CHAPTER 1 BACKGROUND ON THE EPIDEMIOLOGY AND MODELING OF BIV/AIDS

Transcription

1 CHAPTER 1 BACKGROUD O THE EPDEMOLOGY AD MODELG OF BV/ADS Befoe modeling any disease it is cucial to undestand the epidemiological featues of the disease. The fist fou sections of this chapte pesent epidemiological infomation on the human immunodeficiency vius (HV) and the acquied immune deficiency syndome (ADS) including tansmission mechanisms, data, and histoical, biological and clinical aspects. The last thee sections discuss not only advantages and disadvantages of vaious appoaches to modeling HV/ADS, but also the puposes and limitations of epidemiological modeling. 1.1 HV Tansmission Mechanisms The thee known modes of tansmission of HV ae sexual contact, diect contact with HV-infected blood o fluids and peinatal tansmission fom an infected mothe- to he child. Thee is no evidence of spead fom toilet seats, by insects o by casual contact. Thee ae indications that the tansmissibility of HV infection vaies geatly duing the multi-yea couse of infection in an individual (Goedet, Eyste et al., 1987 ; Longini et al., 199). Tansmission may be moe likely duing the ealy flu-ike illness, which occus just afte HV infection and befoe the body develops an antibody esponse. Since the HV vius level in the blood deceases duing the asymptomatic peiod, the tansmissibility appeas to be lowe in this peiod. Then the tansmissibility seems to incease again as the CD4* cell count gets low, the HV vius level in the blood inceases, and symptoms appea. n homosexual patnes the most impotant oute of tansmission seems to be eceptive anal intecouse. The likelihood of tansmission seems to be less fo insetive anal intecouse and fo insetive o eceptive oogenital contact (Kaslow and Fancis, 1989, p. 98). n heteosexual patnes engaging in penile-vaginal intecouse, both male-to-female and female-to-male tansmissions do occu. Howeve, a ecent study in the U.S. suggests that male-to-female HV tansmission is much moe likely than female-to-male tansmission (Padian et al., 1991). Although HV has been isolated in saliva, thee is stong evidence against HV tansmission by kissing (Kaslow and Fancis, 1989, p. 99). The tansfe of blood fom an HV-infected peson to anothe peson has been dealy established as a mode of tansmission. n the U.S. this has been an impotant tansmission mechanism fo the needle-shaing VDUs. n Afica nonsteile needles used fo injection of medications have esulted in numeous HV infections. Befoe blood sceening began in the U.S. in 1985, some HV infections esulted fom blood tansfusions with HV-infected blood and fom HV-infected blood facto concentate fo hemophiliacs. Othe souces of HV infection in this categoy ae tansplanted ogans such as kidneys and semen fo atificial insemination. Tansmission of HV by athopod vectos such as mosquitoes has not been obseved and seems vey unlikely. Peinatal tansmission fom an HV-infected mothe to he child at a time befoe, duing, o afte bith is the thid tansmission mode. The exact times of tansmission and the ole of

2 2 beast feeding have not been established, but appoximately 3% of the childen of HV-infected mothes ae infected. 1.2 Biological and Clinical Aspects of HV nfection The etovius called human immunodef ciency vines (HV) was established in 1983 as the causative agent of ADS. solates of HV ae moleculaly and biologically heteogeneous, with some isolates being moe viulent. Moeove, the HV vius can change apidly, even within an individual. The HV infects a subpopulation of thymus-deived T lymphocytes called CD4+ lymphocytes o T4 cells, which ae helpe/induce cells. These T cells pefom ecognition and induction functions as pat of the immune esponse to foeign stimuli. The HV integates into the CD4+ host cell DA, whee it can emain domant fo a long time. The CD4+ T lymphocytes ae eventually killed by the HV while the HV epoduces itself, so that the numbe of CD4+ cells gadually deceases fom the nomal numbe of about 9/ml (Kaslow and Fancis, 1989). This leads to sevee immunvde6dency in pesons infected with HV. Thus the natual histoy of HV infection is a gadual depletion of CD4+ cells, pogessive unesponsiveness of the immune system and inceased susceptibility to oppotunistic infections such as Pneumocystis Cainii pneumonia and malignancies such as Kaposi's sacoma. Tansmission of HV infection can be though the tansfe of eithe cell-fee H vius o. HV-infected lymphocytes. The pobability of tansmission appeas to depend on the stage of HV infection. n the fist few weeks following infection, moe HV has been isolated fom blood plasma than in the asymptomatic stage, so that people in the pe-antibody stage may be moe infectious than people in the asymptomatic stage. n the late stages of HV infection and ADS, the cellfee HV vius is found moe fequently in blood plasma, so that these people may also be moe infectious. The enzyme-linked immunoassay (ELSA) test fo RV antibodies is inexpensive and useful fo sceening lage numbes of samples ; this test has almost no false negatives, but it is vey sensitive (i.e., it has false positives at the ate of about 2 pe 1). Samples which ae positive by the ELSA test ae then tested by the Westen blot assay. This immunoblot method is vey specific (i.e., false positives occu in no moe than.1% of the samples). Anothe test used is an indiect immunofloescent antibody test involving micoscopic examination of infected cell spots on glass slides. The combination of these and othe tests such as PCR (polymease chain eaction) ae quite accuate in identifying HV-positive individuals (Kaslow and Fancis, 1989). Thee ae many diffeent clinical manifestations of HV infection and ADS. Acute HV infection occus just afte infection in some patients and is chaacteized as an acute febile illness with feve, sweats, lethagy, muscle ache, headache and soe thoat. These symptoms may last 2 to 3 weeks. About 2 months afte infection, the immune system has geneated antibodies which ae ecognized by the ELSA test. Afte an asymptomatic peiod of about 5 yeas, an HV-infected peson may develop some symptoms such as oal candidiosis (thush), haiy leukoplakia, hepes zoste, weight loss, diahea, pesistent genealized lymphadenopathy, neuologic diseases (dementia, myelopathy, polyneuopathy) and tubeculosis. The HV-infected peson may die of these diseases, but most get one o moe of the oppotunistic infections o

