THEORETICAL
POPULATION
39, 148-169
BIOLOGY
(
1991)
A Model of HIV-1 Transmission for Urban Areas of Africa A. MEREDITH Office
JOHN
of Population Research, Princeton Princeton, New Jersey 08540
University,
Received October 31, 1988
A model of the transmission dynamics of HIV-l, appropriate to urban areas of Africa, is presented and its behaviour explored through numerical studies. The model is a two-sex model with age-dependent demographic and behavioural parameters. Adults are classified by age, sex, risk group, and epidemiologic status. HIV-l is transmitted to adults heterosexually, and to infants and children vertically and parenterally. Numerical studies show that, while AIDS will slow population growth, growth rates do not become negative for reasonable parameter values. The sex and age patterns of infection are explored, as is the potential economic impact 0 1991 Academic of changes in the sex and age composition of the population. Press, Inc
1. INTRODUCTION In less than a decade, human immunodeficiency virus type 1 (HIV-l) has spread to more than 130 countries. In its wake, many researchers have turned their attention to the problem of predicting the future course of the temporal and spatial spread of HIV-l. Some have concentrated their efforts upon modeling components of HIV-l transmission [see, for example, Blythe and Anderson 1988, Lui et al. 1986, Lui et al. 1988, Medley et al. 19871 while others have developed population-level models of the spread of HIV-l [see, for example, Mode et al. 1988, Anderson et al. 1987, Dietz 1988, DeGruttola and Mayer 19881. At least two models have been constructed to describe HIV-l transmission and its demographic impact in developing countries [Bongaarts 1989, Bongaarts and Way 1989, May et al. 1988, Anderson et al. 19881 where the principal forms of HIV-l transmission are heterosexual and vertical. Such models are the focus of the present paper, in which a two-sex age-structured model of HIV-l transmission, appropriate to urban areas of Africa, is presented and its behaviour explored through numerical studies. 148 oo40-5809/91 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
HIV-
1 TRANSMISSION
IN AFRICA
149
The model of heterosexual transmission dynamics of HIV-l proposed by Anderson et al. [1988] differs in four substantial ways from the model presented in this paper. First, this assumption of symmetry between males and females in all demographic and epidemiologic parameters, including the efficiency of disease transmission, allows the model to be collapsed to a one-sex model. Second, the demographic structure of the host population is not age-specific; the force of mortality, p, is the same for all ages (p = 0.019), and produces a population in which the expectation of life at birth is 52.6 years, and in which 15% of the population is still alive at age 100 years. Similarly, fertility is expressed in terms of the average birth rate per adult rather than in a set of age-specific fertility rates for women-a distinction which becomes crucial to population dynamics if either sex is disproportionately depleted by the disease. The assumed values of a preepidemic annual population growth rate of 0.04 per annum and p = 0.019 imply a total fertility rate of approximately 12 children per woman, roughly 50% higher than the highest total fertility rates currently observed. Third, the per capita rate at which adults acquire infection is modeled as a property of a couple, independent of the frequency of coitus: 1= @Y/N, where p is the probability of acquiring infection from any one infected partner, c is the average rate of acquiring partners, and Y/N is the probability that any one partner is infected.,Fourth, no distinction is made between adults who are in monogamous unions and those who have multiple sexual partners: the behaviour of all individuals is described by the average rate of acquiring new partners, c. In addition, all adults are assumed to choose sexual partners independently of age. Under this set of assumptions, Anderson ef al. (1988) conclude that AIDS indeed has the potential for turning population growth rates negative, even in populations which were growing extremely rapidly before the HIV-l epidemic. Subsequent refinement of the model [May et al. 19881 by relaxing the assumptions of symmetric heterosexual transmission and homogeneity in sexual behaviour, but retaining the assumption that the risk of infection is a per-couple property rather than a per-coitus property, yields conclusions which are comparable to those derived from the simpler model. Bongaarts [1989] constructed a model for assessing the demographic and epidemiologic impact of HIV-l in urban areas of Africa which is very similar to the model presented in this paper: the two-sex demographic submodel gives a complete dynamic representation of the demographic structure of the population, epidemiologic and behavioural parameters are allowed to vary with age, sexual behaviour is not homogeneous but rather a number of different risk groups are specified for each sex, and the risk of infection is modelled as a per-coitus risk rather than a per-couple risk, with the probability of maleto-female transmission per coitus twice the probability of female-to-male transmission. There are, however, two principal
150
A. MEREDITH JOHN
differences between the model presented here and the Bongaarts model. First, in the present model, the infectivity of an individual is assumed constant from the time of infection, while the Bongaarts model employs an elegant and more realistic convolution of four exponentially distributed subintervals of the infected period, thereby allowing infectiousness to vary with the duration of infection. Second, in addition to heterosexual and vertical transmission-the modes of transmission used in the Bongaarts modelthe model presented below also incorporates blood transfusion and inoculation by contaminated needle as modes of pediatric infection. On the basis of numerical studies employing his models and plausible values for demographic and epidemologic parameters, Bongaarts concludes that, while an AIDS epidemic will certainly reduce population growth rates in urban areas of Africa, the magnitude of the reduction is not substantial enough to drive population growth rates negative or even near zeroa conclusion which is substantiated by the present study. In the next section, the epidemiology of HIV-l in Africa is briefly summarized. The mathematical details of a fully age-structured model of HIV-l transmission dynamics are then presented; the modes of HIV-l transmission in the model are heterosexual intercourse, vertical (mother-infant) transmission, and transmission through blood transfusion and inoculation with contaminated needles. The behaviour of the model is illustrated for a hypothetical urban center in Africa. After it is demonstrated that the model behaves in a demographically plausible manner in the absence of fatal HIV-l infection, the effect of varying one set of epidemiologic parametersthe rate of male-to-female and female-to-male HIV-l transmission per coitus-over a broad range of plausible values is explored.
