Technical description of a mathematical model of the impact of antiretroviral therapy on HIV incidence in South Africa

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1 Technical description of a mathematical model of the impact of antiretroviral therapy on HIV incidence in South Africa Je rey W Eaton, Geo rey P Garnett, Timothy B Hallett March 9, 0 This document describes version of a mathematical of the impact of antiretroviral therapy (ART) on HIV incidence in South Africa used for a systematic model comparison exercise []. This model is identified as the Eaton model in that exercise. The model uses ordinary di erent equations to simulate heterosexual HIV transmission in a two-sex population. The sexually active population is divided into three sexual risk roups which mix semiassortatively. Individuals can move between risk roups and sexual behaviour can chane over the course of the epidemic. The model includes proression throuh five staes of HIV infection accordin to CD4 cell count, and allows for initiation of ART at any level of CD4 count. Parameter values for HIV transmission and proression are based on the best available data from discordant couple studies and lonitudinal HIV cohorts in sub-saharan Africa. The model is calibrated to national HIV prevalence data from South Africa by adjustin sexual behaviour and sexual mixin parameters in a Bayesian framework. Section ives a non-technical description of the model, includin diarams of key model processes. Section ives a mathematical description of the model. Section 3 describes the calibration of the model to the South African epidemic context. Model description. Population structure The model simulates a two-sex adult population ae 5 and older (see diaram in Fiure ). The population is divided into the ae roups 5 to 49, presumed to be sexually active ae roup, and ae 50 and older, who are assumed not to form new sexual contacts. Individuals move from the youner to older ae roup at a rate =/35 per year and die from the ae 50 plus population at an annual rate µ =/.45 per year, calibrated to match the relative sizes of the 5 to 49 year-old population and the ae 50 plus population in the year 990 []. Individuals enter the ae 5 49 population at a rate =0.06 per year, calibrated such that the population rowth over the period 990 to 00 approximately matches the population rowth estimates over that period published by Statistics South Africa []. All new sexual contacts, and hence HIV transmission, occur in the 5 49 ae roup, but the older ae roup is included in the model to assess the total ART need in the intervention scenarios. The sexually active population is divided into three sexual risk roups (termed low, medium, and hih ). As a crude means of simulatin natural variability in individuals sexual risk behaviour, individuals move from hih risk to medium risk, hih to low risk, and medium risk to low risk at a rate. This rate is varied in the model calibration (see Section 3), and is the same for both enders. The proportion of the population in each risk roup before the introduction of HIV into the population is allowed to be di erent for each ender and is estimated in the model calibration. The proportion of the new entrants into the population that enter each risk roup is calculated such that, in the absence of HIV, this proportion of the population in each risk roup would remain constant. The proportion enterin each risk roup remains fixed over the duration of the simulation (meanin that the proportional size of risk roups may chane as a result of the di erential burden of HIV in each risk roup).

2 Males Ae 5-49 (sexually active) Females Hih risk Hih risk α Medium risk Medium risk α Low risk Low risk ν ν Removed Ae 50+ (not sexually active) Removed µ µ Fiure : Diaram of model population structure and sexual mixin. Sexual mixin As described in the previous section, the population is divided into three sexual risk roups ( see Fiure ). The sexual contact rate in each of the risk roups is determined by the overall population averae sexual contact rate, the relative rate of new sexual contacts between the three risk roups, and the size of the risk roup. The underlyin population averae rate of new unprotected sexual contacts, c(t), is allowed to vary over the course of the epidemic in order to model potential behaviour chane in response to the epidemic such as increased condom usae [3] or reductions in the number of new sexual partners [4]. The functional form for the reduction is a loistic loistic function parameterized by the initial contact rate, c 0, the percentae reduction in the contact rate that will occur, c, the start year of the behaviour chane, t c, and the number duration (in years) over which the behaviour chane occurs, d c. Toether, the overall averae contact rate at time t is iven by c(t) =c 0 ( c)+c 0 c +exp t () (tc+d c)/ d c/0 Each of these parameters (the initial contact rate, the percentae reduction in the contact rate, the timin of the start of the reduction, and the duration of the chane) are estimated in the model calibration. The relative contact rates between hih and low risk females and medium and low risk females are also estimated. The relative contact rates for males are calculated based on the relative contact rates for females and the sizes of each of the risk roups for males and females so that the total number of contacts o ered by males and females in the same risk roup is the same. The number of sexual contacts formed between members of each risk roup is determined by the sexual mixin parameter ", as proposed by [5]. A proportion " of sexual contacts are reserved exclusively to be formed with other members of the same risk roup, while the remainin ( ") proportion of partnerships are formed at random. The value of " is estimated in the model calibration. Thus if " = 0 sexual mixin is completely random, while if " =, mixin is fully assortative. As HIV mortality di erentially a ects

