Analysis of a model for the pathogenesis of AIDS

ELSEVIER

Analysis of a Model for the Pathogenesis of AIDS NIKOLAOS I. STILIANAKIS

Theoretical Division, Group T-IO, MS K710, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA KLAUS DIETZ AND DIETER SCHENZLE

Department of Medical Biometry, University of Tiibingen, Westbahnhofstrafle 55, 72070 Tiibingen, Germany Received 8 April 1996; revised 10 March 1997

ABSTRACT According to a previously proposed mathematical model, the pathogenesis of acquired immunodeficiency syndrome (AIDS) could be explained by two phenomena: direct human immunodeficiency virus (HIV) infection of CD4 + T-cell populations and ongoing generation and selection of HIV mutants with increasing replicative capacity. In the present paper, the results obtained with this model are described in more detail. For different values of biologically interpretable parameters, the model predicts very different patterns of CD4 ÷ T-cell decline after primary infection. With the assumption of a variability of 10% to 25% of three parameters between infected individuals, the model yields a realistic distribution curve of the incubation period to AIDS. © 1997 Elsevier Science Inc.

1.

INTRODUCTION

Because the pathogenesis of human immunodeficiency virus (HIV) infection is not yet understood in detail, mathematical modeling provides a way of describing the dynamical mechanisms governing the course of infection and progression to disease, taking into account the complex interactions between the immune system and HIV. This insight into the pathogenesis of the disease could help in the planning and evaluation of treatment. We describe in more detail some results obtained with a previously proposed mathematical model [1] for the course of HIV infection and disease progression. If the hypothetical assumptions underlying this model were true, then it would explain the gradual decline of CD4 ÷ T-cell number and the long and variable incubation period to disease observed among HIV infected individuals. The basic biological assumption of the model is genetic variation of HIV [2]. It is postulated that MATHEMATICAL BIO SCIENCES 145:27-46 (1997) © 1997 Elsevier Science Inc. All rights reserved. 655 Avenue of the Americas, New York, NY 10010

0025-5564/97/$17.00 PII S0025-5564(97)00018-7

NIKOLAOS I. STILIANAKIS ET AL.

28

competitive exclusion between emerging H I V variants leads to an increasing reproductive capacity of the virus over time. This increase is expressed in terms of a higher infection rate of CD4 + T cells. The resulting destruction of infected cells causes the gradual depletion of CD4 ÷ T cells and the ensuing loss of immunocompetence. Simulations show that this mechanism predicts a course of HIV infection characterized by a viremia during the first weeks after primary infection, a long period of relatively low and slightly rising viral burden, and a reappearance of heavy viremia during the last phase of the infection. Such a course of infection is considered typical in the literature [3]. 2.

MODEL EQUATIONS

The model for one individual is deterministic, and it describes how the following five quantities change with time t since infection at time t = 0:

X: number of susceptible CD4 ÷ T ceils; Y: number of infected CD4 ÷ T cells; V: number of free H I V particles; Z: "anti-HIV activity" of the immune system toward removing infected CD4 ÷ T cells and HIV; K: CD4 ÷ T cell infection rate. The nonlinear system of differential equations reads: dX = A - g 0 X - KVX, dt dY

(1)

d'-T = K V X - / x 1 ( 1 + a Z ) Y ,

(2)

dV dt = f l Y - / * 2 ( 1 + bZ)V,

(3)

dZ -d-i- = Og(V) + p ( f ( X ) - Z ) Z ,

(4)

dK d---~-= t°V(gmax - g ) ,

(5)

where

f(X)=

(lq-Cl)(X/Xo) 2 C1..{_(X/Xo) 2 '

V g ( V ) -.~- C23¢_V ,

with X(0) = X0, Y(0) --- Y0, V(0) = V0, Z(0) = 0, K(0) = K0. These model equations are motivated and explained as follows--see [1] for more details.

