Bayesian Projection of Life Expectancy Accounting for the HIV/AIDS Epidemic
BBayesian Pro jection of Life Expectancy Accounting forthe HIV/AIDS Epidemic
Jessica Godwin and Adrian E. RafteryUniversity of WashingtonAugust 25, 2016
Probabilistic projections of mortality measures are important for many applications in-cluding population projection and pension and healthcare planning. Until recently, mostprojections of mortality measures were deterministic, although the UN has recently startedto base its official projections of mortality on probabilistic methods. Most projections, deter-ministic or probabilistic, do not incorporate cause-of-death information or other covariates.Lee and Carter (1992) developed the first method for projecting a mortality measureprobabilistically. The Lee-Carter method projects age-specific mortality rates and is widelyused today. It requires at least three time periods of age-specific death rates, an amount ofdata that is not available in many countries. The method assumes that the logarithm of theage-specific death rates will increase linearly in the future, which may not be optimal forlong term projections (Lee and Miller, 2001). Girosi and King (2008) proposed a Bayesianmethod for smoothing age-specific death rates over both age and time. Though this methodallows for the incorporation of covariates, it has been shown to perform well only for countrieswith good vital registration data. Like the Lee-Carter method, the Girosi and King (2008)method assumes a constant rate of increase. Lutz et al. (1998) developed an expert-basedmethod for probabilistic projections of population that incorporates subjective probabilisticprojections for several demographic measures, including life expectancy.Raftery et al. (2013, 2014) presented a Bayesian hierarchical model (BHM) for projectingmale and female life expectancy probabilistically for all countries of the world to 2100, andthis method is now used by the UN as an input to its official population projections (UnitedNations, 2015). We extend the model in Raftery et al. (2013) to include covariate informationabout generalized HIV epidemic prevalence and coverage of antiretroviral therapy (ART)in each country. A country is said to have a generalized HIV/AIDS epidemic when HIVprevalence is greater than 1% in the general population, and it is not concentrated in atrisk subgroups. While there are many diseases that have a high impact on mortality in agiven country, the generalized HIV epidemic is unusual in that it dramatically increases age-specific mortality rates at prime adult ages. Its demographic impact is therefore different1 a r X i v : . [ s t a t . A P ] S e p Year
Life Expectancy, Years
HIV Prevalence, %
ART Coverage, %
Figure 1: Life expectancy at birth (black), HIV prevalence (red) and ART coverage (blue)for Botswana from 1950-1955 to 2005-2015.from that of other diseases, which tend primarily to affect mortality rates for very youngand/or older ages.Figure 1 shows the rise of the HIV epidemic and the corresponding evolution of lifeexpectancy at birth in Botswana. There was a sharp lowering of life expectancy with therise of the epidemic, and then a rapid recovery to pre-epidemic levels following the widespreadintroduction of ART.To incorporate covariate information into a probabilistic projection model, we must alsohave a method for projecting the covariate of interest into the future. UNAIDS developedthe Spectrum/EPP methodology for projecting HIV prevalence and demographic measures,including life expectancy at birth, while accounting for HIV prevalence and ART coverageamong other things (Stover et al., 2012; Stanecki et al., 2012; Futures Institute, 2014). Themethod is quite complicated and requires fine-grained data on a number of demographic andhealth measures for each country. It is recommended for reconstructing the HIV epidemic,including the time of onset, in a particular country and for projecting the epidemic up to fiveyears into the future. It is not designed for the longer term projections that are importantfor long-term population projections (UNAIDS, 2014, p. 9).We use a version of the EPP package for R for projections of HIV prevalence to 2100(Brown et al., 2010). We develop a simpler model for projecting life expectancy at birthwhile accounting for HIV prevalence and ART coverage that is more practical for long termprojections. 2 Methodology
We use estimates of female life expectancy at birth from the United Nations
WorldPopulation Prospects (WPP) 2015 Revision (United Nations, 2015) for 201 countries. TheUN produces estimates of period life expectancy at birth and age-specific mortality ratesby five-year periods and five-year age groups; these are updated every two years. There areestimates for each country of the world for each five year period from 1950 to 2015. We do notuse life expectancy inputs for Cambodia and Rwanda from the time periods of the genocidesin these countries. To fit the model, we use UNAIDS estimates of past HIV prevalence andART coverage for 40 countries with generalized epidemics. We use 1000 trajectories of HIVprevalence, using the same assumptions as UNAIDS does in their projections. Additionally,we use a single deterministic trajectory of ART coverage from UNAIDS in the projectionstage. We code HIV prevalence as zero for all countries not experiencing a generalized HIVepidemic.