3 males and females of each risk roup, the total number of contacts o ered by males may not balance with the number of partnerships o ered by females. In this case the desired number of contacts desired by males and females are eometrically weihted by the parameter G ranin between zero and one; G = 0 indicates that the females preferences determines the number of contacts while G = indicates that the males desired number of contacts dominates. For this exercise, the value is fixed at G = Natural history of HIV infection HIV transmission rate per 00 PYs 76/ 00 PYs Primary infection.9 mos 7./00 PYs 4.4/00 PYs 7./00 PYs 5./00 PYs CD4 > > CD4 > > CD4 > 00 CD4 < years 4.60 years 4.7 years.03 years Duration of infection: mean 4.6 years (median 3. years) Fiure : Diaram depictin averation duration of and relative HIV transmission rate durin each stae of HIV infection. (Note that vertical axis is not drawn to scale). HIV infection is divided into five staes accordin to the HIV infection is divided into staes accordin accordin to CD4 cell count proression associated with the duration of infection (Fiure ). The staes are:. Primary infection,. CD4 count reater than 350 cells/µl, 3. CD4 count between 00 and 350 cells/µl, 4. CD4 count between 00 and 00 cells/µl, and 5. CD4 count below 00 cells/µl. HIV infected individuals proress from one stae of HIV infection to the next at a rate which is the reciprocal of the averae duration in the stae. The averae duration of primary infection is an.9 months, as estimated by Hollinsworth et al. [6]. The rates of proression to subsequent CD4 cell staes and the overall duration from HIV infection to death is based on estimates of the duration until CD4 cell count thresholds in sub-saharan African cohort from the eart-linc collaboration [7], which estimated mean durations of 4.8 years and 9.4 years to reachin CD4 cell counts of below 350 and 00 cells/µl, respectively. The estimate of an averae of 4.7 years between CD4 count apple 00 cells/µland CD4 apple 00 cells/µlis based on extrapolatin the rate of decline in square-root transformed CD4 count between CD4 apple 350 cells/µlto CD4 apple 00. The overall averae duration from infection to HIV death is 4.6 years, from [7], and the resultin median duration from infection to HIV death is 3. years. The HIV transmission rate of an infected individual varies accordin to these staes of infection. The overall baseline weihted averae transmission rate over the period from the end of primary HIV infection 3

4 to.6 years before death is set to be 0.06 per year, as estimated by Hollinsworth, Anderson, and Fraser [6] usin data from discordant couples in Rakai, Uanda [8]. The relative rates of transmission durin the CD4 staes CD4 > 350, 350 > CD4 > 00, and CD4 < 00 are based on the relative rates of transmission observed for these staes by Donnell et al [], althouh the rate of transmission durin the CD4 < 00 staes has been reduced in accordance with the estimate from Hollinsworth, Anderson and Fraser that no transmission occurs durin the final 9 months of infection [6], presumably because individuals are sick and not very sexually active durin this period. The transmission rate durin primary HIV infection is set at.76 per year, as estimated by Hollinsworth, Anderson, and Fraser [6]..4 ART model Primary Infection CD4 > > CD4 > > CD4 > 00 CD4 < 00 Initiated ART, viral suppressin (mean 3 months) Effective ART, virally suppressed (mean.75 years) Effective ART, virally suppressed (failure rate indicated) 45.6 years years - (.5%) 33 years - (6.7%) 33 years - (8.9%) Treatment failin, viraemia (mean.3 years) Very sick, Reduced transmission (mean 6. months) Fiure 3: Diaram of staes of antiretroviral therapy Individuals may initiate ART from any of the above staes of HIV infection. Antiretroviral therapy is divided into a multistae process (Fiure 3). Upon treatment initiation, all individuals first enter a virally suppressin stae durin which they are on ART but their viral load is not yet fully suppressed. This stae lasts for an averae of 3 months and transmission is assumed to be reduced by half compared to the CD4 stae from which they initiated treatment. After this stae, most patients enter a lon period of e ective ART, while a proportion of patients for whom treatment is not successful o directly to the final stae of the ART model of bein very sick, which lasts for an averae of 6. months before death. The probability of immediately failin treatment depends on the CD4 count stae from which treatment was initiated to allow for hih early mortality when startin treatment at lower CD4 cell counts, but then relatively similar lon-term survival if treatment e ectively suppresses viral load and symptoms are controlled [9, 0, ]. The proportion of patients who fail ART is calibrated to such that mortality in the first year after initiatin ART matches the crude first-year mortality rate observed for each CD4 count stratum in a collaborative analysis of ART cohorts from sub-saharan Africa [0]. 4

