Policies for curbing the HIV epidemic in the United States: Implications of a simulation model

So&-Econ. Ptam. Sci. Vol. 27, No. Printed in Great Britain

3, pp. 153469,

1993

003%0121/93

Xi.00 + 0.00

Pergamon Press Ltd

Policies for Curbing the HIV Epidemic in the United States: Implications of a Simulation Model ALLAN M. SALZBERG’

and DUNCAN

MACRAE JR*

‘Veterans Affairs Medical Center, Bath, NY 14810 and University of Rochester School of Medicine and Dentistry, Rochester, NY 14627, and *University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, U.S.A. Abstract-An 8~om~~rnent simulation model, with monthly steps and with different infectivities for 4 pre-AIDS phases of HIV, simulates the U.S. epidemic from 19762005. The use of AZT or other antiretroviral drugs is introduced under the assumptions that it halves the rate of conversion to AIDS, and that, with behavior change, it reduces transmission coefficients by 2/3. Two policy scenarios are projected. They assume that 66 and 100% of HIV-infected persons whose CD4 count is less than 500 are identified by voluntary testing and are treated with AZT by the end of 1993-which might be approached with full sibsidy of nkiicai treatment and protection against discrimination. Without such pbiicies, we oroiect over 100,000 new AIDS cases and 110.000 HIV infections in 2005, and a $14 billion annual cost. &la&mum use ot’ AZT could reduce these n&nbers to 50,000, 45,00@and $10 billion, respectively. Such policies might draw support from the infected, the uninfected, economizing oilicials and pharmaceutical companies.

The choice of policies for reducing the effects of HIV/AIDS in the United States requires estimation of policy effects, reconciling political conflicts, and coping with budgetary shortages. We here examine policies of subsidized AZT treatment (a concise term for all antiretroviral therapy) for HIV-positive persons, together with this and other incentives for voluntary testing of persons at risk, using a model of the epidemic. We show projected effects of such policies in terms that are relevant for various interested groups, and suggest conditions under which these groups might join in support of such a policy. Possible budgetary savings are also examined. Substantial efforts have been made to aid policy discussions by modeling the epidemic and projecting policy effects [7]. Effects of extensive use of AZT do not, however, seem to have been sufficiently explored. The paper is divided into three sections. The first presents a sim~ation mode1 of HIV that describes and projects the course of the epidemic in some detail. This basic model has been found to be in quantitative agreement with both reported AIDS cases and Centers for Disease Control (CDC) AIDS projections through 1994. The model’s estimates of HIV prevalence are consistent with CDC estimates through 1991 (l-l.5 million in 1991) [25,26,29]. The second section shows and analyzes computed effects of identifying and treating substantial fractions of the HIV-infected population, presuming that this can be achieved by voluntary testing and fully subsidized AZT treatment. The third section summarizes possible sources of political support for such a policy by various interested groups.

THE MODEL We employ a multi-compartmental Markov model with time-dependent infectivity, which is an extension of a recently published model [29]. The structure of the model and the parameters proposed are the result of numerous trials constrained by three criteria: (a) empirical fit to CDC AIDS data and HIV estimates; (b) general conformity of parameters and outputs to independent estimates in the literature, where available; and (c) simplicity where omission of a feature does not 153

ALLAN M. SALZBERGand DUNCAN MACRAE JR

154

appear to lead to quantitatively significant errors. We consider the following eight closed, mutually exclusive, interacting populations that contract and transmit HIV:

(1) high-risk homosexuals;

(2) moderate-risk homosexuals; (3) (4) (5) (6) (7) (81

high-risk intravenous drug users [IVDU]; moderate-risk IVDU; high-risk heterosexual females; high-risk heterosexual males; moderate-risk heterosexual females; and moderate-risk heterosexual males.

We assume that HIV spreads in a “top down” manner from the high-risk to the lower-risk groups; and that transmissions from the lower-risk to the higher-risk groups do not significantly modify the results. By the time HIV has spread sufficiently in the lower-risk groups to alter the epidemic in the higher risk groups, the disease has run its course in the higher risk groups [29]. This issue will be further addressed in the section on model verification. A schematic diagram of assumed compartments and paths of transmission is given in Fig. 1. We do not consider mixing either between IVDUs and gays or between high- and moderate-risk heterosexuals for reasons to be discussed below. We further assume, as do others [12,21], that transmission ceases once clinical AIDS develops. Clearly, this is an approximation as some transmission occurs after AIDS develops. However, this view is reasonable as AIDS is a clinical illness superimposed on a CD4 count of less

0.0084

-7

o~~O~eratc-risk

0.00060 0.0010

High-risk heterosexual males

w

0.016

0.020

* e

) 0.0010

High-risk heterosexual females

0.0053

6 I

I

5

I

Moderate-risk heterosexual males

Moderate-risk heterosexual females

8 0.0010

A I

Hlg”-rWK

IVDU

1

1

A

0.0053 I

0.0075

1. Compartments

roughly proportional

0.0053

lwoaerate-r,slc IVDU

4

t7’ 0.0806 0.059

Fig.

7

A

I

0.0053

-

-

L!Y 0.030 and assumed paths of transmission of HIV infection. Note: area of blocks is to estimated group size. Transmission also occurs within the gay and IVDU compartments, as shown by curved arrows.

Policies for curbing

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than 200; thus, persons with AIDS are not only ill but also have knowledge of their disease state. Further, the number of persons living with clinically defined AIDS is markedly less than that of those with CD4 < 200 but not as yet with a clinically defining AIDS syndrome. We divide the incubation period into four distinct phases [29], having exponentially distributed survivals with means, respectively, of 6 weeks, 2 years, 10 years and 1.5 years. We do this in order both to be consistent with an annual 9-10% exponential drop in CD4 count and to fit the latest data concerning HIV incubation times. This gives a 10 year median time from infection to clinical AIDS, which agrees with the 9.7 year estimate of Bacchetti and Moss [4] and the 11 year estimate of Lemp et al. [20]. Few carriers convert to clinical AIDS in the first 2 years following infection. Figure 2 shows that the distribution of HIV incubation times generated by our model times lies between Lemp’s and Bacchetti’s estimates. Our model projects a leveling of the yearly hazard of developing AIDS for later years in the incubation period, as reported by Bacchetti and Moss, and Taylor er al. [4, 341. The observed leveling hazard argues against the use of a Weibull distribution of incubation times, since the Weibull gives an increasing yearly hazard of developing AIDS. Biologically, the first (6 week) phase corresponds to initial infection with HIV. In this phase, there are high levels of viremia [l 1, 141, which precipitously fall once antibody is produced. This phase has been associated with high infectivity in epidemiological studies of postnatal transmission from mother to child [36]. The second (2 year) phase is characterized by a CD4 count over 500, relatively low infectivity and is usually asymptomatic. The third phase, which we define as the reference (or nominal) phase of infectivity, terminates when the CD4 count drops below 200. The fourth and final infective pre-AIDS phase is characterized by increased antigemenia, viremia and infectivity, which occur once the immune system has been depleted [l&29]. We thus introduce the relative infectivity of each phase of the incubation period compared with a value of unity assigned to phase 3 (the reference phase). We postulate that the relative infectivity takes on the values of 9, 0.2, 1, and 3 for the four phases. Figure 3 presents a schematic diagram of the model’s assumptions about the incubation period. The basic difference equations for the epidemic, in the absence of learning or AZT, are defined in the Appendix. One equation describes the appearance of newly infected cases each month. For each assumed transmission path between compartments (Fig. l), the reference (phase 3) number of infections per month per carrier for gays and IVDUs is estimated by curve fitting their reported AIDS caseload, while for non-IVDU heterosexuals Monte Carlo simulations are used: the transmission coefficients for phases 1, 2, and 4 are obtained by multiplying those of phase 3 by the above relative infectivities. The number of contacts per month is assumed to be proportional both to the number of infectives in a given phase of the incubation period and to the probability of contacting a susceptible in the receiving compartment. The transmission coefficients (Table 1) are then multiplied by the number of contacts to estimate the number of infections through that path, thus combining effects of biological infectivity with the behavior of the carrier. The new infections produced by infectives in the four incubation phases are then summed. An additional 0.7 -