Our methodology builds on the Bayesian hierarchical model for probabilistic projection offemale and male life expectancy used by the UN (Raftery et al., 2013, 2014), which proceedsas follows. First, the Bayesian hierarchical model for female life expectancy is estimated usingMarkov chain Monte Carlo (MCMC), then probabilistic projections of female life expectancyare made from the present day to 2100. Projections of male expectancy are then made basedon the projected values of female life expectancy (Raftery et al., 2014). The model providesa way for estimation and projection for one country to be improved using information fromother countries.At the lowest (observation) level, the Bayesian hierarchical model for female life ex-pectancy at birth is ∆ (cid:96) c,t ≡ (cid:96) c,t +1 − (cid:96) c,t = g ( (cid:96) c,t | θ ( c ) ) + ε c,t +1 , (1) ε c,t ∼ N (0 , ( ωf ( (cid:96) c,t )) ) , (2)where (cid:96) c,t is the female life expectancy at birth for country c in time period t , g ( ·| θ ( c ) ) is theexpected five-year gain in life expectancy, modeled as a double logistic function of current lifeexpectancy and governed by country-specific parameters, θ ( c ) , ε c,t +1 is a random perturbationaround the expected gain, and f ( (cid:96) c,t ) is a smooth function of life expectancy. The doublelogistic function for country c is g ( (cid:96) c,t | θ ( c ) ) = k c (cid:16) − A ∆ c ( (cid:96) c,t − ∆ c − A ∆ c ) (cid:17) + z c − k c (cid:16) − A ∆ c ( (cid:96) c,t − (cid:80) i =1 ∆ ci − A ∆ c ) (cid:17) , (3)3here θ ( c ) = (∆ c , ∆ c , ∆ c , ∆ c , k c , z c ) and A and A are constants. The parameter z c is theexpected country-specific asymptotic five-year gain in life expectancy. The other parametersgovern the maximum value and the pace of rise and fall of expected five-year gains in lifeexpectancy.At the second level of the model, the country-specific parameters θ ( c ) are assumed to bedrawn from the following world distribution:∆ ci | σ ∆ i iid ∼ Normal [0 , (∆ i , σ i ) , i = 1 , . . . , ,k c | σ k iid ∼ Normal [0 , ( k, σ k ) z c | σ z iid ∼ Normal [0 , . ( z, σ z ) . At the third, top level of the model, prior distributions are specified for the world parameters θ = (∆ , ∆ , ∆ , ∆ , k, z, ω ).The Bayesian hierarchical model is estimated using MCMC via Metropolis-Hastings,Gibbs sampling and slice sampling steps, yielding a joint posterior distribution of all modelparameters (Raftery et al., 2013). The smooth function f ( (cid:96) c,t ) specifying the variance of theperturbations is estimated separately and is treated as known in the MCMC algorithm.Once the model has been estimated, projections of life expectancy are made based on eachposterior sample of θ ( c ) and a random perturbation, ε c,t +1 , drawn from a N (0 , ( ωf ( (cid:96) c,t )) )distribution, where ω is drawn from the posterior distribution. After female projections oflife expectancy are made, projections of male life expectancy, (cid:96) mc,t , are made by modeling thegap between the two (Raftery et al., 2014). We expand the BHM to account for generalized HIV/AIDS epidemics by adding a co-variate to the observation level of the model. The covariate is based on
HIV nonART c,t = HnA c,t , defined as follows. Let
HIV c,t and
ART c,t be the HIV prevalence and ART coveragein percent of country c at time period t , respectively. Then HnA c,t = HIV c,t × (100 − ART c,t );it can be viewed as approximating the percentage of the population who are infected but donot receive ART therapy. The covariate we found to best predict change in life expectancywas the change in this quantity, namely ∆