3 3 aeoplasms which chaacteize ADS. These include cytomegalovius infection, Pneumocystis Cainii pneumonia, toxoplasmosis of the bain and Kaposi's sacoma. The oiginal definition of ADS was evised in 1985 (CDC, 1985c) and again in 1987 (CDC, 1987j). ow thee ae plans to change the ADS definition again to include all HV-positive people with CD4+ cell count below 2 ; this would appoximately double the numbe of people satisfying the ADS definition. 1.3 HV and ADS in the United States The histoy of ADS in the United States (U.S.) is inteesting. n June 1981 the Centes fo Disease Contol (CDC) epoted on five homosexual men in Los Angeles with an uncommon Pneumocystis cainii pneumonia (PCP) who had a cellula immune disfunction (CDC, 198a). The next month CDC epoted that 26 homosexual men in ew Yok and Califonia had Kaposi's sacoma (KS), a ae skin cance (CDC, 1981b). n August 1981 CDC epoted moe PCP and KS among homosexual men with immunosuppession (CDC, 1981c). Futhe study of some men with KS evealed that they had geatly educed CD4+ T lymphocytes and that KS cases in these homosexual men wee associated with a lage numbe of sexual patnes and a histoy of sexually tansmitted diseases (Fiedman-Klein et al., 1982). n 1982, the name acquied immunodifficiency syndome (ADS) was used to descibe the people with this new condition. The clinical syndome of ADS was also ecognized in heteosexual intavenous dug uses (VDUs) in _ the ealy 198s (CDC, 1982). The etovius which causes ADS was identified in 1983 and was called the human T-lymphotopic vius (HTLV ) o lymphadenopathy- associated vius (LAV) ; it is now called the human immunodeflciency vius (HV) o HV-1 since an HV -2 has been identified (CDC, 1986h). Thus impotant aspects of HV tansmission and ADS in the U.S. wee identified within a few yeas afte the fist cases wee epoted. amely, that HV was a vius which could be tansmitted sexually and by needle-shaing VDUs, that the immune system deteioated due to the decline of CD4+ cells and that oppotunistic infections and cances occued in the late stage of the disease. ote that the incidence of a disease o condition is the numbe of new cases occuing pe month (o yea o othe time inteval), and that the pevalence is the total numbe of cases which_ exist at a given time. n an equilibium situation fo a disease such as measles, the pevalence is equal to the incidence times the aveage duation of the disease, but that this fomula does not wok fo HV o ADS, since an equilibium has not been eached and the duation of HV infection is long and vaiable. The yealy ADS incidence in the U.S. has inceased evey yea since 198 and was about 44, in 1991 so that 22% of the 2, ADS cases though 1991 occued in The ADS incidence is not unifom thoughout the U.S. Figue 1.1 shows the 1991 annual case ates fo the states. The incidence of ADS is also not unifomly spead among people in the U.S., but is focused in cetain isk goups and occus moe commonly in cetain acial/ethnic goups. So fa, most of the ADS cases in the U.S. have occued in homosexual men with homosexual intecouse as the tansmission mechanism. Thee have also been many ADS cases in intavenous dug uses (VDUs) and some ADS cases in homosexual men who ae also VDUs.

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5 5 Most of the heteosexual ADS cases in the U.S. ae in the isk goup of heteosexual patnes of VDUs. These fou isk goups account fo 94% of the ADS cases in the U.S. in adults with a known isk of HV infection. n childen (<13 yeas old) 78% of ADS cases with known isk of infection have occued in childen whose mothe is an VDU o a heteosexual patne of an VDU (CDC, 1992). Since these ae the majo isk goups in the U.S., they ae the goups analyzed by modeling in this monogaph. Figue 1.2a shows the ADS case ates by age goup and sex in the U.S. The case ate in men is highe than in women because of homosexual and needle-shaing tansmission (appoximately thee fouths of VDUs with ADS ae men). Figue 1.2b shows the epoted adult/adolescent ADS cases by exposue categoy. Blood-tansfusion-acquied HV infections ae now vey ae. Since 1985 all blood in the U.S. has been sceened fo HV antibodies, with positive units being discaded. The isk goup of being a ecipient of a blood tansfusion, blood component o tissue accounts fo about 2% of all ADS cases in the U.S., but this is pimaily the esult of those infected befoe sceening stated. Many hemophiliacs in the U.S. wee infected in the ealy 198s by blood facto concentate which was poduced by pooling 5 to 1 thousand units of blood. Appoximately thee fouths of the 14, hemophiliacs in the U.S. have been infected with HV. Afte HV was identified, the blood facto concentate was heat teated to kill the HV stating in late Hemophiliacs account fo about 1% of ADS cases in adults. Othe categoies with small pecentages of ADS cases ae heteosexual patnes of people bon in patten counties and childen of women bon in patten H counties. Patten H counties as descibed in the next section ae those such as Haiti and many Afican counties, whee heteosexual tansmission pedominates. The pimay souce of data on ADS in the U.S. is the Centes fo Disease Contol (CDC). All 5 states, the Distict of Columbia, U.S. dependencies and possessions and othe elated nations epot ADS cases to CDC using a unifom case definition and case epot fom. CDC compiles the data and epots it monthly in the publication, HV/ADS Suveillance Repot. These epots include Tables of ADS incidence in the states, lage cities, adults, men, women, childen, acial/ethnic goups and age goups. Tables 1.1, 1.2, and 1.3 ae fom the Januay 1992 HV/ADS Suveillance Repot (CDC, 1992) which gives data though Decembe These Tables give the numbes and pecentages of ADS cases in the vaious isk goups. ote that 9% of ADS cases in adults occu in men. Among white men, most ADS cases occu in homosexual men, but among black and Hispanic men, the numbe of ADS cases in homosexual men and in VDUs ae about equal. Among women, most cases occu in VDUs, but many cases also occu in heteosexual patnes of VDUs. Obseve that 29% of all ADS cases occu in blacks, who constitute about 12% of the U.S. population, and 16% of all ADS cases occu in Hispanics, who ae about 8% of the U.S. population. Repoting delays of ADS cases ae significant ; about half of all ADS cases ae epoted within 3 months of diagnosis, but about 15% ae epoted moe than 1 yea afte diagnosis. Vaious methods ae used fo adjusting fo epoting delays, but CDC uses a pocedue developed by Kaon, Devine and Mogan (1989). ADS incidence data must be adjusted fo epoting delays in ode to see tends ove time. Figue 1.3 shows ADS incidences adjusted fo epoting delays

6 6 ACQURED 1MMUODEFCECY SYDROME (ADS) - Annual ates pe 1, adult population, by selected age goup and sex fo epoted cases, United States, Males Females v a , Age Goup.1 P ~ F~ ACQURED 1MMUODEFCECY SYDROME (ADS) - Repoted adult/adolescent cases, by exposue categoy, United States, 199 Homosexual/Bisexual Men (56%) V Dug Use (24%) Homosexual and V Dug Use (5%) Hemophilia Cases (1%) Heteosexual Contact (6%) O Tansfusion Recipients (2%) + Othe/Undetemined (6%)t Figue 1.2 a ADS case ates by age and sex, b) ADS case ates byexposue categoy. Souce: CD (199, p. 16).

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10 1 a. Total cases, cases among homosexual/bisexual men', and cases among women and heteosexual men epoting intavenous (V)-dug use Yea of Diagnosis b. Cases among pesons epoting heteosexual contact with pesons with, o at high isk fo, HV infection 25 Female Male a7 m ao cs U go ~ Yea of Diagnosis Figue 1.3. ADS cases adjusted fo epoting delays by yea of diagnosis. a) total cases, homosexual/bisexual men, and VDUs, b) heteosexual cases. Souce : CDC (1991, p. 36).

11 11 c. Peinatally acquied pediatic ADS cases 8 de / ww 4 - woo MM Yea of Diagnosis 6-2- d. Cases among ecipients of tansfusions of blood o blood poducts 14, '','seas w 8- m , Yea of Diagnosis Figue 1.3. ADS cases adjusted fo epoting delays by yea of diagnosis. c) peinatal cases, d) tansfusion cases. Souce: CDC (1991, p. 361).