2. EPIDEMIOLOGY
OF HIV-l
INFECTION IN URBAN AFRICA
WHO has identified two epidemiologic patterns of HIV-l transmission. The first involves transmission by homosexual activity and by intraveneous drug use, and describes the principal modes of transmission of HIV-l in North and South America, Europe, and the Caribbean. The second pattern involves heterosexual transmission, perinatal transmission, and transmission by contaminated blood, and predominates in Africa. Both symptomatic and asymptomatic infected women can transmit HIV-l to their infants [Piot et al. 19871. Thirty to 65% of the infants born to women infected with HIV-l express HIV-l antibodies at birth; some of these infants will have an active HIV-l infection, while others will simply be expressing passively acquired maternal antibodies to HIV-l [Piot et al. 19881.
HIV-1
TRANSMISSION
IN
AFRICA
151
Parenteral transmission of HIV-l occurs by the transfusion of infected blood or blood products, or by the use of contaminated needles. The risk of HIV-l transmission through blood transfusion depends upon the prevalence of HIV-l in the blood-donor population. In regions where HIV-l is endemic, the proportion of healthy adults who display serological evidence of HIV-l infection ranges from 4 to 30%, thus suggesting that a comparable fraction of blood donors may be infected [Greenberg et al. 1988, Mann 19871. Simple needle-puncture, rather than blood transfusion, appears much less efficient in HIV-l transmission: data on health-care workers sustaining needlestick injuries while treating AIDS patients suggest that the risk of infection is less than 0.5% per puncture [Quinn et al. 19861. Susceptible children in developing countries may acquire HIV-1 infection by intramuscular injection with needles which have not been properly sterilized. In a study in Zaire, seropositive infants (mean age 10.6 months) had received on average 44 intramuscular injections, vaccinations excluded, as opposed to their seronegative counterparts who had received considerably fewer injections [Mann 1986b]. In contrast, there appears to be no significant association between HIV-l seropositivity and childhood immunization [Von Reyn and Mann 1987, Mann et al. 1986a]. The risk of infection from a single heterosexual contact with an HIV-linfected person depends not only upon the type of sexual activity but also on the susceptibility and infectivity of the partners. It appears that, in the absence of other risk factors such as genital ulcers, male-to female sexual transmission of HIV-l may be more efficient than female-to-male HIV-l transmission [Padian 19871. Recent studies have demonstrated age-related differences in the mean (and the median) incubation period, with 1.97 (1.90) years for infants and young children (0 to 4 years old at infection), 8.23 (9.97) years for older children and adults (5 to 59 years old), and 5.50 (5.44) years for elderly patients (60 years or more years old) [Medley et al. 19871. Whether or how cofactors other than age affect the incubation period for developing clinical AIDS is unknown [Curran et al. 19851. Data on HIV-l seroprevalence from Zaire show a bimodal age-specific curve, with a first peak among infants (1 to 2% seropositive) and a second peak (8-10% seropositive) among adolescents and adults aged 16 to 29 years old. Because of the lengthy incubation period of HIV-l, the ageincidence of frank AIDS is somewhat older than that of HIV-l infection. The mean age of frank AIDS cases in Kinshasa is 33.6 years (median, 32 years; range, 1.5 to 64 years); the mean age of men with AIDS is greater than the men age of women with AIDS: 37.4 years and 30.0 years, respectively [Mann et al. 1986a].
152
A. MEREDITH
3.
JOHN
MODEL
In this model of HIV-l infection and subsequent illness individuals are classified by age in single years, sex, behavioural group (pediatric, adolescent but not yet sexually active, sexually active with many partners, sexually active in a stable union, or no longer sexually active), and epidemiologic status (susceptible, infected and infectious, or clinically ill) (Fig. 1). For each behavioural class, sex and age individuals are assumed homogeneous with regard to behaviour, infectivity (if they are infectious) or susceptibility (if they are susceptible). The model is completely agespecified: fertility and mortality vary with age, as do important epidemic and etiologic factors (Table I). Fertility is described by a set of age-specific fertility rates, and mortality is described by a set of age and sex specific mortality rates. For simplicity, it is assumed that the fertility and nonEPIDEMIOLOGIC
CLASS
INFECTED
SUSCEPTIBLE
AIDS
I
FIG.
1.
Schematic
diagram
of the model
of HIV-l
transmission
dynamics.
TABLE
I
Model Notation and Definitions Subscripts a
L
age (in years) calendar time (in years) time spent infected or ill
Superscripts s susceptible infected and infectious (but not presenting clinical illness) a presenting severe HIV-l related illness Demographic parameters age specific fertility rate at age a male and female age specific schedules of transition from not sexually active to &TlF> g1: sexually active with many partners male and female age specific schedules of transition from sexually active with bq, 82: many partners to sexually active in a stable uion male and female age specific schedules of transition from not sexually active to g3:, g3: sexually active in a stable union age and sex specific mortality rate from causes other than AIDS P. number of women age a at time t in the population w,, I number of men age a at time f in the population Ma,, number of births at time t B,
fa
Epidemiologic parameters case fatality rate from HIV-l-related disease at age a, k years after developing ao,k clinical disease rate of developing clinical HIV-l-related disease at age a, k years following Ydf infection with the virus proportion of prostitutes infected at time t if Exogeneous probabilities probability that a seropositive mother bears an infected child PXl probability of female to male heterosexual transmission PX2 probability of male to female heterosexual transmission PX3 probability of infection by innoculation with a contaminated needle PX4 probability of infection from a unit of blood containing HIV-l PXS Endogeneous probabilities probability that a unit of blood contains HIV-l at time t Pl, probability that a needle used for an injection is contaminated at time t P2, probability that a female sexual partner, randomly chosen at time t, is infected P3, probability that a male sexual partner, randomly chosen at time 1, is infected P4, Level of activity proportion of children age a receiving at least one blood transfusion T, average number of blood transfusions per year among children age a receiving Nl, at least one transfusion average number of injections per year among children age a 7% annual number of visits to a prostitute by a male age a, not in a stable union N3, annual number of randomly selected sexual partners of a male age a, not in a N4, stable union annual number of randomly selected sexual partners of a female age a, not in N5, a stable union annual number of visits to a prostitute by a male age a in an otherwise N6, monogamous stable union annual coital frequency of a stable couple with wife age a NT,
153
154
A. MEREDITH
JOHN