Model

0.6 0.5

-

E d

0.4

-

g

0.3

-

0.2

-

0.1

-

+

Bacchetti

x

Lemp

CA

E;

0

+ x /

t

,/:

,

I

I

I

2

4

6

8

10

Years

Fig. 2. Cumulative

probability

of developing

from

I

1

12

14

infection

clinical

AIDS,

by years from HIV infection.

ALLAN M. SALZBERGand DUNCAN MACRAE JR

156

Infectivity by Phase (the actual and frequency of contact).

---Uninfected

LP,. --(0)

rate of rransmission

.rl ---Nominal

S P I K

I I 6 wks.

I I

__-----[comparison

on the product

.

of infectivity

. ---AIDS and death (5)

-Pre-AIDS (4)

base] (3)

10 yrs. Relative

1

___-----

__------

by phase of HIV infection. of infectivity

Note: the actual transmission and frequency of contact.

Fig. 3. Infectivity

1.5 yrs.

1.0 yrs.

3

0

infectivity

0.2

-------

depends

Duration

2 yrs.

9

of infectivity

__----

-----

rate depends

on the product

set of five equations then represents the inflow and outflow of these infected persons for each successive phase of the disease until death. In these computations, we assume that the epidemic began in December 1976 with one infected gay and one infected IVDU. The high and moderate-risk gay and IVDU populations are operationally defined by the initial rates of spread of HIV computed for those populations [29]. The progression of the epidemic was unaffected by learning before the end of 1982 since its causes were not yet known. Further, it could not be affected by AZT until the end of 1987. Thus, we can estimate both the transmission coefficients and the size of the highest-risk gay and IVDU populations by fitting their CDC AIDS data under the assumption that the coefficients were constant within each of the two time periods (19761982 and 1983-1986). Estimation of the high risk populations is based on the fact that saturation effects within these groups modify the rate of spread of HIV [29]. To avoid bias toward interventional strategies, we chose low estimates for the size of the population groups, as shown in Table 2, and conservatively assumed an 87.5% probability that a case of AIDS was reported to CDC [8]. If we assumed a lower reporting probability in the early years, it would give a proportionally higher estimate of HIV prevalence. Other estimates of under reporting are as low as 80%. We also assume that the eight population groups represent 95% of all AIDS cases and HIV carriers [29]. We estimate 5.3 million for the total gay male population, equivalent to stating that about 7% of adult males are bisexual or homosexual. Of these, we Table 1. Transmission coefficient matrix for HIV in the United States (Infections/month/carrier year median incubation period)

I

POPULATION* = (High gay)

I

(Mod

2 3

isay)

(High IVDU) (Mod IVDU) (Hi-female)” (Hi-male)d (Mod-female)e (Mod-male)’

4 5 6 7 8

2

0.0864’ 0.035b 0 0

0.0083’ 0.0084 0.0335 0

0

0

0 0 0 0 0

0 0 0 0 0

3 F 0 0 0.0806’ 0.059’ 0 0 0 0 0

4 0 0 0 0.0075 0.0075 0.030 0 0 0 0

5

for phase 3 of the incubation period; 10 6

0.0006Oq

0

0.0006Oq

0

0.00060q 0.0053q

0 O.OOlOq

0.0053q

0.0010q

0.0053q

0.0010q

0 0.020 0 0

0.016 0 0 0

7

8

0.00060( I -4) 0.00060(1 -4)

0

0.00060(1-q)

0

0.0053(1 -4) 0.0053(1 -4) 0.0053(1 -4) 0 0 0 0.007

0.0010(1 -4) 0.0010(1 -4) 0.0010(1 -4) 0 0 0.005 0

0

*The populations in column I are the infected, the others are the susceptibles. “1976-82; b1983-86; afterwards, an appropriate learning curve is followed. CConservatively set to 0 as these transmissions are not significant. d(High-risk heterosexual)--averages I new partner every I/2 years. c(Moderate-risk heterosexual~averages I new partner every 4 years. Note: q is the fraction of heterosexual infections due to gays or IVDUs that occur in high-risk heterosexuals. 1- q is the fraction to states d,CNon-IVDU male lo female and female to male transmission coefficients 7, 8 (moderate risk). In these calculations we set q = 0.67. include the effects of rate of partner exchange and intrinsic infectivity.

Policies for curbing

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Table 2. Estimated size of the at-risk populations in the United States Sub-group

Population (in millions)

High risk Moderate High risk Moderate High risk Moderate

0.335 5.0 0.09 1.0 10-15~ >lOV

gays risk gays IVDUs risk IVDUs heterosexuals risk heterosexuals

Note: high risk heterosexualsaverage 2 partners per year. Moderate risk heterosexuals average I partner every 4 years. “These populations are so large that the results are independent of their actual sizes.