HnA c,t − = HnA c,t − HnA c,t − . Our expandedobservation equation is then∆ (cid:96) c,t = g ( (cid:96) c,t | θ ( c ) ) + β HnA ∆ HnA c,t − + ε c,t +1 . (4)The parameter β HnA is constant across countries and is estimated by MCMC along withthe other parameters of the Bayesian hierarchical model. It has a diffuse prior distribution,chosen to be spread out enough that it has little impact on the final inference. Specifically,the prior distribution of β HnA is N (cid:18) , . × V ar (∆ (cid:96) c,t ) V ar (∆ HnA c,t ) (cid:19) , where the prior variance isdetermined by the sample variances of observed changes in life expectancy and observed4hanges in HnA . The posterior distribution of β HnA is estimated with the other parametersin the MCMC via Gibbs sampling updates.After estimation, we project female life expectancy in the same manner as outlined inSection 2.2. However, we make a projection based on each posterior sample of ( θ ( c ) , β HnA )and a random perturbation. We account for uncertainty in the HIV trajectories by using1000 yearly trajectories of HIV projections from EPP (Brown et al., 2010). For each country c and year t , we find the median, z t,c , of projected adult HIV prevalence output from EPP.We use a single UNAIDS deterministic projection to 2100 as a baseline reference, and weconstruct 1000 trajectories from the single UNAIDS trajectory by using 1000 multipliers ofthe form z kt,c z t,c , at each time point t for k = 1 , . . . , z kt,c is prevalence at time t incountry c in the k simulated trajectory. Thus the UNAIDS deterministic trajectory servesas the median trajectory of HIV prevalence to 2100, and the EPP trajectories determinethe uncertainty. We construct 5-year averages from the yearly trajectories to be used inthe projection stage. From these, we use a single deterministic trajectory of ART coverageto compute 1000 trajectories of ∆ HnA c,t for all countries. We sample one out of 1000trajectories of ∆
HnA c,t with equal probability to be used in the projection stage.For the country of Liberia, the prevalence is projected to be so low in the future (nearly0) that the multipliers are unrealistically large. We therefore treat it slightly differently. Forthis country we calculate z kt,c − z t,c for each time point t . We then add this distance to theUNAIDS trajectory to yield 1,000 trajectories with the UNAIDS trajectory as the medianand borrow the uncertainty from the EPP trajectories.The methods of Raftery et al. (2013) and Raftery et al. (2014) did not use the generalizedHIV epidemic countries in model estimation. By contrast, our estimation of the BHM doesinclude these countries. The covariate values are set to zero for non-epidemic countries.Thus, the estimation of country-specific parameters and the projection of life expectancy fornon-epidemic countries changes negligibly; we are fitting the model in (1) for these countries.For epidemic countries, the model in (4) allows us to adjust for the effects of HIV on lifeexpectancy in the linear term and to interpret g ( ·| θ ( c ) ) as the expected five-year gain in lifeexpectancy in the absence of the epidemic. Though high AIDS prevalence takes a big toll ona country’s life expectancy at birth in the absence of ART, ART extends an infected person’slife substantially. Several epidemiological case studies show that patients have nearly normallife expectancy when treated with ART (Mills et al., 2011; Johnson et al., 2013). In a countrywhere ART coverage is high, a generalized HIV epidemic affects life expectancy like a chronicdisease (Deeks et al., 2013).In a manner similar to Raftery et al. (2013), the distribution of random perturbations inthe projection stage is ε c,t +1 ∼ N (0 , ( ω × f ( (cid:96) c,t,i ) ), where ω is a model parameter, f ( (cid:96) c,t,i ) isa smooth function and i is an indicator for generalized HIV epidemic. To estimate f ( (cid:96) c,t,i ),we fit the model in (4) using the same function f ( (cid:96) c,t ) as used by Raftery et al. (2013).Then, using mean posterior estimates of g ( (cid:96) c,t | θ ( c ) ), we projected life expectancy forwardfrom 1950-1955 to the 2010-2015 period using only the mean model in (4) with no randomperturbations. We then calculated absolute residuals for these projections.5e fit loess curves to the absolute residuals for non-epidemic countries and for epidemiccountries separately. These curves can be seen in Figure 2. The black dots represent theabsolute residuals from HIV countries, and the red loess curve is fit to these points. Thegrey dots represent the absolute residuals from non-HIV countries, and the