J. Math. Biol. (1992) 3l: 73-99

Journal of

Mathematical

Wology

:g) Springer-Verlag 1992

A modeled time-varying density function for the incubation period of AIDS Marc Artzrouni Department of Mathematical Sciences, Loyola University, New Orleans, LA 70118, USA Received May 14, 1991; received in revised form September 30, 1991

Abstract. Building on the Weibull distribution, we develop a modeled time-varying density function of the incubation time between exposure to HIV infection and full-blown AIDS. This approach leads to a series of cohort-specific density functions that take into account the increasing impact of new therapies such as zidovudine (AZT). The resulting modeled density functions are studied in detail, particularly with regard to their modes and medians. The mode is sensitive to changes in the period incubation time distribution, with even a possibility of a bimodal distribution for certain combinations of the parameters that determine the rate at which the period median incubation time changes. An important substantive result is that when a period median incubation period slowly increases to some leveling off value, say m(xc), then it is surprisingly early on that cohorts of infected individuals have a median incubation period very close to that ultimate value m(xc). words: Human immunodeficiency virus ( H I V ) - A c q u i r e d immunodeficiency syndrome (AIDS) Incubation period -Weibull distribution

Key

1 Introduction

Efforts at the mathematical modeling of the various aspects of the HIV/AIDS epidemic continue at an accelerated pace (Castillo-Chavez 1990). In particular, the distribution of incubation times between infection with HIV and full-blown AIDS has received a lot of attention in the last few years (see Lui et al. 1986, Lui et al. 1988, Medley et al. 1987, Bachetti and Moss 1990, De Gruttola and Lange 1989, Blythe and Anderson 1988, among others). In the early phase of the epidemic it was quite natural to consider that the distribution of incubation times was characterized by one density function that did not change through time. (Different functions or parameter values were considered for different modes of transmission, however.) It is difficult enough to estimate parameters when conditions do not change through time, and making such an assumption was quite reasonable at the beginning of the epidemic. However, one question that is beginning to gain attention is that of a possible lengthening in the incubation period of AIDS. This is occurring as a

74

M. Artzrouni

result of new treatments that delay the progression from H I V infection to AIDS. The example of zidovudine (AZT) is the first one that comes to mind. Epidemiologists have indeed begun to speculate on the impact that A Z T may have on the A I D S epidemic in settings where A Z T is becoming widely available (Anderson et al. 1991, Gail et al. 1990, Brookmeyer and Liao 1990, Brookmeyer 1991, Solomon and Wilson 1990). Some have already noted peculiarities in the incubation distribution for cohorts of infected individuals using A Z T (Schechter et al. 1989). Others have investigated the impact of a variable incubation period in the context of a dynamic model of H I V transmission (Thieme and CastilloChavez 1991). In view of these developments, it seems appropriate to contemplate the question of a time-varying model of incubation times distribution. Concretely, this means that a density function p(x) of incubation times will have to be indexed, in some way, by time. In the simple case of a time-invariant incubation period, the quantity p(x) dx is the probability that an individual will develop AIDS when his infective age is in the interval [x, x + dx]. (The infective age is the length of time an individual has been infected with HIV.) In order to address the question of a time-varying density function p(x) we will make use of the hazard rate Ix(x) of developing AIDS; Ix(x) dx is the conditional probability of converting from H I V infection to AIDS at infective age [x, x +dx] for a person who is known not to have converted before infective age x. The hazard rate is

IX(X) -

p(x)

l" ~

(1.1)

1 - ~o p(s) ds A number of models for p(x) have been proposed (the Weibull, log-normal, normal distributions, etc.). We will focus here on the Weibull, partly because the hazard rate #(x) for the Weibull density has a simple expression. The Weibull density p(x), parameterized by the median incubation period m and the shape parameter//is

p(x) = / / ( I n 2)m ~x ~ i e x p [ - ( l n 2)[x/m] t~]

(1.2)

and the corresponding hazard rate Ix(x) is Ix(x) = / / ( l n 2)m l~x/~ i

(1.3)

For future reference we note that a Weibull density p(x) with median m and shape p a r a m e t e r / / h a s a mode equal to m [ ( / / - 1)/(//ln(2))] ~/l~. We call this latter quantity H(m, //). We now assume that the risk of developing A I D S changes with time; p(x) then becomes a function of the infective age x and of the time v at which an individual became infected with HIV. Equivalently, p(x) is a function of x and of the time t at which the conversion from HIV to A I D S occurs. Indeed, for a fixed infective age x, it is equivalent to index time with respect to the time of infection or with respect to the time of conversion t since t = z + x: given x we know as soon as we know t and vice-versa. In the sequel, the Greek z will always refer to a time of infection whereas t will refer to the time of conversion to AIDS. We now let pc(x, ~) be the density function of the incubation period for a cohort of individuals infected at time ~. (The subscript " c " is to emphasize that pc(x, ~) refers to a cohort of individuals who became infected at the same time ~.)

Incubation period of AIDS

75

We note that the function 1000pc(x, r) is the density of new AIDS cases occurring after x years among a hypothetical cohort of 1000 individuals who became infected at time r. There are now two possible approaches to the modeling of the time-varying density pc(x, 3): the change in the density function may either result from a cohort effect, or from a period effect.

(i) Modeling changes in the density function as a cohort effect One can assume that each cohort of infectives (i.e., each group infected at some time ~) will convert from HIV infection to AIDS with an incubation period that has a Weibull density function where the median m(~) and the shape parameter /~(~) depend on the time of infection ~. In this approach there is a model density function that changes from one cohort to another. In such a case the modeled density function pc(x, ~) is formally identical to the density of Eq. (1.2) except that the constants m and /~ are now functions of T:

p,(x, r) =/~(v)(ln 2)m(r) l~¢~)x/~C¢) 1 e x p { - ( l n 2)[x/m(r)]~¢¢)}.

(1.4)

This approach assumes that changes in incubation period result from a cohort effect: a cohort infected at time r convets to AIDS following a modeled density function, say the Weibull, with parameters m(r) and /~(r) that change from cohort to cohort with z. However, this approach is not satisfactory because changes in the risk of AIDS do not affect one cohort after another, but rather cut across cohorts as new therapies become available. Indeed, at the beginning of the epidemic there was no treatment for HIVinfected persons, and one may assume that all individuals (at least within a given risk group) were subjected to some constant density function for the incubation time. ( F o r example a Weibull with parameters m and /~.) When various therapeutic methods and treatments (such as AZT) become available, they benefit all individuals, regardless of the length of time they have been infected. This results in a lengthening of the incubation period, but in a manner that affects all cohorts of infectives. If these cohorts were previously developing AIDS following a Weibull distribution, they are now subjected to a period effect in the form of a new treatment (or drug) that affects all cohorts at the same period t. This effect results in a change in the median incubation period m and shape parameter/~. Each cohort is no longer subjected to a single Weibull density with a fixed median incubation period, but rather to a whole series of Weibull densities with constantly changing parameters m and ft. The resulting density function for a given cohort is then no longer a Weibull function. This case is treated below, with recourse to the time-varying hazard rate of getting AIDS.

(ii) Modeling changes in the density function as a period effect We suppose that at each period t there is a period Weibull density function p~(x, t) corresponding to the state of medical knowledge prevailing at time t, i.e.,

pp(X, t)

=/~(t)(ln 2)m(t) t~<'~xt~/ i e x p [ - ( l n 2)[x/m(t)]/~')].

(1.5)

76

M. Artzrouni

This density is formally identical to pc(X, z) of Eq. (1.4), but it is fictitious in the sense that no individual or group of individuals will actually experience this particular density function. It is the hypothetical density function that would be experienced by a cohort of individuals who would be subjected during their entire illness to the modeled density pp(X, t) reflecting the state of medical knowledge at time t. The corresponding hazard rate #(x, t) = fl(t)(ln 2)m(t) ~t)x~(')

(1.6)

1

will be the real, actual hazard of developing AIDS at time t for an individual who has been infected x years. This time-varying hazard rate will now be used to derive the actual density function pc(x, z) for a cohort infected at time z. We have

If0

p~,(x, ~) = #(x, T + x) exp -

]

#(s, ~ + s) ds .

(1.7)

The term #(x, z + x) (when multiplied by dx) is the probability of converting to AIDS at time r + x for a person with infective age in the interval (x, x + dx), who is known not to have converted during x years. (This person thus became infected at time r.) The exponential term is the probability of not converting during the first x years of infection for a person who became infected at time r. Substituting p(x, t) of Eq. (1.6) into Eq. (1.7) yields pc(x, r) = (In 2)]3(r + x)m(z + x) -/~(~ + X~xl~ + x)-- l

[

xexp-(ln2)

fo

]

fl(z + s)m(r + s) ~(" + ")s ~(" + ") l ds .

(1.8)

This function pc(X, z) is thus the actual density of incubation times for an individual infected at time r, who is subjected to the time-varying hazard rates #(x, 0 of Eq. (1.6). The remainder of this paper will be concerned with a detailed study of the function pc(x, z) of Eq. (1.8).