12 12 fo vaious isk goups (CDC, 1991b). Fom Figue 1.3a it appeas that ADS cases in homosexual men may be slowing down o leveling off. Also, ADS incidence in VDUs stated late and continues to incease. n Figue 1.3b ADS incidence in people epoting heteosexual contact stated even late, but is now ising vey apidly. Pediatic ADS cases in Figue 1.3c also continue to go up ; the change in the last yea may not be significant and due to epoting changes. The tend in ecipients of tansfusions of blood o blood poducts in Figue 1.3d is clea; ADS cases in this goup ae now declining. This tend is not supising since almost no new HV infections occued as a esult of tansfusions afte blood sceening stated in ealy Although national time tends in ADS cases give some infomation, they often do not tell the whole stoy. Figue 1.4 shows that ADS cases in homosexual men in ew Yok City (YC), San Fancisco (SF), and Los Angeles (LA) appea to be leveling off afte 1987, but ADS cases in homosexual men in all othe egions continue to incease at least though 1989 (Kaon and Bekelman, 1991). Even this gaph does not eveal the complete pictue. Figue 1.5, shows ADS incidence tends in YC, LA, and SF ; ADS incidence appeas to have stated to decease in YC and has slowed down in LA and SF, but may not have leveled off yet in LA and SF. Kaon and Bekelman (1991) also show that the tends in YC, LA and SF ae pimaily due to changes in ADS incidence in white homosexual men with smalle changes in blacks and Hispanics. n many othe egions the ADS incidence is inceasing fo both whites and non-whites. The take-home lesson fom this analysis of ADS in homosexual men is that it is often necessay to look beyond national data into egions, cities, and acial/ethnic goups to discove tends. Data on HV incidence and pevalence is elatively spase and definitely less eliable than the ADS incidence data. Some data on HV incidence and pevalence is available fom blood samples saved fo othe puposes ; see Section on San Fancisco. But geneally, thee is no infomation on HV incidence o pevalence in pevious yeas. Even now it is vey difficult to get infomation on HV incidences o pevalences. Due to nonesponse bias and othe factos, data fom national suveys ae often consideed to be inaccuate. The widely-used value of 1 million HV-infected people in the U.S. is based on a vaiety of estimates such as the back calculation method descibed in Section 1.5 (CDC, 1991b). 1.4 HV and ADS in the Wold The ADS pandemic is now enteing its second decade. Although pojections of its futue ae uncetain, it is dea that the wold-wide epidemic is still in its ealy stages. n the 199s millions of people who ae aleady infected with HV will develop ADS and die. The Wold Health Oganization (WHO) estimates that about 5 to 6 million men and 3 to 4 million women in the wold have been infected with HV. So fa ove 1 million have pogessed to ADS and an equal numbe have HV-elated illness. Woldwide, WHO foecasts that the cumulative numbe of HV-infections will be 3 to 4 million by the yea 2. WHO pojects that thee will be 1 million adult ADS cases and deaths pe yea, with about 1/2 million in Afica and 1/4 million in Asia (WHO, 1991).

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15 15 The WHO has identified thee basic pattens of HV infection and ADS in the wold. Patten fo ADS occus in oth Ameica, Westen Euope, Austalia and ew Zealand. n these aeas, the pedominant modes of tansmission ae homosexual contact and needle-shaing by VDUs, but thee is some heteosexual and peinatal tansmission. The male-to-female atio is 1:1 to 15 :1. Thee is no tansmission by blood tansfusion since blood is now sceened. WHO suggests that the annual incidence of HV infections in these aeas may have peaked in the fist half of the 198s when thee was apid spead among homosexual men and VDUs. The annual HV incidences seem to have now deceased, pesumably due to the eduction of isky behavio in these high isk goups. Heteosexual tansmissions may incease, but WHO pedicts that ADS incidence will emain stable in the 199s (WHO, 1991). Patten fo ADS occus in much of Afica and in pats of the Caibbean. n these aeas the spead is pimaily by heteosexual contact, and the male-to-female atio is about 1 :1. Some HV tansmission occus duing homosexual contact and by blood tansfusion o medical uses of needles and syinges. Peinatal tansmission is significant since thee ae. many infected women. n sub-sahaa Afica the HV epidemic gew in the 197s and 198s. The optimistic pediction of WHO is that the annual HV incidence in this egion will peak in the mid-199s (WHO, 1991). Thei pediction fo Latin Ameica, whee the HV incidence has been less, is simila, although thee is the potential fo high infection ates in Bazil. Patten fo ADS occus in the emainde of the wold, including Asia, the Middle East and oth Afica, whee the HV vius has aived moe ecently. Thee is some HV and ADS in these aeas ; some HV infections wee acquied in othe aeas and some occued within these aeas. The ADS incidence in these egions has been low so fa, but thee is the potential fo inceased incidence. n Asia, HV tansmission began in the late 198s in a few counties, but HV infection has spead apidly since then. WHO pedicts that annual HV incidence in Asia will continue to ise until the ealy pat of the next centuy, and by the end of the 199s the annual incidence in Asia will exceed that in Afica (WHO,1991). Woldwide, heteosexual tansmission is esponsible fo two-thids of all HV infections so fa, and this faction is expected to incease. About one out of thee childen bon to an HV-infected mothe is HV-infected and dies of ADS, usually by age five. The othe two childen become ophans when thei mothe dies of ADS. So fa, thee have been about 1 million HV-infected infants bon, ove half of whom died of ADS. Thee ae now appoximately 2 million uninfected childen who ae o will become ophans. WHO pedicts that thee will be 1 to 15 million ADS ophans by the yea 2, most of them in sub-sahaa Afica (WHO, 1991). The social and economic consequences of ADS in developing counties will be stunning. Eldely people will not only be left without suppot as thei gown childen die, but also will become esponsible fo thei suviving gandchilden. The costs of caing fo people with HV elated diseases will ovewhelm the health cae esouces of many developing counties. Thee will be a decease in the poductivity of the wokfoce as young and middle-aged adults die fom ADS.

16 16 Meeting the challenge of ADS equies extaodinay effots thoughout the wold. Govenments need to make ADS pevention and contol a top pioity. They need to get all agencies and oganizations involved in peventing HV infections. They need to counte discimination against HV-infected pesons and to fight complacency and denial about ADS. The social conditions which put people at isk of HV infection, such as multiple sexual patnes, occu thoughout the wold. o acial and ethnic goups seem to have geate susceptibility o esistance to HV infection. The pospects duing the 199s fo developing a vaccine to pevent HV infection o pevent ADS ae uncetain. Thus the pimay means of pevention is though education of the people about the isks and possible peventative measues. 1.5 Thee Appoaches to Modeling HV/ADS The objective of this section is to pesent desciptions and cite the liteatue on the main appoaches to HV/ADS modeling. Suveys of mathematical models fo HV tansmission and ADS have been given by sham (1988) and Andeson (1988b). An annotated bibliogaphy of statistical methodology fo study of HV/ADS has been published by Fusao et al. (1989). Schwage, Castillo-Chavez and Hetheote (1989) have eviewed both statistical and - mathematical appoaches in HV/ADS modeling. The fist and most diect appoach to pedicting ADS cases in the futue is extapolation (Mogan and Cuan, 1986 ; Kaon et al., 1988, 1989). This method is to fit an assumed fom of the ADS incidence cuve to the ADS incidence data in ecent yeas and then to extend this cuve fo seveal yeas as a pediction of ADS cases in the futue. This method assumes that the_ cuent tends will continue fo at least a few yeas into the futue. Often sepaate cuves and extapolations ae done fo vaious isk goups. Advantages of extapolation ae its simplicity and ease of use. The extapolation method has been a good pedicto of ADS incidence fo a few yeas into the futue, but it is not good fo longe foecasts since it does not conside changes in the HV epidemic due to factos such as behavioal changes o satuation in the high isk goups. Anothe disadvantage is that it does not give any infomation on HV incidence o any undestanding of the HV tansmission mechanisms. Extapolation has been used on United Kingdom data by Healy and Tillett (1988) and on Euopean data by Downs et al. (1987). Extapolation has not woked as well in ecent yeas because thee have been clea changes in tends with a deceasing ate of gowth and the incidence has eached a plateau in some goups. The second appoach to modeling ADS incidence is usually called back calculation (Bookmeye and Gail, 1986 ) 1988 ; Gail and Bookmeye, 1988 ; Bookmeye and Damiano, 1989). The total numbe of cases of ADS at time t is the summation up to time t of the poduct of the HV incidence at time and the pobability of developing ADS within t isk - yeas afte infection. Thus if the HV incidence and the distibution of the ADS incubation peiod wee known up to time t, then the cumulative ADS cases would be calculated in a staight- fowad way using the convolution summation above. Back calculation is a deconvolution pocess ; it uses a given ADS incidence up to time t and an estimated distibution fo the ADS incubation peiod to estimate the HV incidence up to time t. This HV incidence up to time t and its