blue curve is fitto these points. Here one can see that the HIV countries have more variability than thenon-HIV countries. This variability is disseminated into the future for countries currentlyexperiencing a generalized epidemic. For non-HIV countries, f ( (cid:96) c,t,i = nonHIV ) is the blue curvein Figure 2. For HIV countries, f ( (cid:96) c,t,i = HIV ) is the maximum of the blue and red curves up tothe highest observed life expectancy for an HIV country to date, namely 78.1. For projectedfemale life expectancies above 78.1 years, we use the blue curve plus a constant that is thevertical difference between the red and blue curves at 78.1 years. Life expectancy A b s o l u t e R e s i dua l Non-Epidemic
Epidemic
Lowess-non HIV
Lowess-HIV
Figure 2: The black dots represent absolute residuals for epidemic countries; the grey dotsrepresent absolute residuals for non-epidemic countries. The blue line is the loess fit to thenon-epidemic residuals. The red line is the loess fit to the epidemic residuals.
We performed predictive out of sample validation to assess our model. First, we fit themodel in (4) using data from 1950-1955 up to 2000-2005 and projected female life expectancyfor the time periods 2005-2010 and 2010-2015. We also fit the model in (4) using data from6able 1: Predictive Validation Results for Female Life Expectancy. The first column repre-sents the set of countries used in the subsequent calibration calculations. The second columnrepresents the time period of data used to fit the model, and the third column representsthe time periods used in validation. The fourth column represents the number of countriesused in validation. In the fifth column, “No Covariates” represents the model in (1) and“∆
HnA ” represents the model in (4). The sixth column contains the MAE as defined inSection 2.4. The seventh and eighth columns contain coverage metrics for the 80% and 95%predictive intervals respectively.Countries Training Test n Model MAE CoveragePeriod Period 80% 95%HIV 1950-2005 2005-2015 69 No Covariates 3.47 0.49 0.58∆
HnA
HnA
HnA
HnA
HnA
HnA
HnA
HnA
HnA ” refersto the model in (4).The last three columns contain our metrics. The mean absolute error (MAE) is calculatedas 1 n (cid:88) c ∈C (cid:88) t ∈T | ˆ (cid:96) c,t − (cid:96) c,t | , (5)where ˆ (cid:96) c,t is the median projection of female life expectancy for country c in time period t . In(5), C is the set of countries involved in calculating the MAE (either the HIV countries or allcountries), T is the set of five-year time periods involved as shown in the third column, and n is the number of country-time period combinations as shown in the fourth column. Thelast two columns show the proportion of countries whose 80% and 95% posterior predictiveintervals contain the observed life expectancy in the validation period of interest.In all the out of sample scenarios, we saw substantial improvements in coverage for HIVcountries after accounting for HIV prevalence and ART coverage. We broke down the two-period out of sample exercise into the two projection periods to get more detailed informationabout the HIV countries. In 2005-2010, the model with no covariates missed 15 out of 40HIV countries at the 95% level. When we accounted for HIV prevalence and ART coverage,the number of HIV countries missed went down by over half to only 7 at the 95% level. For2010-2015, the model with no covariates missed 14 out of 29 HIV countries at the 95% level,and accounting for HIV prevalence and ART coverage reduced this to only 2 HIV countriesin the time period 2010-2015. In the leave-two-time-periods-out validation exercise, we sawa decrease in MAE for HIV countries in every case, while the MAE remained unchanged fornon-HIV countries. For the non-HIV countries, the addition of the covariate in the modelin (4) changed coverage negligibly.When fitting the model using data from 1950-1955 up to 2005-2010 and projecting femalelife expectancy for 2010-2015, we also saw improvements. Our coverage was closer to nominalfor HIV countries after accounting for HIV prevalence and ART coverage. We missed 