2 The density function pc(x, T) The density function pc(x, r) of Eq. (1.8) is of interest for two reasons: first it is a new model of the time to AIDS that incorporates the now realistic situation in which the risk of converting from HIV infection to AIDS changes with time. Second, we alluded earlier to the fact that the density pc(x, z) (after multiplication by some radix, say 1000) represents the density of new AIDS cases within a cohort of 1000 individuals infected at time z and who are subjected to changing hazard rates of conversion to AIDS. The function 1000pc(x, r) can be viewed as the density of new AIDS cases in a cohort of 1000 individuals who may over time benefit from improvements in therapy. The density pc(x, "r) may thus prove useful in the modeling of the density of AIDS cases in cohort studies of individuals infected at different times z who are subjected to time-varying risks of converting from HIV to full-blown AIDS. One interesting and somewhat intriguing result that will emerge from the analysis is that for certain patterns of changes in the Weibull density function, we can actually obtain a bimodal density pc(X, r) in the variable x: there is first an increase in pc(x, "c), until a first maximum is reached. The function then de-

Incubation period of AIDS

77

creases, only to reach a second maximum later. This is of particular relevance in a cohort study of individuals infected at the same time: a bimodal epidemic curve of AIDS cases in that group can thus be brought about by a changing pattern in the incubation period. The expression for pc(x, 3) in Eq. (1.8) is completely specified as soon as the functions m(t) and /~(t) are known. To simplify matters considerably, we will assume that/~(t) remains a constant/~, and that only m(t) varies; pc(x, 3) is then completely determined by/~ and the form of the function m(t). In order to study the function pc(x, 3) we differentiate it with respect to x and find that Opc(x, v)/Ox is of the form

@~(x, 3) Ox

- E(x, z)[F(x, ~) - 6(x, z)]

(2.1)

where E(x, 3) is an expression that is positive for all (x, ~) and

F(x, ~) = (fl - 1)m(x + z) - ~xm'(x + ~)

(2.2)

G(x, ~) =/~(ln 2)x~/m(x + z) ~ - '

(2.3)

Our goal will be to learn as much as possible from the partial derivative #pc(x, ~)/#x for two special forms of the function re(t): first we will consider that m(t) is a cubic function that either increases or decreases monotonically, then stabilizes. In the case of an increasing m(t) this model corresponds to an incubation period that is short at first, then increases as medical progress extends the duration of the AIDS-free period of infected persons. The decreasing case is less realistic but will be included in the analysis because it presents no difficulty. The second case will be that of a linearly increasing or decreasing function m(t). This is the simplest and most stylized representation of a changing incubation period. (Figure 1 shows two examples of cubic and linear functions m(t).)

(i) Cubic growth of m(t) - general results The median m(t) will be assumed to increase (or decrease) according to a cubic law, from a value of m(0) at time 0 to m(xc) at time Xc. The cubic will have zero derivatives at t = 0 and t = xc, and the function re(t) will be constant for t > x,

m (xc)

....LIIIII............

15.00 v 12.00 E 9.00

J

6.00 0

cubicgrowth I

I

7.50

15.00

I

x¢

22.50

year Fig. 1. Modeled growth of

m(t) between 0 and 18 years

30.00

78

M. Artzrouni 3=3

4.0

3.0

A1

s

20

B2

: I

B2

o

0

0.75

1.50

2.25

3,00

r

Fig. 2. Regions of the (r, s) plane that determine different behavior patterns for the density pc(x, r) (cubic m(x))

(Fig. 1). In terms of the parameters m(0), x,, and m(x,.), the function m ( x ) can be expressed as m ( x ) = a[x 3 - 1.5x,.x 2] + re(O)

(2.4)

where a = 2[m(0) -m(xc)]/X3c . The theorem given below will provide results on the qualitative behavior of the density pc(x, z) as a function of the parameters r = m ( x c ) / m ( O ) and s = x~/m(O). We define the quantity K(/3) = [(/~ - 1)/(/3 ln(2))] w~. We will also make use of the quantity H ( m , #) = m[(/3 - 1)/(/3 ln(2))] lm = m K ( ~ ) discussed earlier that represents the mode of a Weibull density with median m and shape parameter/3. T o facilitate the statement of the theorem we first consider Fig. 2 which represents the (r, s) plane divided into several regions. The functions appearing in Fig. 2 are parameterized by/3 alone. 1. The straight line a~ is defined by the equation s = rK(/3). 2. The area A above al (A = {(r, s ) : s / r >~ K(/3)}) is subdivided into regions Ai and A2 separated by the function s = K(/3)[6r(r - 1)/(/3 - 1)] ~/2 whose graph is denoted 0" 2 : A I ----{(r, s) :(r, s) e A, and if r >~ 6/(7 - / 3 ) then s/> K(/3)[6r(r - 1)/(/3 - 1)] w2} A2 = {(r, s) c A , with r/> 6/(7 - / 3 ) and s < K(/3)[6r(r - 1)/(/3 - 1)] w2}. 3. The area B below 0"~ (B = {(r, s ) : s i r < K(/3)}) is subdivided into two regions B 1 and B2: Bl = {(r, s):(r, s) e B and if 6/(7 - / 3 ) < r
(2.5)

4u212u2(2/3 + 1) - 6u(/3 + 1) + 2.25(/3 + 1)] 1 ~2(r(u) - 1)u2[(2/3 + I)u - 1.5(/3 + 1)] +/3 s(u) = u L ~ ~ ~ - 1)(-1-f~-- u ~ - ~ ] 7 -# l l ' m

(2.6)

where the parameter u is in the interval {([ i +/3]/[2/3 + 1], 1) }. B 2 = {(r, s) :(r, s) ~ B and if r >/6/(7 - / 3 ) then (r, s) is above the parametrically defined curve 0-3 defined in Eqs. (2.5) and (2.6)}.

Incubation period of AIDS When the parameter u increases from [ 1 + / ? ] / [ 2 / ? + 1 ] (r(u), s(u)) are on the curve ~r3 of Fig. 2 with

79 to 1, the points

(i) r(u) decreasing from/?2(5/? + 3)/(/? + 1) 3 to 6/(7 - / ? ) ; (ii) s(u) increasing from s = 0, reaching a maximum and then slightly decreasing to 6K(/?)/(7-/?) (which is the value of s at which the straight line a~ intersects the vertical line r = 6 / ( 7 - / ? ) ) . Theorem 2.1 The density function p~(x, r) has the following characteristics, depending on the position of (r, s) in Fig. 2: Case 1 (r, s) E A, i.e., s/r >~K(/?) (equivalently Xc/m(xc) >~K(/?); in such a case the quantity r,. = x,. -- m(xc)K(/?) is positive). Two subcases: (i) (r, s ) e A j : For any given fixed r <<.~,. the density pc(x, r) has a unique mode that is' less than x,. - v. For r e (r,., x,.) then pc(x, z) has the mode H(m(xc),/?) of the Weibull with median re(x,.) and shape parameter /? (although pc(x, z) is not itself a Weibull distribution). For r >~x,., p,.(x, r) is trivially a Weibull with median re(x,.) and shape parameter/? (and mode H(m(xc), /?)). (ii) (r, s)~A2: As in (i) for any fixed ~ <~r, the density pc(x, r) has a unique mode that is less than xc - r. There is a value z* (z,. <~r* <<,xc) such that for z ~ (re, z*) the density p,.(x, r) will be bimodal: one mode is less than xe - r and the other is equal to the mode H(m(xc),/?) of the Weibull with median m(xc) and shape parameter/?. For r E(r*, x,.) the density pc(x, r) has the mode H(m(x,.),/?) of the Weibull with median m(xc) and shape parameter/? (although pc(x, r) is not itself a Weibull distribution). When r >~x,. then p,(x, r) is trivially a Weibull with median m(x,) and shape parameter/? (and mode H(m(x,.), /?)). Case 2 (r, s)~B, i.e., sir < K(/?) (equivalently xc/m(xc) < K(/?)). Two subcases: (i) (r, s) E BI: For an), fixed z < x,. then pc(x, r) has the mode H(m(x~),/?) of the Weibull with median m(xc) and shape parameter/? (although p,.(x, r) is not itself a Weibull distribution). When r >1xc then p,(x, r) is trivially a Weibull with median re(x,.) and shape parameter/? (and mode H(m(x,.), /?)). (ii) (r, s)~ B2: There exists r + <~x,. such that for r <~r + the density pc(x, r) will be bimodal: one mode is less than x , . - r and the other is equal to the mode H(m(x,),/?) o( the Weibull with median m(xc) and shape parameter /?. For v 6 (z +, x,.), p,.(x, r) has the mode H(m(x,),/?) of the Weibull with median m(x,.) and shape parameter/? (although pc(x, r) is not itsel( a Weibull distribution). When r >~x, then p,(x, r) is trivially a Weibull with median m(xc) and shape parameter /? (and mode H(m(x,.), /?)).

