much change we need from current mortality (low-income based), and slowly adjust using a moving average of 20 percent of current required change and 80 percent of change used in previous years:
Change = Target Mortality – Low Income Mortality
Smooth Change = 0.2 * Change + 0.8 * Last Year Change
Final Mortality = Low Income Mortality + Maximum(Smooth Change, Change)
where Last Year Change = Maximum(Smooth Change(yr-1), Change(yr-1)).
Note that most of the time target mortality is lower than low income mortality, and thus change is negative. Thus, when we find the maximum we are finding the smaller absolute number and smoothing change. <header><hgroup>
Maintaining Increase with Age in Non-Communicable Death Rates
</hgroup></header> In general, both in our initial conditions and forecasts, we try to maintain monotonicity in growth of death rates from chronic causes with increasing age (above 45) by adjusting deaths from a particular cause when initial computations do not illustrate increases with age and compensating in death rates from an alternative cause for which initial computations indicate room for mortality reduction while (a) maintaining monotonicity for that cause also and (b) not changing total mortality in the pair of causes.[1 ] If necessary we also work to make acceptable adjustments in one age category by readjusting the next age category for the same cause of death, decreasing deaths in the younger age category and increasing deaths in the older one, while doing the opposite for the compensating cause of death. Finally, in this overall and quite complicated algorithmic process we try to minimize the adjustments made to the initial calculations of mortality.
To elaborate this process further, when we find for a chronic cause of death a monotonicity problem for a country and a given age category relative to the next younger category, we find the type (H1) with the highest mortality rate (in the 100+ category among non communicable disease),[2 ] then we try to use H1, which often turns out to be cardiovascular disease, to compensate adjustments in other types in order to keep total mortality constant for the same age category.
For each 5-year age category starting at 45 to 49, we compute total mortality as the sum of all types, then for each non-communicable type with non-zero mortality we compute its growth G from the current age category j to the next j+1, for example in the first step from 45-49 to 50-54. Although our emphasis is on avoiding non-monotonicity, we also would like to see some regularity of progression of mortality increase with age, as we find in the quite high-quality data of Sweden. Thus we also look to that progression in Swedish data for a rate of increase of across age categories that we can use as a minimum. Specifically,
Once we start the adjustment procedure we check if there is room to reduce mortality in the current age and type, so we check growth from the previous age category to avoid breaking monotonicity. First we compute Proxy Growth P1 from the previous age category j-1 (40-44 for our example) to the current one j (45-49).
Second we compute the minimum acceptable value for current mortality:
where http://www.du.edu/ifs/help/media/images/health-module/image042.png is the mortality for country r, type d in age category j-1.
Third we compute maximum acceptable value for current mortality, we start with:
But we know that http://www.du.edu/ifs/help/media/images/health-module/image042.png is also going to change to keep the number of deaths constant, so we also consider this adjustment:
And we know that:
Solving for max, we have:
Where Mort is the original mortality for age category j and j+1, country r, and type d. Pop is population for age j and country r and P is the Proxy Growth computed as explained above.
If Min is smaller than Max, then we use Max as the new mortality in age j, in order to keep the adjustment as small as possible, if not that means that Max wouldn’t keep monotonicity from age j-1, so we start trying to adjust going backwards, given that frequently there’s more room in previous age categories. In order to start going backwards we keep track of the first age category that it’s already saturated, i.e. that its growth is already the minimum possible without breaking monotonicity. If we find that the first saturated category is higher than the 45-49 that we started with, that means we have some room going backwards, so we take Max, otherwise we use Min as the new mortality in age j, and keep adjusting forward. The adjustment A is just the difference between original mortality in age j and the new chosen mortality.
Adjust backwards means that we will adjust mortality in age category j2 and j2-1, where j2 goes from j-1 to 10 (which corresponds to 45-49). While going backwards the formulas for min and max change a little bit, given that the adjustment is done in the previous age category.
And substituting the adjustment computed earlier (http://www.du.edu/ifs/help/media/images/health-module/image045.png),
we end up with:
Max gets simplified to:
We then check for room in type H1, and if there’s enough room we adjust mortality for j2. If Min <= Max then we can stop, otherwise we keep going back until we reach the 45-49 category.
Fourth, we verify that, in doing compensation for type H, monotonicity is preserved too. In order to make this verification first we find the potential growth rate GH after applying adjustment A to type H. Then we compute the Proxy Growth PH for type H. If GH is greater or equal than PH then we can apply the adjustments if not we just leave mortality unchanged.