5 For the majority of patients for whom ART is e ective, they are virally suppressed and transmission is reduced by 9% [] compared to the HIV transmission rate in the CD4 count between 00 to 350 cells/µlstae. The period of e ective ART is divided into two staes first a period of early e ective ART lastin an averae of.75 years, and then a lon period of sustained viral suppression. The reduction in transmission is assumed to be the same in both of these staes, but this is implemented as separate ART staes so that the dropout rate from treatment can be varied accordin to the duration on ART, if, for example, we may assume that there is hih dropout in the years followin ART initiation, but patients who remain on treatment for two years are likely to have accommodated treatment and have hih retention thereafter. In addition to the previously described hiher probability for immediate treatment failure and death for those startin ART at low CD4 cell counts, the failure rate for lon-term e ective ART is assumed to be slihtly lower for those who start treatment at hih CD4 cell counts (see Fiure 3), in line with observations that mortality is modestly hiher even several years after treatment initiation for those who start at low CD4 cell counts [] and to ensure that there is no survival benefit in the model from delayin treatment initiation. After patients fail lon-term e ective ART, they enter a stae of treatment-failin in which they are viraemic and are assumed to have the same infectiousness as individuals in the CD4 cell count cateory 00 to 350 cells/µl. The averae duration of this stae is.3 years. Finally individuals enter a stae of bein very sick just before death, which lasts for an averae duration of 6. months. Durin this period of bein very sick, transmission is reduced and assumed to be at the same level as durin the CD4 < 00 cells/µlstae..4. Droppin out from ART Individuals may dropout from any of the first three staes of ART: virally suppressin, early e ective ART, and e ective ART. The model is desined and implemented to permit the rate of dropout from treatment to vary accordin to duration on ART and the CD4 count cateory from which treatment was initiated, in line with data suestin that those startin treatment at hiher CD4 cell counts may have poorer retention in treatment prorammes [3], perhaps because they did not experience AIDS-related symptoms before initiatin ART. But, in accordance with the specification of intervention scenarios for the model comparison exercise [], the application of the model for this exercise assumes that dropout from treatment is constant across all of these strata. Table : CD4 stae after droppin out of treatment CD4 stae at Treatment stae Percentae of droupouts returnin to CD4 cateory ART initiation at dropout CD4 > 350 CD4: CD4: CD4 < 00 Virally suppressin 00% CD4 > 350 Early e ect. ART 00% E ective ART 00% Virally suppressin 00% CD4: Early e ect. ART 00% E ective ART 00% Virally suppressin 00% CD4: Early e ect. ART 50% 50% E ective ART 00% Virally suppressin 00% CD4 < 00 Early e ect. ART 50% 50% E ective ART 00% After droppin out from treatment, the untreated CD4 stae cateory that individuals enter depends on their pre-treatment CD4 cateory and the duration on treatment to simulate reboundin CD4 cell count associated with ART. Individuals who dropout from the virally suppressin stae all return to the same CD4 stae from which they initiated treatment. For those who dropout durin the early e ective ART stae, half move one CD4 stae hiher, while half increase two CD4 staes. Those who dropout from the e ective ART stae all increase two CD4 staes. This is summarized in Table. However, individuals who have dropped out of treatment proress throuh subsequent CD4 staes twice as fast as treatment 5