estimate that 335 thousand are in the highest risk group. These correspond to the bathhouse gays and represent a small fraction of all gays. The outcomes are highly dependent on the size of this sub-compartment; however, the outcomes are relatively insensitive to the assumed size of the lower risk gay population. Changing the size of this sub-population from 4 to 7 million barely influences the course of the epidemic. This is due to the high rate of learned behavior among gays [29]. In the absence of a high rate of learning, this somewhat surprising result would not occur. Indeed, among IVDUs, the results are dependent on the size of the IVDU population. Our estimate of 1.1 million IVDUs does not disagree with CDC estimates. Under these assumptions, we estimated the monthly transmission coefficients for the high-risk groups (note the second set of coefficients in Table 1) by fitting the model to CDC AIDS data for gays and IVDUs from 1983-1987. Although there is some question concerning the completeness of this data set, it is the best available. To account for the less than 7% [I] of AIDS cases who are both gay and IVDUs, and thus have dual risk factors, we allocated these cases year by year to the gay and IVDU populations in proportion to the numbers of gay and IVDU AIDS cases in that year. Since curve fitting was less useful for heterosexuals, we used Monte Carlo simulations [29] to obtain the heterosexual male-to-female and female-to-male coefficients. We first introduced estimates of an average per vaginal contact transmission probability of 0.0018 male-to-female and 0.0014 female-to-male. These numerical values were obtained by analysis of cohort data, which did not indicate a marked difference between these two transmission probabilities [29]. Also, for transfusion infected partners, the 50 AIDS cases in non-IVDU females and 30 AIDS cases in heterosexual non-IVDU males reported in 1990 [l] do not argue for markedly different heterosexual transmission probabilities. Then, assuming an average of 12 intercourses per month, the monthly transmission coefficients for heterosexuals averaging 2 partners per year and one partner every four years were found to be approximately 0.017 and 0.006, respectively, for both sexes together (Table 1). Increasing the number of partners to 32 increases the monthly transmission coefficients only to O.O20/month/carrier due to the low per-contact heterosexual transmission probability. Thus, increasing the number of sexual partners for heterosexuals from 2/year to 32/year has a negligible impact on the spread of HIV provided the average total number of intercourses remains fixed at 144 per year. Similarly, we can estimate the effect of the frequency of intercourse on the monthly transmission coefficients. These results, based on 500 Monte Carlo replications (500 is a number required to reduce the standard error of the mean to less than 15% of the mean), are depicted in Fig. 4. The monthly transmission rates from gays and IVDUs to heterosexuals were then obtained by fitting 1987 heterosexual caseload data under the approximation that in 1987 70% of all heterosexual AIDS cases were due to IVDUs. The assumed effects of learning for the gay and IVDU populations are derived from the changes in transmission coefficients estimated by curve fitting (Table 1). This gives estimates of 15-30% per year decreases for gays and 5% for IVDUs. In this paper, we allow for learning by lowering the monthly transmission coefficients by a fraction 22.5% per year for gays, which also agrees with estimates derived from decreases in rectal gonorrhoea and syphilis [23,24]. After 1992, we decrease the learning effect to 10% per year. We use a 5% annual decrease in the transmission coefficients for IVDUs, and a 5% decrease for heterosexuals although this seems optimistic in view of the recent increase in the syphilis rate among heterosexuals [29].

ALLAN M. SALZBERGand DUNCAN MACRAE JR

158 0.020 & 8 ”

0.018

-

0.016

-

5 ‘Z .v; 0.014

-

E g 2

0.012

-

;

0.010

-

Z 2

0.008

-

0.006

-

.

.

_--- .-----500 repli&tions / / _z *

*

. .

. ,_-/ / ./ ’

*

*

*

/

-----

./ / ,/‘*

2

,/;

120 contactslyr .

144 contacts/yr

*

96 contacts/yr

,, .’

0.004

*

’

’ ’ ’ ’ ’” 1

0.1

I

I I I1llll

1

I

I I I I I II

10

Mean # of partner exchanges

per

100 year

Fig. 4. Effects of frequency of intercourse and partner exchange rate on male-to-female HIV transmission.

COMPUTING Modeling

EFFECTS

OF AZT TREATMENT

eflects of the use of AZT

Since 1987, the use of AZT has led to a decrease in the rate of increase of AIDS incidence [28]. Since policies designed to slow the epidemic can lead to increased use of AZT and case identification, we must extend the previously discussed simulation procedure. Formally, we should introduce three additional binary states that further subdivide the population as to whether the individual has tested positive for HIV, is still sexually active, and is on AZT (or other antiretroviral therapy). This would increase the number of population compartments eightfold. Also, neglecting new recruitment and losses from the carrier state for causes other than AIDS can lead late in the epidemic to systemic discrepancies of up to 20% for IVDUs, who can lose 10% of their population per year. However, when differences between policies are addressed, the discrepancies are reduced. Although this marked increase in complexity is being addressed, we here use a simpler approximation method to estimate the results. AZT therapy decreases the rate of conversion to clinical AIDS by about 50%, which is equivalent to doubling the time to AIDS [28]. The absolute effect of AZT on the AIDS-free interval thus depends on when the drug is begun. If started after the CD4 count falls below 200, the expected increase in AIDS-free time is only 1.5 years. If it is started when the CD4 count is greater or equal to 500, then the expected increase could approach 10 years. Disease state knowledge can also alter behavior, a possibility that has been modeled by Gail et al. [17]. Moreover Anderson et al. [ 131 recently reported that “Zidovudine [AZT] therapy was associated with decreased detection of HIV-l in semen (adjusted [odds ratio] OR = 0.04; 95% CI, 0.0&0.63).” In addition, not only did p-24 antigenemia drop by an average of over 70% with AZT therapy [18], but the amount of infectious HIV RNA dropped by over 85% during antiretroviral therapy [ 191. Similar effects of treatment have also been found when provirus counts are measured [lo, 161. Since HIV develops high level resistance to AZT in a sequential, and possibly reversible, multistep process, and since resistance develops more quickly with advanced disease 16,271, it is likely that this problem can be contained by combination or sequential drug therapy with different agents coupled with earlier intervention. Furthermore, the baseline probability of transmission of HIV by one act of needle sharing or through a single act of sexual intercourse is less than 1% [29], which is equivalent to saying that the amount of virus is less than the 1% infectious dose ID,.,,. We thus assume, conservatively, that being on AZT (which we here connect closely with disease state knowledge due to testing) decreases the already low monthly transmission coefficients by 66%, as compared to untreated carriers who are also unaware of the fact they are infected with HIV. In this approximate method we keep the basic equations but modify both the time constants and the infectivity based on the effect of AZT and the fraction of the infected population taking AZT at a given point in time. For the base case (without policy changes) we use estimates based on

Policies for curbing

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Rosenberg [28] that 35% of gays whose CD4 count is less than 200 and 8% whose count is above 200 take AZT. For IVDUs and heterosexuals, our estimated fractions are 10% and l%, respectively. This approach may slightly underestimate the effects of intervention. Figure 5(a) shows the projected annual incidence of HIV infections of gays, IVDUs and heterosexuals for the base case. In 1989-90, we estimate about 150 thousand HIV infections. Extrapolation of military data to the entire population yields a lower bound for national HIV infections in 1989-90 of 40 thousand [13,25] since the military actively discourages gays and IVDUs. The prevalence of HIV in the military is less than 25% the national average and the incidence appears to be approximately proportional to prevalence [13]. Thus, our estimate of 150,000 and the 40,000 lower bound are consistent. To further show the applicability of this model, we apply it to San Francisco gays. For 1992, we thus project 1600 new AIDS cases, 600 HIV infections and 30 thousand HIV carriers. The corresponding numbers from the San Francisco AIDS office, based on a completely different consensual methodology, are 1563 new AIDS cases, 650 infections and 25 thousand carriers [31]. For gays we see a twin peaked infection curve with wide early swings in the annual infection rate. Prior to 1981, 20% of the HIV carriers were in the first incubation phase of high infectivity, effectively doubling the overall infectivity [29]. Then, before learning could have been involved, 10% of the highest-risk populations were infected and the proportion in phase 1 declined, leading to a corresponding decrease in rate of spread. A positive feedback loop resulted and the effect of the first (highly infective) phase rapidly damped out. By 1984, the infection wave saturated the small

(a) 8 Gays

120 -

+ IVDUs

. 100 -

+ Heterosexuals

80 60 -

0 1975

1980

1985

1995

1990 Year

2000

2005

end

. Hi-gay/gay

0” 5

+ Hi-IVDUIIVDU

0.8

* Fern-het/het

P 0.7 .-E” > z ;;i

0.6 0.5

.g 0.4 ti =i 0.3 LL I

1990 Year

Fig. 5. Projected

HIV incidence

for subgroups. in compartments

(a) Three major population within population groups.

groups.