Fifth, applying the adjustments to type d by subtracting the adjustment from the original mortality in age j for type d, and adding it up adjusted for deaths to the original mortality in age j+1 for type d:
Sixth, applying the adjustments to type H by adding the adjustment to original mortality in age j for type H, and subtracting it adjusted for deaths from the original mortality in age j+1 for type H:
Seventh, if Min is greater than Max and we couldn’t go backwards means that we took Min as the new mortality for age k, and it means that we still don’t have monotonicity because we haven’t changed age j+1 yet. Then we need to find the new mortality value for j+1 using Proxy Growth:
Eighth, we check that this new adjustment doesn’t break monotonicity in death type H, if it doesn’t we apply it as we did for age j, if does break it, we just leave mortality unchanged.
Ninth, applying this adjustment is the same as step 5 and 6, but using ages j+1 and j+2 instead of j and j+1. The only difference here is that when we get to the second to last age category (j = 20, 95-99),[4 ] then the compensatory adjustment for deaths is done in the first age category of the loop (j=10, 45 to 49), and we restart the process for a second and final check of monotonicity.
We have added check limits along the process to avoid mortality going above 1000 per 1000 and below 0 per 1000 at all times, and if the limits are reached then mortality is left unchanged.
 The changes described here for monotonicity with age do not guarantee monotonicity of changes in rates over time within an age category. In fact, they could contribute to some small transients or irregularities in rates over time. In general, however, we believe that they will make such behavior less likely. For such irregularities, see the longitudinal curve in Afghanistan for cancer at 80-84 males; these are most likely to appear, as they would in the real world, when total mortality is not changing much over time.
 In specialized work looking at low senescent aging the last category is 200+.
 An example can illustrate. Say male mortality is 0.5 for Cancer at 45-49, and 1.1 for 50-54, then P is 140% (these are numbers from Sweden). Say total male mortality is 10.1 for 45-49 and 14.5 for 50-54, so total growth is 43%. (These are numbers for any country, say Afghanistan). We can’t use P of 140% on Afghanistan for male cancer, given that for those ages Total Mortality grows only 43%, so we use a Proxy (P) of 11% (43*.25), for male cancer in Afghanistan between ages 45-49 and 50-59. G is the actual growth in mortality for male cancer in Afghanistan between ages of 45-49 and 50-54; say for example: 0.7664, 1.2876 respectively, so G is 68%. In this example, there is no need to change anything (68 > 11). Only if G is smaller than P does the adjustment process begin.
 In specialized work looking at low senescent aging the last category is 195-199. <header><hgroup>
Elasticity of Child Mortality with Health Spending
</hgroup></header> For countries that have a GDP per capita in the initial year of less than $15,000 an elasticity factor with health spending (elhlmortspn ) of -0.06 will affect mortality of children under 5. That is, each 1 percent change in health spending as a percentage of GDP will lower mortality by 0.06 percent; an increase of 100 percent (doubling) would produce an automatic reduction of 6 percent in mortality. We have implemented a limit on the reductions to be at most 80 percent of mortality.
The GBD project’s distal driver formulation does not take public health spending into account. However, we add a term to the basic GBD distal driver formulation to incorporate public health spending as a proximate driver to account for the relatively consistent inverse relationship between total public health expenditures and child mortality rates in poor countries (Anand and Ravallion 1993; Bidani and Ravallion 1997; Jamison et al. 1996; Nixon and Ulmann 2006; Wagstaff 2002). For countries having a GDP per capita (at PPP) of $15,000 or less, our model applies a simple elasticity for the effects of government health expenditure as a percentage of GDP on all-cause mortality (except HIV/AIDS) for the age 0-4 group from the distal driver formulation (the base calculation that health expenditures adjust):
where http://www.du.edu/ifs/help/media/images/health-module/image058.JPG is the mortality rate for age 0-4.
In IFs this formalized version becomes
GDS is government expenditure; elhlmortspn is the elasticity of mortality with health spending, j is age category; r is country/region; d is cause (1 is other communicable); t is time step.