AIDS related mortality rates do not change during the period studied. In the absence of fatal HIV-l infection, the model reduces to a standard population projection model based on a Leslie matrix. In the form presented below, all transition probabilities are expressed as annual probabilities. The model can easily resealed so that transition probabilities are expressed as monthly or weekly probabilities. Largely for reasons of tractability, the behavioural assumptions in the model are extremely simple. There are four behavioural classes of adolescents and adults: not yet sexually active, sexually active with many partners (including, for men, with prostitutes), sexually active in a stable union with one partner (and with prostitutes), and no longer sexually active because of illness or widowhood. Many, but not all, adolescents in the population experience a period of sexual activity when partners are chosen at random from the population of people who are sexually active adolescents. All adolescents eventually progress to a stable union if a marriage partner of the appropriate age is available. Once a union is dissolved by the death of an individual, the remaining partner does not remarry and is no longer sexually active. The etiologic assumptions which govern the progression from susceptible to infected to ill to dead are also very simple. People exist in only three distinct epidemiologic states: susceptible, infected and infectious but without clinical manifestation of infection, and overtly ill and hence no longer sexually active and no longer blood donors. The eventual transitions from infected to ill, and from ill to dead are assumed to be inevitable. 3.1. Pediatric HIV-1
Infection
and Illness
In this model there are two modes of HIV-l transmission vertical (perinatal) transmission, and parenteral transmission. a. Perinatal transmission of HIV-l. In the absence of HIV-l the population, the annual number of births at time t is
Bt= :
f, W,,r>
for children: infection in
(1)
a=15
where f, is the fertility rate of women age a, and W,,, is the number of women age a at time t in the population. In the presence of HIV-l, it is assumed that women who have developed overt AIDS are no longer sexually active and thus are no longer bear children, so that children are borne only by susceptible women ( WI,,) and infected women ( Wi,,) at the same rate, f,. The distribution of births between susceptible and perinatally infected infants is given by
HIVB,=B;+Bf=
1 TRANSMISSION
5
f,W:,,+(l-px,)
a=15
+Pxl
f k=l
2 a=15
f,
155
IN AFRICA
i
;
k=l
a=15
f.W;,t,k] (2)
W;,t,k,
where px, is the probability that an infected mother bears an infected child, and is independent of the duration k of the mother’s infection. b. Parenteral transmission of HIV-l. There are two principal forms of parenteral transmission of HIV-l: infection by transfusion of HIV-l contaminated blood, and infection by inoculation with an HIV-l contaminated needle. The probability that a child of age a is infected by transfusion of HIV-l contaminated blood is given by 71,A T,{ 1 - [l - px, . plJNla},
(3)
where T, is the proportion of children of age a receiving at least one blood transfusion at age a, Nl, is the mean number of transfusions received in a year by children age a who receive at least one transfusion, pl, is the probability that at time t a unit of blood contains HIV-l, and px, is the probability that infection results from transfusion of contaminated blood. Blood donors are chosen randomly from the apparently healthy adult population; the probability that the unit of blood contains HIV-l is thus the probability that a donor between the ages of 15 and 60 is infected, c;=
pl,A ~?==I,
1 c:“=
15 W:,t,k
[w:,,+M:,,+C;=,
+ %,t,k
(4)
w:,,,k+“:,,,kl’
The probability that a child age a is infected by inoculation HIV-l contaminated needle is given by
with an
where N2, is the mean number of injections received by children age a, p2, is the probability that at time t, an HIV-l contaminated needle is used, and px, is the probability that infection results from an inoculation with a contaminated needle. It is assumed that all children under age 5 are inoculated at least per year; the probability that the needle is contaminated with HIV-l is equal to the probability that the last child inoculated with that needle, chosen randomly from the population of children age 0 to 4, was infected, c;=,
p2,A cd=,
[w:.,+M:,,+C;=,
c:=O
w:,t,k+“:,,,k
(6) W:,r,k+“b,r,kl’
156
A. MEREDITH
JOHN
where Mz,t and MA,, are the numbers of susceptible and infected boys age a at time t, and Wz,, and WA,, are the numbers of susceptible and infected girls age a at time t. Thus Z7,, the probability that a susceptible child age a at time t will be infected by blood transfusion or needle-puncture, is
c. Incubation period, clinical illness, and death. In this model, the progression of children from infected to ill is governed by the function ya,k which describes the transition from subclinical infection to overt AIDS of children age a who were infected k years ago, at age a-k. The rate at which ill children die is given by the function ~l,,~ where the HIV-l related case fatality rate depends not only on the child’s age but also on the length of time k since the child developed clinical illness. 3.2, Adult HZV-1 Transmision,
Infection,
and Illness
In this model, adolescents and adults belong to four behavioural classes which differ in the risk of exposure to HIV-l: aldolescents who are not yet sexually active, adolescents who are sexually active but not in stable unions, adults who are sexually active within stable unions, and adults who are no longer sexually active. It is assumed that thoughout their sexually active lives, whether they are in stable and otherwise monogamous unions or not, men continue to visit prostitutes, who serve as a reservoir of the virus. Hence a man whose wife is not infected with HIV-l may become infected after marriage; a woman is not protected from HIV-l infection by marrying an uninfected man, since he may subsequently become infected by a prostitute. Women are not at risk of infection by sexual intercourse while they are pregnant or during a period of postpartum sexual abstinance which, in many societies, can be quite lengthy. People who are widowed, who develop overt AIDS, or whose spouses develop overt AIDS cease sexual activity. a. Movement among behavioural classes. The age-specific movement of adolescent females from sexually inactive to sexually active with many partners is governed by the female age-specific transition schedule, g 1,“; age-specific movement of adolescent males from sexually inactive to sexually active with many partners follows the male age-specific transition schedule, g 1,“. Sexually active adolescent males also visit prostitutes. It is assumed that any adolescent who is not yet sexually active is not infected with HIV-l. The transition to a stable union is governed by four schedules: two for men (g2,M, g3:) and two for women (g2,W, g3:). It is assumed that susceptible and infected individuals are equally likely to enter stable unions; however, individuals who have already developed clinical illness
HIV-l
TRANSMISSION
IN AFRICA
157
will not enter stable unions and will become sexually inactive. Individuals select their spouses randomly with respect to previous sexual history and current epidemiologic status. Thus, for example, the probability that a man marries an infected woman is given by the share of eligible infected women in the population. The stable unions thus formed are of four types indexed by the epidemiologic status of husband and wife at the time of marriage: (i) (ii) (iii) (iv)
husband husband husband husband
susceptible, wife susceptible infected, wife susceptible susceptible, wife infected infected, wife infected.