(b) Incidence

fraction

160

ALLAN

M.

SALZBERG

and

DUNCAN

MACRAEZJR

number of highest risk gays and IVDUs and spread into the larger moderate risk populations. An additional cause of the slowdown in the gay population was learned behavior after 1982. Figure 5(b) shows how the infection wave moves through compartments that are subdivisions of the populations shown in Fig. 5(a): high and moderate risk gays and IVDUs and female and male heterosexuals. Prior to 1982, almost all the gay and IVDU infections were in the higher risk compartments; by 1986, most infections were among those with lower risk. This is consistent with the decreasing proportions of new infections observed in metropolitan epicenters. Among those infected by heterosexual intercourse before 1983, about 80% were female. By 198687, this computed percentage dropped to 70%. By the year 2000, we project that the percentage will approach 50%. By comparison, 80% of all reported heterosexual AIDS cases in 1986 were in females, whereas by 1990 the percentage fell to about 67% [26]. Today, learning among the gays and early saturation among the IVDUs are slowing the spread of HIV. After 1998, non-drug-using heterosexuals could be the group with the highest number of new HIV infections?. This expected trend, often discussed, suggests the need for non-drug-addicted heterosexuals to join with others in serious consideration of means for curbing the epidemic$. Our projections for HIV incidence are discordant with those of Brookmeyer [8]. He employs an extension of back-calculation, and does not use a biologically oriented model to deconvolute the integral: a(t) =

‘Z(s)J’(t -s Is) ds s0 where a(t) = the cumulative number of AIDS cases at time c; F is the distribution of incubation times, and Z(s) is the number of HIV infections in year s. Rather, he uses smoothing splines both to solve for Z and to force Z(s) not to have “implausible oscillations” which we found to be the solution of our nonlinear equations. Furthermore, his estimate of 30,000 new infections in 1990 is less than the previously discussed lower bound based on extrapolation of military data. We treat the effect of policy changes as altering the fractions on AZT, starting in the year when the policy is adopted. In particular, we consider three cases. The first, or base case, continues present policy. Case 2 assumes that 2/3 of carriers whose CD4 count is below 500 are on AZT starting in 1993; and case 3 assumes, as a limiting case, that all carriers with CD4 below 500 are on AZT by 1993. All patients with diagnosed clinical AIDS are assumed to be treated. We fully realize that it would be more difhcult to treat IVDUs than gays; however, the above cases illustrate the impact of intervention. These are optimistic scenarios and may not be attainable without a combination of monetary incentives (e.g. free AZT), effective guarantees against discrimination, widespread effective counseling, and broad public education. The results for the scenarios shown can, however, be used to interpolate approximately between these and the base scenario in order to estimate effects of lower treatment proportions. We use short time horizons (near the year 2000) because of the greater uncertainty of longer-term projections and the possibility that other factors, such as the availability of new drugs or vaccines, can emerge. This leads to a conservative estimate of the benefits of treatment, as even early reduction of new infections cannot greatly affect AIDS incidence by that time. We show time paths, not simply overall sums, to allow a view of various effects of possible policies over time.

Figure 6 projects reported AIDS incidence for the three scenarios from 1985 to 2005. In 1989 there was a transient decrease in the rate of increase in AIDS incidence [33] due to the use of AZT, largely by gays whose CD4 count was below 200. After 2 years, the rate of increase closely parallels the original. This is because, in the present (base case), use of AZT blocks the outflow from the fourth phase of the incubation period but does not decrease inflow. Thus, the population of the fourth phase increases and soon compensates for the reduced rates of transition between phases. Our projected values for AIDS incidence for the base case agree quantitatively with CDC reports [2, 331, as shown in Fig. 6 for 1985-91. IThese results agree with Harris’ projections Iprivate communication (1991)]. $On a per capita basis, the incidence rate among non-IVDU heterosexuals would still be lower than for gays or IVDUs.

Policies for curbing

z 2 z

110 100 -

-

zz _

‘,:

.E 4 e 2;

60-

-

.50-

n ;2 m e, r

40 30-

2

66% AZT

*

100% AZT

q

CDC + *’ +

,*’ VQ VP EB’

20 8’ lo/ 1985

161

States

70 -

z_z WC

EL

in the United

Baseline +

2

the HIV epidemic

I

I

I

I

1990

1995

2000

2005

Year

Fig. 6. The effect of AZT usage on AIDS incidence.

After 1993, using AZT in a high fraction of the carriers with CD4 below 500 markedly reduces the projected AIDS incidence. Since the medical treatment of clinical AIDS is continually improving, we assume that before 1987 the mean survival time with AIDS was 12 months, but that this increased to 20 months starting in 1988 due to the use of AZT [16]. By 1992, we again increased the mean survival time with clinical AIDS to a 3 year mean. By the year 2005, we project a cumulative total of almost l/2 million fewer deaths if all carriers are treated than under the present use of AZT. This decrease is due largely to prolongation of the lives of those already infected; lessened infectivity (or a vaccine introduced after 1995) has minimal effect on AIDS incidence before the year 2000. Model verljication A necessary condition for model validity is consistency with known facts over the domain for which it was developed. A common statistical method for verifying a well-behaved linear model is to specify a reasonable range for each of its computed parameters, and observe how the output varies as they are perturbed. In a nonlinear system such as the HIV epidemic, this approach is less useful, as small changes in pre-1982 transmission coefficients both between high risk gays and between high risk IVDUs seriously perturb the results. We set to zero the transmission coefficients that connect gays and IVDUs for several reasons. First, the number of AIDS cases common to both risk groups is far smaller than the number among either gays or IVDUs; also, we accounted for these by allocating them proportionally to the two groups. Second, as time goes on the percentage of cases with both risk characteristics decreases [l]. Third, allowing HIV spread between gays and IVDUs would necessitate modeling additional sub-populations, increasing the number of parameters to be estimated to the point that they could not be estimated uniquely from the limited extant data. The model’s output provides further justification for setting the transmission coefficients from low to high risk groups [7’R(low, high)] equal to zero. Figure 5(a) and (b) shows that by the time a significant number of lower risk gays, IVDUs and heterosexuals became infected the epidemic had run its course in the higher risk groups whom they could infect. Quantitatively, TR (low, high) must be significantly less than TR (high, low) since the act of transmission is symmetric and the low risk populations are far larger than the high risk ones. This, in turn, reduces the low-tohigh transmission coefficients below those of the high-to-low. Relaxing the assumption that TR (low, high) = 0 with the above constraint leads to results virtually identical to those obtained when the coefficients are set to zero. Next, we postulated the relative infectivities. Due to the nonlinearity of the epidemic, the transmission coefficients for the highest risk gays (and IDVUs) were fixed. The reasonableness of the former coefficient for 197682 (0.0864) may be evaluated as follows. Since high-risk gays prior to 1982 had a large number of partners and about 15 unprotected intercourses per month, the probability of transmission through one act of anal sex, referenced to the third phase of incubation