AIDS MODEL

29

Although several types of cells may become infected, HIV preferentially enters T cells expressing surface CD4 ÷ receptors, to which HIV attaches with very high affinity. Therefore, in the model, HIV is assumed to reproduce exclusively within an individual's pool of CD4 ÷ T cells. Prior to infection with HIV, an individual's CD4 ÷ T-cell pool is assumed to be in a steady state, in which all cells die at a constant rate /z0, being replenished through a constant influx of A cells per day. This gives an initial CD4 ÷ T-cell pool size X 0 = A / P,0. This number is to be equated with the clinically used "normal" total CD4 ÷ T-cell count. Any CD4 ÷ T cell is considered to be either intact and susceptible to HIV infection or infected and actively producing HIV. Although latently infected cells are known to exist, they will not be included here [4]. Thus, Y denotes the current number of cells, each of which is assumed to produce HIV at a constant rate /3. Given at time t a number of susceptible CD4 ÷ T cells and a number V of free HIV particles, the average number of newly infected cells per day is given by the mass action term KVX, where K denotes the current average cell infection rate. The quantity K represents a kind of overall rate for HIV infection that includes binding to a CD4 ÷ receptor and actual cell entry. As explained later, this average cell infection rate is considered to be a dynamical variable. Each infected cell is assumed to be destroyed at an intrinsic r a t e / z 1. In addition, infected cells are assumed to be removed by the immune system at a rate atZlZ proportional to the current anti-HIV activity Z. Then the average number of HIV particles produced by one infected cell is given by the time-dependent quantity /3//z1[1+ aZ(t)] if one neglects the (slow) variability of Z(t) during the short lifetime of an infected cell. Each free HIV particle is assumed to have an intrinsic death rate /-~2 and to be neutralized or removed by the immune system at a rate b/x 2Z. For simplicity, we ignore the small fraction of free virus particles removed upon infection of cells. The immune system response, which acts by destroying infected cells as well as by neutralizing and eliminating HIV particles, is only crudely modeled by an "anti-HIV activity" Z, such that, as stated earlier, al~lZ and bl~2Z give the current rates at which infected cells and HIV particles are removed, respectively. The equation for Z derives from the idea that, upon primary infection, HIV will stimulate specific antibodyproducing and cytotoxic cells to multiply at an intrinsic rate p[f(X)- Z] until the immune system itself puts a certain limit (almost) independent of the concentration of HIV particles and infected cells. The function f(X) is used to incorporate the fact that the activity of the immune system becomes reduced if there are not enough CD4 ÷ T cells. The cellular immune response has been characterized as the dominant

30

NIKOLAOS I. STILIANAKIS ET AL.

effective part of the immune response from the very beginning and throughout the infection course in H I V infection [5]. In contrast, although the presence of neutralizing antibodies has been shown, there is no indication about any effective contribution of the humoral immune response [6,7]. Thus, our anti-HIV-activity equation models the effective immune response that is predominantly the cellular immune response. The equation for K serves to mimic the effect of an ongoing increase in the H I V cell infection rate by mutation and selection of H I V mutants within an infected individual. An increase in H I V replication kinetics (virulence) in the course of disease progression (CD4 ÷ T-cell decline) has been well documented by a number of experimental studies [8-12]. As shown in [1], the model implies competitive exclusion between H I V mutants with different cell-infection rates. Therefore, through generation of new H I V mutants, the average cell-infection rate K should increase with time with a speed depending on the mutation rate, which is taken proportional to the number V of viral particles. With higher values of K, the increase of K slows down, because H I V probably cannot improve its cell-infection rate beyond a certain maximal value

gmax3.

QUALITATIVE FEATURES OF THE MODEL The dynamics of the model [Eqs. (1)-(5)] predicts three steady states:

(i) X * = X o,

Y*=0,

V*=0,

Z*=0,

K*=Ko,

(6)

where X 0 = A / / x 0. (ii) X** = Xo,

Y** = 0,

V** = 0,

Z** = 1,

K** = K*, (7)

where r 0 < K* ~< Kmax(iii) X*** = ~',

Y*** = ? ,

Z*** = Z,

K*** = Kmax.

V*** = P,

(8)

The "virgin" state is unstable because, in the model, any minute initial amount of virus activates the immune system to the level Z** = f(X o) = 1, which is then maintained even in the absence of virus. The

AIDS MODEL

31

resulting model dynamics depend on R* =

flKoXo

/z 1/.t2(1 + aZ** )(1 + bZ**)'

(9)

the reproduction number of the HIV causing the primary infection. Parameter values such that R* < 1

(10)

lead to a maximal eigenvalue of zero for the virgin steady state. This suggests a complicated dynamical behavior depending on the initial values Y(0) and V(0). A fast enough increase in the infection rate K may lead to an increase in the reproduction number beyond 1, and then the model asymptotically approaches the third equilibrium state, as described later. If the increase in the infection rate does not allow a crossing over of the threshold value 1, the model predicts HIV elimination leading to the second steady state. Under the condition R* > 1,

(11)

numerical work suggests that the course of an HIV infection always approaches the third steady state. This equilibrium is given by the following equations:

K ----

= Kmax,

(12)

~KmaxX° /z,/z2(1 + a Z ) ( l + b Z ) '

(13)

. =Xo -if, ~ = t%(1 - 1 / / ~ ) X o I&l(1 + a~7) ' V=

/3/'~°(1 - 1 / R ) X ° + bZT) '

(14) (15)

(16)

~1 ~ 2 ( 1 "1"-a Z ) ( 1 D

where the steady-state value of Z can be obtained from the quadratic equation

og(P) + p[f(R) - 2 ] 2 = o.

(17)

32

NIKOLAOS I. STILIANAKIS ET AL.

The model predicts a decrease in the CD4 ÷ T-cell population to a level that is lower than 20% of the normal value. This level is the threshold at which an HIV-infected individual is diagnosed in the United States as having AIDS [13]. Thus, in the model, we define the "incubation period to AIDS" as the time from primary infection until the total CD4 ÷ T-cell count falls below 20% of the normal value. During the last stage of the infection, the model shows a fast increase of viral load and of infected cells. The remaining CD4 + T cells continue to decrease reaching a minimal value, and the anti-HIV activity is reduced to a very low level. 4.