17 17 extapolation fo a few yeas ae then used to foecast the ADS incidence fo a few yeas. The distibution of the incubation peiod fo ADS can be estimated paametically o nonpaametically. The back calculation pocedue is often applied to sepaate isk goups. Back calculation has been used fo foecasting ADS incidence fo a few yeas and does have the advantage that it also yields estimates of HV incidence ; howeve, thee ae seveal disadvantages. t does not yield any infomation on the HV tansmission dynamics o estimates of paamete values. Estimated distibutions fo the ADS incubation peiod ae uncetain and the back calculation pocedue is vey sensitive to the distibution used (Bookmeye and Damiano, 1988 ; Hyman and Stanley, 1988). The instability of the back calculation pocess implies that the confidence intevals of the estimates of HV incidence and futue ADS incidence ae vey wide. The thid appoach to modeling ADS is to use HV tansmission dynamics models which include the pogession to ADS. These models often have the population divided into compatments consisting of those who ae susceptible, in each of the infectious stages, o in the ADS phase. n deteministic tansmission models, the movements between these compatments by becoming infected, pogessing to the next stage o ADS, migating o dying ae specified by systems of diffeence o diffeential equations. The models of this type used in this monogaph ae fomulated in Chaptes 3 and 7. Dynamic models and compute simulations ae expeimental tools fo compaing egions o isk goups, testing theoies, assessing quantitative conjectues and answeing questions. Fo example, in Chapte 6 we detemine the elative impotance of satuation in the high isk goup and behavio changes in changing the ADS incidence in homosexual men in San Fancisco. Modeling can also be used to theoetically evaluate, compae o optimize vaious detection, pevention, intevention o contol pogams. A detailed discussion of the puposes and limitations of dynamic epidemiological modeling is given in the next section. Some HV tansmission dynamics models ae stochastic with pobabilities of moving to the next stage at each time step. Mode et al. (1989) investigated the elationship between Monte Calo simulations of a stochastic HV/ADS model and solutions of a deteministic model using the expected values. Tan (1991) looks at geneal stochastic models fo the simulation of H tansmission and ADS. n the book (Tan, 1992) the stochastic analog of the deteministic model used in Chaptes 5 and 8 is analyzed and the esults ae compaed with those fo the deteministic model. A special issue (vol 17, o 2, Decembe 1991) of the jounal Mathematical Biosciences contains six papes on stochastic models fo HV/ADS. Both deteministic and stochastic dynamic epid emiological models have been used to analyze many diseases including measles, ubella, mumps, chicken pox, poliomyelitis, smallpox, gonohea, schistosomiasis, malaia, cholea and abies. Fo examples, see the books by Bailey (1975), Andeson (1982), Andeson and May (1982, 1991) and Hethcote and Yoke (1984). Many dynamic models have been analyzed mathematically ; see Hethcote (1989a) fo an intoduction to mathematical epidemiology o Hethcote et al. (1981) and Hethcote and Levin (1989) fo suveys of mathematical esults. n ecent yeas thee has been a temendous numbe of modeling papes which use dynamic models of HV tansmission and pogession to ADS. Suveys of these modeling papes ae cited

18 18 at the beginning of this section. Two authos who have contibuted to HV/ADS modeling ae Andeson and May (e.g., Andeson, 1988 ; Andeson and May, 1988 ; May, 1988) ; some of thei modeling esults ae incopoated into thei encyclopedic book (Andeson and May, 1991), which is built fom thei numeous epidemiological papes duing the past ten yeas. Thei models have coveed not only homosexual and heteosexual populations in the United States and United Kingdom, but also heteosexual populations in Afica. They have estimated epoduction numbes (contact numbes), doubling times and demogaphic consequences. Some of the multigoup models fo HV/ADS ae simila in fomulation to the multigoup models developed fo gonohea, which is anothe sexually tansmitted disease (STD) (Lajmanovich and Yoke, 1976 ; Hethcote and Yoke, 1984). Multigoup models ae appopiate fo STDs because people ae heteogeneous in thei behavios and contact ates with othes. The contacts can be homosexual, heteosexual o needle-shaing contacts. The models in this monogaph in Chaptes 3 and 7 ae multigoup models. Popotionate mixing in multigoup models was used in gonohea modeling by old (198) and Hethcote and Yoke (1984). They also used a convex combination of intenal contacts and popotionate mixing, which has been named pefeed mixing by Jacquez et al. mixing by Hyman and Stanley (1988, 1989). Jacquez et al. (1988) and biased (1988) used multigoup compatmental models fo HV with constant ecuitment into susceptible classes and vaiable infectivity in the infectious stages to analyze the effects of diffeent mixing pattens. This analysis has been extended to moe complicated mixing stuctues by Jacquez et al. (1989) and Koopman et al. (1989). Castillo--Chavez and coauthos (Blythe and Castillo-Chavez, 1989 ; Busenbeg and Castilio--Chavez, 1989 ; Castillo-Chavez et al., 1989) have focused on the theoetical analysis of divese mixing stuctues in HV/ADS models. Hyman and Stanley (1988, 1989) have consideed both continuous and discete HV/ADS models with heteogeneity and diffeent mixing stuctues. They have analyzed the spead fom high to low isk goups, the effects of vaiable infectivity and the instability of the back calculation pocedue. An innovative appoach to dynamic modeling of HV/ADS has been developed by Dietz and Hadele (Dietz, 1988 ; Dietz and Hadele, 1988 ; Hadele et al., 1988). Thei models explicitly conside the dynamics of pais of individuals and the duation of thei patneships. These pai fomation and dissolution models ae distinctly diffeent fom the contact ate mixing matix models used by many othe modeles. We have not attempted to give a complete suvey of HV/ADS modeling ; othe papes ae cited in late chaptes. Fo moe infomation the eade is efeed to the suvey aticles cited ealie o the collection of papes edited by Castillo---Chavez (1989). The pimay featues which distinguish the appoach in this monogaph ae the use of the deteministic dynamic simulation models as a famewok fo the intepetation of data, the effots to estimate paamete values fom data, the fitting of the model to HV and ADS incidence data and compaisons obtained by fitting the same model in fifteen diffeent subegions of the U.S.