ALLAN M. SALZBERGand DUNCAN MACRAE JR

162

for this coefficient, would be 0.0864/15 (=0.006). Since the precise nature of gay sexual behavior before 1982 is uncertain, we estimate a probability range of 0.004-0.008, which is approximately three times that for male-to-female vaginal intercourse (0.0018). In addition, a phase 1 relative infectivity of 9 accounts for the effect of partner exchange early in the epidemic [30]. Increasing the relative infectivity to 16 would decrease this estimate to 0.003, while decreasing it to 4 would both increase the per contact probability to 0.012 and reduce the effect of partner exchange. Thus, we chose a mid-range of reasonable estimates and found the resulting projections to be consistent with the known epidemiology [29]. After 1984, the time-dependent infectivity can be replaced by a single “average” value for phases 14, wherein the system behaves linearly. To further show the stability of the model, we adjusted the relative infectivities to 12, 0.2, 1.0, and 4, respectively, in order to increase the importance of both the “spike” and pre-AIDS highly infectious phases while decreasing the effect of phases 2 and 3, which is biologically plausible. Increasing the height of phase 4 had virtually no impact on the projections prior to 1988. Figure 7 shows that these model outputs are but marginally affected from 1980 through 2004. Alteration of the learning curve parameters (from 15 to 30% for gays) does alter the number of carriers in the later years. The resulting variations in the number of carriers ranges from a decrease of 25% to an increase of 50% by the year 2004. However, as the changes are systemic, this does not greatly alter the differences between the policy and non-policy time paths. We disregard infection paths between high- and moderate-risk heterosexuals, as the greatest uncertainty in the heterosexual spread of HIV arises from estimating the probability that a gay or IVDU will transmit HIV to a high- or moderate-risk heterosexual in a month [29]. We denote this probability as “q” in Table 1 and set q = 0.67 in our calculations. Since there are 10 times as many moderate-risk as high-risk heterosexuals, any individual moderate-risk heterosexual has less than l/10 the chance of an encounter, and thus less than 5% (=0.5 x 0.1) of the risk of a high-risk one. The variations due to mixing heterosexual risk groups are but a fraction of this effect. The effects of IVDU gender (not represented by separate compartments) are implicitly accounted for by the different transmission coefficients of IVDUs to heterosexual males and females (see Table 1). Figure 6 shows that the model’s estimates of AIDS incidence four years (1987-91) beyond the time for which the parameters were determined (1976-86), remain in excellent agreement with reported AIDS incidence. In addition, our estimates for cumulative heterosexual AIDS cases by October 1991 and the number of heterosexual cases in the preceding 12 months are within 7% of the reported values of 12,000 and 3100, respectively [l]. These results lend credence to our implicit treatment of IVDU gender, as a large fraction of heterosexual infections are due to contact with infected IVDUs [29]. In addition, Fig. 8 shows that the employment of AZT profoundly decreases the penetration of HIV into the non-IVDU heterosexual community for either a 66 or 90% assumed reduction in transmissivity due to taking AZT.

0.80 1980

I

I

I

I

I

1985

1990

1995

2000

2005

Year Fig. 7. The effect of changing

the relative infectivities on the course of the HIV epidemic shows ratios of outputs depicted in Figs 5, 6, 9 and 11).

(the vertical

axis

Policies for curbing

the HIV epidemic

l

Baseline

+

100% AZT-66%)

*

100% AZT-90%

in the United

AZT

States

163

./’

INFEC + INFEC

/

/ .’

. +-+ /+/+--+ -.L -*------*--*-*~* + */

1992

I

I

I

1994

1996

1998

I 2000

I

I

I

2002

2004

2006

Year

Fig. 8. Changes in the time path in heterosexual HIV prevalence with variations in assumed drop in infectivity due to AZT usage (for two estimates of AZT efficacy on reducing the infectivity). Note: AZT is used on 100% of HIV carriers whose CD4 count is less than 500.

While the above discussion does not “prove” the model (indeed, a model can never be proved) it does enhance its credibility. Furthermore, the nonlinearity of the system makes it highly unlikely that the boundary conditions of population size, sexual behavior, and transmissivity will ever be known with sufficient precision to allow a forward computation of the spread of HIV from empirical estimates of the boundary conditions. Thus, the “ideal treatment” of HIV is likely unattainable. Disaggregated

eflects of widespread

AZT

treatment

We next consider a number of somewhat separable effects of widespread AZT treatment. We treat them separately in order to call attention to distinct processes that we have modeled and to the incidence of these effects on various affected and concerned groups. These are as follows: (1) Effects of testing. (a) Cost of testing?. (b) Behavior change due to knowledge of infection. (2) Direct cost of providing fully subsidized AZT. (3) Effects due to treatment of infected persons. (a) Lengthened life through extension of the incubation period. (b) Postponement of AIDS treatment costs for infected persons. (4) Effects due to modifying the number of new infections. (a) Mixed effect of AZT treatment on new infections by increasing the incubation period but decreasing infectivity. (b) Saving AIDS treatment costs through prevented infections. (c) Saving life years through prevented infections. (5) Combined effect on direct monetary costs of testing, treatment and care [the sum of (la), (2), (3b) and WN. (la) Cost of testing. Assumptions will be discussed below. (lb) Behavior change due to knowledge of infection. As noted above, this is incorporated in modified monthly transmission coefficients. (2) Cost of subsidized AZT. Figure 9 shows the computed time path of the incidence of new HIV infections under our three scenarios after 1993. The small dip of HIV incidence from 1988 to 1991 is due to the limited effect of AZT and case identification that has already occurredj. The substantial drops in 1994 in the curves for 66 and 100% treatment reflect the corresponding tFor all the monetary costs involved, analysts should distinguish between budgetary and social costs. $We estimate

1.15 million

carriers

at the end of 1991.

ALLAN M. SALZBE~Gand DUNCAN MACRAE JR

164

~

Baseline 66% AZT 100% AZT

1985

1980

1990

1995

2000

2005

Year Fig. 9.