During the marriage, the couple may move from one subcategory to another as the epidemiologic status of either partner changes: from (i) to (ii) by intercourse with a prostitute, from (ii) to (iv) by marital intercourse, or from (iii) to (iv) by marital intercourse or by intercourse with a prostitute. b. Movement among epidemiologic classes. (i) Transmission among sexually active adolescent who are not in a stable union. Men are infected with HIV-l either by sexual intercourse with an infected prostitute or by sexual intercourse with an infected woman chosen randomly from the sexually active, nonprostitute population. If a man of age a has intercourse N3, times per year with prostitutes, a fraction [, of whom are infected, then the probability that he will be infected by a prostitute during the year is given by
where px, is the probability of female-to-male transmission of HIV-l per coitus, and (1 -p~,)~~~ is the probability of not being infected by a prostitute in N3, coital acts. The probability that a man of age a is infected by intercourse with a woman chosen randomly from the population of sexually active women is given by 173
A I-
{ [ 1 - (p3,. px,)]N4.},
where N4, is the number of sexual partners per year selected randomly with replacement from the sexually active female population or, equivalently, the number of coital acts with randomly chosen female partners per year, and px, is again the probability of female-to-male transmission per
158
A. MEREDITH
JOHN
coitus. The probability that a randomly chosen sexually active women is infected with IV-l, p3,, is determined endogeneously: p3,G
c;=, CZ=”K,,,, c:=, cw:,,+cL,
wtJ.kl
(10) sexually
actwe
wth
multiple
partners
Because p3, varies over time as the proportion of infected women in the population changes, the probability that a sexually active male will become infected will also vary over the course of the epidemic. The model for women is symmetric to that for men. The probability that a woman of age a is infected by a man chosen at random from the population of sexually active men who are not in stable unions is rr, A 1 - {[l - (p4,. pXj)y”},
where N5, is the number of sexual partners per year, selected randomly with replacement from the sexually active male population or, equivalently, the number of coital acts with randomly chosen male partners per year, and px, is the probability of male-to-female transmission per coitus. The probability that a randomly chosen sexually active man is infected with HIV-l, p4,, is determined endogeneously:
c;= 1c:=ow,,, p4f G C$=,
’ sexually
active
with
multiple
(ii) Transmission among adults in a stable union. It men in marital unions will visit prostitutes, a fraction infected. Hence the probability that a man of age a will HIV-l has two components: l7,, the probability of prostitute, ~,~l-{C1-(i,.PX,)IN6.},
(12)
partners
is assumed that {, of whom are be infected with infection by a (13)
where N6, is the annual coital frequency of a married man with prostitutes, and 176, the probability of infection by the man’s wife, age a’, z76~1-([1-(z.pX*)]N’~~},
(14)
where N7,. is the annual coital frequency of the couple and z is an indicator variable which equals one if the wife is infected with HIV-l and zero if she is not infected. It is assumed that women who are married have sexual intercourse only with their husbands. Hence the probability that a woman age a will be infected is (15)
HIV-l
TRANSMISSION IN AFRICA
159
where N7, is the annual coital frequency of the couple and z is an indicator variable which equals one if the husband is infected, and zero if he is not. c. Incubation, illness, and death. The rate at which adults pass from infected to clinically ill with AIDS is given by the distribution of incubation times Yo,k where the transition rate depends not only on the aldult’s age but also on the length of time k since the adult was infected. The rate at which ill adults die is given by the distribution of lengths of illness aa,k where the HIV-l-related case fatality rate depends not only on the adult’s age a but also on the elapsed length of time k since the aldult developed clinical illness.
4. NUMERICAL
STUDIES
The numerical implementation of the model presented above is based on a standard cohort projection model of population growth and structural evolution for a population closed to migration. In addition to moving individuals forward in time along an age dimension with decrements to the population from mortality, the model incorporates movements by sex, age, and time among epidemiologic and behavioural classes by the rates derived in the preceeding section. The model has been tested with projection periods of 3, 6, and 12 months; there are slight variations in the numerical results as a function of projection period length, but the qualitative results are similar. The results reported below are based on projections by 6-month increments for 50 years. 4.1. Parameter Specification
Within this projection framework, four groups of parameters are defined: (i) demographic parameters, specifying the fertility and non-HIV-l related mortality patterns in the population; (ii) behavioural parameters, specifying the age-specific rates of movement between behavioural classes and age-specific coital frequency patterns; (iii) epidemiologic parameters, specifying the spread of the disease through the population; and (iv) etiologic parameters, specifying the time pattern of progression of the disease within an individual. Values for the parameters used below have been gleaned from the demographic and epidemiologic literature, and as such are not intended to represent the course of the disease in any particular locale. Furthermore, there is little certainty about the values of many of the epidemiologic and etiologic parameters; for the purpose at hand-illustrating the behaviour of the model-values which represent, in some cases, little more than best guesses have been employed.