Proof See Appendix I. The results of Theorem 2.1 are summarized in Table 1. We make two observations concerning the results of Theorem 2.1. 1. We note that the qualitative results are homogeneous in the parameters m(0), m(x,) and x,. in the sense that they depend only on r =m(x,~)/m(O) and s = x,./m(O): if the three parameters are multiplied by the same constant k, the point (r, s) remains unchanged and so is the qualitative behavior of the density pc(X, r). This suggests that it is the relative values of m(0), m(xc) and x,. that will determine the growth pattern of the density pc(x, r). For example, if for some set of parameters m(0), m(x,) and x,. we know from the position of (r, s) that pc(x, O)

80

M. Artzrouni

Table 1. Characteristics of pc(x, z) as functions of z and of the position of (r, s) in Fig. 1 (cubic m(x))

(r,s)i n ~

TC

A1

Onemode~

A2

One mode ~; x

x - r

A

B1

T+

x

Xc

One m~de = Hl(m(x c), Two modes: one s x - r ether = H(m(Xc), c l~)

One mode = H(m(Xc), Two modes: one ~ x c - x

B2

1:*

other = H(m(Xc) ,

~)

One mode = H(m(x c),

Weibull with median m( x c..), shape parameter #, mode = H(m(Xc), I~)

~)

One mode = H(m(xc),

13)

~,)

will have two modes, then the theorem insures that if the three parameters are multiplied by the same constant (say k) then pc(x, 0) will still have two modes, although they will not have the same values; the second mode for example will be equal to H(m(xc), fl) in the first case and H(km(xc), fl) in the second. Of course this result hinges crucially on the cubic assumption for the growth of the median m(x). A different functional form for re(x) may not yield the same neat homogeneity result. 2. The results of Theorem 2.1 lead to natural interpretations. Let us look for example at the values of r -- m(xc)/m(O). Figure 2 suggests that these values fall within three intervals, corresponding to three different types of behavior for the density pc(x, 0): r may be less than 6 / [ 7 - f l ] (r "small"); r may be between 6/[7 - fl] and f12(5fl + 3)/(fl + 1) 3 (r "intermediate"); and r may be larger than f12(5fl + 3)/(fl + l) 3 (r "large"). The case "r small" includes the situation in which r is less than 1 that corresponds to a decreasing cubic function m(x). (i) r ~< 6 / [ 7 - fl] (r "small"): Figure 2 shows that the qualitative behavior of the density pc(x, z) will depend on the value of s: if s is less than rK(fl) (i.e., (r, s) e B~ ~ xc/m(xc) <~K(fl)) then the theorem insures that the density pc(x, O) will have only one mode H(m(Xc), fl) equal to that of the "stabilized Weibull" prevailing at time t > xc. This is because with relatively small values of r and x++ the period Weibull distribution prevailing at any time t rapidly becomes close to the "stabilized Weibull" with median m(xc). It is therefore not surprising that pc(x, 0) will have a mode equal to that of the Weibull with median m(xc). If other parameters remain unchanged, then for a larger xc the median m(x) increases more slowly, and the influence of the early Weibull distributions has more time to make itself felt: as soon as xc/m(x~) > K(fl) the point (r, s) is in A l and the density pc(x, 0) will have a mode less than xc. (This mode will in fact approach H(m(O), fl) as x,+ --* oo. This can be seen from a careful examination of the intersection of the functions F(x, 0) and G(x, 0) when x+. is large.) Again, the interpretation of this result is natural: as Xc becomes large, the function re(x) stays close to m(0) longer and therefore for a longer period of time a cohort infected at time 0 will be subjected to period Weibull distributions that will differ only little from the "initial Weibull" with median m(O).

81

Incubation period of AIDS

(ii) r/>/~2(5fl + 3)/(fl + 1) 3 (r "large"): if s is less than rK(fl) (i.e., (r, s)~B2 xc/m(xc) <~K(/3)) then contrary to the previous case the density pc(x, 0) will have two modes: one reflecting the influence of the early lower values of m(x) and the other reflecting the impact of the "stabilized Weibull" with median m(xc). Hence there are two modes, regardless of how small s is; with a small s (i.e., a small value of xc) one could have expected like in the previous case that the effect of the early low values of m(x) would not have had the time to manifest itself. However, the theorem insures that because r is large, there always will be an early mode. (iii) 6/[7 -/3] < r
consider the values/3 = 2 and 3 that are of particular interest for two reasons. First because a statistical study has yielded estimated values for/~ of 2.396 and 2.571 for homosexuals and transfusion-associated AIDS cases, respectively (Lui et al. 1988). Thus a range from 2 to 3 seems reasonable, and includes a fairly skewed Weibull (for /3 = 2) as well as a more symmetrical one (for /3 = 3). (Figure 3 depicts these two densities for a median of 8 years.) Another reason why this range from 2 to 3 is of interest is that the density p,(x, T) can be calculated in closed form for/3 = 2 and/3 = 3. (This is the object of the next section.) The regions Am, A2, BI, B2 for the two values of/3 are depicted in Fig. 4. (The figure for/3 -- 3 is therefore the same as that of Fig. 2.) The intersection of the three curves al, a2, a3 is at [6/(7 -/3), 6K(/3)/(7 - 3)] which is (1.2, 1.02) for /3 = 2 and (1.5, 1.48) for/~ = 2: this intersection moves toward the north-east from (1.2, 1.02) as /3 increases from 2 to 3; K(/3) is an increasing function of/3, so as /3 increases the line ~r~ moves upward, and the region /3~ expands. At the same time the curve a2 comes closer to the line al. 0.140

=

0.105

0.070

0.035

0

I

i

i

5.00

10.00

15.00

20.00

Fig. 3. Weibull density functions for a median of 8 years and two different shape parameters fl = 2 and fl = 3

82

M. Artzrouni

,0]

4.0' ,=2 3.0'

s 3°1

2it

2.0" 1.0

B2

1.

0 0 8

I

I

i

0.75

1.50

2.25

3.00

r

0

i

0.75

i

f

1.50

2.25

3.00

b

Fig. 4. Regions A l, A2, B~, B 2 for fl = 2, 3, with a rectangle of "most likely" values for the parameters (r, s)

The study of the sensitivity of the regions A1, A2, BI, B 2 raises the question as to which areas of the (r, s) planes are the most likely in terms of the actual position of the point (r, s). We indicate on the graphs a rectangle of "most likely" values that includes values of r and s both between 1 and 2. Suppose m(0) is 10 years, which is a realistic value of the median incubation period at the beginning of the epidemic. Then s in the (1, 2) interval means that xc/m(O) = xc/ 10 is between 1 and 2: thus the time xe at which the median would reach its maximum would be between 10 and 20 years after the beginning of the epidemic. This appears to be a reasonable range for the time needed to bring the median up to some maximum attainable value, through various interventions, such as the use of AZT, for example. If the parameter r is also in the range (1, 2), then m(Xc) is between 10 and 20, which covers a range from no improvement to a doubling of the median incubation period. Again, this seems a reasonable range of values for r. It is then of some interest to note that the two rectangles cover for both values of fl the region around the intersection of the three functions a~, a2, a3. Thus for plausible values of the parameters m(0), m(Xc), and xc the behavior of the density pc(x, ~) can conform to either one of the four patterns corresponding to the regions A~, A2, B~, B 2 - a n d a small shift in the parameter values can move the density from one pattern to another. These results and observations will be illustrated below with actual examples of the density pc(x, ~) that can be obtained in closed form for when/3 = 2 or 3.

(ii) Cubic growth of m(t) - closed form expressions for/3 = 2, 3 For a constant shape parameter/3 and the cubic m(x) = a[x 3 - 1.5xc] + m ( 0 ) for the time-varying median incubation period, the density pc(x, ~) of Eq. (1.8) is

I

/3(ln 2)x ~- 1 exp -/3(ln 2) pc(X, ~) =

io

s,,.s

1

[a[(z + x) 3 - 1.5xc(z + x) 2] + m(0)] t3

[a[(z + x) 3 -- 1.5x,.(r + x) 2] + m(0)] ~ (2.7)

In general pc(x, ~) cannot be calculated in closed form because of the exponents/3 and/3 - 1 in the integrand. (This is why only the qualitative results of the previous section could be given in general.) However, for fl = 2 and/3 = 3

Incubation period of AIDS

83

the integral I(x, ~) appearing in Eq. (2.7) can be calculated in closed form using elementary functions. In order to calculate I(x, ~) for/~ = 2 or 3 it is necessary to obtain a partial fraction decomposition of the integrand. In order to do that the cubic m(x) must be written as

m(x) = (x - d)(ax 2 + bx + c).

(2.8)

The parameter a is the same as above and b, c, d are also functions of m(0), x,., and m(x,.); d is the real solution of the cubic equation x, = 2/3 x [d + m(O)/(ad2)] and is

The parameter b is then equal to -m(O)/d 2 and c = bd. The integral I(x, ~) becomes

For /~ = 2 or / / = 3, it can be seen, using standard calculus, that for properly chosen constants Ao, A1, A2, A3, A4, As, Bo, and B~, the integral I(x, r) in Eq. (2.10) can be expressed as

where the first term on the right-hand side of Eq. (2.11) is the value of the expression in the brackets taken at u = z + x minus the same expression at u = ~. The constants Ao, A1, A2, A3, A4, As, Bo, and B~ depend on the parameters a, b, c, d, a n d / / , and are given in Appendix II. The integral remaining on the right-hand side of Eq. (2.11) can now be expressed using elementary functions and we obtain

84

M. Artzrouni

where

p,=b/a

q,=c/a

S = [ B o + B,d]/[ad2 + b d +c]

R1 = - a S

Ro = BI - S(ad + b).