PARAMETER VALUES

Most model parameter values cannot be estimated in the usual statistical sense from currently available data. Therefore the values proposed below and listed in Table 1 as "standard values" for model simulations should be considered more or less reasonable guesses. Some of the model parameters are strongly interrelated and may be scaled in combination without affecting the dynamics of the model. The total number of CD4 ÷ T cells of a healthy individual is estimated to be X 0 = 1011 [14]. The lifespan of noninfected CD4 ÷ T cells depends on their function. Memory cells live much longer than do naive and precursor cells. Estimates of the lifespan vary between a few days and 2 or more years [15-17]. In this model, we do not distinguish between the different CD4 ÷ T-cell subpopulations. Thus, the chosen rate /~0 = 0.004/day is an average value. This rate corresponds to an average lifespan of 250 days. If the total number of CD4 ÷ cells is to be maintained by a constant input A of new cells, one must have A = 4.0 x 108/day. The lifespan of actively infected C D 4 - T cells varies between a few days and 1 week, Studies indicate a lifespan between viral expression and cell death of 2 - 4 days [18,19]. Therefore, we choose /z I = 0.3/day. Setting a = 1, we assume that infected cells die at a rate of 0.6/day in the presence of an anti-HIV activity Z** = 1. Free virus loses infectivity with a half-life of less than 1 day [20]. Therefore, we put /x2 = 1 / d a y and b = 1, thus assuming that free virus is removed at a rate of 2 / d a y in the presence of an anti-HIV activity Z** = 1. Together, we have a total rate of 2 / d a y for the removal of virus and 0 . 6 / d a y for the death rate of infected cells, indicating a lifespan of about 2 days. These values are in line with recent estimates for the half-life of free virus in HIV-infected patients [21-23]. Several studies about the number of infectious viral particles released by an infected cell give estimates between 50 and 1000 particles [19]. We choose as standard value /3 = 1000 particles/day. This means

AIDS MODEL

33

that an infected cell during its life can produce f l / g l =3333 viral particles if it escapes the immune response. The parameters 0 and p quantify the activation rate of the immune response and are taken as p = 0.5/day and 0 = 10-6/day. Because immune reactions are expected to vary between infected individuals, the parameter p should be considered an individual specific parameter. The initial value for the T cell infection rate is taken as K0 = 1.35 × 10-14/day. This value gives an initial basic reproduction number for HIV particles of R* = 1.125, given a maximum anti-HIV activity Z** = 1. The parameter co determines the speed at which the cell infection rate K increases through generation and selection of HIV mutants. Here we set co = 10 -16 per viral particle per day. Moreover, we assume that the rate K cannot increase beyond the value Kmax = 20 K0. 5.

SIMULATIONS

5.1. THE BASELINE BEHAVIOR OF THE MODEL With the parameter values of Table 1, the model yields the curves shown in Figure 1. Three phases can be distinguished: (1) initial, or viremia, phase until about 6 months after infection; (2) asymptomatic phase until about 10 years; and (3) progression to disease. The initial phase of the infection is characterized by a loss of about 30-40% of CD4 + T cells during the first 2-3 months after primary infection. The anti-HIV activity ( Z ) reaches its maximum value 1 month after infection. This causes a reduction of viral production and some recovery of the CD4 + T-cell number. The viral population (V) grows exponentially during that first phase of infection, reaches a maximum in the 9th week, and then decreases again. During the asymptomatic phase, the CD4 + T-cell number shows damped oscillations of about 70-85% for about 2-3 years and continues to decline almost linearly. Because the immune response is functionally coupled to the depletion of the CD4 + T-cell number, the immune function slowly weakens, too, over the asymptomatic period. After an oscillatory viremia, free virus begins to increase monotonically with a slow rate throughout the asymptomatic period. Perhaps, compared with the initial peak virus load, the model in its present form predicts a somewhat too high virus load during the asymptomatic phase. 5.2. THE EFFECTS OF THE RATE OF REPRODUCTION INCREASE Figure 2 shows the CD4 + T-cell courses generated by the model for different values of parameter co, keeping all the other parameters as in Table 1. The upper course in Figure 2 is produced with a very small co

34

NIKOLAOS I. STILIANAKIS E T AL

TABLE 1 Parameter Values Used in Model Dependent variables X Y V Z K

Initial values Number of noninfected CD4 ÷ T cells Number of infected CD4 ÷ T cells Number of H I V particles Anti-HIV activity of the immune system Average infection rate of CD4 + T cells

