CHAPTER 7 THE HIV TRANSMISSION DYNAMICS MODEL FOR FIVE MAJOR RISK GROUPS

Transcription

1 CHAPTER 7 THE HIV TRANSMISSION DYNAMICS MODEL FOR FIVE MAJOR RISK GROUPS Chapters 2 and 3 have focused on odeling the transission dynaics of HIV and the progression to AIDS for hoosexual en. That odel is now expanded in Section 7.1 to include five ajor risk groups : hoosexual (and bisexual) en, hoosexual (and bisexual) en who are intravenous drug users, intravenous drug users (IVDUs), heterosexual partners of VDUs and neonates of feale IVDUs and heterosexual partners. In this odel the hoosexual en have hoosexual partnerships within their own group and also with hoosexualivdus. The IVDUs have needle-sharing partnerships with other IVDUs and also with hoosexual-ivdus. Obviously, the heterosexual partners of VDUs have sexual partnerships with people who are VDUs. Perinatal transission occurs fro the feale IVDUs and the feale heterosexual partners. Figure 7.1 suarises the risk groups and possible routes of transission of HIV infection. The fitting procedure for the five-group odel is outlined in Section 7.2. This odel with the first three risk groups is used for New York City in Chapter 8. The full odel is used for regional coparisons in Chapter 10. The five risk groups above are chosen since they account for 94% of the adult/adolescent cases and 78% of the perinatal pediatric cases with known source of infection reported through Deceber, 1991 in the United States (CDC : 1992). See Tables 1.1 to 1.3. The odel does not include other heterosexual contacts, heophiliacs, blood transfusion recipients, and children with other perinatal transission. It is not reasonable to include ore risk groups in an initial odel for a local population. Blood transfusion recipients (2% of AIDS cases) could be included, but blood banks are regional instead of local. Siilarly, the incidence in heophiliacs (1% of all AIDS cases) is quite unifor throughout the United States, so this is a nationwide phenoenon instead of a local phenoenon, particularly since the blood factor concentrate is prepared in a few places and distributed nationwide. It is probably not necessary to include a separate population of bisexual en since heterosexual contacts with bisexual en accounts for only 651 out of 206,392 AIDS cases in the United States through Deceber It is not possible to include people or sexual partners born in pattern II countries since the iigration ties and HIV status of these people is not known. 7.1 The General Model Recall that the hoosexual en are subdivided into active and very active subpopulations. In the odeling of hoosexual en in San Francisco (SF), the 10% of the hoosexual en in the very active class were ten ties as sexually active as those in the active class (the paraeter values are F = 0.10 and R = 10). Here the IVDU population is also subdivided into these two activity level classes based on their nuber of needle-sharing partners per onth. The needle-sharing contacts are odeled in a way analogous to the hoosexual contacts in that there is an average nuber of new needle-sharing partners per onth and there are probabilities of transission per new needle-sharing partner that are dependent on the stage of

2 84 hoosexual en hoosexual IVDUs 1 I VDUs neonates infected perinatally heterosexual partners of VDUs Figure 7.1 Risk groups and infection transission connections. infection of the partner just as for hoosexual partners. As indicated previously the hoosexual VVDUs have both hoosexual partners and needle-sharing partners. The progression of HIV infecteds to AIDS has been developed carefully as a sequence of stages in Chapter 2 and the sae progression is used for all adult risk groups in this general odel. The progression for children has also been described in Chapter 2. The progression variables and paraeters in Figure 3.1 and Table 3.1 now exist for all five risk groups: the hoosexual en with suffix H, the hoosexual VDUs with suffix B for both, the IVDU population with suffix D for drug user, the population of heterosexual partners of IVDU with suffix P for partners, and the neonates with suffix C for children. Thus the odel consists of five sets of finite difference equations for the nubers in each copartent, which are siilar to those in Figure 3.2. All populations except the heterosexual partners and children are subdivided into very active and active subgroups where the activity is either sexual or needle-sharing partnership foration. For exaple, the quantities for hoosexual en are QVH, SVH, XH(1),..., XH(8), QAH, SAH, YH(1),..., YH(8), ZH(1), *.' ZH(6) and ZAIDSH. The suffixes H, D, B, P and C are also used for other paraeter values. For exaple, QHH, QHD, QHP and QHC are the probabilities of transission of HIV infection by an infective in an asyptoatic stage during a hoosexual partnership, needle-sharing partnership, heterosexual partnership, and childbirth, respectively. The quantities PAH, PAD and PAP are the average nuber of new hoosexual, needle-sharing and heterosexual partnerships per onth. As in the odel for hoosexual en, the fraction I of new partnerships are distributed by proportionate ixing and the fraction l-t of new partnerships occur internally to each group. The total incidences in the active and very active subpopulations are the sus of the internal incidences and the external proportionate ixing incidences. The incidences with suffixes H and