19 Puposes and Limitations of Epidemiological Modeling Epidemiology is the study of the distibution and deteminants of disease pevalence in humans. Epidemiologists study both infectious diseases and chonic diseases such as cance and cadiovascula disease. One function of epidemiology is to descibe the distibution of the disease, i.e., find out who has bow much of what, whee and when. Anothe function is to identify the causes o isk factos fo diseases in ode to find out why eveyone doesn't have the same thing unifomly. A thid function of epidemiology is to build and test theoies. A fouth function is to plan, implement and evaluate detection, contol and pevention pogams. Epidemiological modeling can play an impotant ole in the last two functions above. Hee we focus on modeling infectious diseases in human populations and do not conside models fo chonic diseases such as cance. n most of this section "epidemiological modeling" efes to dynamic modeling whee the population is divided into compatments based on thei health status (e.g. susceptible, infectious, ecoveed) and the movements between compatments by becoming infected, pogessing, ecoveing o migating ae specified by diffeential o diffeence equations. Fist, fifteen puposes of epidemiological modeling ae descibed and then thee limitations ae consideed; they ae all summaized in Table 1.4. One cannot infe fom the numbes of puposes and limitations that the advantages ovewhelm the limitations since the fist limitation is vey boad. A pimay eason fo infectious disease modeling is that it leads to clea statements of the assumptions about the biological and social mechanisms which influence disease spead. The model fomulation pocess is valuable to epidemiologists and modeles because it foces them to be pecise about the elevant aspects of tansmission, infectivity, ecovey and enewal of susceptibles. Epidemiological modeles need to fomulate models clealy and pecisely using paametes which have well-undestood epidemiological intepetations such as a contact ate o an aveage duation of infection (Hethcote, 1976, 1989a). Complete statements of assumptions ae cucial so that the easonableness of the model can be evaluated in the pocess of assessing the conclusions. An advantage of mathematical modeling of infectious diseases is the economy, claity and pecision of a mathematical fomulation. A model using diffeence, diffeential, integal o functional diffeential equations is not ambiguous o vague. Of couse, the paametes must be defined pecisely and each tem in the equations must be explained in tems of mechanisms, but the esulting model is a definitive statement of the basic pinciples involved. Once the mathematical fomulation is complete, thee ae many mathematical techniques available fo detemining the theshold, equilibium, peiodic solutions, and thei local and global stability. Thus the full powe of mathematics is available fo analysis of the equations. Moeove, infomation about the model can also be obtained by numeical simulation on digital computes of the equations in the model. The mathematical analyses and compute simulations can identify impotant combinations of paametes and essential aspects o vaiables in the model. n ode to choose and use epidemiological modeling effectively on specific diseases, one must undestand the behavio of the available fomulations and the implications of choosing a paticula fomulation.

20 2 Table 1.4. Puposes and Limitations of Epidemiological Modeling Fifteen Puposes of Epidemiological Modeling 1. The model fomulation pocess claifies assumptions, vaiables and paametes. 2. The behavio of pecise mathematical models can be analyzed using mathematical methods and compute simulations. 3. Modeling allows exploations of the effect of diffeent assumptions and fomulations. 4. Modeling povides concepts such as a theshold, epoduction numbe, etc. 5. Modeling is an expeimental tool fo testing theoies and assessing quantitative conjectues. 8. Models with appopiate complexity can be constucted to answe specific questions. 7. Modeling can be used to estimate key paametes by fitting data. 8. Models povide stuctues fo oganizing, coalescing and coss-checking divese pieces of infomation. 9. Models can be used in compaing diseases of diffeent types o at diffeent times o in diffeent populations. 1. Models can be used to theoetically evaluate, compae o optimize vaious detection, pevention, theapy and contol pogams. 11. Models can be used to assess the sensitivity of esults to changes in paamete values. 12. Modeling can suggest cucial data which needs to be collected. 13. Modeling can contibute to the design and analysis of epidemiological suveys. 14. Models can be used to identify tends, make geneal foecasts, o estimate the uncetainty in foecasts. 15. The validity and obustness of modeling esults can be assessed by using anges of paamete values in many diffeent models. Thee Limitations of Epidemiological Modeling 1. An epidemiological model is not eality ; it is an exteme simplification of eality. 2. Deteministic models do not eflect the ole of chance in disease spead and do not povide confidence intevals on esults. 3. Stochastic models incopoate chance, but ae usually hade to analyze than the coesponding deteministic model.

21 21 Thus mathematical epidemiology povides a foundation fo the applications (Hetheote et al., 1981 ; Hethcote and Levin, 1989). Modeles ae able to exploe and examine the effects of diffeent assumptions and fomulations. Fo example, they can compae an odinay diffeential equations model, which coesponds to a negative exponential distibution fo the infectious peiod, with a delay-diffeential equations model, which assumes that eveyone has the same fixed, constant-length infectious peiod. They can examine the impact of assuming homogeneous mixing instead of heteogeneous mixing. They can study the influence of diffeent mixing pattens between goups on the spead of a disease in a heteogeneous population (see Jacquez et al., 1988 ; Castillo-Chavez et al., 1989). They can decide if models with and without an exposed class fo people in the latent peiod behave diffeently. The esults of exploing diffeent fomulations may povide insights fo epidemiologists and ae cetainly useful to modeles who ae choosing models fo specific diseases. Fo example, if the essential behavios (thesholds and asymptotic behavios) ae the same fo a fomulation using odinay diffeential equations and a fomulation using delay diffeential equations, then the modele would pobably choose to use the odinay diffeential equations model since it is simple (Hethcote and Tudo, 198). Epidemiologists need to undestand the concept of a theshold which detemines whethe the disease pesists o dies out. They need to know that the epoduction numbe is a combination of paametes which gives the numbe of seconday cases "epoduced" by a typical infective duing the infectious peiod in a population whee eveyone is susceptible. They need to be awae that this epoductive numbe is the same as the contact numbe, which is the aveage numbe of adequate contacts of an infective peson duing the infectious peiod (Hethcote and Van Ak, 1987). They need to ealize that the aveage eplacement numbe is one at an endemic equilibium because, in an aveage sense, each infective eplaces itself by one new infective in an equilibium situation. Pobably the most valuable contibutions of mathematical modeling to epidemiologists ae concepts like those above. Mathematical models seve as a famewok to explain causes, elationships and ideas. Mathematical models ae vey useful in obtaining conclusions that have an easily undestood intepetation. Helping epidemiologists intenalize conceptual and intuitive undestandings of disease pocesses, mechanisms and modeling esults is an essential aspect of the wok of mathematical modeles. Epidemiological modeling is an impotant pat of the epidemiologist's function to build and test theoies. Mathematical and compute simulation models ae the fundamental expeimental tools in epidemiology. Expeiments with infectious diseases in actual populations ae often unethical o vey expensive o impactical. Thus modeling is essential fo exploatoy wok. The only data usually available ae fom natually occuing epidemics o fom the natual incidence of endemic diseases ; unfotunately, even these data ae not complete since many cases ae not epoted. Since epeatable expeiments and accuate data ae usually not available in epidemiology, mathematical and compute simulation models must be used to pefom needed theoetical expeiments with diffeent paamete values and diffeent data sets. t is easy in a compute simulation to find out what happens if one o seveal paametes ae changed.