160
A. MEREDITH
JOHN
The numerical model uses a set of fertility a. Demographic parameters. rates based on the period age-specific fertility rates for Kenya, 197771978 [World Fertility Survey 19841, scaled to correspond to a total fertility rate of 6.5 children per woman, with the age pattern of fertility preserved. The age- and sex-specific mortality rates used to model the age pattern of nonHIV-l related mortality are based Coale-Demeny West model life tables, level 16 [Coale and Dmeny 19841, and correspond to male and female life expectancies of 54.1 years and 57.5 years, respectively, and male and female infant mortality rates of 99 and 82 deaths per thousand live births, respectively. The population growth rate, in the absence of HIV-l mortality, is endogeneously determined by the fertility and mortality rates to be 0.034 per annum. The starting susceptible population is the stable population corresponding to the fertility schedule life tables used in the numerical model. b. Behavioural parameters. The age-specific schedule of entry into sexual activity for girls, g 1y, is defined as
gl,w=
0.00 a19 035 1. lS
POPULATION
39, 148-169
BIOLOGY
(
1991)
A Model of HIV-1 Transmission for Urban Areas of Africa A. MEREDITH Office
JOHN
of Population Research, Princeton Princeton, New Jersey 08540
University,
Received October 31, 1988
A model of the transmission dynamics of HIV-l, appropriate to urban areas of Africa, is presented and its behaviour explored through numerical studies. The model is a two-sex model with age-dependent demographic and behavioural parameters. Adults are classified by age, sex, risk group, and epidemiologic status. HIV-l is transmitted to adults heterosexually, and to infants and children vertically and parenterally. Numerical studies show that, while AIDS will slow population growth, growth rates do not become negative for reasonable parameter values. The sex and age patterns of infection are explored, as is the potential economic impact 0 1991 Academic of changes in the sex and age composition of the population. Press, Inc
1. INTRODUCTION In less than a decade, human immunodeficiency virus type 1 (HIV-l) has spread to more than 130 countries. In its wake, many researchers have turned their attention to the problem of predicting the future course of the temporal and spatial spread of HIV-l. Some have concentrated their efforts upon modeling components of HIV-l transmission [see, for example, Blythe and Anderson 1988, Lui et al. 1986, Lui et al. 1988, Medley et al. 19871 while others have developed population-level models of the spread of HIV-l [see, for example, Mode et al. 1988, Anderson et al. 1987, Dietz 1988, DeGruttola and Mayer 19881. At least two models have been constructed to describe HIV-l transmission and its demographic impact in developing countries [Bongaarts 1989, Bongaarts and Way 1989, May et al. 1988, Anderson et al. 19881 where the principal forms of HIV-l transmission are heterosexual and vertical. Such models are the focus of the present paper, in which a two-sex age-structured model of HIV-l transmission, appropriate to urban areas of Africa, is presented and its behaviour explored through numerical studies. 148 oo40-5809/91 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
HIV-
1 TRANSMISSION
IN AFRICA
149
The model of heterosexual transmission dynamics of HIV-l proposed by Anderson et al. [1988] differs in four substantial ways from the model presented in this paper. First, this assumption of symmetry between males and females in all demographic and epidemiologic parameters, including the efficiency of disease transmission, allows the model to be collapsed to a one-sex model. Second, the demographic structure of the host population is not age-specific; the force of mortality, p, is the same for all ages (p = 0.019), and produces a population in which the expectation of life at birth is 52.6 years, and in which 15% of the population is still alive at age 100 years. Similarly, fertility is expressed in terms of the average birth rate per adult rather than in a set of age-specific fertility rates for women-a distinction which becomes crucial to population dynamics if either sex is disproportionately depleted by the disease. The assumed values of a preepidemic annual population growth rate of 0.04 per annum and p = 0.019 imply a total fertility rate of approximately 12 children per woman, roughly 50% higher than the highest total fertility rates currently observed. Third, the per capita rate at which adults acquire infection is modeled as a property of a couple, independent of the frequency of coitus: 1= @Y/N, where p is the probability of acquiring infection from any one infected partner, c is the average rate of acquiring partners, and Y/N is the probability that any one partner is infected.,Fourth, no distinction is made between adults who are in monogamous unions and those who have multiple sexual partners: the behaviour of all individuals is described by the average rate of acquiring new partners, c. In addition, all adults are assumed to choose sexual partners independently of age. Under this set of assumptions, Anderson ef al. (1988) conclude that AIDS indeed has the potential for turning population growth rates negative, even in populations which were growing extremely rapidly before the HIV-l epidemic. Subsequent refinement of the model [May et al. 19881 by relaxing the assumptions of symmetric heterosexual transmission and homogeneity in sexual behaviour, but retaining the assumption that the risk of infection is a per-couple property rather than a per-coitus property, yields conclusions which are comparable to those derived from the simpler model. Bongaarts [1989] constructed a model for assessing the demographic and epidemiologic impact of HIV-l in urban areas of Africa which is very similar to the model presented in this paper: the two-sex demographic submodel gives a complete dynamic representation of the demographic structure of the population, epidemiologic and behavioural parameters are allowed to vary with age, sexual behaviour is not homogeneous but rather a number of different risk groups are specified for each sex, and the risk of infection is modelled as a per-coitus risk rather than a per-couple risk, with the probability of maleto-female transmission per coitus twice the probability of female-to-male transmission. There are, however, two principal
150
A. MEREDITH JOHN
differences between the model presented here and the Bongaarts model. First, in the present model, the infectivity of an individual is assumed constant from the time of infection, while the Bongaarts model employs an elegant and more realistic convolution of four exponentially distributed subintervals of the infected period, thereby allowing infectiousness to vary with the duration of infection. Second, in addition to heterosexual and vertical transmission-the modes of transmission used in the Bongaarts modelthe model presented below also incorporates blood transfusion and inoculation by contaminated needle as modes of pediatric infection. On the basis of numerical studies employing his models and plausible values for demographic and epidemologic parameters, Bongaarts concludes that, while an AIDS epidemic will certainly reduce population growth rates in urban areas of Africa, the magnitude of the reduction is not substantial enough to drive population growth rates negative or even near zeroa conclusion which is substantiated by the present study. In the next section, the epidemiology of HIV-l in Africa is briefly summarized. The mathematical details of a fully age-structured model of HIV-l transmission dynamics are then presented; the modes of HIV-l transmission in the model are heterosexual intercourse, vertical (mother-infant) transmission, and transmission through blood transfusion and inoculation with contaminated needles. The behaviour of the model is illustrated for a hypothetical urban center in Africa. After it is demonstrated that the model behaves in a demographically plausible manner in the absence of fatal HIV-l infection, the effect of varying one set of epidemiologic parametersthe rate of male-to-female and female-to-male HIV-l transmission per coitus-over a broad range of plausible values is explored.