11 HIVcountries out of 29 in the model with no covariates, but only one HIV country after accountingfor the HIV epidemic. The MAE also decreased when accounting for HIV prevalence andART coverage.Predictive validation results for male life expectancy are shown in Table 2 in AppendixB, and the conclusions are broadly similar.The current method used by the UN to project life expectancy in the presence of theHIV epidemic is the Spectrum/EPP package (Futures Institute, 2014; Stanecki et al., 2012;Stover et al., 2012). However, Spectrum is a complicated model with heavy data demandsand is intended only for short-term projections up to five years into the future. An importantquestion is therefore whether our simpler method can produce short-term projections similarto those of the more complex Spectrum method.To answer this, we fit our model in (4) with WPP 2012 estimates of female life expectancyfrom 1950-1955 up to 2005-2010 (United Nations, 2013). Then we projected female lifeexpectancy to 2010-2015. We compared the projections from our simpler model designed to8 Projection Comparisons for 2010-2015
WPP 2012 Δ non A R T Absolute Deviations from WPP2015 Estimate 2010-2015
WPP 2012 Δ non A R T Figure 3: Comparison between short-term projections in WPP 2012 using Spectrum, andour simpler method. On the x -axis are projections of female life expectancy in 2010-2015produced using Spectrum and published in WPP 2012. On the y -axis are projections offemale life expectancy in 2010-2015 produced by fitting the model in (4) with WPP2012data up to 2005-2010. The projections are mostly similar and remain close to the y = x line.make long-term projections to the WPP 2012 projection for 2010-2015 made using Spectrum.In the left panel of Figure 3, we see that the five-year projections from our simpler methodare similar to the projections from the more complicated Spectrum. In fact, the correlationbetween the projections from our proposed model and those published in WPP 2012 is 0.89.The right panel of Figure 3 shows the absolute deviation from the WPP 2015 estimateof female life expectancy in 2010-2015 for our projections and the projections published inWPP 2012. The WPP 2012 projections have a mean absolute error of 2.70 years using theWPP 2015 estimate for comparison. The projections produced with the model in (4) havea mean absolute error of 2.17 years.The large outlier in the right panel of Figure 3 is the country of Botswana, and corre-sponds to the largest deviation seen in the left panel of Figure 3. Botswana has the highestHIV prevalence in the world, at 24.3% in 2010-2015, and has had a recent scale up of ARTcoverage. The boost in ART coverage yielded a rapid recovery in life expectancy of nearly 20years from the WPP 2012 estimate for the 2005-2010 time period. Our model captures thislarge jump in life expectancy. In summary, our model produces similar projections to thecurrent methodology designed for short-term projections, but using a much simpler modelwith smaller data requirements. We now give specific results for five countries that illustrate specific aspects of the method.Results for all countries we consider as having generalized epidemics are given for female life9
950 2000 2050 2100
Nigeria
Year L i f e E x pe c t an cy ● ● ● ● ● ● ● ● ● ● ● ● ● No Covariates ∆ HIVnonART
Nigeria
Year H I V P r e v a l en c e ● ● ● ● ● ● ● ● ● ● ● ● ● Median80%95%
Figure 4: The left panel shows projections of female life expectancy in Nigeria under themodel in equation (1) in blue and equation (4) in red. The solid lines represent medians,and the dashed lines are the 95% intervals. After accounting for HIV prevalence and ARTcoverage, we see slightly more uncertainty about the future life expectancy in Nigeria andslightly higher median projections. The right panel shows the single trajectory of pastestimates of HIV we use in model fitting in black. In red we have the median, 80% intervaland 90% interval of probabilistic trajectories of HIV prevalence from EPP we use in ourprojections.expenctancy in Appendix A and for male life expectancy in Appendix B.