22 22 Anothe advantage of epidemiological modeling is the availability of models to assess quantitative conjectues. Fo example, models can be used to check the claim that a two-dose vaccination pogam fo measles would lead to hed immunity wheeas a one-dose pogam would not (Hethcote, 1983). With a model one could see if the ADS vius would eventually disappea in a population of homosexual men if half of them became celibate o pacticed safe sex. Although it is undestood how an infective can tansmit the infection by contacts with othes and how a disease speads though a chain of infections, it is not easy to compehend the lage scale dynamics of disease spead in a community o to evaluate quantitative theoies without the fomal stuctue of a mathematical model. An epidemiological model connects the micoscopic desciption o ole of an infectious individual to the macoscopic spead of the infection in a community. Mathematical models ae a fomal, quantitative way of building and testing theoies. When fomulating a model fo a paticula disease, it is necessay to decide which factos to include and which to omit. This choice will often depend on the paticula question that is to be answeed. Simple models have the advantage that thee ae only a few paametes, but have the disadvantage of possibly being naive and unealistic. Complex models may be moe ealistic, but they may contain many paamete values fo which estimates cannot be obtained. The at of epidemiological modeling is to make suitable choices in the model fomulation so that it is as simple as possible and yet is adequate fo the question being consideed. t is impotant to ecognize both the capabilities and limitations of epidemiological modeling. Many impotant questions cannot be answeed using a given class of models. The most difficult poblem fo a modele is to find the ight combination of available data, an inteesting question and a mathematical model which can lead to the answe. Fo ADS one can ask whethe needle-shaing is moe impotant fo tansmission than sexual activity. How impotant ae homosexual intavenous dug uses in linking the epidemics in the homosexual/bisexual men and intavenous dug uses? s thee a "coe" goup fo ADS as thee was fo gonohea and is it the same coe goup? How impotant ae bisexuals and postitutes in the tansmission pocess? How apidly does HV infection spead fom high isk goups to low isk goups? Why do ADS cases in many populations incease initially as a cubic function of time? Hyman and Stanley (1988) suggest that this cubic gowth could be explained by the spead of infection fom high contact-ate goups down though lowe contact-ate goups. s sustained tansmission likely in the heteosexual population with fequent patne changes? What effect will theapy with zidovudine (AZT) have on the aveage suvival time with ADS and on the tansmission? Thee ae many elevant questions to ask about ADS. By fitting the incidence pedicted by a model to actual incidence data (numbe of new cases pe month), it is possible to estimate key paametes such as contact numbes o aveage duations. Contact numbes (epoduction numbes) ae estimated fo vaious diectlytansmitted diseases in Andeson and May (1982, 1991). Hethcote and Van Ak- (1987) pesent methods fo estimating contact numbes fo the goups in a multigoup model fo disease tansmission. n ADS modeling, fitting the pedicted HV pevalence and ADS incidence to data on both can lead to estimates of the aveage numbe of new patnes pe month and the pobability

23 23 of HV tansmission pe new patne (see Chapte 6). Since these paametes have aleady been estimated fom the liteatue, the modeling povides a check on these a pioi estimates. Thus the model is a way of coalescing paamete estimates and othe pieces of data to see if they fit into a consistent famewok. The model can seve as the conceptual and quantitative unifie of all the available infomation. Compaisons can lead to a bette undestanding of the pocess of disease spead. t may be enlightening to compae the same disease at diffeent times, the same disease in diffeent populations o diffeent diseases in the same population. One method fo compaing diseases is to estimate paamete values fo the diseases and then compae the paamete values. Fo example, in Andeson and May (1982, 1991) and Hethcote (1989a) estimates of contact numbes (also called epoduction numbes) ae given fo vaious diseases such as measles, whooping cough, chicken pox, diphtheia, mumps, ubella, poliomyelitis and smallpox. These numbes lead to factions of the population which must be immune in ode to achieve hed immunity fo these diseases. Childhood diseases ae compaed in ual and uban aeas in Andeson (1982). The level of HV infection in homosexual men in San Fancisco is compaed at vaious times and the changes in behavio in this population ae given in Chapte 6. A vey impotant eason fo epidemiological modeling is the value of models fo theoetical evaluations and compaisons of detection, pevention, theapy and contol pogams. Epidemiologists and politicians need to undestand the effects of diffeent policy decisions. Qualitative pedictions of infectious disease models ae always subject to some uncetainty since the models ae idealized and the paamete values can only be estimated. Howeve, quantifications of the elative meits of seveal contol methods o of contol vesus no contol ae often obust in the sense that the same conclusions hold fo a boad ange of paamete values and a vaiety of models. Contol stategies fo gonohea such as sceening, esceening, infectee tacing, infecto tacing and potential vaccines fo both women and men wee compaed in Hethcote and Yoke (1984). Vaious vaccination stategies fo ubella and measles have been compaed using modeling (Hethcote, 1983, 1989b ; Andeson and May, 1982). Sometimes an optimization method can be used on some aspect of an epidemiological model. Hethcote and Waltman (1973) used dynamic pogamming to find an optimal vaccination stategy to contol an epidemic at least cost. Longini et al.(1978) detemined the best age o social goups to vaccinate fo Hong Kong and Asian influenza when thee was a limited amount vaccine available. Hethcote (1988) found theoetical optimal ages of vaccinations fo measles in thee geogaphic locations. See Wickwie (1977) fo a suvey of modeling papes on the contol of infectious diseases. Pedictions o quantitative conclusions of a model ae said to be sensitive to a paamete if a small change in the paamete causes a lage change in the outcome. A model is insensitive to a paamete if the outcomes ae the same fo a wide ange of values of that paamete. An impotant eason fo epidemiological modeling is the detemination of the sensitive and insensitive paametes since this infomation povides insight into the epidemiological pocesses. Once the sensitive paametes ae identified, it may be possible to make special effots to gathe data to obtain bette estimates of these paametes. Thus the modeling pocess can help identify cucial of