2. EPIDEMIOLOGY
OF HIV-l
INFECTION IN URBAN AFRICA
WHO has identified two epidemiologic patterns of HIV-l transmission. The first involves transmission by homosexual activity and by intraveneous drug use, and describes the principal modes of transmission of HIV-l in North and South America, Europe, and the Caribbean. The second pattern involves heterosexual transmission, perinatal transmission, and transmission by contaminated blood, and predominates in Africa. Both symptomatic and asymptomatic infected women can transmit HIV-l to their infants [Piot et al. 19871. Thirty to 65% of the infants born to women infected with HIV-l express HIV-l antibodies at birth; some of these infants will have an active HIV-l infection, while others will simply be expressing passively acquired maternal antibodies to HIV-l [Piot et al. 19881.
HIV-1
TRANSMISSION
IN
AFRICA
151
Parenteral transmission of HIV-l occurs by the transfusion of infected blood or blood products, or by the use of contaminated needles. The risk of HIV-l transmission through blood transfusion depends upon the prevalence of HIV-l in the blood-donor population. In regions where HIV-l is endemic, the proportion of healthy adults who display serological evidence of HIV-l infection ranges from 4 to 30%, thus suggesting that a comparable fraction of blood donors may be infected [Greenberg et al. 1988, Mann 19871. Simple needle-puncture, rather than blood transfusion, appears much less efficient in HIV-l transmission: data on health-care workers sustaining needlestick injuries while treating AIDS patients suggest that the risk of infection is less than 0.5% per puncture [Quinn et al. 19861. Susceptible children in developing countries may acquire HIV-1 infection by intramuscular injection with needles which have not been properly sterilized. In a study in Zaire, seropositive infants (mean age 10.6 months) had received on average 44 intramuscular injections, vaccinations excluded, as opposed to their seronegative counterparts who had received considerably fewer injections [Mann 1986b]. In contrast, there appears to be no significant association between HIV-l seropositivity and childhood immunization [Von Reyn and Mann 1987, Mann et al. 1986a]. The risk of infection from a single heterosexual contact with an HIV-linfected person depends not only upon the type of sexual activity but also on the susceptibility and infectivity of the partners. It appears that, in the absence of other risk factors such as genital ulcers, male-to female sexual transmission of HIV-l may be more efficient than female-to-male HIV-l transmission [Padian 19871. Recent studies have demonstrated age-related differences in the mean (and the median) incubation period, with 1.97 (1.90) years for infants and young children (0 to 4 years old at infection), 8.23 (9.97) years for older children and adults (5 to 59 years old), and 5.50 (5.44) years for elderly patients (60 years or more years old) [Medley et al. 19871. Whether or how cofactors other than age affect the incubation period for developing clinical AIDS is unknown [Curran et al. 19851. Data on HIV-l seroprevalence from Zaire show a bimodal age-specific curve, with a first peak among infants (1 to 2% seropositive) and a second peak (8-10% seropositive) among adolescents and adults aged 16 to 29 years old. Because of the lengthy incubation period of HIV-l, the ageincidence of frank AIDS is somewhat older than that of HIV-l infection. The mean age of frank AIDS cases in Kinshasa is 33.6 years (median, 32 years; range, 1.5 to 64 years); the mean age of men with AIDS is greater than the men age of women with AIDS: 37.4 years and 30.0 years, respectively [Mann et al. 1986a].
152
A. MEREDITH
3.
JOHN
MODEL
In this model of HIV-l infection and subsequent illness individuals are classified by age in single years, sex, behavioural group (pediatric, adolescent but not yet sexually active, sexually active with many partners, sexually active in a stable union, or no longer sexually active), and epidemiologic status (susceptible, infected and infectious, or clinically ill) (Fig. 1). For each behavioural class, sex and age individuals are assumed homogeneous with regard to behaviour, infectivity (if they are infectious) or susceptibility (if they are susceptible). The model is completely agespecified: fertility and mortality vary with age, as do important epidemic and etiologic factors (Table I). Fertility is described by a set of age-specific fertility rates, and mortality is described by a set of age and sex specific mortality rates. For simplicity, it is assumed that the fertility and nonEPIDEMIOLOGIC
CLASS
INFECTED
SUSCEPTIBLE
AIDS
I
FIG.
1.
Schematic
diagram
of the model
of HIV-l
transmission
dynamics.
TABLE
I
Model Notation and Definitions Subscripts a
L
age (in years) calendar time (in years) time spent infected or ill
Superscripts s susceptible infected and infectious (but not presenting clinical illness) a presenting severe HIV-l related illness Demographic parameters age specific fertility rate at age a male and female age specific schedules of transition from not sexually active to &TlF> g1: sexually active with many partners male and female age specific schedules of transition from sexually active with bq, 82: many partners to sexually active in a stable uion male and female age specific schedules of transition from not sexually active to g3:, g3: sexually active in a stable union age and sex specific mortality rate from causes other than AIDS P. number of women age a at time t in the population w,, I number of men age a at time f in the population Ma,, number of births at time t B,
fa
Epidemiologic parameters case fatality rate from HIV-l-related disease at age a, k years after developing ao,k clinical disease rate of developing clinical HIV-l-related disease at age a, k years following Ydf infection with the virus proportion of prostitutes infected at time t if Exogeneous probabilities probability that a seropositive mother bears an infected child PXl probability of female to male heterosexual transmission PX2 probability of male to female heterosexual transmission PX3 probability of infection by innoculation with a contaminated needle PX4 probability of infection from a unit of blood containing HIV-l PXS Endogeneous probabilities probability that a unit of blood contains HIV-l at time t Pl, probability that a needle used for an injection is contaminated at time t P2, probability that a female sexual partner, randomly chosen at time t, is infected P3, probability that a male sexual partner, randomly chosen at time 1, is infected P4, Level of activity proportion of children age a receiving at least one blood transfusion T, average number of blood transfusions per year among children age a receiving Nl, at least one transfusion average number of injections per year among children age a 7% annual number of visits to a prostitute by a male age a, not in a stable union N3, annual number of randomly selected sexual partners of a male age a, not in a N4, stable union annual number of randomly selected sexual partners of a female age a, not in N5, a stable union annual number of visits to a prostitute by a male age a in an otherwise N6, monogamous stable union annual coital frequency of a stable couple with wife age a NT,
153
154
A. MEREDITH
JOHN