(2.13)

With the integral I(x, ~) given in closed form by Eq. (2.12) the density pc(x, ~) of Eq. (2.7) is /3 In 2x a - 1 e x p [ - f l In 2 + I(x, ~)] pc(x, ~) = [a[(r + x) 3 - 1.5x,.(~ + x) 2] + m(0)] t~"

(2.14)

It should be remembered that for x > xc the function m(x) is constant and equal to m(xc). Therefore, strictly speaking Eq. (2.14) is the expression for p,.(x, r) only for x + r < x,.. We recall that I(x, r) is the integral from 0 to x of s ~ 1/m(z + s)/3. If r > Xc, then trivially I(x, ~) = x~/flm(Xc)~; the denominator appearing in Eq. (2.14) is m(xc) ~ and p,:(x, r) is of course the Weibull with median m(xc) and shape parameter/3. If x + r > xc (with r < Xc) the denominator appearing in Eq. (2.14) is m(Xc) ~ and I(x, ~) is really

I(xc - ~, r) +

f[

se- ~

m-~;~ ds = I(xc - z, r) + c

x~ _ (xc - ~)~ m(xc)t~/3

(2.15)

T

where I(xc - ~, ~) is given by Eq. (2.12). In Fig. 5 we depict typical densities pc(x,r) for xc = 18, m ( 0 ) = 10, m(x,.) = 19 (r = 1.9, s = 1.8) and for three cohorts of HIV-infectives: those infected at time r -= 0 (i.e., when the period median incubation period was then m(0) = 1 0 ) ; those infected at time z =3.5; and those infected at time T = xc = 18). When/3 = 2 the point (r, s) (r = 1.9, s = 1.8) is in A2 (see Fig. 4). The density pc(x, 0) has one early mode and is thus rather under the influence of the early low median m(0) = 10 years. We recall from Table 1 that for ~ larger than rc = xc - m(xc)K(fl) (here % = 1.9) and r less than some r* (that cannot be calculated in closed form) the density will have two modes. In Fig. 5 the density is plotted for z = 3.5, and the two modes are apparent, although not very pronounced. (This was always the case for all realistic values of (r, s).) When r reaches Xc = 18 the density pc(x, 18) is exactly the Weibull with median m(x,) = 19 (dashed line). These three densities illustrate how different cohorts are subjected to different influences: cohorts infected early have an early mode reflecting the lower early median and the fact that early on the median m(x) does not change rapidly. Intermediate cohorts are subjected to more rapidly changing medians m(x) and to both lower and higher period median incubation period. As a result of these influences we see that the cohort infected at r = 3.5 has a density with a long plateau of about 15 years (i.e. from 5 to 20 years). If we recall the interpretation of say 1000p(x, 3.5) as the density of new AIDS cases for a cohort of 1000 individuals infected at time ~ = 3.5, we see that this particular cohort will have a long and protracted epidemic. When/3 = 3 the point (r, s) is in B 2 (see Fig. 4). The density p,:(x, 0) has already two modes. The first corresponds to the influence of the early low median incubation period m(0) = 10. The second is equal to 18.8, the value of the mode of the "stabilized" Weibull with median 19 years - even though the period Weibull density does not reach its median of 19 years for another 18 years (x, = 18). As increases pc(x, ~) slowly changes shape and loses its first mode (for some ~ = r +) as in Fig. 5 where the density p,(x, 3.5) is graphed, as well as the Weibull pc(x, 18). We note how all three densities have the same mode H(m(x,),/3) = 18.8.

Incubation period of AIDS

85

0.060

[3=2

0.045 x" 0.030 c£

0.015 ~~,/,

,~k/,Mode=H(m(Xc),~) =16.1 ' 10.00

20iO0

a

' 30.00

...... 40.00

x (years)

0.060

[3=3

.-/'f" -tl "'" ""-. t=Xc=l 8 "~"

0.045

~

,

,

,

.

x

0.030

\

0.015 "~,

o

10.00

20.00

b

30.00

40.00

x (years)

Fig. 5. Densities pc(x, ~) of Eq. (2.15) for x~ = 18, m(0) = 10, m(x~) = 19 (r = 1.9, s = 1.8) and for three cohorts of HIV-infectives: z = 0, z = 3.5, and ~ = x,. - 18

(iii) Linear growth of m(x) - general results W h e n the m e d i a n i n c u b a t i o n time m(x) g r o w s l i n e a r l y f r o m m(0) for x = 0 to m(x,.) for x = x, (see Fig. 1), the e q u a t i o n for m(x) is

re(x) = m(O) + x[m(x,.) - m(O)]/x,,

x <~x,

(2.16)

a n d m(x) = m(x,) w h e n x > x,.. A g a i n , m(x,) c a n be either larger o r s m a l l e r t h a n m(0), the f o r m e r case b e i n g the m o r e realistic one. W i t h // c o n s t a n t a n d for x < x,. the d e n s i t y pc(x, z) o f Eq. (1.8) is (In 2)fix ~ i exp

p c ( x , ~) =

I

- (ln 2)//

I

m(0) +

m(0) -~

s/J l ds J ] m(x,.) -- re(O) (z + s)~ ~ Xc

n(x(.) - m(0) X(.

(r + x ) l ~ (2.17)

86

M. Artzrouni

When x > x,. the value of m(z + s) or (m(z + x)) is m(0) + (z + s)[m(xc) m(O)]/xc (or m(O) + (z + x)[m(xc) - m(O)]/x,.) as in (2.17) as long as the argument z + s (or ~ ÷ x) is less than Xc ; when this argument is larger than x,. then m(3 + s) or (m(3 + x)) is m(xc). A linearly changing function m(x) is a stylized representation of the change in the median incubation period that may be less realistic that any smoothly increasing function such as the cubic. There are two main reasons for studying this case, however. First we consider this case in order to assess whether the shape of the function m(x) plays an important role. It is indeed of interest to compare results between the cubic growth and the linear growth, this latter case representing a "neutral" pattern of change - neutral in the sense that in the absence of any knowledge concerning the concavity of a "real" function m(x), a linear growth is the least compromising assumption one can make. A second reason for studying the linear growth pattern for m(x) is that this case is obviously more tractable than the cubic growth case. Furthermore, there is one specific feature about the linear growth pattern that makes this case appealing. Let us write for a minute the density pfix, 3) of Eq. (2.17) as p,.(x, 3;m(O),m(Xc),X,.) to indicate explicitly the dependence on the three parameters m(0), re(x,.), x,.. It is easy to see that pc(x, 3; m(O), m(xc), x,) pc(X, 0; m(0) + 3[m(xc) - m(O)]/xc, m(xc), x,. - z): this identity simply expresses the fact that for a given set of parameters (m(0), m(xc), x,.), the cohort infected at time 3 will have the same density function as a cohort infected at time 0 with parameters m(0) + 3[m(x,,) - m(O)]/xe, m(x,), x~. - 3. (This property did not hold in the previous case of a cubic m(x); even though the density p,.(x, 3) involved for > 0 a cubic function m(r + x) that was translated from the original cubic m(x), the function m(z + x) was not a cubic that was reaching a minimum for x = 0 and a maximum for x = x,.. Therefore the density pfix, 3) for z > 0 was not the same as a density pc(x, 0) for different parameters of the cubic m(x).) In the previous section, r denoted the ratio m(xc)/m(O) and s the ratio x,./m(O). Here, for fixed (m(0), m(xc), x~.), we know that a cohort infected at time 3 will have the same density as a cohort infected at time 0 with parameters m(O) + ~[m(x,.) - m(O)]/x,., m(xc), xc - 3. The corresponding ratios r and s must now be indexed by 3: r(3) =m(x,.)/[m(0) +z[m(xc) -m(O)]/x,,] and s(3) = Ix,. - ~]/[m(0) + r[m(x,.) - m(O)]/x,.]. In short, for fixed (m(0), m(Xc), xc) and any z ~> 0 the density pc(x, 3) is the same as the density p,.(x, 0) with parameters r = r(3) and s = s(3). We note that the Jr(3), s(z)] point, which is parameterized by z, satisfies I s ( z ) - s ( O ) ] / [ r ( z ) - r ( 0 ) ] = s ( O ) / [ r ( O ) - 1]. In the r(3), s(3) plane the points Jr(z), s(3)] will therefore connect [r(0), s(0)] and [1, 0] for 3 increasing from 0 to x,.. The theorem below now mirrors the one given when m(x) was a cubic. Theorem 2.2 With a linear function m(x), the behavior of the density function p,(x, O) depends on the position of (r(0), s(0)) in the (r, s)plane (Fig. 6) (given as an example .for fl = 3). Case 1 I f s(O) <<,r(O)K(fl) (i.e., [r(0), s(0)] is in A1 or Az above the line al with equation s(r) = r(r)K(fl)) then pc(x, O) has a single mode smaller than xc. (i) I f r(O) < 1 (i.e., [r(0), s(0)] is in A1 on the left on the vertical line a2 with equation r(z) = 1) then [r(3), s(3)] will for 3 = 3,. = x,. - m(x,.)K(fl) enter the region B1 in which p,.(x, z) has one mode equal to H(m(Xc), fl). (An example of such a case

Incubation period of AIDS

87

4.0

[8=3

A1

s.(o):2 v. O m2.

^ ~2

0-2

,'{ s~e-)

.

/

Wr )a ~ ...~.~.~o'a'= J's(O)=l.67 / ~ 4 " ~ t / / " /#]'r(0)=3 B2

s,(xo):o ~ \

/~_.__3.

i

0

u1

~l'r(O)=0"7 ~1~%__,\

/ / 0

fs(0)=3.33 "~r(0)=3 ~ j ~

~"

1.00

-"

"

~r"

B1 i

2.00 r (*)

-

rs(,~ o3~., 13 i

3.00

4.00

Fig. 6. Regions of the (r(~), s(z)) plane that determine the different behavior patterns for the density p,(x, r). (linear m(x))

is depicted in Fig. 6 with s(O)= 2 and r ( 0 ) = 0.7; the dotted line shows the trajectory of the point Jr(r), s(r)] from Jr(0), s(0)] to (1, 0)).

(ii) I f r(O)>1 (i.e., [r(0),s(0)] is in A2) the [r(~),s(v)] point will for r = z< = xc - m(xc)K(fl) enter the region B 2 in which the density pc(x, r) has one mode smaller than x< - ~ and another mode equal to H(m(xc), fl). When r reaches the value ~* (where z* satisfies the equation s(r*)/r(r *) = [(fl/r(r *) - 1)/(fl In 2)] 11/~ then the point [r(z), s(v)] enters the region B] in which the density pc(x, r) has only the mode equal to H(m(x<.), fl). In the [r(Q, s(~')] plane the frontier between B] and B2 is thus the locus 0-3 of points [r(v), s(r)] satisfying the equation s(r)/r(r) = [(fl/r(r) - 1)/(fl In 2)] ]m.