6 naïve individuals (rates described in Fiure ). After individuals have dropped out of treatment, they are eliible to restart treatment aain once. Upon restartin treatment individuals proress throuh the same staes of ART as when first initiatin ART. The dropout rate may be di erent, but in the implementation for the model comparison they were assumed to be the same as dropout from first ART initiation. Because individuals may restart treatment, but only once, the model separately tracks treated and untreated people accordin to the number of times which they have initiated ART. To summarize this, Table lists all of the staes of antiretroviral treatment throuh which infected individuals can proress, and the subscript identifyin each staes in the technical model description that follows. Table : Staes of antiretroviral therapy. Subscript Stae Duration Infectiousness 0 Untreated, no access to treatment Untreated, access to ART Virally suppressin 3 months 50% lower than prev. stae 3 Early e ective ART, virally suppressed.75 years 9% lower than CD4 < E ective ART, virally suppressed (Fiure 3) 9% lower than CD4 < Treatment failin, viraemic.3 years Same as 00 > CD4 > 00 6 Very sick 6. months Same as CD4 < 00 7 Untreated, dropped out after first initiation, eliible to restart ART. 8 Re-initiated ART, virally suppressin 3 months 50% lower than prev. stae 9 Re-initiated ART, early e ective ART.75 years 9% lower than CD4 < Re-iniated ART, e ective ART (Fiure 3) 9% lower than CD4 < 350 Re-initated ART, treatment failin, viraemic.3 years Same as 00 > CD4 > 00 Re-initated ART, very sick 6. months Same as CD4 < 00 3 Untreated, dropped out after second initiaton, not eliible to restart..5 HIV transmission The probability of transmission in a contact between a susceptible with an infected individual depends on the annual transmission rate m,u in HIV stae m, the intensity of a contact apple rm,r F between a male in risk roup r M and female in risk roup r F, and a factor G that is the relative transmission rate for males to females compared to females to males. The intensity parameter accounts for factors that a ect the probability of transmission in di erent types of partnerships such as the partnership duration, coital frequency, and condom usae. The per-contact female-to-male transmission probability based on these parameters is e m apple rm,r F and the male-to-female transmission probability is e m apple rm,r F G where m is the HIV stae of the infected partner. 6

7 Model equations We divide the population into four cateories accordin to their infection and treatment status: S,r : HIV uninfected and sexually active (susceptible) individuals of ender in risk roup r. I,r m,u: HIV infected and sexually active individuals of ender in risk roup r and HIV infection stae m. Thesubscriptu = 0 indicates untreated individuals who do not have access to treatment, u = indicates untreated individuals who do have access to treatment, u = 7 indicates individuals who have dropped out of treatment and are eliible to restart, and u = 3 indicates those who have dropped out from treatment a second time and are not eliible to restart. T,r m,u: HIV infected and sexually active individuals on ART (treated) who bean treatment in HIV infection stae m and are in treatment stae u (see Table ). R m,u: Individuals removed from the sexually active population of ender, in HIV infection stae m, and treatment stae u. (Uninfected individuals are indicated by m = 0 and untreated individuals by u {0,, 7, 3}). In the above and throuhout the followin mathematical description, the superscript {M,F} corresponds to ender; superscript r {H, M, L} corresponds to sexual risk roup for the sexually active population; subscript m {0,...,5} corresponds to HIV infection stae with 0 representin uninfected and to 5 correspondin to the staes of infection from primary infection to CD4 < 00; and subscript u {0,...3} corresponds to ART status with 0 indicatin untreated individuals without access to treatment, indicatin untreated individuals with access to treatment, and the remainin staes indicatin di erent staes of ART as indicated in Table. 7

8 The followin di erential equations define the dynamics of the roups: ds,r di,r,0 di,r, = +,r S, + I, + T,, + X,r S,r f,r (t)+ + X r, =( )f,r (t)s,r + X,r I,r, X r, = f,r (t)s,r + X,r I,r, X r, di,r m,0 = m I,r m,0 + X,r I,r m,0 m + + X r, di,r m, = m I,r m, + X,r I,r m, m + m + + X r, dt,r m, = mi,r m, + X,r T,r m, dt,r m,3 =( m ) m, T,r m, + X,r T,r m,3 dtm,u,r = m,u T,r m,u + X,r Tm,u,r dt,r m,6 = m, T,r m, + m,5 T,r m,5 + X,r T,r m,6 di,r m,7 = X 5X m m 0,u 0,r m 0,u0Tm 0,u 0 + m m 0 u 0 = X m, + m, + + X r, I,r, I,r,0 S,r I,r m,0 for m I,r m, for m T,r m, m,3 + m,3 + + X r, m,u + m,u + + X I,r m,7 +,r I,r m,7 m + m + + X r, dt,r m,8 = m I,r m,7 + X,r T,r m,8 dt,r m,9 =( m ) m,8 T,r m,8 + X,r T,r m,9 dtm,u,r = m,u T,r m,u + X,r Tm,u,r dt,r m, = m,8 T,r m,8 + m, T,r m, + X,r T,r m, di,r m,3 = X X m 0 X u 0 =8 m m 0,u,r 0 m 0,u0Tm 0,u 0 + m r, T,r m,3 m,6 + m,6 + + X r, T,r m,u for u {4, 5} T,r m,6 I,r m,7 for m m,8 + m,8 + + X r, T,r m,8 m,9 + m,9 + + X r, m,u + m,u + + X I,r m,3 +,r I,r m,3 m + m + + X r, r, T,r m,9 m, + m, + + X r, T,r m,u for u {0, } T,r m, I,r m,3 for m 8

