Variable Infectiousness in HFV Transmission Models

IMA Journal of Mathematics Applied in Medicine & Biology (1988) 5, 181-200

Variable Infectiousness in HFV Transmission Models S. P. BLYTHE AND R. M. ANDERSON

Parasite Epidemiology Research Group, Department of Pure and Applied Biology, Imperial College, London University, London SW7 2BB, UK [Received 29 December 1987]

Keywords: AIDS; HIV transmission; variable infectiousness; multi-stage models; mechanistic models; antigenaemia; parameter estimation; incubation period; probability of transmission. 1. Introduction

models of the transmission dynamics of the human immunodeficiency virus (HTV), the aetiological agent of the acquired immunodeficiency syndrome (AIDS), typically assume that the infectiousness of an infected person during a sexual contact with a susceptible individual remains constant throughout the incubation period of the disease AIDS (Anderson & May, 1986; Anderson et al., 1986, 1987; Bailey, 1986; Knox, 1986; Pickering et al., 1986; Blythe & Anderson, 1988; May & Anderson, 1987). However, recent epidemiological and clinical studies of HlV-infected patients and their sexual partners suggest great variability in the likelihood of transmission from an infected to a susceptible person during sexual contact (Burger et al., 1986; Darrow et al., 1987; Gaines et al., 1987; Padian et al., 1987; Winkelstein et al., 1987a, 1987b). Such variability appears to depend on a wide variety of factors including type of sexual activity and the length of time the infected partner has been harbouring HIV infection. Ginical and immunological studies of infected patients are beginning to reveal that antigenaemia and loss of antibodies to core HTV antigens are linked to the development of immunological abnormalities and clinical disease (Pedersen et al., 181 MATHEMATICAL

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Two different approaches to the encapsulation of temporal variation in the infectiousness of HTV-infected persons, and variability in the incubation period of the disease AIDS, in simple homogeneous mixing models of viral transmission in male homosexual communities are described. The first approach is based on the division of the infected population into a series of subclasses with differing levels of infectivity and different durations of occupancy. The second approach is more mechanistic in character and is based on an attempt to relate changes in viral abundance within an infected person to the likelihood that the disease AIDS develops. Variable incubation is induced by variation in the rate of change of viral abundance in the infected population. Numerical projections of changes in the incidence of AIDS through time, generated from both types of model, are compared with projections based on the assumption of constant infectivity throughout the incubation period of AIDS. Model formulation highlights areas in which more detailed quantitative epidemiological studies are required. Methods of parameter estimation and future research needs are discussed.

182

S. P. BLYTHE AND R. M. ANDERSON

Time since infection (years) FIG. 1. An impression of variation in individual infectiousness /9(u) may vary with time u since infection. The initial phase (u < 2 year io this example) represents the growth and decay of free virus from the moment of infection, under a growing antibody response. The second phase (u s>2 year) shows a steady increase towards a maximum value, as a result of free viral concentration (titre) build-up as the immune system loses competence (based on Pedersen et al., 1987).


1987; Zolla-Pazner et al., 1987). Over the long and variable incubation period of the disease (defined as the length of time between infection and the appearance of disease symptoms: see Lui et al., 1986; Medley et al., 1987, 1988), immunological abnormalities appear to occur in a sequential pattern following infection. The sequence may be defined as (a) seropositivity for HIV with no immunological abnormalities, (b) seropositivity for HTV with a depressed T4/T8 cell ratio, (c) seropositivity for HIV with a depressed T4/T8 cell ratio and T4 cell depletion, and finally (d) seropositivity with a decline in antibodies to core HIV antigens, a depressed T4/T8 cell ratio, T4 cell depletion, and lymphopenia (overt disease) (Zolla-Pazner et al., 1987). Studies on the temporal association, from point of infection to the appearance of symptoms of disease, between HIV antigen in blood serum, antibody concentration profiles, and the immunological and clinical state of patients suggest that there are two peaks in antigenaemia (Pedersen et al., 1987). A schematic diagram of this pattern appears in Fig. 1. Following an early rise in the concentration of detectable antigen after infection, which may last from a few months to a year, antigenaemia falls to a low level and then begins to rise again as symptoms of disease, that is, AIDS related complex (ARC) or AIDS itself, appear (Lange et al., 1986, 1987; Goudsmit et al., 1987). Recurrence of HTV antigenaemia seems to indicate a poor prognosis with respect to the development of AIDS: in the study of Pedersen et al. (1987) of 34 infected