(2.18)

(See example in Fig. 6 with s(0) = 3.33 and r(0) = 3.)

Case 2 I f s(O) > r(O)K(fl) (i.e., [r(0), s(0)] is below the line 0-]) then there are two subcases depending on whether [r(0), s(0)] is in B, or B2. (i) [r(0), s(0)] ~ B1, i.e. [r(0), s(0)] is below the curve 0-3. In such a case pc(x, O) has a single mode equal to H(m(x<.), fl). As r increases the point [r(~), s(r)] approaches (1, 0), always staying in Bl, and pc(x, ~) keeps the single mode H(m(x~), fi). (ii) [r(0), s(0)] EB2, i.e. [r(0), s(0)] is above the curve 0-3. In such a case p,.(x, O) has two modes: one less than &. and one equal to H(m(xc) , fl); pc(x, z) loses its first mode and keeps only the mode H(m(x<), fl) when ~ reaches r +, the value of ~ for which [r(z), s(v)] is on the locus 0-3. (See example in Fig. 6 with s(0) = 1.67 and r(0) = 3.)

Proof. The proof is similar to that of Theorem 2.1 and is elementary. It will be

omitted. An examination of Figs. 6 and 2 is of interest when comparing the effect of a cubic and of a linear growth of m(x) on p,(x, 0). In both figures the regions A], A2, B~, B 2 and the curves 0-1, 02, 0-3 have the same interpretation. The line 0-] is the same and the curve 0-3 has the same general shape in both cases. One important difference however is that the curves ~r] and 0-3 always intersect at

88

M. Artzrouni

r = 1 for m(x) linear, but intersect at 6/[7 -/~] for m(x) linear. Table 1 for a cubic can therefore be used for the interpretation of the linear case - only the shapes of the regions A~, B~ and of the curves cr~ change. Contrary to the cubic case, it is possible to follow on Fig. 6 the "qualitative" evolution of the density p~(x, ~) as ~ increases. Given an initial point [r(0), s(0)], the points [r(r), s(r)] moves on a straight line toward (1, 0). For every ~ the position of [r(~), s(~)] determines the behavior of p,(x, ~). The question of the sensitivity to/~ of the different regions in the (r(~), s(z)) plane is fairly straightforward and can be dealt with as in the case of m(x) cubic (Fig. 4). As before, the slope K(/~) of the line ~r~ is an increasing function of /~. For all values of /~ the curve ~r3 intersects the line ~ at r(~) = 1; for increasing/~ the curve ~r3 grows toward the right since ~3 intersects the r(r) axis at r(~) =//.

(iv) Linear growth of m(x) - closed form expressions for t~ = 2, 3 As before, we let I(x, ~) denote the integral appearing in Eq. (2.17); I(x, ~) can be calculated in closed form for/3 = 2, 3. We let 2 = [(m(x,.) - m(O)]/xc. (i) /~ = 2. For x
T +x I(x, r) = A ln(m(0) + 2u) 2 + m(0)q- 2uJ~

(2.19)

where A = 1/22 ~ and B = (m(0) + r2)/22. (ii) ~ = 3 .

Forx
[ Au+B ]~+x I(x, ~) = [ m ~ + 2u] 2 + C ln(m(0) + 2u) ,

(2.20)

where C = 1/23, A = [2C22m(0) + 2r]/2, B = [Am(0) + C2rn(0) 2 - rz]/22. With I(x, ~) given by Eq. (2.19) for/~ = 2 or Eq. (2.20) for/? = 3, the density pc(x, ~) of Eq. (2.17) is, for x < x~ - r (ln 2)/~x t~ i exp[-/~ in 2 x I(x, ~)] pc(X' "r)= [m(0) + m(xc)-m(O)xc

(z + x) 1 B"

(2.21)

As before, when x >~ xc - z the integral I(x, ~) in (2.21) should be replaced by I(xc-r,r)+the integral from x c - r to x of s ~ ~/m(x,.) ~ which is

[x # - (x,. - r)~]/[flm(x,.)~]. Figure 7 depicts the densities p~(x~, ~) with m(x) linear for the same parameters x c, m(O), m(xc), and r as in the cubic case of Fig. 5. (The densities in dotted lines, which are the Weibull distributions, are therefore the same in Figs. 5 and 7.) There is a discontinuity in the derivative #pc(x, r)/Ox at x = x c - ~ since m(r + x) has a sharp turn at x = x~, - r. This discontinuity is visible in Fig. 7. Other than this discontinuity, the densities appearing in Fig. 7 for re(x) linear are surprisingly similar to those of Fig. 5 for m(x) cubic. This example suggests that there is nothing essential about the shape of the function m(x): for the same parameters x~, m(O), m(x,.), the densities pc(x, ~) will be quite similar, whether m(x) is cubic or linear.

Incubation period of AIDS

89

0.060

T=0

0.045

,~=Xc=18 x~, 0.030

cZ

0.015

i

10 i00

' 20.00

a

' 30.00

40.00

x (years)

0.060

~=3

0.045

f

.,'~.T=Xc=18 "\

T=0

',,

x,~ 0.030

,£ 0.015

0

b

i

i

i

10.00

20.00

30.00

40.00

x (years)

Fig. 7. Densities pc(x, r) o f Eq. (2.21) ( w i t h linear m(x)), x , = 18, m ( 0 ) = 10, m(x~) = 19 (r - 1.9; s = 1.8) a n d f o r t h r e e c o h o r t s o f HIV-infectives: r = 0, r = 3.5, a n d ~ = x , = 18

Other growth patterns for m(x) were investigated, particularly an S-shaped logistic-type function. However, in this case it is harder to bring the parameters of the model into the analysis for fairly straightforward results such as those given in Theorems 2.1 and 2.2. Also, there are no closed form expressions for p~.(x, ~) when m(x) is a logistic function. Finally, we make one remark in the contexts of Figs. 5 and 7 representing the densities pc(x, T) in closed form for m(x) cubic and linear. Although x, is 18 years (i.e., the Weibull density stabilizes after 18 years) the density p,.(x, 3) becomes quite close to the Weibull p,(x, x,.) even for fairly small values of 3; p,.(x, 3.5) is represented in the figures; pc(x, 5) is quite close to pc(x, xc) and p,.(x, 10) is almost indistinguishable from pc(x, x,.). This example shows that even though the median incubation period m(x) will still be increasing for another 8 years, any density pc(x, 3) for ~ ~> 10 will be essentially equal to the stabilized Weibull p~.(x, x,). This result will be further investigated (and confirmed) in the next section which is devoted to a sensitivity analysis of the density p,(x, 3) to the parameters x,., m(O), m(x,) and 3.

90

M. Artzrouni

3 The median incubation period of each cohort

In the previous section we were able to study in some detail the density p,(x, ~) of incubation times for a cohort infected at any time ~. This was done by looking at the mode of the distribution and its general form - which can be investigated in closed form when m(x) is cubic or linear and the shape parameter fl is either 2or3. It is however desirable to study other characteristics of the density p~(x, ~) such as its mean or median. It turns out that the median of pc(x, ~) is easy to determine numerically when pc(x, ~) can be calculated in closed form (i.e., when fl = 2 or 3). Indeed, by elementary considerations one can see that the median of the density function p,(x, ~) is the solution m of the equation

I

~ m(r

+ s)-~s ~ l ds

fl1

(3.1/

The integral on the left-hand side is I(m, ~); m is thus the solution of I(m, z) = 1~ft. As a verification it is easy to see from Eq. (3.1) that if re(x) is a constant m(0) then the solution m is of course m(0). The function I(x, ~) is a monotonically increasing function of x, since it is the integral from 0 to x of a positive function. When I(x, ~) is available in closed form it is therefore a very simple matter to find numerically the solution m of the equation I(m, ~) = 1~ft. (Of course all calculations, including those of the mean, can always be done numerically for values of fl other than 2 or 3, but with considerably more difficulty.) Table 2 contains the results of a sensitivity analysis of the median m to the parameters x~, m(O), m(x~,) and ~ (for a cubic m(x); "xc" appearing in the table is x~, the time at which m(x) becomes constant). The median m(0) prevailing at time 0 was kept equal to 10 throughout - a reasonable assumption concerning the median incubation period at the beginning of the A I D S epidemic. There are two panels, one corresponding to xc = 10 and the other to xc = 20: the median m(x) reaches its maximum after 10 or 20 years; m(xc), this ultimate "leveling-off" value of the median, is taken equal to 15 and 20: a moderate and a dramatic improvement. The median m of the distribution pc(x, ~) is calculated for even values of ~ from 0 to 16, and m/m(xc) (multiplied by 100) is also given. For example with xc = 10, m(xc)= 15, and f l - - 2 the median incubation period m or the cohort infected at time ~ = 4 is 14.9 years, which represents 99.1 percent of m(xc) (14.9 is a rounded value, which explains why 14.9/15 is not 99.1). This is a typical example of the most striking result that emerges from the table: it is surprisingly early on that cohorts of infectives have a median incubation period very close to the stabilized median prevailing after x c. When Xc = 10, the median m reaches 99 percent of m(xc) at the latest for the cohort infected at time ~ = 4. Even though the median m(x~) will still be increasing for another six years the actual median incubation period m is essentially equal to m(xc) for cohorts infected at time T = 4 and beyond. Similarly, when x~ = 20, the 99 percent mark is reached at the latest for ~ = 12. It is normal that the 99 percent m a r k should be reached later since re(x) increases for 20 years, instead of 10. We also note that for the same values of x~., m(xc) and ~, the median m is closer to m(xc) when fl = 3 than when fl = 2. For example, with xc = 20 and