9 dr 0,0 = X r S,r µr 0,0 dr m,0 = m R m,0 + X r I,r m ( m + µ)r m,0 for m dr m, = m R m, + X r dr,r m, = mr,r m, + X T,r m, r dr,r m,3 =( m ) m, R,r m, + X r drm,u,r = m,u R,r m,u I,r m ( m + m + µ)r m, for m R,r m,u m, + m, + µ R,r m, T,r m,3 + X r dr,r m,6 = m, R,r m, + m,5 R,r m,5 + X r dr,r m,7 = X 5X m m 0,u 0 m 0,u 0R,r m 0,u 0 + m m 0 u 0 = dr,r m,8 = m R,r m,7 + X r T,r m,8 dr,r m,9 =( m ) m,8 R,r m,8 + X r drm,u,r = m,u R,r m,u m,3 + m,3 + µ R,r m,3 m,u + m,u + µ Rm,u,r for u {4, 5} T,r m,6 R,r m m,6 + m,6 + µ R,r m,6,7 + X r m,8 + m,8 + µ R,r m,8 T,r m,9 + X r dr,r m, = m,8 R,r m,8 + m, R,r m, + X r dr,r m,3 = X X m 0 u 0 =8 T,r m,u m m 0,u 0 m 0,u 0R,r m 0,u 0 + m I,r m,7 m,9 + m,9 + µ R,r m,9 m + m + µ I,r m,7 for m m,u + m,u + µ Rm,u,r for u {0, } T,r m, R,r m m, + m, + µ R,r m,,3 + X r I,r m,3 m + m + µ R,r m,3 for m In the above equations, the parameter,r is the proportion of new susceptible individuals of ender that should enter risk roup r in order to maintain a constant proportion,r in risk roup r in the absence of HIV infection. Solvin the above equations with f,r (t) = 0 ives that,r = P r,,r P,r,r0 +. () The function f,r (t) is the force of infection for the roup S,r which depends on the contact rate c,r (t) at time t in that roup, the probability that a contact is formed with an infectious partner, and the probability of transmission in that contact. The contact rate for a risk roup depends on the averae underlyin contact rate c(t) which chanes over time accordin to equation, the size of the risk roups at the beinnin of the epidemic,r and the relative contact rates of the hih and medium risk roups to the low risk roup,r (where,l := ). The weihted averae of the relative contact rate by the size of the initial risk roup yields the annual contact rate for each risk roup at time t: c,r (t) = c(t),r. P,r0,r0 9