3 85 D on the variables and paraeters are used for hoosexual partners and needle-sharing partners, respectively. Thus the incidence for hoosexual en is the su of the internal and external (proportionate-ixing) hoosexual incidences given in Section 3.2, where each paraeter and variable now has the suffix H. For the IVDUs, the incidence is the su of siilar internal and external needle-sharing incidences, with suffix D. Thus the incidence in the active subpopulation of IVDUs is [ (1-ETAD)x pi x PHD x to i x QHD x YD(i} x QAD-Y SAD D (+l) + SAD,, ETAD s pi x PHD x vi x QHD x [R x XD(i)+YD(i)] x PHD i where CD = PHD x (QAD - YD(+1)} + R x PHD x (QVD-XD(+1)), and the incidence in the very active subpopulation of IVDUs is ~ (1 ETAD) x pi x R x PHD x vi x QHD x XD(i}1 x Sgy+ + ETAD x pi x PHD x u1 x QHD x (R x XD(i)+YD(i)] z.r x PHD x SVD The hoosexual-vdus are considered to have both hoosexual partnerships and needle-sharing partnerships. The incidence in the hoosexual-ivdus with suffix B is the su of both the hoosexual and needle-sharing incidences. The incidence for the heterosexual partners of IVDUs is not given by the incidences above, but is given by pt x PAP x of x QHP x [XD(I)+YD(I)] x -SAP Note that the IVDUs do not have two activity levels in ixing with heterosexual partners and the heterosexual partners have only one activity level. This incidence ter for the heterosexual partners of IVDUs is a very siple version of a coplicated process. Although heterosexual partnerships with IVDUs are constantly being fored and dissolved, it is difficult to odel this pairing process. The incidence ter above assues that people in the population of heterosexual partners of IVDUs are ixing with the IVDUs and foring partnerships. It is helpful to think that this population is a changing aalgaation consisting of those who have been or currently

4 86 are or will becoe heterosexual partners of IVDUs. The incidence expression above sees to be the siplest way to odel heterosexual partners of IVDUs. The incidence of perinatal HIV infections is the product of the fecundity rate FC (children born per feale per onth), the probability QHC of transission by an asyptoatic other to a neonate and the weighted su of infected feales. The weighted su of feale IVDUs is the su of the products of the relative weights of transission, the fraction PIW of IVDUs who are woen and the nuber of infected IVDUs. The weighted su of feale heterosexual partners is the su of the relative weights of transission, the fraction PHW of heterosexual partners who are feales and the nuber of infected heterosexual partners. The su of these two is the weighted su of infected feales. Thus the incidence of perinatal HIV infections is FC x QHC x pt x tit x PIW x [XD(I)+YD(I)j + pt x &i x PHW x YP(I) 7.2 The Fitting Procedure The fit criteria given in Section 6.1 for the odel for hoosexual en in San Francisco ust be expanded for this ore general odel. As before, the first criterion for the siulations is that the paraeter values ust be consistent with the a priori paraeter estiates obtained frodata. Since estiates of HIV incidence are generally not available, the second criterion is now that the yearly AIDS incidences in the risk groups in the siulations ust be close to the yearly AIDS incidence data supplied by the Centers for Disease Control (CDC). Many of the paraeter values in the odel are-fixed at their estiated values, but soe are allowed to vary to give the best fit. For exaple, the population sizes of the hoosexual en, IVDUs and heterosexual partners of VDUs are all fixed, but the population size of the hoosexual-wdus is varied in order to obtain the best fit to the AIDS incidence data in this population. The igration rates and natural ortality rates are fixed at the estiated values. The paraeters related to the progression through the stages ( and 7i for i = 1,. -.,7) are all fixed and equal to the values found in Chapter 2. Other fixed paraeter values are the probabilities (or proportions) of transission QHH, QHD and QHP per asyptoatic partner, the paraeters wk which ultiply these QH values to deterine the probability of transission by a stage k infective and the paraeters p k which are the fractions still active in stage k. The predicted AIDS cases in the five risk groups in the NYC odel ust be close to the delay-adjusted AIDS incidence. The nuber of AIDS cases per year are not linearly independent so that a hypothesis of the chi-square goodness of fit test is not satisfied. Nevertheless the chi-square value of the su of the squares of the observed inus the expected divided by the expected values is coputed as a easure of the fit. Separate chi-square sus are also coputed for each of the five risk groups. The fit to the AIDS incidence data in the population of hoosexual en is obtained by varying the initial average nuber PASH of partners per onth, the year STRH in which reduction in partners starts and the yearly reduction factor RDNH. Siilarly, the fit to the AIDS incidence data in the IVDU population is obtained by varying the analogous paraeters