Incubation period of AIDS

91

Table 2. Sensitivity of the median m of the density pc(x, r) for different values of the parameters x c, m(O), m(xc) and r (for a cubic m(x)) x,. = 10

"c 0 2 4 6 8 10 12 14 16

/~ = 2; m(x,) = 15

,6 = 3; m(x~) = 15

/3 = 2; m(x,.) = 2 0

/~ = 3; m(xc) = 2 0

m 14.0 14.6 14.9 15.0 15.0 15.0 15.0 15.0 15.0

m/m(xc) 93.4% 97.3% 99.1% 99.8% 100.0% 100.0% 100.0% 100.0% 100.0%

m 14.6 14.9 15.0 15.0 15.0 15.0 15.0 15.0 15.0

m/m(xc) 97.0% 99.1% 99.8% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0%

m 18.6 19.5 19.8 20.0 20.0 20.0 20.0 20.0 20.0

m/rn(xc) 93.0% 97.3% 99.2% 99.8% 100.0% 100.0% 100.0% I00.0% 100.0%

m 19.5 19.9 20.0 20.0 20.0 20.0 20.0 20.0 20.0

m/m(xc)

m 11.6 12.5 13.3 14.0 14.5 14.7 14.9 15.0 15.0

m/m(x~.) 77.6% 83.3% 88.8% 93.4% 96.4% 98.3% 99.3% 99.8% 99.9%

m 12.1 13.0 13.8 14.4 14.7 14.9 15.0 15.0 15.0

m/m(x~.) 80.4% 86.5% 92.0% 96.0% 98.2% 99.3% 99.8% 99.9% 100.0%

m m/m(x,,) 14.1 70.3% 16.1 80.6% 17.7 88.7% 18.7 93.7% 19.3 96.7% 19.7 98.5% 19.9 99.4% 20.0 99.8% 20.0 100.0%

m 15.5 17.6 18.8 19.4 19.8 19.9 20.0 20.0 20.0

m/m(x~.) 77.4% 87.9% 94.0% 97.2% 98.8% 99.5% 99.8% 100.0% 100.0%

97.6% 99.3% 99.8% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0%

x, = 20 "c 0 2 4 6 8 10 12 14 16

m(x,.) = 20 the median m for a cohort infected at time r = 4 is 17.7 w h e n / / = 2 and 18.8 when /~ = 3; for /~ = 2 m has reached 88.7 percent of the ultimate value 20, but a full 94 percent of that value was reached when/~ = 3. This result suggests that when the underlying shape parameter is 3 (i.e., the "stabilized Weibull" pc(x, xc) is rather symmetric, see Fig. 3) then the density pc(x, ~) will more rapidly approach the Weibull with median m(x,.) than when the shape parameter is 2 (i.e., when the "stabilized Weibull" pc(x, x,) is more skewed). This phenomenon must have something to do with the skewness of the Weibull, although it is difficult to interpret and arguably counter-intuitive. The opposite result could also have been expected. Indeed, Fig. 3 shows that a more skewed Weibull with/~ = 2 gives more weight to individuals who develop AIDS later on. The more individuals remain uninfected a long time, the stronger should be the influence of the later incubation period m(x) that is also closer to the stabilized median m(xc); intuitively, when fl = 2, one could then expect a given cohort to have a density with a median closer to m(xe) than if /~ = 3. Such an interpretation is questionable though, as we have a fairly complex mechanism at work with cohorts, subjected to constantly changing incubation times m(x). In any event, the substance of the result is clear: a more symmetric "stabilized Weibull" leads to a density pc(x, ~) that will approach p,(x, xc) more quickly than when the "stabilized Weibull" is more skewed. (This can be seen in Fig. 5, where pc(x, 3.5) appears somewhat closer to p,.(x, xc) for/? = 3 than for/~ = 2.)

92

M. Artzrouni

4 Discussion This paper discussed the impact of a changing regime of incubation time distributions on the actual density function pc(x, ~) experienced by different cohorts of HIV-infectives. We saw that the density pc(x, ~) could take quite a variety of shapes, depending on the parameters of the cubic function m(x) used to model a changing median incubation time: from one early mode when the early influence of m(x) was strongest, to two modes and/or a long plateau when both the early and late influences were strong, to a single late mode corresponding to the stabilized Weibull prevailing for cohorts infected after time x,. One important substantive result is the fact that the impact of the stabilized Weibull is strong quite early on. Indeed, for relatively small values of z the density pc(x, r) will be close to the Weibull p,(x, x,), both qualitatively (in terms of the shape) and quantitatively (in terms of the median). Qualitatively, Table 1 shows that there are values of r less than x,. (i.e., cohorts still subjected to a changing regime of incubation times) for which the density will have a unique mode H(m(x,), ~) equal to that of the Weibull p~(x, x,). For example if (r, s) is in A~, then p~(x, r) will have that single mode as soon as r reaches r,. This will occur when z reaches z* for (r, s) in A2, and when r reaches r + when (r, s) is in B2. When (r, s) is in Bl, the density p,(x, r) will have the single mode H(m(x,), [3) for all values of r. Let us take an example of this latter situation. From Fig. 2 it can be seen that if r = 2 and s = 1 then (r, s) is in Bl (for /3 = 3); r will be 2 and s will be 1 if x, = 10, re(x,) = 20, m(0) = 10. Therefore, the results insure that if the median m(x) starts at 10 years, increases slowly to 20 years over the next 10 years according to a cubic law, and stabilizes thereafter, then in fact every cohort of HIV-infectives will have the single mode H(m(x~ ), /3) ( = 19.7) of the Weibull p~(x, 20). Even though m(0) is 10 (with mode 9.9), from the very first cohorts on, the actual mode will be 19.7. Quantitatively, Table 2 and Figs. 5 and 7 show that for relatively small values of ~ the median incubation time to AIDS of an actual cohort will often be quite close to the median that will eventually be observed when m(x) stabilizes. In the previous example, Table 2 told us that for z --0 the median is already 19.5 years: the median incubation time for the first cohort infected at time 0 is already 19.5, even though the current period median incubation time m(0)is only 10. This numerical illustration provides an example of results that can be of some relevance in cohort studies of HIV-infectives who may be subjected to changing regimes of incubation time distributions. One may start a cohort study of 1000 individuals today, when current conditions suggest a median incubation time of 10 years. Yet, if the median starts to increase, as in the example above, it will be 19.5 years until 500 HIV-infectives have converted to full-blown AIDS. One might add that the median is a particularly attractive statistic in the context of a cohort study, given that by definition the median is determined as soon as 50 percent of the cohort has experienced the event, namely the conversion from HIV infection to full-blown AIDS. The median is therefore determined fairly early in a cohort study, and it is of course not sensitive to outliers. Finally we note the obvious next step in the line of research concerned with changing regimes of incubation time distribution. When the density pc(x, ~) is a time-invariant function p(x), the density of AIDS cases a(t) is given by the

Incubation period of AIDS

93

well-known convolution equation a(t) =

h(t - u)p(u) du

(4.1)

do where h(x) is the density of new HIV infections and ~o is the maximum possible incubation time. In the present context of a time-varying density of incubation time, the function p(u) in Eq. (4.1) is replaced with pc(u, t - u) since h(t - u)p(u) du is the number of conversions to AIDS from h ( t - u)du new infections u years ago during the time interval ( t - u, t - u + du). Equation (4.1) thus becomes a(t) =

h(t - u)p,.(u, t - u) du

(4.2)

and a(t) is therefore the density of new AIDS cases in an epidemic characterized by a density of HIV infections h(t) and a time-varying density function of incubation times pc(x, r). Equation (4.2) shows why it is desirable to have p,.(x, r) in closed form whenever possible. Indeed, in such a case, for a given h(t), the function a(t) of Eq. (4.2) can be calculated numerically using standard techniques. Having to calculate pc(x, r) numerically would considerably complicate matters since the calculation of a(t) would then involve the numerical calculation of an integral with an integrand that itself has to be calculated numerically.

Appendix I Proof of Theorem 2.1 We recall that the sign of @,.(x, z)/c?x is the same as that of F(x, ~) - G(x, z) where F(x, ~) = (fl - 1)m(x + z) - flxm'(x + ~), G(x, ~) = fl ln(2)x~/m(x + z) ~-- i and m(x) is the cubic m(x) = a[x 3 - 1.5xcx 2] + m(0). In order to get a feel for the functions F(x, r) and G(x, ~) we display in Figs. 8a, 8b, and 8c, three typical pairs of functions F(x, r), G(x, r) for three increasingly larger values of r (z --= 0, 6.555, and 9.5) and the same values of the parameters fl, m(O), m(x,) and x c (fl = 3 , m ( 0 ) = 8, m ( x c ) = 12, x c = 18.4). In other words, the median incubation period is assumed to increase following a cubic function from 8 years at time 0 to 12 years after 18.4 years. After that, the median remains unchanged and equal to 12 years. The three graphs will thus provide information on three cohorts of individuals (infected at times 0, 6.555, and 9.5) and subjected to the changing regime of the median incubation period defined by the parameters fl, re(O), m(x,.), and xc that were specified in this particular example. We note that the function F(x, ~) goes from ( f l - 1)m(0 to ( f l - 1)m(xc) when x increases from 0 to the value x* = xc - r; x* is the value of x for which the function m(x + ~) reaches its maximum and remains constant. For x/> x* the function F(x, z) also remains constant and equal to ( f l - 1)m(xc). The function G(x,T) is an increasing function of x that is equal to G(x, ~ ) = fl l n ( 2 ) x ~ / m ( x y ~ 1 when x ~> x*. Clearly, when z ~>xc the function F(x, ~) is constant and equal to (fl - 1)m(x,), and G(x, r) is equal to fl ln(2)x~/ m(xc) ~ ~ for all x: such a cohort of individuals infected at some time ~ ~> x,. is