10 The total number of contacts desired to be formed by members of risk roup r of ender is thus J,r (t) =c,r (t) S,r + X Im,r + X X Tm,u,r m m u (3) The number of these contacts J,r desired to be formed with each risk roup of the opposite ender 0 depends on the value of the assortativity parameter ". A proportion " of the partnerships are desired only to be formed with the members of the same risk roup r =, while the remainin " proportion of the partnerships are formed with each risk roup of the opposite ender proportionally to the number of partnerships J 0, o ered by those risk roups. Formally, the proportion of contacts desired to be formed by ender and risk roup r that are formed with the risk roup of the opposite ender is defined as Q r,r = " 0 r, +( ") J 0, Pṙ J (4) 0,ṙ where r, is the Kronecker delta defined as r, =ifr = r0 and 0 otherwise. In the case that males in risk roup r M and females in roup r F do not aree on the number of partnerships to be formed between the risk roups, i.e. Q M r M,r F J M,r M 6= Q F r F,r M J F,r F, the discrepancy is balanced accordin to the parameter G as G Q M r M,r F = Q M Q M r M,r F J M,r M r M,r F Q F r F,r M J F,r F Q F r F,r M = Q F Q M r M,r F J M,r G (5) M r F,r M Q F r F,r M J F,r F The probability that transmission occurs in a contact between a susceptible and an infected individual depends on the stae of infection and treatment status of the infection accordin to the transmission rate parameter m,u and on the value of the partnership intensity multiplier apple rm,r F for a partnership between the male in risk roup r M and the female in risk roup r F. The force of infection is then calculated by summin over the probability that each contact is with an infectious individual and the probability that transmission occurs accordin to the infection stae and treatment status of the partner: f,r (t) =c,r (t) X " Q,r Pm Pu 0 I0 m p 0,,r m,u + P m Pu T # 0, m,u p 0,,r m,u r,r (6) 0 S 0, + P m Pu I,r m,u + P m Pu T,r m,u where p,r,r0 m,u is the probability of transmission per contact by an infected individual of ender in risk roup r, HIV stae m and treatment status u to a susceptible individual of the opposite ender in risk roup, defined as n p,r,r0 m,u = exp m,u apple rm,r F G ( =F ) o. (7) 3 Model calibration The model is calibrated to nationally representative HIV prevalence data and ART scale-up data from South Africa. The eneral stratey for model calibration is that parameters related to the natural history of HIV infection, the e ect of antiretroviral therapy on individual infection, and patterns of access and retention in the existin ART proram are fixed and informed from the literature as described in Section. Parameters relatin to sexual behaviour and mixin, the start time of the epidemic, the timin and manitude of sexual behaviour chane, and the timin and rate of existin ART scale-up are estimated usin a Bayesian approach. This yields a joint distribution of parameter combinations representin di erent sexual mixin patterns consistent with the observed epidemic. 0

11 Table 3: Model parameters Parameter Description Value Population rowth rate (in absence of HIV) 0.03 per year Rate of proression from 5-49 ae roup to 50+ /35 per year µ Mortality rate out of the 50+ ae roup /.45 per year,r Proportion of the ae 5-49 population of ender in risk roup r in absence of HIV estimated,r Proportion of new entrants of ender enterin risk roup r derived from and r,r Rate of movin from risk roup r to 0 r0 for ender estimated c(t) Population mean contact rate per year at time t estimated,r Relative contact rate between risk roup r and low risk roup for ender estimated " Deree of assortative mixin estimated G Balance between male and female partner preference 0.5 applerm,rf intensity of partnership between male in risk roup rm and female in rf estimated m,u Annual HIV transmission rate in stae m and ART status u (see Fiure 3, 3) m Rate of proression from HIV stae m to stae m + m Rate of proression from HIV stae m to stae m + after treatment dropout Proportion of HIV infected population with access to ART m Rate of ART initiation for ender in stae m m Rate of re-initiatin ART after dropout for ender in stae m m,u Rate of proression from ART stae u to u + when initiatin ART in HIV stae m for ender (Fiure 3) 3 m Probability of immediate treatment failure if initiatin ART in stae m m,u Rate of droppin out of ART if initiated in stae m and currently in stae u m,u Rate of droppin out of ART after re-initiatin if re-initiated in stae m and currently in stae u m m 0,u Probability of enterin CD4 stae m after droppin out of stae 0 (m0,u 0 ) (Table ) t0 date at which HIV epidemic is seeded into population estimated,r seed HIV prevalence for ender and risk roup r year