VARIABLE INFECTIOUSNESS IN HIV TRANSMISSION MODELS

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2. General model

2.1 Formulation and Assumptions We base our general model of fluctuating infectiousness during the long and variable incubation period of AIDS on the simple homogeneous mixing model of HIV transmission in a male homosexual community introduced by Anderson et al. (1986) and considered in detail by Blythe & Anderson (1988) in the context of distributed incubation period. Let X{t) and Y(t) denote the numbers of individuals who are susceptible (= uninfected) and infected (= infectious, but AIDS not yet developed), respectively. We assume that all infected individuals Y eventually proceed to develop AIDS on a long and variable time-scale from the point of infection. We define A as the rate of recruitment of new susceptibles to the population of sexually active male homosexuals (an 'open' population), n as the per capita death rate of susceptible and non-HTV-infected individuals (assumed to be the same for the X and Y populations), and c as the average number of new sexual partners per unit time. The per capita 'force' or rate A(r) of infection is defined as (1)


homosexual male patients, 49% developed AIDS within two years of reappearance of antigenaemia. The pattern of two peaks in antigen concentration in the serum of infected persons suggests the hypothesis that the infectiousness of infecteds may similarly have two peaks during the long (and variable) incubation period of AIDS. This hypothesis is based on the assumption that peaks in circulating HTV antigens are positively associated with the infectiousness of an infected person for a susceptible (i.e. uninfected) sexual partner. As yet, this assumption is unsubstantiated for HTV infection, owing to the practical and ethical difficulties surrounding both the study of temporal fluctuations in antigen concentration in infected persons and the likelihood of transmission of HTV to susceptible contacts. However, the fact that, for many viral infections of humans, the concentration of circulating antigen is often associated with infectiousness (Roitt, 1986) provides some justification for the hypothesis being extended to HTV infection (Anderson, 1988). In this paper, we explore different ways of encapsulating the two episodes of infectiousness into simple mathematical models of HTV transmission. Our aims are twofold. First, we seek a method of inclusion that minimizes the computational burden of adding this biological complication to existing models, mirrors observed changes in antigen concentration, and permits parameter estimation. Our second aim is to explore how fluctuations in infectiousness influence predictions of the pattern and shape of the AIDS epidemic, as well as the estimation of summary epidemiological parameters such as the basic reproductive rate Ro of HIV infection.

184


where B(t) is given by

fe-'"V('-")du.

(2)

Here, /3(u) is the probability of transmission (the infectiousness of an infected individual) per partner, dependent upon the time since infection u, and C(u) is given by

Here, J{i) represents the initial 'invasion' of newly infected people that starts the epidemic (J(t) is zero for all time, except for a small interval just after t = 0, and is usually such that the integral of /(/) over all t equals one individual). The incidence V{t) of AIDS through time (the rate of leaving the Y class) is defined as V(t) = /(«)e"""r(r - u) du. (3) Jo The differential equations for the rates of change of X{t) and Y(t) with respect to time t are given by (4a) = k(t)X(t) + W - V(t) - fiY{f),

y(0) = 0.

(4b)

2.2 Analysis and Numerical Solution The model defined in equations (4) is complex and difficult to evaluate for the general case of variable /J(u). One approach is the direct evaluation of the integral defined in equation (2) at each time-step in the numerical procedure employed to solve the integrodifferential equations (4). However, this method imposes a considerable burden of computation, which we wish to minimize on the grounds that any characterization of variable infectiousness which we adopt is likely to be incorporated at a later date into more complete and complex models of HIV transmission in the population as a whole. Alternative approaches include the following methods. A form for the probability density function f(u) can be chosen, such as an Erlang distribution, for which the linear chain technique can be applied as detailed in MacDonald (1978), Gurney et al. (1985), and Blythe & Anderson (1988). Following the successful application of this method, one of three approaches can be adopted. (i) Subdivide the range of u in equation (2) into m intervals, and assign a


where /(u) is the probability density function of the incubation period (the duration of stay in the Y stage). Note that, if /3(u) takes a constant value /}>0 for all u, then the term B(t) reduces to /3Y(r). The term r(f) is the rate of recruitment of new infectious individuals (i.e. to the Y stage), and is given by

VARIABLE INFECTIOUSNESS IN HTV TRANSMISSION MODELS

185

3. Mofti-dass model

The single class of infectious individuals (Y) can be replaced by a series of N subclasses in which the length of stay is assumed to be exponentially distributed (a constant per capita removal rate ajt and hence a mean duration of stay 1/oj for the ;th subclass; theCT/Sare distinct). We also assume that in each subclass individuals have a different magnitude of the infectiousness parameter denoted by Bj, which is constant within the yth subclass. The equation for Y(t) in the model defined in equation (4b) is replaced by

- ( a , + fi)Yj(t)