24 24 data that should be collected. f the data and infomation ae insufficient to estimate paametes in a model o to piece eveything togethe, then the model fomulation pocess can identify the missing infomation which must be gatheed. Modeling can enhance an epidemiological suvey by poviding a famewok fo the analysis of the suvey esults. n the design phase modeling can help identify impotant questions that need to be asked in ode to have adequate data to obtain esults. The statistical aspects of a suvey design ae now analyzed befoe the suvey is conducted. Similaly, those design aspects elevant to paamete estimation in models should be analyzed befoe the suvey is conducted to see if all significant data ae being collected. Anothe pupose of epidemiological modeling is to make foecasts about the futue incidence of a disease. Although people often think of pediction as the fist o only pupose of epidemiological modeling, the puposes given above ae moe impotant. Accuate foecasts ae usually not possible because of the idealization in the models and the uncetainty in the paamete values. Howeve, possible foecasts unde vaious scenaios can sometimes be given o tends can be identified. One pupose of epidemiological modeling is to educe the uncetainty in futue incidence pojections. Fo example, impoved estimates of the numbe of HV infections and ADS cases in futue yeas ae needed in ode to povide appopiate public health pogams and adequate medical cae fo patients with HV-elated medical conditions and ADS. Although simple extapolations fo 2 o 3 yeas of the time tends fo numbes with ADS o in a final stage of HV infection ae often adequate, bette estimates ae obtained with models which conside population changes due to the disease tansmission pocess. The validation of models and modeling esults is difficult since thee ae aely enough good data to adequately test o compae diffeent models with data (Andeson and May, 1991). Howeve, when the same concept o esult is obtained fo a vaiety of models and by seveal modeles, then the confidence in the validity of the esult is inceased. Fo example, thee diffeent papes using diffeent appoaches to compaing two ubella vaccination stategies agee on the citeia detemining situations when each stategy is best (Hethcote, 1989b). Similaly, when a quantitative esult holds fo a wide ange of paamete values, then this esult has some obustness. Fo example, if the same elative values of a vaiety of contol pocedues ae obtained fo wide anges of paamete values, then the elative values may be obust even though the absolute values may vay (see Hethcote and Yoke, 1984). Afte discussing the puposes and advantages of epidemiological modeling, it is impeative to also discuss the limitations. Epidemiologists and policy makes need to be awae of both the stengths and weaknesses of the epidemiological modeling appoach. The fist and most obvious limitation is that all epidemiological models ae simplifications of eality. Fo example, it is often assumed that the population is unifom and homogeneously mixing ; this is always a simplification, but the deviation of eality fom this simplification vaies with the disease and the cicumstances. This deviation fom eality is aely testable o measuable ; howeve, it can sometimes be estimated intuitively fom an undestanding of the disease epidemiology. As descibed in the intoduction, pat of the eason fo the diffeences between the model s behavio and the actual spead of the disease is that tansmission is based on

25 25 the inteactions of human beings, and homo sapiens have highly vaiable behavio and inteactions. People do not behave in easonably pedictable ways like molecules o cells o paticles. Because tansmission models ae simplifications with usually unknown elationships to actual disease spead, one can neve be completely cetain about the validity of findings obtained fom modeling such as conceptual esults, expeimental esults, answes to questions, compaisons, sensitivity esults and foecasts. Even when models ae made moe complicated in ode to bette appoximate actual disease tansmission, they ae still abstactions. Fo example, multigoup models based on isk goups o sexual activity levels o age stuctue still assume that the distinctions between goups ae clea and shap and that the mixing between goups follows some assumed patten. The modele must always execise judgment in deciding which factos ae elevant and which ae not when analyzing a specific disease o question. The modele needs to know that an SR model is usually adequate since the behavio is essentially the same fo the SER model whee E is the latent peiod class. The notation SER fo a model means that susceptibles (S) move to an exposed class (E) (when they ae in a latent peiod ; i.e., they ae infected, but not yet infectious), then move to an infectious class (), and finally to a emoved class (R) when they ecove and have pemanent immunity. The modele must ealize that assuming that the infectivity is constant duing the infectious peiod is easonable fo diseases like measles with a shot infectious peiod (about 8 days), but is not easonable fo a disease like the human immunodeficiency vius (HV) whee the aveage time fom HV infection to ADS is about 1 yeas. To incopoate age stuctue, the modele must decide whethe to use a model with time and age as continuous vaiables o to use discete compatments coesponding to age backets ; the fome equies the estimation of bith, death and aging ate coefficients as functions of time while the latte equies the estimation of bith, death and tansfe ate constants. The modele needs to decide whethe the disease is epidemic in a shot time peiod so that vital dynamics (biths and deaths) can be excluded o the disease is endemic ove a long time peiod (many yeas) so that vital dynamics must be included. The modele must be awae that as the complexity of a model is inceased so that it bette appoximates eality, the numbe of paametes inceases so that it becomes inceasingly difficult to estimate values of all of these paametes. Thus the modele must (consciously o unconsciously) make many decisions egading the elevant aspects in choosing a model fo a specific disease o question. When descibing the puposes fo epidemiological modeling, we have been thinking pimaily of deteministic models, but the puposes also apply to stochastic epidemiological models. Deteministic models ae those which use diffeence, diffeential, integal o functional diffeential equations to descibe the changes in time of the sizes of the epidemiological classes. Given the stating conditions fo a well-posed deteministic model, the solutions as a function of time ae unique. n stochastic models, thee ae pobabilities at each time step of moving fom one epidemiological class to anothe. When these models ae simulated with the pobabilities calculated using andom numbe geneatos, the outcomes of diffeent uns ae diffeent so that this appoach is often called Monte Calo simulation ; conclusions ae obtained by aveaging the

26 26 esults of many compute simulations. Simple deteministic models fo epidemics have a pecise theshold which detemines whethe an epidemic will o will not occu. n contast, stochastic models fo epidemics yield quantities such as the pobability that an epidemic will occu and the mean time to extinction of the disease. Thus the appoach, concepts and appopiate questions ae quite diffeent fo stochastic models (Batlett, 196 ; Ludwig, 1974 ; Bailey, 1975 ; asell, 1985 ; Becke, 1989). Both deteministic and stochastic epidemiological models have othe limitations besides being only appoximations of eality. Deteministic models do not eflect the ole of chance in disease spead. Sometimes paamete values in deteministic models ae set equal to the mean of obseved values and the infomation on the vaiance is ignoed. A set of initial conditions leads to exactly one solution in a deteministic model ; thus no infomation is available on the eliability of o the confidence in the esults. When quantities such as the contact numbe, eplacement numbe, aveage infectious peiod, etc. ae estimated by fitting output to data, confidence intevals on these estimates ae not obtained. Confidence intevals also can not be found fo compaisons and foecasts. Some undestanding of the dependence on paamete values is obtained though a sensitivity analysis whee the effects of changes in paamete values on esults ae found. f the vaiance of the obseved value of a paamete is high and the esults ae sensitive to that paamete, then the confidence in the esults would be low. Howeve, the lack of pecise confidence intevals implies that the eliability of the esults is uncetain. Stochastic epidemiological models incopoate chance, but it is usually hade to get analytic esults fo these models. Moeove, computational esults ae also hade since Monte Calo simulations equie many compute uns (25 to 1 o moe) in ode to detect pattens and get quantitative esults. Even fo stochastic epidemiological models whee paametes ae estimated by fitting the mean of the simulations to data, it may not be possible to find confidence intevals on these paamete estimates. 1.7 Expectations of Modeling HV Tansmission Dynamics and ADS n June 1988 the Pesidential Commission on the HV Epidemic ecommended that "the eseach community should be actively engaged in developing innovative models that bette descibe and explain the tansmission of HV within the population." On July 25-29, 1988 the ADS Modeling and Epidemiology Wokshop was held in Leesbug, Viginia. inety scientists in the fields of mathematics, statistics, epidemiology, data management, biology, behavioal sciences and social sciences paticipated. "The pupose of the Wokshop was to examine the cuent status of ADS and HV modeling, to assess the potential benefits fom mathematical and statistical analysis, and to make ecommendations fo a pogam of eseach." The conclusions fom this meeting given below ae fom the epot of the Office of Science and Technology Policy, Executive Office of the Pesident (OSTP, 1988). "The maio findings of the wokshov ae 1. "Mathematical modeling and statistical analysis must be bought to bea fully on the ADS epidemic. The success of