12 3. Data The model is calibrated usin HIV prevalence data from two sources. The first is HIV prevalence amonst 5 49 year-old prenant women from the annual national antenatal prevalence surveys from 990 to 008 [4]. The second is national HIV prevalence amonst 5 49 year-old males and females from the three nationally representative household surveys conducted by the Human Sciences Research Council in 00, 005, and 008 [5, 6, 7]. Fiure 4 shows the mean and 95% confidence intervals for each of these estimates. The discrepancy between the level of HIV prevalence in the antenatal surveillance and the household survey based prevalence is reconciled by incorporatin a bias parameter in the antenatal prevalence compared to prevalence amonst the eneral 5 49 year old population from the household surveys. This is described in detail below in Section 3.3. South Africa HIV prevalence data 30% 5% ANC Prevalence Females, HSRC Survey Males, HSRC Survey HIV Prevalence 0% 5% 0% 5% 0% Mean and 95% CI Year Fiure 4: Ae 5 49 HIV prevalence from three nationally representative HSRC household surveys and HIV prevalence amonst antenatal care attendees. The model is also calibrated to the percentae of the adult population on antiretroviral therapy. This is calculated by dividin the total number of adults reported to be on ART by the South Africa Department of Health in June of each year from 005 to 00 [8] by the annual mid-year population size estimate of those over 5 years old from Statistics South Africa []. The resultin estimates for the proportion of the adult population on ART fo05 to 00 to which the model is calibrated are shown in Fiure Estimated model parameters A vector of 0 parameter values are estimated, and are either used directly or to derive a number of the model inputs in the equations described in Section and Table 3. The mathematical model parameters which are estimated from fittin to HIV prevalence data from South Africa are iven in Table 4 alon with the prior distribution from which they are estimated. This collection of model parameters estimated from the data will be referenced toether as the parameter vector.

13 % of adults on ART 5% 4% 3% % % 0% South Africa ART scale up Year Fiure 5: Percentae of adult population (ae 5+) on ART at midpoint of each year (South Africa Department of Health) Table 4: Esimated model parameters Parameter Description Prior t 0 Start date of the epidemic Unif(983, 988) M,L Proportion of males not in the low risk roup Unif(0., 0.7) M,H Proportion of males in hih risk roup of those not in low risk roup Unif(0., 0.8) M,L F,L Proportion of females not in the low risk roup Unif(0.05, 0.5) F,H Proportion of females in hih risk roup of those not in low risk roup Unif(0., 0.8) F,L Rate of movement from hiher to lower risk roups Unif(0.0, 0.5) c 0 Mean annual contact rate at start of the epidemic Unif(0.5, 3.0) c Proportion reduction in averae contact rate Unif(0.0, 0.7) t c Year behaviour chane start Unif(990, 000) t c + d c Year behaviour chane ends Unif(00, 0) F,M Relative contact rate between medium risk and low risk females Unif(, 50) F,H F,M Additional relative contact rate for hih risk women Unif(0, 50) " Assortativity of sexual mixin Unif(0., 0.8) apple H Partnership intensity for partnership involvin a hih risk partner apple M Partnership intensity for partnership between medium and low risk apple L Partnership intensity for partnership between low risk partners t A0 Date of ART scale-up Unif(003.5, 005.5) A 0 ART initiation rate durin initial ART scale-up Unif(0.0, 0.6) t A t A0 Time between initial ART scale-up and intensified ART scale-up A ART initiation rate durin intensified ART scale-up Unif(0.5,.5) These parameters have a joint uniform prior distribution such that 0 <apple H <apple M <apple H <. 3

14 3.3 Statistical methods We use a Bayesian analysis to estimate probability distributions for the unknown parameters iven the model and available HIV prevalence data, W. Let denote the set of parameters to be estimated from section 3. and M denote the mathematical model and the associated fixed parameter values. For a iven set of parameter values, the model outputs a set of predicted HIV prevalence values = M( ) correspondin to the data. Then we can produce a likelihood function p(w M( ) for the probability of the data iven the model and parameter values. If we let p( ) denote a prior distribution on the unknown parameters, usin Bayes theorem the posterior distribution of iven the model and data is iven by p( W,M) / p( )p(w M( )) Likelihood function We now derive the combined likelihood for the national seroprevalence survey data and antenatal clinic data. First we define the likelihood for an individual datum. For the national survey estimates, let W F,t and W M,t be the HIV prevalence in the ae roup 5-49 reported by a survey in year t. As above, define,t to be the predicted population HIV prevalence for ender at time t by the model M( ). We assume that national survey prevalence provides an unbiased estimate of the true population prevalence, but loit-transform the data to stabilise the error variance, so that W,t,t lo = lo +,t (8) W,t,t where,t Normal(0,,t ) and the errors are conditionally independent iven. Thus the likelihood function for an estimate from a national prevalence survey is p(w,t M( )) = q exp,t,t (loit(w,t ) loit(,t )) (9) In calculatin the likelihood, the value of,t is replaced by an estimate ˆ,t based on the confidence intervals reported by the HSRC which account for the complex samplin stratey. The confidence intervals in the HSRC reports are on the inverse-loit scale, so the error variances are estimated by ˆ,t = loit(cimax,t ) loit(ci min (.975),t ) (0) where is the inverse of standard normal cumulative distribution function. For the antenatal clinic data, similar to [9, 3], we assume that HIV prevalence W amonst antenatal clinic attendees at time t is linearly related to prevalence in the eneral female ae 5 to 49 population on the loit scale and that this a ect is fixed over time. To model this, we introduce an additional parameter such that W F,t lo = lo + +, () W F,t where the error term Normal(0, ) and recallin that F,t is the HIV prevalence amonst females aed 5 to 49 predicted by the model at time t. Thus ( p(w M( )) = q exp (loit(w ) (loit( F,t )+ ) )) () Once aain, when evaluatin the likelihood, we replace with an estimate ˆ computed from the confidence intervals published by the South African Department of Health. In the case of the antenatal clinic data, the reported confidence intervals are symmetric, suestin that they have been estimated on the untransformed scale rather than loit-transformed prevalence. Derivin an estimate of the samplin 4