( ; = 2 , 3 , . . . . N),


particular value of B(u) to each interval. Equation (2) can then be evaluated by solving a system of delay-differential equations. (ii) Subdivide the range of u in equation (2) into m intervals, and make B{u) in any given interval vary linearly over that interval. This approach gives a piecewise linear approximation to B{u). Again, B(t) is evaluated by solving a system of delay-differential equations. Hyman & Stanley (1988) have proposed a piecewise linear approximation to B(u) in a model of the general form defined in Section 2.1, but they employed a Weibull distribution for f(u), and hence had to generate temporal solutions from a partial differential equation model. The linear chain method is not applicable with this probability density function. (iii) Choose a function (or sum of functions) to approximate B(u) over the entire range of u that facilitates the application of the linear chain method. The function B(t) can then be evaluated by solving a system of ordinary differential equations. Unfortunately, for all the methods outlined above, the number of equations that must be solved to evaluate B{t) is large. Typically, (i) and (ii) will require m x (n + 2) and m x (n + 4) delay-differential equations, respectively, where n is the order of the Erlang function used to approximate /(u). Thus, even for quite modest m (say 5) and n (say 1 or 2), some fifteen or twenty delay-differential equations must be manipulated in the numerical solution of the model equation (4). With method (iii), the interaction of the component functions of B(u) with f(u) will mean that the system of auxiliary equations to be solved will have the order of NxpXn members, where N is the number of functions comprising B(u), and p their 'average' order in the sense of the Erlang order. This process could easily give rise to ten or more equations. From the preceding discussion, it is clear that a great deal of computation is required to generate time-dependent solutions of the general model defined in Section 2.1, owing to the interaction between the infectiousness B(u) and incubation period f(u) functions. In an attempt to reduce the magnitude of the computational problem, in the following section we consider two alternative methods of formulating the mathematical description of temporal fluctuations in the infectiousness of infected individuals.

186


with Yj(0) = 0 for all j . The 'force' of infection can then be calculated using equation (1), with

2

2

i-i

(6)

l-i

while incidence is given by V(t) = oNYN{t).

V(s) = r(s)f[

°'

>

(7)

5 + CT + /i

where V(s) and r(s) are the Laplace transforms of the AIDS incidence function V(t) and the rate of recruitment of infecteds r(t), respectively. Taking the inverse Laplace transform of equation (7) recovers equation (3) for V(i), with N

j j

(8)

,N

'"N

fly=nvn /"I

("/-«>) (/=i>2 /

AO-

/-I

It can also be shown that

Hs) = 2 fiA') = W t^{\ —^—, l-\

(9)

/ =ia y/ _ 1 s + a/ + /i

where B(s) is the Laplace transform of B{t). Taking inverse transforms of equation (9) gives

5(0 = where

Here, bn = ou bNi = a,, and

bii=t\oi/t\(ak-al)

(j = 2 , . . . , N \ i = l , 2 , . . . , j ) .

k+i

From equations (2) and (9), we see that S(u) = /8(u)C(u).

(11)


Equations (5) and (6) are very straightforward to solve numerically. One special case is worth noting: if all os — a, we recover a probability density function f(u), which is Erlang in form and of order N, for the duration of the incubation period. It can be shown that

VARIABLE INFECnOUSNESS IN HTV TRANSMISSION MODELS

187

Hence, for f(u) as defined by equation (8), we have

Thus, equation (5) is revealed as a special case of our general model, with the probability density function/(u) of the incubation period defined as the sum of N exponential distributions and /J(u) given by (from equation (11)) /8(«) = 5(u)/C(«).

(12)

0.08

0.03 2

4

6

8

10

Time since infection (years) Flo. 2. Infectiousness /?(«) as a function of time u since infection for the multi-class model. N = 3, 7, = 0-9 year, T 2 = 6-0 years, T 3 = M years, and 0 2 = O. (a) /J^O-08, 0 3 = 0-298; (b) 0 , = O-O5, 8 0 for both. 0-323.


Figure 2 shows an example of P(u), as given by equation (7), for N = 3, with at = 1/0-9, o2 = 1/6-0,CT3=1/1-1 year' 1 , and (a) ft = 0-08, p2 = 0-0, ft = 0-0298 and (b) ft = 0-05, ft = 0-0, ft = 0-323. The pattern is broadly consistent with the schematic shown in Fig. 1, although a prolonged period of very low infectiousness is not reproduced. The Appendix outlines a scheme for estimating the parameters /Jy using least-squares methods; the successful application of this method depends on the availability of observations on the value of the infectiousness parameter in samples of patients at various time intervals from the point of infection. The basic reproductive rate Ro of infection is denned as the number of secondary cases of infection generated on average by one primary case in a susceptible population (Anderson et ai, 1986; May & Anderson, 1987). In

188


homogeneous mixing models in which /3 is constant and independent of the time interval from the point of infection (see Blythe & Anderson, 1988), Ro is given by (13) where Q

-f' Jo

For the endemic persistence of the infection, Ro> 1. In the multi-stage model, we have

(14)

For n~»0, equation (13) reduces to R0 = ficT, where T is the mean incubation period. In equation (14), when /i = 0 we have Ro=2

py ?},

(15)

where 7J = 1/oy is the mean duration of the yth substage.