In this calculation we use health expenditure as a percentage of GDP, rather than health expenditure per capita, to avoid any confounding with the distal driver for GDP per capita. We established this coefficient for all-cause mortality in the 0-4 age category on the basis of multivariate regressions using the GBD distal driver specifications as a base. <header><hgroup>
Several important differences in our approach to forecasting health relative to that of the Global Burden of Disease (GBD) project required development of algorithms for computation of initial conditions and small multiplicative adjustments to formulations. Specifically, we forecast by country, we begin forecasts in the base year of 2010, and we maintain 5-year age categories. Moreover, we have our own sources of data for GDP per capita and education attainment level, which we forecast using our own models. The initial data we obtained from the GBD project provided country, sex, and cause specific mortality, but from the year 2008 (subsequently updated to 2010) and in slightly different age categories. This section details our approach to reconciling differences in initial data. <header><hgroup>
Normalization and Scaling Factors
</hgroup></header> IFs initializes the base year (2010) data using age, sex, cause, and country-specific mortality data for 2010 provided by the Global Burden of Disease (GBD), courtesy of Colin Mathers at the World Health Organization. Those data are for infants and then for 5-year age categories up to 85+. To fill holes we used the same base rates for 1-4 as for infants and for all 5-year age categories above 85, subject to normalization to total mortality for the age category. The model then computes normalization and scaling factors which reconcile the results of the forecast regression models with these initial data.
The normalization factor helps match the sum of all mortalities in the health module to the mortality computed in the population module in the base year (2010). This process assures that we have initial conditions consistent with UNPD mortality data in our base year (i.e., the sum of all deaths will be the same as the UNPD mortality data for each 5-year age and sex category for the year 2010). The normalization factor uses all types of mortality except for AIDS in the numerator and denominator:
The scaling factor sets the historic proportions across the different causes of mortality, assuring consistency of total deaths forecast using the GBD formulations and our 2010 values of driving variables with the cause-specific mortality data in the GBD’s detailed death file. The scaling factor uses distal driver regression results for the denominator and GBD 2010 data for the numerator:
These adjustments mean that, except for the total mortality by age and sex of the UN, our numbers in the 2010 base year will not match other data precisely, but that the overall pattern of deaths by cause should be quite close to the GBD data.[1 ] In the forecasts themselves, we keep the multiplicative scaling and normalization parameters constant over time because there is no clear reason for changing them in the base, but have added parameters to control convergence in scenarios: hlgbdconvdown , hlgbdconvup , hlscaleconvdown , where the first parameter controls the normalization factor when is greater than 1, the second one controls the normalization factor when is smaller than 1, and the last one controls the scaling factor when is greater than 1.
 Complicating initialization further, the UNPD presents its data in 5-year ranges, including 2005-2010 and 2010-2015. The age- and sex-specific survivor-table values in those ranges therefore do not correspond to specific years like our base of 2010. After correspondence with Kirill Andreev of the UNPD, which we acknowledge appreciatively, we decided to average the mortality values in the two 5-year ranges ending and beginning with 2010. <header><hgroup>
Cause-specific Mortality: Infants and the Elderly
</hgroup></header> The detailed deaths data file (by cause, sex, and age) that we have obtained through the generosity of Colin Mathers at the World Health Organization does not include cause-specific infant or old-age (85+) mortality (greater than 85). Because IFs forecasts both infant mortality and 5-year age categories to 100 years, we incorporate detailed mortality data from Sweden (as a proxy, thanks mainly to availability of data) in order to initialize Group II (excluding mental health) cause-specific mortality for these missing populations.[1 ]
The first step is to find the weights per age category for Sweden as follows:
Where j is the smaller IFs age category a (for example infants), JJ is the bigger corresponding GBD age category (for example children < 5, which implies the addition of infants plus children 1-4), p is gender, and d is mortality type.
The second step is to check the monotonicity of growth in the existing mortality data for each country and type of mortality (from age 45 forward). If monotonicity is not found (i.e., mortality rates do not rise in step with increasing age groups) then the initialization data is left unchanged for this country and mortality type combination. If initial mortality does increase monotonically, we further adjust mortality:
Where j, JJ, p and d are the same as in the previous equation and Pop is the population vector.
This option has been currently disabled because it was producing too much NCDs compared to CDs for countries like Mali for people over 90 years of age.
 This may be an issue also for Groups 1 and 3, but we have used the procedure with Sweden only for Group 2.