X 0 = 1011 Y0 = 1.0 V0 = 1.0 Z(0) = 0.0 K0 = 1.35 X 1 0 - 1 4 / d a y

Parameters and constants A /~0 /x I a/x I /x2 b/x 2 /3 0

/9 to K~na~ C1 C2

Standard values Renewal rate of CD4 + T ceils Death rate of noninfected CD4 + T cells Death rate of infected CD4 ÷ T cells Maximum death rate of infected cells due to anti-HIV activity Death rate of virus Maximal death rate of virus due to anti-HIV activity H I V production rate by an infected CD4 ÷ T cell Rate of immune activation due to viral presence Rate of the autonomous activation of the immune response Rate of the reproduction increase per virus particle Maximal infection rate Constant of function f(X) Constant of function g(V)

4.0 × 108/day 4.0 × 10-3/day 0.30/day 0.30/day 1.0/day 1.0/day 103/day 10-6/day 0.50/day 10-16/day 20/day 0.04 103

Derived parameters

R0 R*

Initial value of the basic reproduction 4.50 n u m b e r at primary infection Value of the basic reproduction n u m b e r 1.125 of H I V in the presence of an anti-HIV activity Z** = 1

AIDS MODEL

35

1.0

le+12

0,8"

8e+11

0.6

6e+11

0.4.

4e+11

0.2

2e+11

0.C

0

1

2

3

4

5 6 7 TIME (YEARS)

8

9

10

11

0 12

FIG. 1. Typical model simulation of the course of HIV infection with parameter values as in Table 1. Right-hand scale: viral population V (thin solid line); left-hand scale ( X + Y ) / X o (thick solid line), Z(dashed line); The simulation describes the first viremia phase after initial infection and the decline of the CD4 + T cells as well as that of the anti-HIV activity during infection. The constant (dotted) line marks the 20% level of ( X + Y ) / X o, which denotes AIDS diagnosis.

and shows extremely slow H I V infection dynamics, where the CD4 ÷ T cells remain at a high level after infection for many years. In this case, viral levels after initial viremia remain low throughout the whole period. This course can account for a very slow chronic H I V infection [24]. The lowest curve in Figure 2 shows very fast H I V infection dynamics where the CD4 ÷ T cells decrease to a level of 2 - 3 % of the original with only a very short and small rebound after 5 months. The 20% level is already passed after 4 months. This curve could describe a fast disease progressor [25]. The simulations o f Figure 2 give an impression of the wide range of possible CD4 ÷ T-cell courses that can be predicted from the model. 5.3.

THE EFFECTS OF THE INFECTIOUS DOSE

Because primary infection starts with an unknown number of viral particles and infected cells, we investigate the model behavior for two different initial conditions with Y(0) = 1, I1(0) = 1 and II(0) --- 100, II(0)

NIKOLAOS I. STILIANAKIS ET AL.

36 1.0

0.8-

= 0.6

8

F4-

a o 0.4-

0,2-

~-

0.0

0

' i

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

' 2

3

4

~i

6

'7

8

9

1'0 ' 1'1 1 2

TIME (YEARS)

FIG. 2. Three courses of CD4 ÷ T cells [(X + Y ) / X o] generated by the model showing the range of possible outcomes. Parameter values as in Table 1 except the value of ¢o, being 1 0 - 1 8 / d a y for the very slow course (thick solid line) and representing a chronic infection with very slow progression, 10-~6/day for the typical course (thin solid line), which is an average of the majority of HIV infection, and 10-15/day for a fast course leading to fast disease progression (dashed line).

= 1000, using the m o d e l values of Table 1. As is evident f r o m Figure 3, the predicted courses of infection are very similar. Hence, the long-term model dynamics is largely i n d e p e n d e n t of the initial infectious dose. 5.4. INFECTIONS WITH DIFFERENT H1V STRAINS T h e effects of infection with H I V strains with different replicative capacities and therefore different basic r e p r o d u c t i o n n u m b e r s are presented in Figure 4. Assuming that the a n t i - H I V activity has the same effect on the death o f C D 4 ÷ T cells and virus particles [a = b in eqs. (2), (3), and (9)], the r e p r o d u c t i o n n u m b e r s without and with a n t i - H I V activity Z** = 1 are given as

flKoXo

Ro= - /.rl be2

(18)

AIDS M O D E L

37

1.0

le+12

0.8

8e+11

0.6

6e+11

0.4

4e+11

0.2

2e+11

0.0

0

1

2

3

4

5 6 7 TIME (YEARS)

8

9

10

11

12

0

FIG. 3. Model simulations for V0 = 1, Y0 = 1 (thick solid line) and V0 = 1000, Y0 = 100 (thin solid line) with all other parameters as in Table 1. The courses of ( X + Y ) / X o , V, and Z (as described in Figure 1) are very similar in both cases.

and R* =

R°

( 1 + a ) 2"

(19)

With a = 1 and R o = 3.0, 4.5, and 6.0, we obtain the curves of CD4 ÷ T-cell counts in Figure 4, the curve for R o = 4.5 being the same as in Figure 1. For R o = 3.0, there is an initial viremia, but, because of R* = 0.75, the virus is cleared after appearance of the a n t i - H I V activity. The n u m b e r of CD4 + T cells is virtually unaffected. The case R o = 6.0(R* = 1.5) represents a m o r e aggressive H I V variant. Here, one obtains a large decrease in CD4 + T cells during viremia in the first 2 months, and, after 4 years, the n u m b e r of CD4 + cells is less than 20% of normal. 5.5.