Projected effects of A2T on HIV incidence.

projected decrease in HIV transmission due to both decreased infectivity and behavior improvement of treated carriers. These projections assume that the increase in AZT usage begins in 1993 and continues from that time. This leads to a necessary increase in treatment costs for the first year or so. The costs will drop relatively as AIDS and HIV incidence drops. An approximate indication of these costs over the lifetime of the persons treated (for the 100% treatment case) is given by the horizontally shaded area at the lower right in Figure 9. We assume here that each new HIV case could require AZT treatment over an average extended incubation interval double the present median of 10 years, with a monetary cost on the order of $3000 a year (see below). (3a) lengthened life through extensiun of the ~nc~batiun period. The cases that are treated with AZT {again, the horizontally shaded area at the lower right of Fig. 9) are each projected to have an additional period of active life averaging up to 10 years, based on our assumptions. (3b) Postponement of AIDS treatment co.sts. Prolongation of the incubation period also leads to postponement of the social costs of AIDS treatment, when this disease finally occurs. If we were to discount such future costs, then for each case treated (horizontally shaded areas at lower right in Fig. 9) the AZT treatment costs would be offset to some extent by this savings due to postponement. A more direct indication of this effect is given by the vertically shaded area in Fig. 6, above, since the reduction in AIDS cases over the period shown is due largely to postponement. (4z,l Mixed efict of AZT treatment on carriers’ infection of others. May and Anderson [22] have suggested that a longer duration of the incubation period would correspond to an increased infective period and greater net transmission of the virus. This, in turn, would lead to increased costs from new HIV cases attributable to AZT therapy. We assume, however (as noted above), that the infectivity of persons treated is greatly reduced. Monte Carlo simulations indicate that if an additional S%/year of known infected carriers desist from activities that can transmit HIV, even with no change in infectivity, then the effect suggested in 1221will not occur. (#I Saving AIDS treatment costs through prevention. AIDS treatment costs are not only postponed (for infected persons given AZT), but also prevented when a case of HIV is prevented (see vertically shaded areas in Fig. 9). (4) Saving life years through prevented infecrions. If each prevented case is associated with an average extension of active life from the 10 year median incubation period to the longer period of life expectancy or working life, then the total number of HIV cases prevented (the vertically shaded areas at the upper right in Fig. 9) can be multiplied by this average increase in life years, yielding total years saved (over the time interval until death). Since AIDS patients are generally young, this average increase is particularly significant. Figure 10 expresses in monetary terms the cumulative effect, for each of the three scenarios, on life years saved up to the year specified through avoidance of premature death. These effects are due primarily to an extended incubation period and secondarily to decreased HIV incidence. We choose a conservative value of a year of lost life due to premature death ($25,000) comparison with customary estimates [37, p. 2973. It

Policies 2.50

r

200

v) g 2 0

165

for curbing the HIV epidemic in the United States

0

Basecase

m

66% on AZT

m

100% on AZT

150

.c

2000 Year

end

Fig. 10. Cumulative dollar cost due to premature death from AIDS (billions of undiscounted 1990 dollars).

is possible that, over the longer run, these are the dominant social benefits from the proposed policies. (5) CombinedJinancial e&&s on direct costs. For those decision makers who are concerned with the net financial effects of our two “policy” scenarios, Fig. 11 projects these effects, composed of costs of testing, of AZT treatment, and of treatment of AIDS. The cost of a test, including its administration, is assumed to be $8, based on the military cost of S4 for the complete test itself [5], rather than higher figures, such as $35, that are sometimes mentioned; we assume efficient administration. The projected annual cost of testing in the base case then becomes $50 million per year in 1991; this increases to $300 million per year in scenarios 2 and 3 after 1992. In all cases, the cost of testing is a small fraction of AIDS treatment costs, as shown in Fig. 11. The sensitivity of the test is assumed to be 100% for all phases of the disease other than initial infection, and zero there. This gives, today, a sensitivity of over 98%. Based on Army data, the false positive rate should be less than 1 in 170,000 [5]. We assume that the combined annual treatment and social support costs per AIDS case are $50,000. For AZT treatment, we assume a cost of $5500 per year per HIV carrier undergoing treatment whose CD4 count is below 200, and $2000 per year for carriers with CD4 over 200. (More intensive therapy is required for lower CD4 counts). This presupposes a modest decrease in the cost of antiretroviral drugs. Figure 11 shows the base scenario as a smoothly ascending curve; for it, annual costs will rise from about $4 billion in 1992 to $14 billion in 2003 unless significant changes occur. Considering all possible uncertainties, our use of conservative costs and the fact that we neglect pediatric AIDS, our estimates are in reasonable agreement with Hellinger’s estimates of $5.8 billion in 1991 and $10 billion in 1994 [35]. 14 12 z ; ._ P .c z : -z z 4

lo-

n

Baseline

+

66% AZT

*

100% AZT

86 4.-. 2-

,-.

0 1986

._./

./.

I

I

I

I

I

I

I

1988

1990

1992

1994

1996

1998

2000

I 2002

1 2004

Year Fig. 11. Effects of AZT on total annual HIV and AIDS treatment costs.

I 2006

ALLAN M. SALZBERGand DUNCAN MACRAE

166

JR

100 r

“‘\+
n :“-:--:-:+

l_

II--50

-

40 0.4

Fig. 12. Effect of assumed

~:~s~~~~o”s

I

I

I

I

I

0.5

0.6

0.7

0.8

0.9

Fraction

CD4.z.500

proportion

‘I 1.0

on AZT

using AZT (for CD4 < 500) on 1997 AIDS

and HIV incidence

The two curves corresponding to additional AZT treatment first rise above the base curve. This initial increase in cost is real and occurs because we begin immediately to treat a large number of carriers, while the number with AIDS decreases slowly over the following 3 year period to a new quasi-steady state that numbers about three times AIDS incidence. At the same time, the distribution of infected individuals in later phases increases. In 1988, for example, 62% of HIV carriers had CD4 counts less than 500; by 1995 this fraction could exceed 75%

~91. Figure 11 also suggests that interventional strategies can have a payback after the first few years. After the treatment curves cross below the base curves, areas of net benefit are generated. The more treated, the faster is the payback. The payback is primarily due to the lessened AIDS caseloads; the effect of HIV prevention does not come into play until the year 2000. In fact, allowing the decrease in infectivity due to intervention to range from 67% by f33% changes the annual cost of treatment in 2005 and annual AIDS incidence by less than 15%. The 100% scenario, although mathematically pure, is somewhat unrealistic (a more realistic upper bound is 85%). Figure 12 suggests an essential linearity of HIV and AIDS incidence in response to the fraction of carriers taking AZT (from 40 to 100% treated) so that simple linear interpolation can be used to estimate mid-range use of AZT.