5 87 PASD, STRD and RDND. The fit to the AIDS incidence data in the hoosexual IVDU population is obtained by adjusting the population size NSIZEB. The fitting procedure involves choosing paraeter values sequentially with those subject to the ost uncertainty chosen last or adjusted to fit the AIDS incidence data. Fixed values are used for any paraeters such as population sizes, igration rates, natural ortality rates, nuber of stages, stage transition rates, probabilities of transission by asyptoatic infectives, ultiplying factors for probability of transission and fractions still active in the stages, activity level fractions and ratios, and the transfer rate constants between the activity levels. After the paraeters above are fixed, the reduction starting years STRH and STRD are chosen and the size NSIZEB of the hoosexualivdu population is chosen. Initial guesses are ade for four paraeters : the average nubers of partners per onth (PASH and PASD), and the yearly reduction factors (RDNH and RDND). A Fortran coputer progra uses the IMSL subroutine BCONF to find the values of these four paraeters which iniize the su of the chi-square values for the hoosexual en, hoosexual--ivdus and IVDUs. This progra is siilar to FIT10.FOR listed in the Appendix. After the siulation odel has been fit to the IVDUs, then the odel is fit to the heterosexual partners of IVDUs by varying only one paraeter, the average nuber PAP of new heterosexual partners per onth. After both the IVDUs and their heterosexual partners have been fit, then the odel is fit to the perinatal cases by varying only one paraeter, the fecundity rate FC. This progra is siilar to IVDUIO.FOR listed in the Appendix. 7.3 Discussion of the Model Epideiological odelers attept to capture the essential features of an epideiological process in a precisely stated odel. Of course, the odels are extree siplifications of the coplex huan and biological processes that contribute to the spread of an infectious disease, but they can be very useful in attepting to understand the basic epideiological echaniss. The art of epideiological odeling is to forulate a odel so that it has enough coplexity to be consistent with all observations, but is not ore coplicated than necessary to fit the data. Thus there is a balance between the goals of siplicity and realis. The odel forulated here for the spread of HIV in the five risk groups shown in Figure 7.1 attepts to capture the essential aspects in a relatively siple odel. This odel consists of five sets of nonlinear difference equations siilar to those in Figure 3.2 for copartents siilar to those in Figure 3.1. The odel incorporates the ajor transission echaniss of hoosexual and heterosexual intercourse, needle-sharing by IVDUs and perinatal infection. Heterogeneity in sexual and needle-sharing behavior is reflected by the division of each population of hoosexual en, hoosexual IVDUs and IVDUs into two subgroups consisting of those who are active and those who are very active. The odel includes igration into and out of the risk populations, noral, non-hiv related deaths, and changes in sexual (or needle-sharing) behavior by soe individuals. Because there is no solid data on changes in the sizes of the risk groups over tie, the odeling here assues that the population sizes reain constant. The subdivision into only two sexual (or needle-sharing) activity levels is reasonably siple, but it is