DEPENDENCY ON THE EFFECT OF IMMUNITY ON THE DEATH R A T E O F I N F E C T E D CD4 ÷ T C E L L S

Figure 5 shows the courses of the CD4 + T cells for different values of the strength of the anti-HIV activity a = 0.8, 1.0, and 1.2 for a constant R 0 = 4.5 (all the other values as in Table 1). For a reduced by

N I K O L A O S I. S T I L I A N A K I S E T AL.

38

1.0-

0.8

-= 0 . 6 I-÷

o 0.4-

0.2

0.0 0

• J

'

~' '

3

'

4

'

5

'

I~ '

7

'

8

" §'

I'0'

I'1 ' 12

TIME (YEARS) FIG. 4. O u t c o m e s of infections with different H I V strains in t e r m s of C D 4 + T-cell counts. F o r R 0 = 3.0, K0 = 0 . 9 × 1 0 - 1 4 / d a y , R* = 0.75 ( d a s h e d line), there is n o change visible, because the small n u m b e r of H I V particles p r o d u c e d are rapidly eliminated and do not affect the n u m b e r of C D 4 ÷ T cells. F o r R 0 = 4.5, K0 = 1.35 X 1 0 - 1 4 / d a y , R* = 1.125 (thick solid line), the course r e p r e s e n t s the typical HIV-infection case. F o r R 0 = 6.0, K0 = 1.8× 1 0 - 1 + / d a y (thin solid line), R* = 1.50 C D 4 + T cells d r o p fast, leading to disease progression. In all cases, a = 1 with the o t h e r p a r a m e t e r s as in Table 1.

20% of the standard value, the strength of the anti-HIV activity causes a large drop in CD4 ÷ T cells to about 30% of the normal during the first phase of the infection and a faster progression to disease. On the other hand, for a value of a = 1.2, the anti-HIV activity is able to keep the basic reproduction number R* of the virus below 1. Consequently, the virus can be eliminated. The CD4 ÷ cells remain almost unaffected. Hence, the strength of the host anti-HIV activity is predicted to strongly determine the outcome of an HIV infection. 5.6.

DEPENDENCY OF THE IMMUNE RESPONSE ON THE ACTIVATION RATE

The results depend not only on the strength a of the anti-HIV activity, but also on its speed of activation during primary infection. This is determined by the parameter p, and Figure 6 shows curves of CD4 ÷

AIDS MODEL 1 ,O

39

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

m. . . . . . . .

0.8

~

0.6-

O 0.4-

0.2

0.0

0

J

"

'~

'

:3

'

4'

'

5' 6' '7' T I M E (YEARS)

8"

§'

1'0' 1'1' 12

FIG. 5. Simulation o f the C D 4 + T-cell depletion in t h r e e infections o f H I V variants with the s a m e basic r e p r o d u c t i o n n u m b e r R 0 = 4.5 in the p r e s e n c e of a n t i - H I V activity a but at different strengths: a = 0.8 (thin solid line), a = 1.0 (thick solid line), and a = 1.2 (dashed line). All o t h e r p a r a m e t e r s as in Table 1.

T-cell counts for p = 0 . 7 / d a y and p = 0.3/day. The curve for p = 0 . 7 / d a y is almost identical with the curve for p = 0 . 5 / d a y shown in Figure 1. If the speed of immune activation is reduced to p = 0.3/day, then the CD4 + T cells undergo a large decrease to 23% of the normal cell number. Although the number rebounds to as much as 88% of the normal cell number, this rebound is followed by a rapid decline in cell number. These results indicate that the timing of the immune-system activation may play an important role in the course of the infection because delayed activation may shorten the incubation time to AIDS. 5.7.

DEPENDENCY ON THE PRODUCT OF THE REPRODUCTION RATE A N D THE R E P R O D U C T I O N I N C R E A S E

For a given value of R 0 and R*, H I V infection dynamics still depends on the parameters /3 and to. The larger these parameters are, the faster more virulent H I V mutants are generated and selected. Two combinations of to and /3 were chosen [(to = 0.8× 10 -16,/3 = 800) and (to = 1 . 2 × 1 0 - 1 6 , / 3 = 1 2 0 0 ) ] and the courses of the CD4 + T-cell counts

40

NIKOLAOS I. STILIANAKIS E T AL. 1.0

0.8

_0.6

0.4

0.2

0°0"

0

1

2

3

4

5 6 7 TIME (YEARS)

8

9

10

11

12

FIG. 6. C D 4 + T-cell courses of H I V infection for two different values of the activation rate for the anti-HIV activity p = 0.3 (thin solid line) and 0.7 (thick solid line), with all other parameters as in Table 1.

are shown in Figure 7, compared with the course using the values of to and /3 from our standard simulation of Figure 1. There is almost no difference in the courses of the T cells at the first phase of infection for all three combinations of to and /3. However, the combination representing a fast increase in the infection rate and viral production yields a shorter asymptomatic phase, with a crossing of the 20% threshold after 7 years. Thus, increasing the viral replication and infection rate by about 20% leads to a faster progression to disease (after about 3.5 years) compared with the typical case (Figure 1). On the other side, the combination representing a slow viral-replication and infectivity increase leads to a slow decline in the CD4 ÷ T cell counts, which even after 12 years has reached only about 40% of the normal value. Thus in Figure 7, one can see the difference that a fast or slow replicative virus can make on the infection course. 5.8.