FEASIBILITY

OF ADOPTION

AND

IMPLEMENTATION

Figures 6, 9, 10 and 11, showing effects of the proposed policies, are not merely indications of their potential effectiveness, but can also be used by various concerned parties to assess particular effects of interest to them. We shall now discuss these particular interests, in view of the possibility of combined? support for such policies. Four key groups may all gain from such policies: (a) persons infected with HIV; (b) persons not now infected, who may be saved from infection; (c) public officials concerned with reducing the fiscal problems associated with AIDS; and (d) manufacturers of AZT and similar drugs. Znfictedpersons. Victims of HIV have sought remedies but have feared that their opponents aim to control them rather than to relieve their ailment. Free treatment would dispel some of this fear. Effective protection against discrimination might also lead more persons at risk to be tested and treated voluntarily. Further public education about the need for testing and treatment might also help. In an atmosphere that bred genuine confidence in health authorities and providers, persons at risk in a free society might be encouraged to be tested and treated, probably prolonging their own lives and reducing risks to others. Coordinated services to infected persons, and expert counseling to those at risk, could also contribute. tSuch a coalition would be analogous broadly, for anti-poverty strategies

to the “social [32].

contract”

discussed

in the late 1980s for welfare

reform

and, more

Policies for curbing

the HIV epidemic

in the United

States

167

Susceptibles who are not infected. Those who are at risk but not infected can also benefit from this policy. The policy appears to add to the effects of education by a drastic reduction in infectivity of persons under treatment with AZT. This effect contributes to the reduction in the epidemic projected in the above figures. It does not, of course, mean that either infected persons or susceptibles can ignore the precautions needed in sexual or drug-using behavior. Public oficials concerned with theflow offundr. We have suggested that the net effect of this policy on annual direct costs of the epidemic to public and medical authorities might (at best) shift from negative to positive after a few years. Further, its cumulative effect on (undiscounted) cash flow might be positive several years later-if every infected person was tested beginning in 1993. With fewer tested, the shift would occur at approximately the same time, but the net payoff would be less. Although the effects of such a policy on social costs-including the extension of life and the concomitant increase in productivity-would be far more positive than this, effects of such policies on cash flow alone might be turned from negative to positive if the government was able to convince enough infected persons to be voluntarily tested and treated. Drug producers. Much of the cost of such a policy results from the use of AZT or similar drugs. Cutting this cost should make the net cash flow more positive. Possibly the price can be reduced through competition or via the development of new drugs. Even if not, economies of scale should benefit producers and result in lower costs. To develop such a policy in detail, enact it, and implement it expeditiously is indeed a complex task. Problems of implementation are of the greatest importance. We believe that carriers can be most effectively identified and treated, consistent with individual rights, if they are induced to be tested voluntarily. It is not certain, however, what proportion will respond even if AZT is free and effective guarantees against discrimination are in place. CONCLUSION A simulation model can be used to estimate the course, over time, of various benefits and costs from increased free AZT treatment. These estimates suggest that further study of such effects can be valuable in view of potential shared interests in the outcomes. On this highly controversial issue, any viable proposal requires substantial expert consensus and broad political support. To develop expert consensus, we hope that our conclusions can be subjected to critical scrutiny [9, Chap. 93. The effects we have projected can be examined with other simulation programs. If there are significant differences in projections, they may be traced by sensitivity analyses. In addition to review of the model itself, three important questions require attention as a result of their potential for controversy: (a) The effectiveness of AZT treatment on prolonging active life; (b) Preventive effects of AZT treatment (through either medical or educative effects); and (c) The relative effectiveness of various policies in inducing people at risk to be tested and treated. We hope that with study of realistic possibilities for policies involving subsidized AZT treatment, and consideration of shared group interests, conditions for a broadly supported effective policy will be enhanced. Acknowledgements-We are indebted to Heather Bannerman, Allan M. Brand& Peter S. Bearman, David Jolly, F. Marc Laforce, Lester W. Lee, Rick Small, David L. Weimer, Dale Whittington and the reviewers for helpful suggestions. An earlier version of this paper was presented at the Association for Public Policy Analysis and Management meeting, 28 Oct. 1991 in Bethesda, Md.

REFERENCES 1. AIDS statistics update. The AIDS Reader 1, 184186 (1991). 2. AIDS statistics update. The AIDS Reader 1, 103-105 (1991). 3. D. Anderson, T. O’Brien, J. Politch, A. Martinez et al. ElTects of disease state and Zidovudine therapy on the detection of HIV-I in semen. JAMA 267, 2769-2776 (1992). 4. P. Bacchetti and A. Moss. Incubation period of AIDS in San Francisco. Nature 338, 251-253 (1989).

ALLAN M. SALZBERG and DUNCAN MACRAE JR

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5. D. E. Bloom and S. Glied. Benefits and costs of AIDS testing. Science 252, 1798-1804 (1991). 6. C. A. Boucher, E. O’Sullivan, J. Mulder et al. Ordered appearance of Zidovudine resistance mutations during treatment of 18 HIV positive subjects. J. Infect. Dis. 165, 105-110 (1990). 7. M. L. Brandeau, H. L. Lee, D. K. Owens, C. H. Sox and R. M. Wachter. Policy analysis of human immunodeficiency virus screening and prevention: An overview of modelling approaches. AIDS Pub. Policy J. 5, 119-131 (1990). 8. R. Brookmeyer. Reconstruction of future trends of the AIDS epidemic in the U.S. Science 253, 3742 (1991). 9. C. F. Citro and E. A. Hanushek (Editors). Improving Information for Social Policy Decisions: The Uses of Microsimulation Modeling, Vol. 1, Review and Recommendations. National Academy Press, Washington, D.C. (1991). 10. A. Clark, H. Holodniy, D. Schwartz et al. Decrease in HIV provirus in peripheral blood mononuclear cells during ZDV and human rIL-2 administration. J. AIDS 5, 52-59 (1991). 11. S. Clark. M. Sang, W. Decker, G. Shaw et al. High titers of cytopathic virus in plasma of patients with symptomatic primary .HIV-1 infection. N. Engl. J. Med. 324, !%4-960 (1991). 12. S. Coleate and J. Hvman. A risk based model for HIV. Proc. N&l. Acad. Sci. 86. 4793-97 (1989). 13. D. Coian, R. Pomeiantz, Z. Wann et al. HIV infection among members of the reseive componentsof the U.S. Army. J. Infect. Dis. 162, 827-836 (1990). 14. E. Daar, T. Moudgh, S. Meyer and D. Ho. Transient high levels of viremia in patients with primary HIV-I infection. N. Engl. J. Med. 324, 961-964 (1991). 15. I. Devincenzi, R. Ancille, J. Baptista et al. (European study group). Risk factors for male to female transmission of HIV. Br. Med. J. 298, 411415 (1989). 16. R. Dickover, R. Donovan, E. Goldstein et al. Decreases in unintegrated HIV DNA are associated with antiretroviral therapy in AIDS patients. J. AIDS 5, 31-37 (1991). 17. M. H. Gail, D. Preston and E. Piantadosi. Disease prevention models of voluntary confidential screening for human immunode&iency virus. Statist. Med. 8, 59-81 (1989). 18. C. Hendrix. P. Volberdine. R. Chaisson. HIV antigen variability. J. AIDS 4. 847 (1991) 19. M. Holodniv and T. Mer&n. Reduction in plasma HIV RNA cody number foliowing didebxynucleoside as determined by PCR. 3isr Interscience Conf. on Antimkrobials and Chemotherapy (ICAAC), Chicago, III. Abstract 704 (1991). 20. G. Lemu. S. Pavne. G. Rutherford er al. Proiections of AIDS morbidity and mortality in San Francisco. JAMA 263, 1497-1501 (199i)). ’ 21. R. M. May and R. M. Anderson. The transmission dynamics of human immunodeficiency virus. Phil. Trans. R. Sot. 21, 239-281 (1988). 22. R. M. May and R. M. Anderson. Potential of community wide chemotherapy and immunotherapy to control the spread of HIV-l. Nature 350, 356-359 (1991). 23. Morbidity and Mortality Weekly Reports (MMWR-CDC). Continuing increase in infectious syphilis in the U.S. MMWR 37, 35-36 (1988). 24. Trends in gonorrhea in homosexually active men. MMWR 38, 762-763 (1989). 25. HIV prevalence estimates. MMWR 39, RR16 (1990). 26. AIDS-the first 10 years. MMWR 40, 358-369 (1991). 27. D. Richman, J. Grimes and S. Lagakos. Effect of stage disease and drug dose on ZDV susceptibilities of isolates of HIV. J. AIDS 3, 743-746 (1990). 28. P. Rosenberg, M. Gail, L. Shrager, S. Vermund et al. National AIDS incidence trends and the extent of ZDV therapy. J. AIDS 4, 39241 (1991). 29. A. Salzberg, S. Dolins and C. Salzberg. A multiperiod compartmental model of the HIV pandemic in the U.S.A. Socio-Econ. Plann. Sci. 25, 167-178 (1991). 30. A. Salzberg, S. Dolins and C. Salzberg. Effects of a variable infectivity on HIV risk factors. 30th Interscience Conf. on Antimicrobials and Chemotherapy (ICAAC). Atlanta, Ga. (1990). 31. San Francisco Dept of Public Health. Surveillance Branch, AIDS Office, HIV Incidence and Prevalence in San Francisco in 1992: Summary Report From an HIV Consensus Meeting (1992). 32. I. V. Sawhill. Antipoverty Strategies for the 1980~ Urban Institute, Washington, D.C. (1986). 33. E. Steel and H. Haverkos. Increasing incidence of reported AIDS. N. Engl. J. Med. 325, 6546 (1991). 34. J. Taylor, J. Kuo and R. Detels. Is the incubation period of AIDS lengthening? J. AIDS 4, 69-75 (1991). 35. United States Public Health Office: Research Activities, Agency for Health Care Policy and Research (F. Hellinger) USPHS # 146. Government Printing Office, Washington, D.C. (1991). 36. P. Van de Pere, A. Simonon, P. Msellati, D. Hitimana et al. Postnatal transmission of HIV-l from mother to infant. N. Engl. J. Med. 325, 593-598 (1991). 37. D. L. Weimer and A. R. Vining. Policy Analysis: Conceprs and Practice. Prentice-Hall, Englewood Cliffs, N.J. (1992).