AIDS related mortality rates do not change during the period studied. In the absence of fatal HIV-l infection, the model reduces to a standard population projection model based on a Leslie matrix. In the form presented below, all transition probabilities are expressed as annual probabilities. The model can easily resealed so that transition probabilities are expressed as monthly or weekly probabilities. Largely for reasons of tractability, the behavioural assumptions in the model are extremely simple. There are four behavioural classes of adolescents and adults: not yet sexually active, sexually active with many partners (including, for men, with prostitutes), sexually active in a stable union with one partner (and with prostitutes), and no longer sexually active because of illness or widowhood. Many, but not all, adolescents in the population experience a period of sexual activity when partners are chosen at random from the population of people who are sexually active adolescents. All adolescents eventually progress to a stable union if a marriage partner of the appropriate age is available. Once a union is dissolved by the death of an individual, the remaining partner does not remarry and is no longer sexually active. The etiologic assumptions which govern the progression from susceptible to infected to ill to dead are also very simple. People exist in only three distinct epidemiologic states: susceptible, infected and infectious but without clinical manifestation of infection, and overtly ill and hence no longer sexually active and no longer blood donors. The eventual transitions from infected to ill, and from ill to dead are assumed to be inevitable. 3.1. Pediatric HIV-1
Infection
and Illness
In this model there are two modes of HIV-l transmission vertical (perinatal) transmission, and parenteral transmission. a. Perinatal transmission of HIV-l. In the absence of HIV-l the population, the annual number of births at time t is
Bt= :
f, W,,r>
for children: infection in
(1)
a=15
where f, is the fertility rate of women age a, and W,,, is the number of women age a at time t in the population. In the presence of HIV-l, it is assumed that women who have developed overt AIDS are no longer sexually active and thus are no longer bear children, so that children are borne only by susceptible women ( WI,,) and infected women ( Wi,,) at the same rate, f,. The distribution of births between susceptible and perinatally infected infants is given by
HIVB,=B;+Bf=
1 TRANSMISSION
5
f,W:,,+(l-px,)
a=15
+Pxl
f k=l
2 a=15
f,
155
IN AFRICA
i
;
k=l
a=15
f.W;,t,k] (2)
W;,t,k,
where px, is the probability that an infected mother bears an infected child, and is independent of the duration k of the mother’s infection. b. Parenteral transmission of HIV-l. There are two principal forms of parenteral transmission of HIV-l: infection by transfusion of HIV-l contaminated blood, and infection by inoculation with an HIV-l contaminated needle. The probability that a child of age a is infected by transfusion of HIV-l contaminated blood is given by 71,A T,{ 1 - [l - px, . plJNla},
(3)
where T, is the proportion of children of age a receiving at least one blood transfusion at age a, Nl, is the mean number of transfusions received in a year by children age a who receive at least one transfusion, pl, is the probability that at time t a unit of blood contains HIV-l, and px, is the probability that infection results from transfusion of contaminated blood. Blood donors are chosen randomly from the apparently healthy adult population; the probability that the unit of blood contains HIV-l is thus the probability that a donor between the ages of 15 and 60 is infected, c;=
pl,A ~?==I,
1 c:“=
15 W:,t,k
[w:,,+M:,,+C;=,
+ %,t,k
(4)
w:,,,k+“:,,,kl’
The probability that a child age a is infected by inoculation HIV-l contaminated needle is given by
with an
where N2, is the mean number of injections received by children age a, p2, is the probability that at time t, an HIV-l contaminated needle is used, and px, is the probability that infection results from an inoculation with a contaminated needle. It is assumed that all children under age 5 are inoculated at least per year; the probability that the needle is contaminated with HIV-l is equal to the probability that the last child inoculated with that needle, chosen randomly from the population of children age 0 to 4, was infected, c;=,
p2,A cd=,
[w:.,+M:,,+C;=,
c:=O
w:,t,k+“:,,,k
(6) W:,r,k+“b,r,kl’
156
A. MEREDITH
JOHN
where Mz,t and MA,, are the numbers of susceptible and infected boys age a at time t, and Wz,, and WA,, are the numbers of susceptible and infected girls age a at time t. Thus Z7,, the probability that a susceptible child age a at time t will be infected by blood transfusion or needle-puncture, is
c. Incubation period, clinical illness, and death. In this model, the progression of children from infected to ill is governed by the function ya,k which describes the transition from subclinical infection to overt AIDS of children age a who were infected k years ago, at age a-k. The rate at which ill children die is given by the function ~l,,~ where the HIV-l related case fatality rate depends not only on the child’s age but also on the length of time k since the child developed clinical illness. 3.2, Adult HZV-1 Transmision,
Infection,
and Illness
In this model, adolescents and adults belong to four behavioural classes which differ in the risk of exposure to HIV-l: aldolescents who are not yet sexually active, adolescents who are sexually active but not in stable unions, adults who are sexually active within stable unions, and adults who are no longer sexually active. It is assumed that thoughout their sexually active lives, whether they are in stable and otherwise monogamous unions or not, men continue to visit prostitutes, who serve as a reservoir of the virus. Hence a man whose wife is not infected with HIV-l may become infected after marriage; a woman is not protected from HIV-l infection by marrying an uninfected man, since he may subsequently become infected by a prostitute. Women are not at risk of infection by sexual intercourse while they are pregnant or during a period of postpartum sexual abstinance which, in many societies, can be quite lengthy. People who are widowed, who develop overt AIDS, or whose spouses develop overt AIDS cease sexual activity. a. Movement among behavioural classes. The age-specific movement of adolescent females from sexually inactive to sexually active with many partners is governed by the female age-specific transition schedule, g 1,“; age-specific movement of adolescent males from sexually inactive to sexually active with many partners follows the male age-specific transition schedule, g 1,“. Sexually active adolescent males also visit prostitutes. It is assumed that any adolescent who is not yet sexually active is not infected with HIV-l. The transition to a stable union is governed by four schedules: two for men (g2,M, g3:) and two for women (g2,W, g3:). It is assumed that susceptible and infected individuals are equally likely to enter stable unions; however, individuals who have already developed clinical illness
HIV-l
TRANSMISSION
IN AFRICA
157
will not enter stable unions and will become sexually inactive. Individuals select their spouses randomly with respect to previous sexual history and current epidemiologic status. Thus, for example, the probability that a man marries an infected woman is given by the share of eligible infected women in the population. The stable unions thus formed are of four types indexed by the epidemiologic status of husband and wife at the time of marriage: (i) (ii) (iii) (iv)
husband husband husband husband
susceptible, wife susceptible infected, wife susceptible susceptible, wife infected infected, wife infected.