V A R I A B I L I T Y O F THE I N C U B A T I O N TIME TO A I D S

A distinct feature of the HIV infection process is the extreme variability of the incubation time period to AIDS [25]. This variability can be explained in the context of our model by host variability.

AIDS MODEL

41

1.0

0.8

0.6

%%% %%%% %%%%% %%%%%%

I+

a o

0.4

0.2 ~ 0.0

o

i

3

'

,4

'

5 6 ' "7 TIME (YEARS)

'

8

........ '

§

1'0

1'1

12

FIG. 7. Dependency of the CD4 + T-cell counts on the average increase in infection and viral-production rate. Given R* = 1.125 and R 0 = 4.5, we determine three pairs of to,/3 [0.8 X 10 -16, 800 (thin solid line) 10 -16, 10 3 (thick solid line); and 1.2×10 -16, 1200 (dashed line)], with K0=1.69×10 -14, 1.35×10 -14, and 1.13x 10-14/day, respectively. All other parameters as in Table 1.

Assuming that some of the p a r a m e t e r s of the model are host specific, a model simulation with a specific set of p a r a m e t e r s refers to one infected individual. One of these individual-specific p a r a m e t e r s is the cell-infection rate K0 of the H I V variant causing primary infection. With other model p a r a m e t e r s fixed, the value of K0 determines the basic reproduction n u m b e r R 0 of H I V at initial infection. A n o t h e r p a r a m e t e r that may be individual specific is the speed to at which H I V increases its cell-infection rate through generation and selection of m o r e virulent mutants. Moreover, the activation rate p of the immune response also can vary from host to host. Therefore, here we consider K0, to, and p host-specific p a r a m e t e r s that may explain the variability in the time span between primary infection and the decline of the CD4 ÷ T-cell n u m b e r to below 20% of the normal value. We p e r f o r m e d a series of simulations by using the p a r a m e t e r values of Table 1 except for choosing r a n d o m values for K0, to, and p from independent Gaussian distributions with the following means and stan-

42

NIKOLAOS I. STILIANAKIS ET AL.

dard deviations: K0 = (1.35 + 0.225) × 1 0 - 1 4 / d a y , to -----(1.0 + 0.25) × 1 0 - 1 6 / d a y , and p = (0.5_+0.05)/day. E a c h m o d e l simulation was continued for at m o s t 25 years but was stopped if the n u m b e r of C D 4 ÷ T cells fell below 20% o f the n o r m a l value. This resulted in a series of 500 "incubation times" with the cumulative distribution function shown in Figure 8. T h e m e d i a n o f the incubation time to A I D S is about 10 years. By taking into account an average of a n o t h e r 1 - 2 years of life in this disease stage, we obtain a life expectancy o f about 1 1 - 1 2 years. This is close to the estimated average lifespan o f H I V - i n f e c t e d individuals f r o m primary infection to death [24]. T h e cumulative value of the distribution after 25 years is 90.4%. This m e a n s that, in 9.6% of all simulated infections, the virus was either eliminated or it persisted at a very low level without reducing the n u m b e r of C D 4 ÷ T cells below the 20% threshold even after 25 years. 6.

DISCUSSION

T h e aim o f this work was a m o r e detailed analysis of a dynamical model describing the pathogenesis of H I V infection. T h e structure of the m o d e l presented here is similar to that of well-known models 1.0

0.8-

0.6-

0.4-

0.2

0.0

0

~,

6

8

1 ' 0 12 '-I~4 ' I'6 ' I'8 ' 2'0 ' 2'2 ' 2'4 TIME (YEARS)

FIG. 8. Distribution function of the time point at which the CD4 ÷ T-cell count reaches the 20% level of the total CD4 + T-cell number during HIV infection denoting diagnosis of AIDS.