6 88 consistent with the survey data and it is adequate to fit the HIV and AIDS incidence data. Based on four surveys of hoosexual behavior, it is estiated in Section 5.2 that for two sexual activity levels, ten percent of the population has ten ties as any new sexual partnerships per onth. The corresponding needle-sharing paraeters are estiated in Section Heterogeneity in behavior is an iportant factor in the transission of sexually transitted diseases. Hethcote and Yorke (1984) explored the effects of heterogeneity in the transission of gonorrhea and found that a "core" group of sexually very active, highly-efficient transitters were central to the transission and persistence of gonorrhea. One of the ajor barriers to a better understanding of the HIV transission process is the lack of inforation on the social, sexual and needle-sharing ixing which occurs between subgroups of the risk groups. Many authors have used a odeling approach to explore the effects on HIV transission and AIDS of behavioral heterogeneity and different ixing structures. Attepts have been ade to estiate the entries in contact or ixing atrices fro survey data. Papers exploring sexual ixing and heterogeneity include : Ahlgren et al. (1990), Anderson (1988a, 1988b), Anderson and May (1991), Blythe and Anderson (1988), Blythe and Castillo--Chavez (1989), Busenberg and Castillo-Chavez (1991), Cardell and Kanouse (1989), Casti lo--chavez et al. (1989), Hyan and Stanley (1988, 1989), Jacquez et al. (1988, 1989), Knox (1986), Koopan et al. (1989), Pickering (1986), Sattenspiel and Sion (1988) and Sion and Jacquez (1992). Papers odeling HIV transission in IVDUs include Blower et al. (1991) and Kaplan et al. (1989, 1990). Staged progression has becoe the standard way to odel the long infectious period leading to AIDS. Here the progression fro HIV infection to AIDS is odeled by a sequence of seven infectious stages with AIDS as the seventh stage and death due to AIDS as the eighth stage. The ean ties in each stage are given in Chapter 2 for both clinical staging and T4-cell count staging. The infectivity and aount of sexual activity vary with the stage of infection. The relative infectivity and relative sexual activity copared to those in the asyptoatic stage are estiated in Sections and The HN incidence ters are all based on the principle of ass action ; i.e., the incidence is directly proportional to the product of the nuber of susceptibles and the nuber of infectives in the stages. The siplest forulation for the interactions between subgroups of a risk group is based on the assuption of proportionate ixing. In this case the activity levels are specified for each subgroup and then the new partnerships of a person are distributed in proportion to the activity levels and population sizes of the subgroups (Hethcote and Yorke, 1984). Thus a person is ore likely to have a new partner fro a ore active subgroup than fro a less active subgroup. This assuption of proportionate ixing has the iense advantage that the entries in the n x n contact atrix can be estiated fro n pieces of inforation (Hethcote and Van Ark, 1987). Unfortunately, this proportionate ixing assuption does not see to be realistic since very active people see to be uch ore likely to for new partnerships with other very active people. The next siplest forulation for subgroup interactions sees to be a convex cobination of proportionate ixing and internal ixing (in which people ix only with their own subgroup). This forulation, which was used in odeling gonorrhea (Hethcote and Yorke, 1984, p. 83), is now often called preferred ixing (e.g., Jacquez et al., 1988, 1989). In the n2

7 89 odeling here, the incidences are forulated with preferred ixing and the convex cobination paraeter q is adjusted to fit the data. As indicated in Section 7.1, the incidence ters for HIV transission between needle--sharing IVDUs are also forulated with preferred ixing. It ay see strange to use the sae for for the incidence ters for hoosexual en and for IVDUs ; however, this sae for is based on the following siilarities. People in both groups do for partnerships that consist of one or ore contacts (hoosexual or needle-sharing). Bath houses or gay bars ight be analogous to shooting galleries as places where ultiple partnerships occur. The nuber of contacts and duration of partnerships vary greatly in both populations so that the very active and active categories are a siplification for both populations. People in both populations can have ultiple siultaneous partnerships and the incidence ters account for this soewhat since this behavior could correspond to frequent partner change. Thus the incidence ter above is a siplification for both populations, but is reasonable for both populations. There are data fro sexual behavior surveys which show that the average_ partnership rate of hoosexual en in soe cities has changed over tie (see Section 5.3.2). There are also soe indications that the needle-sharing behavior of IVDUs ay have decreased over tie (see Section 8.2.4). Here changes in behavior are odeled by specifying the initial average nuber of new partners per onth, the starting and stopping dates for changes in the partnership rate and the yearly reduction factor. Modeling HIV transission and AIDS in heterosexual partners of IVDUs is handled with the siplest possible forulation. The HIV incidence for these heterosexual partners is proportional to the nuber of new partnerships per onth of heterosexuals with IVDUs ; consequently, the only paraeter to be adjusted in the fitting process is the average nuber of IVDU partners per onth of heterosexuals. Although there is soe evidence that the probability of heterosexual transission is different depending on whether the infected person is ale or feale (Padian et al., 1991), this possible asyetry is not included in our odel. If this asyetry were included, then the odel would need separate classes for ale and feale IVDUs and for ale and feale heterosexual partners of IVDUs. This ore coplicated odel would have ore paraeter values to be estiated, and this ight be difficult since the yearly AIDS incidence is often sall and erratic in the risk groups of feale IVDUs and their ale heterosexual partners. The siple odel used here works well in fitting the AIDS incidence data for heterosexual partners. The odeling of perinatal transission fro HIV-infected others to their newborn children before, during or just after birth is also odeled in the siplest way possible. The perinatal HIV incidence involves only the fecundity rate (birth rate), the probability of transission during birth and the nuber of HIV-infectious feales. The progression of perinatally-infected children to AIDS is odeled using the two-track staging syste given in Section 2.3. For further clarification of the incidence ters or other aspects of the odel, see the saple coputer progras in the Appendix.