 Beginning with discovery of a problem for Mali, the use of this distribution procedure in IFs ran into some logic problems with poor behavior. As of September, 2013, the spread of initialization data for infants 1-4 and for ages 85+ is disabled. That spread, although desirable, is not necessary and at this point the code generates more problems than it solves. This section of the documentation is maintained should we want to revisit the issue. <header><hgroup>
Incorporating Proximate Drivers
Although the distal driver approach serves us well in forecasting health, thinking about intervention and leverage points in order to achieve alternate health futures necessarily involves the inclusion of proximate health drivers into the model. Therefore we need to have an approach that can layer specific risk analysis onto the underlying distal driver specifications. We need to discuss our general approach to doing that, which uses population attributable fraction or PAF formulations, and we need to provide equations of the models used to forecast proximate drivers in IFs and to build the relative risk estimates we used in the model. <header><hgroup>
Adjusting Mortality Due to Changes in a Single Risk Factor
</hgroup></header> We build our approach on an understanding of two basic concepts used in the Comparative Risk Assessment (CRA) project (Ezzati et al. 2004), specifically relative risk (RR) and population attributable fraction (PAF). An RR is a “measure of the risk of a certain event happening in one group compared to the risk of the same event happening in another group”. We follow the approach taken by the CRA study, comparing our forecast population at risk to an “ideal” population with a “theoretical minimum” level of risk. For example, the WHO estimates that children under five who are moderately or severely underweight are almost nine times more likely to die of communicable causes than a population of “normal-weight” children (Blössner and de Onis 2005).
As its name suggests, a PAF or population attributable fraction reflects the degree to which a specific risk factor is associated with the occurrence of a specific health outcome. Formally, it is the proportional reduction in disease or death rates for the total population (including those with and without the risk factor) that we would expect if we reduced a particular risk factor to a theoretically minimum level (Ezzati et al. 2004). The further the current situation is from the ideal, the closer the value of the PAF will be to 1.
A PAF is calculated as:
(∑RR(x)P(x)-∑RR(x)P’(x)/∑RR(x)P(x)) = 1 - ∑RR(x)P’(x)/∑RR(x)P(x)
RR(x) is relative risk at exposure level x; and
P(x) is the population distribution in terms of exposure level, i.e. the shares of the population exposed to each level of exposure;
P’(x) is the theoretical minimum population distribution in terms of exposure level; for certain risks this is defined as no exposure; where this is not realistic, the WHO defines an international reference population
Following this definition, multiplying the mortality from a particular disease by the PAF yields an estimate of the number of people who would not have died had the risk factor been at its theoretical minimum level. If we assume that the values of RR(x) and P’(x) for particular risk factors and diseases do not differ across countries or change over time, then changes in the PAF are solely a function of changes in P(x), the exposure of the population to the particular risk factor. Thus, it is necessary to be able to forecast the future levels of the risk factors. Other Help topics describe how this is done for specific risk factors such as undernutrition, obesity, smoking, and indoor air pollution. Since our forecast of health outcomes from distal drivers implicitly suggests certain proximate driver levels, we are really interested in the effect of a difference in (1) estimates of the future levels of a risk factor based only on distal drivers (representing an “expected” value for a country given those distal drivers), and (2) estimates based upon a more complete set of drivers (representing our best forecast for a country using initial conditions and therefore path dependency, the additional and/or alternative drivers, and potential scenario interventions) . We therefore calculate two versions of the PAF, namely PAFDistal and PAFFull. Defining MortalityDistal as the mortality calculated using only the distal drivers and MortalityFinal as the mortality after accounting explicitly for the risk factor, we can state that:
- MortalityDistal * PAFDistal represents the number of people who would not have died had the risk factor been at its theoretical minimum level using the distal formulations for mortality and the proximate risk factor; and
- MortalityFinal * PAFFull represents the number of people who would not have died had the risk factor been at its theoretical minimum level using a more complete formulation for mortality and the proximate risk factor
If we assume that no other factors influence the difference in total mortality between the distal formulation and that using the full model, then:
MortalityFinal - MortalityDistal = MortalityFinal * PAFFull - MortalityDistal * PAFDistal
MortalityFinal = MortalityDistal * ((1-PAFDistal) / (1-PAFFull))
= MortalityDistal * ∑RR(x)PFull(x)/∑RR(x)PDistal(x)
The adjustment factor is the ratio of the weighted average relative risks based on the distributions using the distal-only versus the full formulations for estimating the value of the risk factor. A higher weighted-average relative risk based on the full formulation implies that the distal drivers overestimate our anticipated improvement (or underestimate the deterioration) in the risk factor. Thus, the mortality forecast needs to be adjusted upwards. Alternatively, if the weighted-average relative risk is lower based on the full formulation than on the distal formulation, the mortality forecast will be adjusted downwards. Note that this property of the calculation actually obviates the need to know the theoretical minimum population.
 Dictionary of Cancer terms, National Cancer Institute; accessed online, January 2010. http://www.cancer.gov/dictionary/.
 This is very reasonable for P’(x) by its definition. With respect to RR(x), we assume these to be the same for all countries unless otherwise specified in the CRA reports. Any change over time is likely to be picked up in other parts of our model dealing with changes in technology and the efficiency of health care systems.