27 27 mnathematical techniques in othe fields of application and the pesence 7f modem computational capability ague stongly fo this added ippoach to ADS and HV analysis and pojection. The modeling of the gonohea epidemic ealie this decade, cost-benefit analysis of ubella in England, and pedictions of the spead of Hong Kong flu have exhibited that modeling and statistical pojections of epidemics ^an make key contibutions." 2. "Mathematical and statistical methods that have been used fo sexually tansmitted diseases and othe epidemics aeonlypatially licable to ADS and HV. The basic stuctue will be simila but empiical paametes and functions that chaacteize the new disease ae equied : inta- and inte-isk goup mixing, patne elationships and duation, and biologi cal tansmissibility and infectivity duing the peiod between HV and ADS. Pogess has been made in fomulating these paametes, but moe data ae equied and a divese set fo modeling techniques and models ae appopiate." 3. ".Given a supply of biological and behavioal data of good guality. modeling and advanced statistical analysis an povide bette estimates of ADS and HV_ pevalence than ae now available and eventually of HV incidence. Cost-benefit assessments of pevention and intevention stategies and estimates of health cae costs can be made. All of these depend on the accuacy and the coveage of the data. The mathematical studies can, in tun, point towad data needs and to insufficiencies in data quality." 4. "ADS incidence data allow shot-tem pojections (3 to 5 yeas). HV pevalence is difficult to estimate. Statistical extapolations have shown that ADS appeas to be inceasing as the thid powe of the time. Dynamic model analysis has been used to exhibit the subgoup behavio accounting fa this dependence. HV pevalence has been obtained by back-calculation fo special isk goups and the geneal population. Howeve, eo anges ae lage fo this kind of calculation. HV incidence can only be measued in special goups." 5. "The quality of ADS case dataisgood. Some selected isk goup data ae of special value. mpovements in the aualitv of HV. pevalence data eauiethatthe cuently planned seopevalence guveys, including those of the geneal population and the special family of suveys, poceed if they ae found to be feasible." 6. "Pogess towad long-ange pedictions fo ADS and HV pevalence eauies new biological and behavioal data and collaboative effots between modeles. statisticians. epidemiologists, and behavioal, social and data management scientists." 7. "Collaboation has inceased in the past yea. and this has begun to incease the numbe of scientific esults. n the past yea thee have been a numbe of meetings in which mathematical scientists, modeles, and statisticians have woked togethe with epidemiologists. Collaboative wok has been funded by the ational nstitute fo Dug Abuse (DA) on the advanced statistical analysis of ADS among V dug uses at seveal univesities." 8. "Thee is a good supply of high-duality, motivated mathematical eseaches, applied mathematicians, and statisticians to conduct the modeling effot. Thee ae cuently about 2 people involved in all of the mathematics wok, seveal of whom wok in PHS. The wok to be done will equie a total of 6 people in the next seveal yeas."

28 28 9. "Howeve. funding fo the modeling effot is not adequate. Few of those now doing modeling ae suppoted. Funding equied to suppot those now woking on modeling and the additional scientists equied to cay out the needed eseach would amount to $6 million by 1991." 1. "Access to ADS and HV data must be impoved.. As data ae collected, they must be made available to eseaches. A compehensive diectoy of data souces is equied to bing modeles, epidemiologists, and behavioal and social scientists to the data. Lage-population, public-use databases must continue to be made moe available in tems of the kinds of data eleased. Smallpopulation datasets need to be made easie to access. A eseach envionment that facilitates shaing of data needs to be pomoted. Howeve, all of these measues must be taken in a way that peseves confidentiality and maintains individual anonymity." The ten conclusions of this 1988 Wokshop ae quite inteesting. Despite the fact that the past contibutions of epidemiological modeling cited in conclusion 1 ae pimaily conceptual, conclusion 3 emphasizes the use of mathematical modeling fo foecasting. Although we ae enthusiastic about the value of dynamic tansmission modeling fo conceptual undestanding, we ae moe cautious about its value fo foecasting futue ADS incidence and HV pevalence and incidence. t may be difficult to do bette foecasting than done by the extapolation and back calculation techniques mentioned in conclusion 4 and descibed in moe detail in the pevious section. Dynamic tansmission modeling can exploe possible futue incidences unde diffeent scenaios, but. the pedictions ae likely to be simila to those obtained by shot-tem foecasting using simple diect methods such as extapolation. We ae disappointed that the Wokshop findings put so much emphasis on the use of mathematical modeling to give foecasts. Most of the othe findings seem valid since they addess such issues as data and pesonnel availability, collaboation and funding. Two moe ecent publications have concluded that mathematical models fo HV tansmission and ADS have geat potential fo foecasting, but that the data ae not cuently available fo paamete estimation in these models. n ADS: Sexual Behavio and ntavenous Dug Use (Tune et al., 1989) published by the- ational Academy Pess, it is stated on p. 74 that "...pomising mathematical models of the dynamics of the spead of HV infections equie data on a wide ange of sexual behavios ; these data cuently ae not available.... Thee ae cuently no eliable data on sexual contacts fo the national population ; thee ae also no such data fo goups with elevated isks of tansmitting o contacting HV infections (e.g., men who have sex with men, V dug uses, heteosexuals with many sexual patnes). ndeed, thee is no eliable infomation on the size of the nonmonagamous heteosexual population. The lack of such data makes pedictions about the futue spead of ADS extemely uncetain." n ADS Foecasting, published by the Geneal Accounting Office (1989), fou types of foecasting models ae pesented and analyzed. n thei conclusions, they state on pp that

29 mico-simulation models wee judged most compehensive in the sense that they measued six components of the epidemic. These models explicitly conside HV tansmission and the development of ADS and show how these contibute to past, cuent, and foecasted levels of the epidemic. n addition, pojected futue occuences fo individual components can be adjusted by the foecaste, allowing assessments of the effect of potential changes. But, unfotunately, the empiical basis of these models was extemely weak, due to lack of available infomation on specific components of the ADS epidemic. f knowled about ADS tansmission wee impoved, the most compehensive mico-simulation) models would be deemed supeio to the othes in t e sense that mico-simulation models make explicit the foces that contibute to the foecast." We agee with the two quotations above ; cuently thee ae inadequate data to estimate the paametes in dynamic tansmission simulation models well enough so that they will povide accuate foecasting. As bette data become available, the foecasts of futue HV and ADS incidences will impove, but it will be vey difficult to estimate the accuacy of these foecasts. The social mixing, sexual inteactions and needle-shaing behavio in ou society ae vey complicated so that some simplifications ae necessay in a model. The foecasts depend on the model used and it is difficult to decide how to choose a model with the appopiate amount of complexity. One guiding philosophy is to include the minimum stuctue to give easonable fits to the data. Once the model is specified, it is then necessay to estimate the values of the paametes in the model. Even if the complete cuent and past sexual and needle-shaing behavio wee known fo each peson in the United States, the poblem would still emain of deciding on a easonable model and then deciding how to summaize, goup, categoize and condense the data in ode to estimate paamete values in the model. Tansmission dynamics models can be used to help identify tends, make geneal foecasts o estimate the uncetainty in the foecasts, but they ae not moe likely than othe appoaches to give accuate ADS foecasts in the U.S. o elsewhee in the wold. Both of the quotations above have focused on foecasting as the eason fo ADS modeling. We believe that the othe benefits of epidemiological modeling descibed in the pevious section ae vey impotant and ae the aspects whee mathematical modeling and compute simulations of HV/ADS can make the biggest contibutions. n this monogaph we emphasize conceptual undestanding by the econstuction of HV/ADS epidemics instead of foecasting futue ADS cases.