AIDS MODEL

43

employed to describe the transmission dynamics of infections in host populations. The host-parasite dynamics considered here is between target immune-system cells (CD4 ÷ T lymphocytes) and infectious HIV particles. Especially, the model accounts for the long and variable period from primary infection to development of AIDS observed in HIV-infected individuals [2,3,25]. On the basis of two fundamental features of HIV infection--namely, genetic variation of the virus and direct CD4 ÷ T-cell killing by the virus--the model provides an explanation for the HIV-infection process. In the model, we suggest that ongoing mutation and selection between HIV variants lead to a slowly but steadily increasing cell-infection rate. Simulations performed here show some interesting features inherent in the model. The model can reproduce all three phases observed in typical HIV infections, and it does not depend on the initial dose of primary infection. Moreover, one can also see that rare cases of HIV infection, such as fast progression and long-term infection, are easily reproduced. With respect to the parameter values, some of them remain unknown; but, for some of them, such as the half-life of the virus and infected cells, there are realistic estimates, which are used in the model. The dependency of the model dynamics on some of the most important parameters delivered interesting information about the dynamics of the infection process. The simulations show the effects of an infection with different HIV strains on the infection course. Different strengths of the immune response of HIV-infected persons, different immune-response activation rates, or different replication and infection rates between HIV-infected individuals may substantially affect the spread of the infection within the host and the progression to disease. Host variability in terms of specific alleles of the host's human leukocyte antigens (HLAs) may contribute to the variable rate of disease progression and outcome in individuals infected with HIV. Because the disease process includes dysregulation of the immune system, whose regulatory elements are encoded by the HLA genes, one may suggest that some of the model parameters are host specific. As such, their effects on the course of the infection may vary from host to host. Simulations in which these parameters are varied showed that variability of "incubation times" to AIDS can be explained by differences in the replicative capacity of the HIV strain infecting an individual, the speed of the underlying mutation and selection process, or the host-specific immune-response activation. Experimental and clinical work support the association between genetic variation and infection as well as disease progression [8-12, 26, 27]. Cells from different infected individuals show different replicative

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NIKOLAOS I. STILIANAKIS ET AL.

capacity [8-12, 26, 27]. Host-specific variability, as already mentioned, also plays an important role in infection progression [28-30], which explains not only why "incubation times" to A I D S vary, indicating within-host evolution for H I V , but also why an increase in the reproduction n u m b e r of H I V variants does not necessarily m e a n an increase of infectivity of H I V between individuals. The interesting features predicted by the model need to be verified by more data. More precise information about the lifespan of CD4 ÷ T cells, viral replication rates, and death rates of infected CD4 ÷ T cells and virus would be of great interest. The behavior of the model (by appropriate modification) in drug treatment would also be an adequate test for the model's flexibility. In summary, analysis of a mathematical model for the pathogenesis of A I D S shows that direct killing of CD4 ÷ T cells by H I V and ongoing generation and selection of H I V mutants with increasing replicative capacity are sufficient to cause the clinical picture of H I V infection observed, including the long and variable incubation period.

This work was supported by the U.S. National Institute on Drug Abuse (NIDA) through a grant to the Societal Institute of the Mathematical Sciences (SIMS) (NIDA Grant No. DA-04722). This work was completed at the Los Alamos National Laboratory, USA, where N. L Stilianakis is a postdoctoral fellow supported by the German Cancer Research Centre.

REFERENCES 1 D. Schenzle, A model for AIDS pathogenesis. Stat. Med. 13:2067-2079 (1994). 2 R.A.Weiss, How does HIV cause AIDS? Science 260:1273-1279 (1993). 3 G. Pantaleo, C. Graziosi, and A. S. Fauci, The immunopathogenesis of human immunodeficiency virus infection. N. Engl. J. Med. 328:327-335 (1993). 4 A.N. Phillips, Reduction of HIV concentration during acute infection: independence from a specific immune response. Science 271:497-499 (1996). 5 R.A. Koup, J. T. Safrit, Y. Cao, C. A. Andrews, G. McLeod, W. Borkowsky, C. Farthing, and D. D. Ho, Temporal association of cellular immune responses with the initial control of viremia in primary human immunodeficiency virus type 1 syndrome. J. Virol. 68:4650-4655 (1994). 6 J. E. Groopman, P. M. Benz, R. Ferriani, K. Mayer, J. D. Allan, and L. A. Weymouth, Characterization of serum neutralization response to the human immunodeficiency virus (HIV). AIDS Res. Hum. Retroviruses 3:71-85 (1987). 7 T. Harrer, E. Harrer, S. A. Kalams, T. Elbeik, S. I. Staprans, M. B. Feinberg, Y. Cao, D. D. Ho, T. Yilma, A. M. Caliendo, R. P. Johnson, S. P. Buchbinder, and B. D. Walker, Strong cytotoxic T cells and weak neutralizing antibody responses in a subset of persons with stable nonprogressing HIV type 1 infection. AIDS Res. Hum. Retroviruses 12:585-592 (1996).