15 error variance on the loit scale involves two steps: first estimatin the untransformed error variance ˆ, and then usin the delta method to approximate the variance ˆ of the loit-transformed distribution, which depends on the estimated antenatal clinic prevalence at time t, W. The equations for this are ˆ = loit(cimax ) (.975) loit(ci min ), and (3) ˆ = ˆ W ( W ) (4) To arrive at the likelihood for the full data W, we assume that the data points are conditionally independent iven the predicted prevalences = M( ) and the antenatal clinic bias parameter. Definin T N to be the set of years when national survey prevalence estimate are available and T C the set of years where antenatal clinic estimates are available, the full likelihood is p(w M( ), )= Y Y Y p(w,t M( )) p(w M( ), ) tt N {M,F} tt C = Y Y q exp tt N {M,F},t,t (loit(w,t ) loit(,t )) (5) ( ) Y q exp (loit(w ) (loit( F,t )+ )) tt C 3.3. Priors The prior distributions for the estimated model parameters are iven in Table 4. The antenatal bias parameter is assumed to have Uniform(0, ) prior distribution. This amounts to assumin that the odds ratio of ANC prevalence to adult females prevalence is between and.7. Restrictin the prior value of to be positive ensures that the model does not mistakenly interpret the ANC prevalence as the true population prevalence when fittin the model Estimatin the posterior distribution Multiplyin the likelihood and prior distribution yields the joint posterior distribution of the parameters and iven the model and data up to a scalin constant: p(, W,M) / p(w,,m)p(, ) (6) We are principally interested in the values of the model parameters. The posterior distribution for W,M can be calculated by interatin out the parameter. Observe that Z p( W,M) / p(w,,m)p(, )d Z = p( )p( ) Y Y Y p(w,t M( )) p(w M( ), )d tt N {M,F} tt C = Y Y Z p(w,t M( ))p( ) p( ) Y p(w M( ), )d, tt C tt N {M,F} (7) so if we can e ciently evaluate R p( ) Q tt C p(w M( ), )d, then we can e ciently. Before 5

16 attackin this, let us define three useful quantities: S = X (8) tt C D = S X F,t (9) D = S X W tt C (W F,t) tt C The first can be thouht of as the pooled variance of the ANC prevalence estimates, the second as the precision-weihted mean di erence between ANC data prevalence and the model predicted female prevalence, and the third as the precision-weiht mean squared distance between the ANC data and the predicted female prevalence. Now, aain consider our interal. We will do a bit of rearranin to show that the interal can be evaluated as a normal cumulative distribution function. For brevity, denote Ẇt = loit(w ) and t = loit( F,t ), and all sums and products are over the set T C. Z Z ( Y p( )p(w C M( ), )d = q exp 0 t Z ( X = q Q exp t 0 t Z ( X = K exp = K = K 0 Z 0 Z 0 exp exp t ) Ẇt ( t + ) d Ẇ t (Ẇt t + (Ẇt ) t ) d /S D/S + D /S d S D + S D = K e (D D )/(S ) p S Z = s S Q t e (D D )/(S ) 0 p S e ( apple D p S D d D) /(S ) d ) t ) d 0 D p S Usin this expression, we can e ciently evaluate the posterior density function p( W,M)uptoa constant. We use the incremental mixture importance samplin (IMIS) alorithm to approximate and sample from the posterior distribution [0]. (0) () 6

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