Nigeria in West Africa is the most populous country in Africa. It has a relatively smallepidemic, with prevalence of 3.6% in 2010-2015. Figure 4 shows a comparison betweenprojections of life expectancy under the model (1) with no covariates in blue, and the model(4) with the HIV covariate in red. The median projections of female life expectancy are higherthan those projected when not accounting for the HIV/AIDS epidemic, and accounting forthe epidemic leads to more uncertainty about the future trajectory of female life expectancyin Nigeria.
Kenya in East Africa has a medium-sized epidemic with HIV prevalence 5.7% in 2010-2015. Figure 5 shows that Kenya has already recovered to pre-epidemic life expectancylevels. After accounting for HIV prevalence and ART coverage, we project a slightly highermedian female life expectancy to 2100 with more uncertainty at all time periods.10
950 2000 2050 2100
Kenya
Year L i f e E x pe c t an cy ● ● ● ● ● ● ● ● ● ● ● ● ● No Covariates ∆ HIVnonART
Kenya
Year H I V P r e v a l en c e ● ● ● ● ● ● ● ● ● ● ● ● ● Median80%95%
Figure 5: The left panel shows projection of female life expectancy in Kenya under the modelin equation (1) in blue and equation (4) in red. The solid lines represent medians, and thedashed lines are the 95% intervals. After accounting for HIV prevalence and ART coverage,we see a higher median projection of life expectancy with more uncertainty. The right panelshows the single trajectory of past estimates of HIV we use in model fitting in black. Inred we have the median, 80% interval and 90% interval of probabilistic trajectories of HIVprevalence from EPP we use in our projections.11 .3 South Africa
South Africa has the largest HIV/AIDS epidemic in the world in absolute numbers. Theestimated prevalence in the 2010-2015 time period is 17.5%. Figure 6 shows a comparisonbetween projections of life expectancy under the model in (1) in blue and the model in (4)in red. Figure 6 reflects the clear impact of ART coverage on recovery in life expectancyunder a large epidemic. After accounting for HIV prevalence and ART coverage, we projectan initial recovery to pre-epidemic life expectancy levels with a steady rise through theend of the century. When not accounting for the HIV epidemic and, particularly, ARTcoverage, the model projects median end of century life expectancy only slightly higherthan South Africa’s life expectancy before the HIV/AIDS epidemic. This is contrary tothe epidemiological literature referenced in Section 2.3 that shows life expectancy recoversquickly after a scale-up of ART coverage.As mentioned in Section 2.4, there are a number of countries for which the UN did nothave up-to-date life expectancy data at the time of publication of the WPP 2015, includingSouth Africa. The UN estimates the female life expectancy for the period 2010-2015 as 59.1years (United Nations, 2015). Statistics South Africa has published mid-year estimates oflife expectancy for each calendar year (Statistics South Africa 2010, 2011, 2013, 2014, 2015),and averaging these gives an estimate for the five-year period 2010-2015 of 60.7 years. Whenwe fit our model with data up to 2005-2010 and project forward five years to 2010-2015,our 95% interval for 2010-2015 is (56.6, 65.1). When we fit our model with data only upto 2000-2005 and project forward ten years to 2010-2015, our 95% interval for 2010-2015 is(61.3, 63.7). In both cases, our interval captures the outcome, whether measured by the UNor Statistics South Africa, and in particular the rapid increase in life expectancy due to thewidespread rollout of ART.
Due to the early and rapid rise of HIV prevalence in Botswana, the ART scale-up wasalso quick. This led to the rapid recovery in life expectancy in the 2005-2010 time period.After accounting for the epidemic and antiretroviral therapy, we project a slightly slowerrise in median life expectancy with more certainty than the model that does not account forthe epidemic. This is in agreement with the epidemiological literature cited in Section 2.3suggesting that HIV/AIDS will affect life expectancy as a chronic disease would after ARThas become pervasive.