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45

8 B.L. Asj6, L. Morfeldt, J. Albert, G. Biberfeld, A. Karlsson, K. Lidman, and E. M. Feny6, Replicative capacity of human immunodeficiency virus from patients with varying severity of HIV infection. Lancet 2:660-662 (1986). 9 P . T . A . Schellekens, M. Tersmette, M. Th. L. Roos, R. P. Keet, F. de Wolf, R. A. Coutinho, and F. Miedema, Biphasic rate of CD4 + cell count decline during progression to AIDS correlates with HIV-1 phenQtype. AIDS 6:665-669 (1992). 10 M. Koot, A. H. V. Vos, R. P. M. Keet, R. E. de Goede, M. W. Dercksen, F. G. Terpstra, R. A. Coutinho, F. Miedema, and M. Tersmette, HIV-1 biological phenotype in long-term infected individuals evaluated with an MT-2 cocultivation assay. AIDS 6:49-54 (1992). 11 S.A. Bozzette, J. A. McCutchan, S. A. Spector, B. Wright, and D. D. Richman, A cross-sectional comparison of persons with syncytium- and non-syncitium-inducing human immunodeficiency virus. J. Infect. Dis. 168:1374-1379 (1993). 12 R.I. Connor and D. D. Ho, Human immunodeficiency virus type 1 variants with increased replicative capacity develop during the asymptomatic stage before disease progression. J. V/rol. 68:4400-4408 (1994). 13 Centers for Disease Control, 1993 revised classification system for HIV infection and expanded surveillance case definition for AIDS among adolescents and adults. Morb. Mortal. Wk~y. Rep. 41:RR-17 (1992). 14 J. Kuby, Immunology. W. H. Freeman, New York, 1992. 15 C. A. Michie, A. R. McLean, C. Alcock, and P. C. L. Beverley, Lifespan of human lymphocyte subsets defined by CD45 isoforms. Nature 360:264-265 (1992). 16 A.A. Freitas and B. B. Rocha, Lymphocyte lifespans: homeostasis, selection and competition. Immunol. Today 14:25-29 (1993). 17 A.R. McLean and C. A. Michie, In vivo estimates of division and death rates of human lymphoctyes. Proc. Natl. Acad. Sci. U.S.A. 92:3707-3711 (1995). 18 M. Somasundaran and H. L. Robinson, Unexpectedly high levels of HIV-1 RNA and protein synthesis in a cytocidal infection. Science 242:1554-1557 (1988). 19 R. Kierman, J. Marshall, R. Bowers, R. Doherty, and D. McPee, Kinetics of HIV-1 replication and intracellular accumulation of particles in HTLV-1 transformed cells. AIDS Res. Human Retrovirus 6:743-752 (1990). 20 S.P. Layne, J. L. Spouge, and M. Dembo, Quantifying the infectivity of HIV. Proc. Natl. Acad. Sci. U.S.A. 86:4644-4648 (1989). 21 X. Wei, S. K. Ghosh, M. E. Taylor, V. A. Johnson, E. A. Emini, P. Deutsch, J. D. Lifson, S. Bonhoeffer, M. A. Nowak, B. H. Hahn, M. S. Saag, and G. M. Shaw, Viral dynamics in human immunodeficiency virus type 1 infection. Nature 373:117-122 (1995). 22 D . D . Ho, A. U. Neumann, A. S. Perelson, W. Chen, J. M. Leonard, and M. Markowitz, Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection. Nature 373:123-126 (1995). 23 A . S . Pereison, A. U. Neumann, M. Markowitz, J. M. Leonard, and D. D. Ho, HIV-1 dynamics in vivo: virion clearance rate, infected cell lifespan, and generation time. Science 271:1582-1586 (1996). 24 L. K. Schrager, J. M. Young, M. G. Fowler, B. J. Mathieson, and S. H. Vermund, Long-term survivors of HIV-1 infection: definitions and research challenges. AIDS 8:$95-$108 (1994). 25 J.P. Phair, Variations in the natural history of HIV infection. AIDS Res. Hum. Retroviruses 10:883-885 (1994).

46

NIKOLAOS I. STILIANAKIS ET AL.

26 M. Koot, I. P. M. Keet, A. H. V. Vos, R. E. Y. De Goede, M. Th. L. Roos, R. A. Coutinho, F. Miedema, P. Th. A. Schellekens, and M. Tersmette, Prognostic value of HIV-1 syncytium-inducingphenotype for rate of CD4 ÷ cell depletion and progression to AIDS. Ann. Intern. Med. 118:681-688 (1993). 27 F. Miedema, M. Tersmette, and R. A. W. van Lier, AIDS pathogenesis: a dynamical interaction between HIV and the immune system. Immunol. Today 11:293-297 (1990). 28 R. A. Kaslow, R. Duquesnoy, M. Van Raden, L. Kingsley, L. Marrari, H. Friedman, S. Su, A. J. Saah, R. Deteis, J. Phair, and C. Rinaldo, A1, Cw7, B8, DR3, HLA antigen combination associated with rapid decline of T-helper lymphocytes in HIV-I infection. Lancet 335:927-930 (1990). 29 D. L. Mann, M. Carrington, M. O'Donell, T. Miller, and J. Goedert, HLA phenotype is a factor in determining rate of disease progression and outcome in HIV-1 infected individuals. AIDS Res. Hum. Retroviruses 8:1123-1124 (1992). 30 R.A. Kaslow and D. L. Mann, The role of the major histocompatibility complex in human immunodeficiency virus infection: ever more complex? J. Infect. Dis. 169:1332-1333 (1994).