94

M. A r t z r o u n i 60.0

xU_

G(x,~)

(.9 30.0 F(x,x)

15.o

15.0 IJ_

X* 7.50

a

15.00

o

|

22.50

30.00

i

0

b

X

60.0

7.50

15.00

22.50

30.00

x

'r~,5 G(×,~)

x~ 4 5 . 0

30.0

x~ 15.0 E

"r=6.555

x~ 45.0

(x,~

0

(.9

60.0

"t=O

x~. 45.0 (_9 30.0

---'~'__---~.' x* 0 C

F(x,~)

i

i

I

7.50

15.00

22.50

30.00

X

Fig. 8 a - c . Three pairs o f functions F(x, r) and G(x, z), for three different cohorts of HIV-invectives w h e n ( r , s ) eA~ of Fig. 2; fl = 3; x c = 18.4; m(xc) = 12; m(0) = 8; r = 1.5; s = 2.3

thus obviously subjected to a Weibull with median m(xc) and shape p a r a m e t e r fl (since m(x) = m(x~) for x ~> xc). The relative values of the functions F(x, z) and G(x, z) tell us whether the density pc(x, z) increases or decreases: pc(x, z) increases at x if F(x, T) > G(x, z) (and vice-versa). F o r example in Fig. 8a we see that for z = 0 the m a x i m u m of the function pc(x, z) is reached for x a r o u n d 8 years (i.e., F(x, 0) is larger than G(x, 0) for x less than a b o u t 8, and after that G(x, 0) is larger than F(x, 0)). Let us examine the case when the two functions will intersect at a value of x larger than x* (for example in Figs. 8b and 8c for z = 6.555 and z = 9.5). If x~>x*=xc-z this means that G(x,z)=flln(2)x~/m(xc) B 1 and F(x,z)= ( f l - 1)m(xc). Equating F(x, z) and G(x, z) yields a point of intersection at x = m(xc)K(fl) where K(fl) -- [(fl - 1)/(fl In 2)]l/~. This m o d e turns out to be the m o d e of the Weibull distribution with median m(x~.) and shape p a r a m e t e r fl; furthermore, this m o d e is independent of r. (This does not mean however that pc(x, z) is a Weibull - it simply means that the density has a m o d e equal to that of the Weibull with median m(xc) and shape p a r a m e t e r ft.) F o r this m o d e to exist the quantity m(xc)K(fl) must be larger than x c - Z which means that z must be larger than the critical value zc = x c - m(x,.)K(fl). Here m(xc)= 12, K(fl)= K(3) = 0.9871, and xc = 18.4, so zc = 18.4 - 12(0.9871) = 6.555. F o r any z >~ 6.555 the intersection of the two curves will therefore occur at x = m(xc)K(fl) = 12(0.9871) -- 11.8. F o r z = Zc = 6.555 this intersection will occur just at the value of x for which the function F(x, T) becomes constant (Fig. 8b). F o r z larger than 6.555 but less than x,. = 18.4 the density will thus have one m o d e equal to that of the "stabilized Weibull" prevailing after time x,, namely the Weibull with median m(Xc) and shape p a r a m e t e r fl (Fig. 8c). F o r >~ xc the density pc(x, "r) will be a Weibull with median m(xc) and shape p a r a m e t e r ft.

Incubation period of AIDS

95

We now attack the formal proof, which will rely on the graphs of the functions F(x, 3) and G(x, ~). The example of Fig. 8 tells us what to look for as far as the relative behavior of the two functions F(x, 0) and G(x, 0) are concerned. The key to the case in Fig. 8a is the relative values of F(x*, 0) and G(x*, 0): if F(x*, O) <~G(x*, 0) then as in Fig. 8a the density pc(x, 0) will have one mode ~~K(fl)m(x,.). In terms of the variables r =m(xc)/m(O) and s = x,/m(O) this means that s/r >~k(~) (i.e., (r, s) is in the region A = Al w A2). We now distinguish between the two cases, depending on whether (r, s) is in the region A, or its complement B = B 1 w B2.

Case 1 (r,s) eA, i.e., s/r>~K(B ) which means that F(x*,O)<~G(x*,O) and x,. >1K(fl)m(xc).

(i)

~ ,

Figure 8 shows that as ~ increases the point x* = Xc - r moves to the left and the function F(x, 3) becomes flatter; F(x*, 3) remains smaller than G(x*, 3) until reaches the critical value ~c for which the two functions intersect at x* (Fig. 9b). Indeed, if we recall that F ( x * , r ) = ( ~ - l ) m ( x c ) and G ( x * , ~ ) = / ~ ( l n 2 ) ( x c - O#/m(x,) ~-1 it can be seen that F(x*, 3) remains smaller than G(x*, 3) as long as ~ ~ r c and that G(x*,rc)=F(x*, 3,). (This is because x,. >1k(~)m(Xc).) In short, we see that for any fixed r ~ rc the density p~(x, 3) will have one mode less than x * - - x , - 3. (ii) r > ~, Figure 8b shows the relative position of the two functions F(x, 3) and G(x, ~) for ~ ~ rc will depend critically on the partial derivatives 8F(x, rc)/Sx and 8G(x, T,)/Sx at x = x*. (To be precise the partial derivative of F(x, ~) will be the partial derivative from the left, since the partial derivative from the right is 0.) Indeed, if at x* the partial derivative of G(x, ~c) is larger than that of F(x, re) (as in Fig. 8b) then the two functions will continue to have only one intersection as x* slides to the leR (i.e., as T increases beyond ~c). As we saw in the example above, that unique intersection will occur at x = m(xc)K(~) and is the mode of the Weibull with median m(xc) and shape parameter ft. Elementary calculations show that 8F(x*,r,.)/Ox = 6/~K(/~)(r-l)r/s 2 and 8G(x*, rc)/Sx =/~(/~ - 1)/K(/~). Setting 8G(x*, r~)/Sx >~OF(x*, ~)/Ox yields s >>.K(~)[6r(r - 1)/(/~ - 1)] ~/2. We now let 0"2 denote the curve s =K(~)[6r(r- 1 ) / ( f l - 1)] ~/2. (See Fig. 2 in the text.) The curve 0"1 (i.e., the straight line s = rK(~)) and the curve a2 intersect at r = 6 / [ 7 - / / ] , s = 6K(/~)/ [7-/~]. The theorem is thus proved for (r,s) eA1: indeed, in such a case sir >>.K(fl) and s >~K(~)[6r(r - 1)/(/~ - 1)]1/2 which shows that for ~ ~< ~ ~ x,. the density is trivially a Weibull with median m(x,.) and shape parameter/~.) If s 8F(x*, r,.)/Sx. Figure 9 illustrates such a situation in which (r, s)~A2: Fig. 9a shows that two functions F(x, 3) and G(x, 3) for r = 0 , and Fig. 9b shows the functions when z = rc = Xc - m(x,)K(~). When r is larger than 3, but smaller than some value 3" (for which the two functions will be tangent)

96

M. A r t z r o u n i

60.0 %vx" 45.0

%-

(.9

c~

%u.

60.0

30.0

F(x,'0

~. 30.0

15.o

[a_

7.50 a

15.00

150 X~

0

i

22.50

i

30.00

7.50

15.00 X

x

/

60.0 %` x" 45.0 (.9

'~=1,817

45.o

30.0

60.0

"~--2..5

'-D 30.0

x~ 15.0

~Z 15.o t.L

o

22.50

30.00

X

C

30.00

'r--7

F(x,,)

/

. ~ ~

o

i

15.00

-

%-

t.l.

7.50

o,x,/

x~ 45.0 F(x,~)

22.50

I X* i

i

i

7.50

15.00

22.50

d

30.00

x

Fig. 9 a - d . F o u r pairs o f functions F(x, z) and G(x, r) for four different cohorts of HIV-infectives w h e n (r,s) ~ A 2 of Fig. 2; fl = 3; x,. = 18.4;

m(xc)

= 16.8; m(0) = 8; r = 2.1; s = 2.3

then because OG(x*, r,)/Sx > OF(x*, T,)/SX there will be two modes (one for x less than x* and another at x =m(x,)K(fl)). Figure 9c represents the two functions for a value of r for which the two functions are about tangent; the value r* for which the two functions will be tangent cannot be found in closed form but can always be calculated numerically. For r > T* (but ~ x C.) This proves the theorem for (r, s) e A2.

Case 2 (r, s)eB, i.e., sir G(x*, 0) and x,. < K(fl)m(xc). There are two distinct subcases, depending on how many points of intersections the graphs of the functions F(x, 0) and G(x, 0) will have. (i) Figure 10a shows an example in which F(x, 0) and G(x, 0) have only the point of intersection at H(m(xc), fi).

/

60.0 "G" x- 45.0

1:=01

r3

%`

-

30.0

g

15.o

kL

~G(x,0

0 a

7.50

i

i

22.50

x

(.9_ 30.0 ' %J/] x" 15.0. u_ X*l 0.

F(x,0

15.00

60.0. %x- 45.0.

30.00

G(x:0/ /

b

F(x"0

,

7.50

"c=2

15.00

i

22.50

30.00

x

Fig. 1 0 a - b . T w o pairs o f functions F(x, T) a n d G(x, T) for two different cohorts of HIV-infectives w h e n (r, s) ~ B I of Fig. 2; /~ = 3; % = 12.8; m(xc) = 22.4; m(0) = 8; r = 2.8; s = 1.6

I n c u b a t i o n period o f A I D S

.~- 60.0 x" 45.0

97 60.0

F(x,~) /

F(x,'{) 45.0

(.9 30.0 30.0

vX- 15.0 "

g 15o

0

J

,,

/

/

It.