Germany is an example of a country that does not have a generalized epidemic. As canbe seen in Figure 8, projections of life expectancy under the model in (1) and the model in(4) differ negligibly in both median and uncertainty.12
950 2000 2050 2100
South Africa
Year L i f e E x pe c t an cy ● ● ● ● ● ● ● ● ● ● ● ● ● No Covariates ∆ HIVnonART
South Africa
Year H I V P r e v a l en c e ● ● ● ● ● ● ● ● ● ● ● ● ● Median80%95%
Figure 6: The left panel shows projection of female life expectancy in South Africa underthe model in equation (1) in blue and equation (4) in red. The solid lines represent medians,and the dashed lines are the 95% intervals. After accounting for HIV prevalence and ARTcoverage, we see an initial recovery of life expectancy to pre-epidemic levels followed by asteady rise through the end of the century. The right panel shows the single trajectory ofpast estimates of HIV we use in model fitting in black. In red we have the median, 80%interval and 90% interval of probabilistic trajectories of HIV prevalence from EPP we use inour projections. 13
950 2000 2050 2100
Botswana
Year L i f e E x pe c t an cy ● ● ● ● ● ● ● ● ● ● ● ● ● No Covariates ∆ HIVnonART
Botswana
Year H I V P r e v a l en c e ● ● ● ● ● ● ● ● ● ● ● ● ● Median80%95%
Figure 7: The left panel shows projection of female life expectancy in Botswana under themodel in equation (1) in blue and equation (4) in red. The solid lines represent medians,and the dashed lines are the 95% intervals. After accounting for HIV prevalence and ARTcoverage, we see a dampened, slow-rising median projection of life expectancy with lessuncertainty. Germany
Year L i f e E x pe c t an cy No Covariates Δ HIVnonART
Figure 8: This figure shows projection of female life expectancy in Germany under the modelin equation (1) in black and equation (4) in red. The medians are identical, and the predictiveintervals differ negligibly. 14
Discussion
We have developed a probabilistic method for projecting life expectancy while accountingfor generalized HIV/AIDS prevalence and antiretroviral therapy coverage. Our method hasrelatively modest data requirements. Through predictive validation we have shown that ourmethod improves upon life expectancy projections for HIV/AIDS countries using the methodin Raftery et al. (2013), while leaving projections for non-epidemic countries essentiallyunchanged. Our projections improve in terms of both the mean absolute error of pointpredictions and the calibration of predictive intervals. Our method produces similar short-term projections to the UNAIDS Spectrum/EPP package, with a simpler model that requiresmuch less data. Moreover, the method can produce long-term projections out to 2100.Our model reflects the literature consensus mentioned in Section 2.3 that HIV prevalencewill have large impacts on life expectancy only in the absence of antiretroviral therapy. OnceART covers a large proportion of the infected population, there is a one-time gain in lifeexpectancy to pre-epidemic levels and the effects will be modest afterwards.One limitation of our method is the quality of the ART coverage data and projections.As ART coverage is relatively new and hard to measure, the data we have are noisy. Im-provements in ART data quality would likely result in improvements in projections for thegeneralized HIV epidemic countries. Given good quality data, our method could also beextended to account for other covariates that explain changes in life expectancy. The datawould need to be available for every country used in model fitting back to 1950. Methodologyfor projecting the covariates would also be required.
The project described was supported by grants R01 HD-054511 and R01 HD-070936 fromthe Eunice Kennedy Shriver National Institute of Child Health and Human Development.The authors are grateful to Le Bao, Samuel Clark, Yanjun He and Hana ˇSevˇc´ıkov´a for helpfuldiscussions and sharing data and code.