During the marriage, the couple may move from one subcategory to another as the epidemiologic status of either partner changes: from (i) to (ii) by intercourse with a prostitute, from (ii) to (iv) by marital intercourse, or from (iii) to (iv) by marital intercourse or by intercourse with a prostitute. b. Movement among epidemiologic classes. (i) Transmission among sexually active adolescent who are not in a stable union. Men are infected with HIV-l either by sexual intercourse with an infected prostitute or by sexual intercourse with an infected woman chosen randomly from the sexually active, nonprostitute population. If a man of age a has intercourse N3, times per year with prostitutes, a fraction [, of whom are infected, then the probability that he will be infected by a prostitute during the year is given by
where px, is the probability of female-to-male transmission of HIV-l per coitus, and (1 -p~,)~~~ is the probability of not being infected by a prostitute in N3, coital acts. The probability that a man of age a is infected by intercourse with a woman chosen randomly from the population of sexually active women is given by 173
A I-
{ [ 1 - (p3,. px,)]N4.},
where N4, is the number of sexual partners per year selected randomly with replacement from the sexually active female population or, equivalently, the number of coital acts with randomly chosen female partners per year, and px, is again the probability of female-to-male transmission per
158
A. MEREDITH
JOHN
coitus. The probability that a randomly chosen sexually active women is infected with IV-l, p3,, is determined endogeneously: p3,G
c;=, CZ=”K,,,, c:=, cw:,,+cL,
wtJ.kl
(10) sexually
actwe
wth
multiple
partners
Because p3, varies over time as the proportion of infected women in the population changes, the probability that a sexually active male will become infected will also vary over the course of the epidemic. The model for women is symmetric to that for men. The probability that a woman of age a is infected by a man chosen at random from the population of sexually active men who are not in stable unions is rr, A 1 - {[l - (p4,. pXj)y”},
where N5, is the number of sexual partners per year, selected randomly with replacement from the sexually active male population or, equivalently, the number of coital acts with randomly chosen male partners per year, and px, is the probability of male-to-female transmission per coitus. The probability that a randomly chosen sexually active man is infected with HIV-l, p4,, is determined endogeneously:
c;= 1c:=ow,,, p4f G C$=,
’ sexually
active
with
multiple
(ii) Transmission among adults in a stable union. It men in marital unions will visit prostitutes, a fraction infected. Hence the probability that a man of age a will HIV-l has two components: l7,, the probability of prostitute, ~,~l-{C1-(i,.PX,)IN6.},
(12)
partners
is assumed that {, of whom are be infected with infection by a (13)
where N6, is the annual coital frequency of a married man with prostitutes, and 176, the probability of infection by the man’s wife, age a’, z76~1-([1-(z.pX*)]N’~~},
(14)
where N7,. is the annual coital frequency of the couple and z is an indicator variable which equals one if the wife is infected with HIV-l and zero if she is not infected. It is assumed that women who are married have sexual intercourse only with their husbands. Hence the probability that a woman age a will be infected is (15)
HIV-l
TRANSMISSION IN AFRICA
159
where N7, is the annual coital frequency of the couple and z is an indicator variable which equals one if the husband is infected, and zero if he is not. c. Incubation, illness, and death. The rate at which adults pass from infected to clinically ill with AIDS is given by the distribution of incubation times Yo,k where the transition rate depends not only on the aldult’s age but also on the length of time k since the adult was infected. The rate at which ill adults die is given by the distribution of lengths of illness aa,k where the HIV-l-related case fatality rate depends not only on the adult’s age a but also on the elapsed length of time k since the aldult developed clinical illness.
4. NUMERICAL
STUDIES
The numerical implementation of the model presented above is based on a standard cohort projection model of population growth and structural evolution for a population closed to migration. In addition to moving individuals forward in time along an age dimension with decrements to the population from mortality, the model incorporates movements by sex, age, and time among epidemiologic and behavioural classes by the rates derived in the preceeding section. The model has been tested with projection periods of 3, 6, and 12 months; there are slight variations in the numerical results as a function of projection period length, but the qualitative results are similar. The results reported below are based on projections by 6-month increments for 50 years. 4.1. Parameter Specification
Within this projection framework, four groups of parameters are defined: (i) demographic parameters, specifying the fertility and non-HIV-l related mortality patterns in the population; (ii) behavioural parameters, specifying the age-specific rates of movement between behavioural classes and age-specific coital frequency patterns; (iii) epidemiologic parameters, specifying the spread of the disease through the population; and (iv) etiologic parameters, specifying the time pattern of progression of the disease within an individual. Values for the parameters used below have been gleaned from the demographic and epidemiologic literature, and as such are not intended to represent the course of the disease in any particular locale. Furthermore, there is little certainty about the values of many of the epidemiologic and etiologic parameters; for the purpose at hand-illustrating the behaviour of the model-values which represent, in some cases, little more than best guesses have been employed.
160
A. MEREDITH
JOHN
The numerical model uses a set of fertility a. Demographic parameters. rates based on the period age-specific fertility rates for Kenya, 197771978 [World Fertility Survey 19841, scaled to correspond to a total fertility rate of 6.5 children per woman, with the age pattern of fertility preserved. The age- and sex-specific mortality rates used to model the age pattern of nonHIV-l related mortality are based Coale-Demeny West model life tables, level 16 [Coale and Dmeny 19841, and correspond to male and female life expectancies of 54.1 years and 57.5 years, respectively, and male and female infant mortality rates of 99 and 82 deaths per thousand live births, respectively. The population growth rate, in the absence of HIV-l mortality, is endogeneously determined by the fertility and mortality rates to be 0.034 per annum. The starting susceptible population is the stable population corresponding to the fertility schedule life tables used in the numerical model. b. Behavioural parameters. The age-specific schedule of entry into sexual activity for girls, g 1y, is defined as
gl,w=
0.00 a19 035 1. lS