APPENDIX

Dlrerence Equations For HIV Spread in the United States The relevant 1. Ds(l,j,

2a. Ds(2,j,

difference

k) = &.,,[V(&

k)

2b. Ds(3, j, k) 2c. Ds(4, j, k) 2d. Ds(5, j, k) 2e. Ds(6, j, k)

equations

are:

N)TR (kj) x AZT(L, N, k - l).s(N, L, k - l)s(O,j,

-s(5,j,k =s(l, j, k - 1)/(12t,) = s(2, j, k - 1)/(12t,) = s(3, j, k - 1)/(12t,) = s(4, j, k - 1)/12t,) = s(5, j, k - 1)/12t,);

1)-s(6,j,

- l)] -s(l,j,

-

1)/(12t,);

-

k s(2, j, k s(3, j, k s(4,j, k s(S,j, k -

1)/(12t,); 1)/(12&); 1)/12t,);

k - l)/[s(O,j, k - 1)/(12t,)

0)

Policies

for curbing

the HIV epidemic

in the United

States

169

The equations hold for all N,j, k. Where: D = an operator for “first difference over time”; thus, Ds(N,j, k) = s(N,j, k) -s(N,j, k - l), where s(i,j, k) = the number in phase i in population j, k months into the epidemic. Thus, eqn (1) represents the increment to the newly infected phase 1 for population j from time k - 1 to k. V(L, N) = the relative infectivity of phase N of the incubation period for the L th population; N ranges from 1 to 4 since phases 0 (uninfected), 5 (AIDS), and 6 (death) do not (or are assumed not to) transmit infection. The infectivity is expressed relative to V (L, 3) which is set as unity; values of TR (L,j) are defined for phase 3. TR (L,j) = the mean number of infections/month a person in the Lth population group transmits to the jth population group, defined for the third phase of the incubation period. The values of TR (L,j) are the elements of the matrix of transmission coefficients shown in Table 1. These transmission coefficients are defined in relation to their use in eqn (1) which includes the third following term, the infected’s probability of encountering an uninfected person. AZT (L, IV, k - 1) is a function to account for the effect of AZT. Before 1987, AZT (L, N) = 1. Afterwards, it is the amount of average residual infectivity in population L in phase N. The decrease in infectivity from baseline is proportional to the assumed fractional decrease in infectivity times the fraction of carriers in population L, phase N who are on AZT in month k - 1. s(O,j, k - l)/[s(O,j, 0) - s(5,j, k - 1) - s(6,j, k - l)] = the proportion, among those available as partners, at month k - 1 who are uninfected: In the model, this is used as the probability that a targeted individual in the jth population has not been previously infected with HIV by month k - 1. The denominator excludes persons who have AIDS or who have died and cannot be selected as partners in activities that spread HIV. t, to t, = mean number of years to move from a given phase. In exponential distributions, this is the reciprocal of the rate of flow out of a given state into the next. In the above equations, a factor of 12 was used to convert from years to months. Equations (2a)-(2e) consist entirely of terms involving these rates: a positive term representing inflow from the previous term and, in all but the last, a negative outflow to the next term. Once AZT is used, for CD4 counts less than 500, the numerical values of t, and t4 are increased where t,(AZT) = (1 -f,,)t4 + 2f4,Lt4; f4,L= the fractions of persons on AZT in population L and phase 4. Similar modifications occur for phase 3 but are not made for phases 1 and 2 since AZT is assumed to be used only after the CD4 count is less than 500. If these equations are added together with the expression for Ds(O,j, k), we would find that X,,OS (N,j, k) = 0. Equations (2a)-(2e) are linear equations; however, eqn (1) is nonlinear and can, under the conditions found for the epidemic in the U.S., lead to highly nonlinear behavior in HIV incidence prior to 1984. After 1984, the nonlinear effects damp out and the epidemic behaves in a linear manner; the time-dependent infectivities can then be approximated by “average infectivities” [29]. These effects complicate the application of statistical smoothing techniques to times before 1984 to estimate HIV incidence. They also render forward calculations questionable. The known history of the epidemic defines the boundary conditions. We assume that it began in December 1976 (k = 0), and that all the s(N,j, 0) = 0 except N = 1,j = 1 and j = 3, for which s(N,j, 0) = 1. This is equivalent to stating that at the beginning of the epidemic, one high-risk gay and one high-risk IVDU were the only infected individuals and were in the first phase of HIV incubation.