-15.00

l,X*l 7.150 15.00 22150 30.00 X

0 a

0 7.50

b

15.00

22.50

30.00

x

Fig. l l a - b . Two pairs of functions F(x, T) and G(x, T) for two different cohorst o f HIV-infectives when (r, s ) E B 2 of Fig. 2; fl = 3; x c = 5.6; m(xc) = 16; m(0) = 8; r = 2; s = 0.7

As T increases, the point x* = x c - ~ slides to the left, and the point of intersection remains unchanged (since the functions F(x, T) and G(x, T) are the same for any x > x* (Fig. 10b).) As before, when ~ reaches xc and beyond, the density pc(x, T) is trivially the Weibull with median H(m(xc), fl)) and shape parameter ft. This corresponds to the region ill. (We will determine below the border between B1 and B2.) (ii) Figure l l a shows an example where the two functions have three intersections (Therefore the density p~(x, 0) has two modes, one less than x * = x , . - T, the other equal to H(m(xc), fl).) As ~ increases, x* slides to the left. For some ~ = T + (T + <,Xc) the two curves are tangent and the first mode becomes a point of inflection with a horizontal tangent. As T increases beyond ~+, there is only the mode at H(m(x,.), fl) (Fig. lib). As before, when T reaches x C and beyond, the density pc(x, T) is trivially the Weibull with median H(m(x~), fl) and shape parameter ft. This situation in which the density A.(x, 0) has two modes corresponds to the region

B 2.

The border between the two cases B~ and B 2 corresponds to the situation in which the functions F(x, 0) and G(x, 0) will be tangent for some x + ~< x*. The parameters m(xc), xc, and m(0) for which this occurs will be those for which there will exist such a point x +, which is characterized by F(x +, O) = G(x +, 0), and c~F(x +, O)/(~x = c3G(x +, O)/gx. If we consider the change of variable u = x/x~ the function m(x) can be written as m(x) = m(0)[(2r - 2)( - u 3 + 1.5u 2) + 1]. As functions of u, F(x, 0) and G(x, O) can now be written F(u, 0) = m(0)[(2r - 2)(u3{2/~ + l} - 1.5u2{/Y + 1} +/~ + 1] and G(u, 0) = m(0)/~(ln 2)(su)P/[(2r - 2 ) ( - u 3 + 1.5u 2) + 1]. We note that F(x +, O) = G(x ~, 0) and 8F(x +, O)/Ox = OG(x +, O)/Ox imply that for some value u + of the variable u we have F(u +, O) = G(u +, 0), and OF(u +, O)/(~u = OG(u +, 0)/ ~3u. If we solve this system of two equations in the two unknowns r and s, we obtain for r a value denoted r(u) (to emphasize the dependence on u) given by

r(u) = 1 +

2u(5 - 2/~) + 3(/9 - 2) - 3x/-4u2(3 - 2fl) + 4u(3/3 - 4) + 5 - 4-fl 4ue[2u2(2/~ + 1) - 6u(fl + 1) + 2.25(/9 + 1)]

(1) Given the solution

r(u) of Eq. (1) for r, the solution s(u) for s will be

s(u) = 1 F2(r(u) - 1)uZ[(2fl + 1)u - 1.5(;9 + 1)] +/~ - 1.T/~"

J

(2)

98

M. Artzrouni

In short, the parameterized locus of points (r(u), s(u)) (curve a3 in Fig. 2) is the set of values of (r, s) for which the two functions F ( x , O) = G(x, 0) will be tangent - and the parameter u is the fraction of x~. at which the point of tangence will occur. Indeed, for fixed x c, a given u less than 1 (and r = r(u), u = s(u)), the functions F ( x , O) = G ( x , 0) will be tangent at the point x + = ux,.. (Once u and xc are fixed, m(0) is Xc/S(U) and m ( x c ) = r ( u ) m ( O ) = r ( u ) x c / s ( u ) . ) It is a simple matter to see that the region B~ under the curve a 3 is the area of the (r, s) plane for which the functions F ( x , ~) = G ( x , T) have a single intersection for all z (see Fig. 10). The region B 2 above the curve a3 is the area for which the functions F ( x , ~) = G ( x , z) will have two intersections for z less than some ~ + and one for larger than r + (see Fig. 11). F o r u = 1 the point of tangence is at Xc which is then also the mode of the density pc(x, 0), The corresponding point (r, s) obtained from Eqs. (1) (2) is ( 6 / ( 7 - / 3 ) , K(/3)6/(7-/3)) which is at the intersection of the curve a3 and the straight line a~. As u decreases from 1, the point (r(u), s(u)) moves down on the curve 0"3 and reaches the value (/32(5/3 + 3)/(/3 + 1) 3, 0) for u = (/3 + 1)/(2/3 + 1). This shows that when there is a point of tangence of the functions F ( x , O) and G ( x , O) then the fraction of x c at which the point of tangence will occur is at least (/3 + 1)/(2/3 + 1).

A p p e n d i x II

Coefficients Ai, Bg for the closed-form expression o f pc(X, z) [cubic m(x)]. In what follows q = b - da. (i) Coefficients Ai (i = 0, 1. . . . .

5) and B i (i = 0, 1) f o r / 3 = 2 (1)

As= A4= A3=O

B0 =

2qZ"r/dc - 3a - 18aZ'c/q

(2)

4q 2 -- 27a2dc/q - 3az - 2q

(3)

Ao = 4q 2 _ 27aZdc/q.

Given Bo of Eq. (3) the other coefficients are B~ = A2 = [2za/dc - 3Boa]/q

A~ = z / d c - Bo.

(4)

(ii) Coefficients Ai (i = 0, 1 . . . . ,5) and Bi (i = 0, 1) f o r / 3 = 3

We first define the quantities h~ = - [5qZ/a + 27adc/q]dc/4 h4 = - q 3/6a - 9adc

h2 =

-27dcq/4

h 5 = -5ar2/dc

- 4 . 5 d c q + q4/6a

(5)

1 - 3qr2/dc + 3za/q

(6)

h3 = h6 =

D = h l h 4 - h2h 3 •

(7)

The coefficients B0 and B~ are then B1 = [h6h4 - hsh2]/D

B0 = [hi h5 - h3 h 6 ] / D

(8)

Incubation period of AIDS

99

and 1

A o = ~aa[2r + B l d c ( 4 . 5 d c + q3/6a 2) - BodcqZ/6a] A, = dcBo - ,2/dc 1

(9) (10)

A 2 = ~aa[al ( - 7 a d c - q3/3a) + Boq2/3]

(11)

A 3 = qZB1/3a + 2qBo/3

(12)

A4 = 3qB1/2 + aBo/2

(13)

A5 = Bla.

(14)

References Anderson, R. M., Gupta, S., May, R. M.: Potential of community-wide chemotherapy or immunotherapy to control the spread of HIV-1. Nature 350, 356 359 (1991) Bacchetti P., Moss, A. R.: Incubation period of AIDS in San Francisco. Nature 338, 251 253 (1989) Blythe S. P., Anderson, R. M.: Distributed incubation period and infectious periods in models of the transmission dynamics of the human immunodeficiency virus (HIV). IMA J. Math. Appl. Med. Biol. 5(0, 1 19 (1988) Brookmeyer R., Liao, J.: Statistical modelling of the AIDS epidemic for forecasting health care needs. Biometrics 46, 1151 - 1163 (1990) Brookmeyer, R.: Reconstruction and future trends of the AIDS epidemic in the United States. Science 253, 37-42 (1991) Castillo-Chavez, C. (ed.) Mathematical and statistical approaches to AIDS epidemiology. (Lect. Notes Biomath., Vol. 83, Berlin Heidelberg New York: Springer 1990 De Gruttola, V., Lange, N.: Modelling progression of HIV infection. Presented at V. International Conference on AIDS, 4 9 June 1989, Montreal, Abstract p. 151 (1989) Gail, M. H., Rosenberg, P. S., Goedert, J. J.: Therapy may explain recent deficits in AIDS incidence. J. Acq. Immun. Def. Synd. 3, 296 306 (1990) Lui, K-J, Lawrence, D. N., Morgan, W. M., Peterman, T. A., Haverkos, H. W., Bregman, D. J.: A model-based approach for estimating the mean incubation period for transfusion-associated acquired immunodeficiency syndrome. Proc. Natl. Acad. Sci. USA, 83, 3051-3055 (1986) Lui, K-J, Darrow, W. W., Rutherford, G. W.: A model-based estimate of the mean incubation period for AIDS in homosexual men. Science 240, 1333-1335 (1988) Medley, G. F., Anderson, R. M., Cox, D. R., Billard, L.: Incubation period of AIDS in patients infected via blood transfusion. Nature 328, 719 721 (1987) Schechter, M. T., Craib, K. J. P., Le, T. N., Montaner, J. S. G., McLeod, W. A., Elmslie, K. D., O'Shaughnessy, M. V.: Influence of Zidovidome on progression to AIDS in cohort studies [letter], Lancet, i: 1026 1026 (1989) Solomon, P. J., Wilson, S. R.: Accommodating change due to treatment in the method of back projection for estimating HIV infection incidence. Biometrics 46, 1165 1170 (1990) Thieme, H. R., Castillo-Chavez, C.: How many infection-age dependent infectivity affect the dynamics of HIV/AIDS? SIAM J. Appl. Math. (submitted)