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World Population Prospects: The 2015 Revision . United Nations,Department of Economic and Social Affairs, Population Division New York, NY, USA.17 ppendix A: Female Life Expectancy and Adult HIVPrevalence Pro jections
In the left panel of every row below are projections of life expectancy using our model(4) for all countries we treat as generalized epidemic countries. In the right panel, ourobserved trajectories of HIV prevalence and our projections of HIV prevalence made usinga single trajectory of HIV prevalence for each country allowing the EPP projections to giveuncertainty. The format of these figures is the same as that of Figures 4-7 in Section 3.18
950 2000 2050 2100
Angola
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Angola
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Bahamas
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Bahamas
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Belize
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Belize
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Benin
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100 . . . . . Benin
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Botswana
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Botswana
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Burkina Faso
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Burkina Faso
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Burundi
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Burundi
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Cameroon
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Cameroon
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Central African Republic
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Central African Republic
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Chad
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Chad
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Congo
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Congo
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Cote d'Ivoire
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Cote d'Ivoire
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Djibouti
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Djibouti
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Equatorial Guinea
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Equatorial Guinea
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Ethiopia
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100 . . . . Ethiopia
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Gabon
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Gabon
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Gambia
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Gambia
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Ghana
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100 . . . . Ghana
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Guinea
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Guinea
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Guinea−Bissau
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Guinea−Bissau
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Guyana
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Guyana
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Haiti
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Haiti
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Jamaica
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Jamaica
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Kenya
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Kenya
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Lesotho
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Lesotho
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Liberia
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Liberia
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Malawi
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Malawi
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Mali
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100 . . . . . . Mali
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Mozambique
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Mozambique
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Namibia
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Namibia
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Nigeria
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Nigeria
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Rwanda
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Rwanda
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Sierra Leone
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Sierra Leone
Year H I V P r e v a l en c e lllllllllllll Median80%95%
South Africa
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
South Africa
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Swaziland
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Swaziland
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Togo
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Togo
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Uganda
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Uganda
Year H I V P r e v a l en c e lllllllllllll Median80%95%
United Republic of Tanzania
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
United Republic of Tanzania
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Zambia
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Zambia
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Zimbabwe
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Zimbabwe
Year H I V P r e v a l en c e llllllllllllllllllllllllll
Year H I V P r e v a l en c e llllllllllllllllllllllllll Median80%95% ppendix B: Male Life Expectancy and Adult HIV Preva-lence Pro jections
HnA ” represents the model in (4). Thesixth column contains the MAE as defined in Section 2.4. The seventh and eighth columnscontain coverage metrics for the 80% and 95% predictive intervals respectively.Countries Training Test n Model MAE CoveragePeriod Period 80% 95%HIV 1950-2005 2005-2015 69 No Covariates 3.74 0.36 0.51∆
HnA
HnA
HnA
HnA
HnA
HnA
HnA
HnA
950 2000 2050 2100
Angola
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Angola
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Bahamas
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Bahamas
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Belize
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Belize
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Benin
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100 . . . . . Benin
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Botswana
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Botswana
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Burkina Faso
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Burkina Faso
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Burundi
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Burundi
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Cameroon
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Cameroon
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Central African Republic
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Central African Republic
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Chad
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Chad
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Congo
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Congo
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Cote d'Ivoire
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Cote d'Ivoire
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Djibouti
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Djibouti
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Equatorial Guinea
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Equatorial Guinea
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Ethiopia
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100 . . . . Ethiopia
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Gabon
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Gabon
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Gambia
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Gambia
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Ghana
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100 . . . . Ghana
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Guinea
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Guinea
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Guinea−Bissau
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Guinea−Bissau
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Guyana
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Guyana
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Haiti
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Haiti
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Jamaica
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Jamaica
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Kenya
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Kenya
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Lesotho
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Lesotho
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Liberia
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Liberia
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Malawi
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Malawi
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Mali
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100 . . . . . . Mali
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Mozambique
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Mozambique
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Namibia
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Namibia
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Nigeria
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Nigeria
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Rwanda
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Rwanda
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Sierra Leone
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Sierra Leone
Year H I V P r e v a l en c e lllllllllllll Median80%95%
South Africa
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
South Africa
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Swaziland
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Swaziland
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Togo
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Togo
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Uganda
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Uganda
Year H I V P r e v a l en c e lllllllllllll Median80%95%
United Republic of Tanzania
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
United Republic of Tanzania
Year H I V P r e v a l en c e lllllllllllll Median80%95%
950 2000 2050 2100
Zambia
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Zambia
Year H I V P r e v a l en c e lllllllllllll Median80%95%
Zimbabwe
Year L i f e E x pe c t an cy lllllllllllll No Covariates D HIVnonART 1950 2000 2050 2100
Zimbabwe
Year H I V P r e v a l en c e llllllllllllllllllllllllll