20 Time (years) FIG. 3. Comparison of temporal solution of AIDS incidence between constant /) (curve (a), with 0 = 8-0, 7" = 8 0 years) and multi-stage models. Af = 3, r, = 0-9 year, 72 = 6 0 years, T3= 11 years, and 02 = 00. (b) /3J = 0-2, # = 0-2; (c) 0 | = 0 1 , 0^ = 0-282; (d) 0 J = 008, 03 = 0-298; (e) 0,' •= 005, 03 = 0-323. Ro = 8 0 in both cases.


,Ti o, + n

VARIABLE INFECnOUSNESS IN HIV TRANSMISSION MODELS

189

If we wish to compare the predictions of a model with constant /3 with those of a model with variable P(u), a simple relationship provides a means of ensuring that the values of Ro are approximately the same for each model:

P~2PiT,/T.

(16)

4. Mechanistic model

It seems likely that the length of the incubation period of the disease AIDS in an individual patient and his/her infectiousness to susceptible sexual partners are closely associated. Clinical and immunological studies suggest that HTV antigen concentrations in blood serum samples remain high in those individuals who convert to AIDS on a short time-scale (Goudsmit et al., 1987; Pedersen et al., 1987). Conversely, in those who convert to AIDS on a long time-scale, the two episodes of high antigenaemia may be separated by a long period, perhaps of many years duration, during which HIV antigen is difficult to detect by current methods. These observations tentatively suggest that a useful approach in the mathematical description of HIV transmission is to formulate a model in which antigen concentration (assumed to reflect virus abundance in the infected person, and hence his/her infectiousness) is linked to the duration of the incubation period. In this section, we formulate a model which encapsulates a simple mechanistic description of the relationship between the incubation period and infectiousness. We consider two phases of HIV infection, which we shall refer to as the primary and the secondary phases. For reasons that shall become apparent below, we start with the secondary stage. 4.1 The Secondary Infectious Phase We define the secondary infectious phase as the period which follows the initial rise and fall in antigenaemia. During this secondary phase, the patient shows increasing signs as time progresses of immunodeficiency and overt disease, up to


Note that equation (16) is linear, so that there exists a range of values of f$j which have the same equivalent /S and Ro, given the values of 2}. Figure 3 shows a comparison between four incidence curves, (b)-(e), for the N subclass variable infectiousness model (N distinct 0/s, but Ro is the same for all), and the simple model, curve (a), in which /3(u) is assumed to adopt a constant value /3 (from equation (16), the same for all four cases) throughout the duration of stay in the infected (Y) stage. In Fig. 3, n = 0-03125 year"1, c = 20 partners year"1, and the average incubation period T is set at 8 years. For the multi-stage model, we take N = 3 with Tj = 0-9 year, T2 = 6 years, and T3 = 1-1 years, and a variety of values of the /J/s (/32 = 0-0) for which Ro = 8-0. For the simple model, equation (16) yields an estimate of /3 of 005 to obtain the same value of Ro = 80. Clearly, the constant /3 model is only a good approximation of the multi-class model in case (d).

190


= (W/x2)g(W/x).

(17)

We can then write down an equation for the rate of change with respect to time of the number Y2(t) of infected people in the incubating stage, very much in the spirit of the previous section: At

= m{t)-V{t)-nY2{t),

y2(0) = 0.

(18)

Here, m(i) is the rate of recruitment to the secondary phase from the primary phase of infection (which has a population of Yx{i) at time t) and V(t) (as in equation (3), but with F(x) rather than/(;t) and m{t -x) rather than r(t -x)) is the incidence of developing AIDS (i.e. leaving the Y2 or secondary stage). In

FIG. 4. The inverse relationship between a (rate of increase in phase two) and phase-two duration x. a, >a1>oc3 implies *, N points uk (k = 1, 2 , . . . , M)); (iv) vary N and repeat until good fits are obtained to data on the form of / ( « ) and f}(u). Note that all this depends on the availability of data. At present, reasonable data are available for f(u) (see Medley et al., 1987), but information on /J(u) is extremely limited. In the estimation procedure defined above, the evaluation of equation (10) for fi(u) may appear to be somewhat daunting. Note, however, that, if we are fitting the function fi(u) at M points, then N

£(«*) = 2 fat (* = 1, 2, . . . , M),


core antigens to development of clinical disease in HTV infection. Br. Med. J. 295, 567-72.

200


where

Thus, we can obtain estimates of the pys by linear least-squares methods; the vector of fitted values is given by Here, D,j = 2 Win

(.hi =

1,2,...,N)

and A = S ^ob.(«»)

(; = 1, 2 , . . . , N).


k-l