Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol163.Issue2.Feb2000:

A model of HIV-1 pathogenesis that includes an intracellular

delay

Patrick W. Nelson

a,b

, James D. Murray

c

, Alan S. Perelson

b,* a

Department of Mathematics, Duke University, Durham, NC 27708, USA b

Theoretical Biology and Biophysics Group, Los Alamos National Laboratory, Mail Stop K710, Los Alamos, NM 87545, USA

c_{Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA}

Received 20 July 1999; received in revised form 14 October 1999; accepted 18 October 1999

Abstract

Mathematical modeling combined with experimental measurements have yielded important insights into HIV-1 pathogenesis. For example, data from experiments in which HIV-infected patients are given potent antiretroviral drugs that perturb the infection process have been used to estimate kinetic parameters un-derlying HIV infection. Many of the models used to analyze data have assumed drug treatments to be completely ecacious and that upon infection a cell instantly begins producing virus. We consider a model that allows for less then perfect drug e€ects and which includes a delay in the initiation of virus production. We present detailed analysis of this delay di€erential equation model and compare the results to a model without delay. Our analysis shows that when drug ecacy is less than 100%, as may be the case in vivo, the predicted rate of decline in plasma virus concentration depends on three factors: the death rate of virus producing cells, the ecacy of therapy, and the length of the delay. Thus, previous estimates of infected cell

loss rates can be improved upon by considering more realistic models of viral infection. Ó _{2000 Elsevier}

Science Inc. All rights reserved.

Keywords:HIV; Delay; Viral life cycle; T-cells

1. Introduction

In recent years, clinical research combined with mathematical modeling has enhanced progress in the understanding of HIV-1 infection. The introduction of a new class of drugs, protease www.elsevier.com/locate/mbs

*_{Corresponding author. Tel.: +1-505 667 6829; fax: +1-505 665 3493.}

E-mail address:asp@receptor.lanl.gov (A.S. Perelson).

inhibitors, used in combination with previously approved drugs such as zidovudine (AZT) and lamivudine (3TC) has shown much promise in reducing the detectable levels of viral RNA in the plasma. The introduction of these potent antiviral agents has opened the door to de®ning kinetic parameters associated with HIV-1 dynamics in infected individuals. Using mathematical models to interpret clinical data from HIV-1 infected patients treated with these potent antiviral agents has led to an estimate of the half-life of productively infected cells of 1.6 days or less and of the clearance rate of free virions from the plasma of six hours or less [1±3]. These initial estimates were claimed to be minimal estimates because the models used to analyze the patient data assumed the drug was completely e€ective. Re®ning these estimates will improve our quantitative description of viral dynamics and our understanding of the speed at which drug resistance emerges. Also, the dynamics of the models assumed that cells upon infection instantly begin producing virus. Both the assumption of 100% drug ecacy and of instantaneous production of virus upon infection are relaxed in this paper.

Ho et al. [4] and Wei et al. [3] developed models that were applied to data from an experiment where patients received an antiretroviral drug. Before drug therapy was initiated the patients were determined to have a quasi steady state concentration of virus in their plasma, suggesting that the rate of viral production equaled the rate of virus clearance. After administration of the drug, blood samples were taken for several days and plasma viral RNA levels were measured. The results showed that when virus production was interfered with by drugs the virus declined approximately exponentially. The data, when analyzed in the context of the model, allowed an estimate to be made of the quasi-steady state rate of viral production. The estimated production level was much higher then previously thought and provided an explanation for the ease by which HIV could mutate and escape therapy when only a single drug was used. Extensions of this work showed that after drug is applied there is a delay before virus begins to decline [1]. This delay is due to pharmacological events, the mechanism of protease inhibitor action that allows pre-existing virus particles to continue infecting cells and delays in the process of producing virus particles [1,5,6]. The viral decline is at ®rst very rapid and then after about two weeks slows [2]. Thus there is a rapid ®rst phase followed by a slower second phase of viral decline. Analysis of the data in the context of models showed that the viral decline during the ®rst, rapid, phase could be used to estimate bothc, the virus clearance rate constant andd, the decay rate of productively infected T-cells [1]. In the model, the asymptotic rate of the exponential decline of virus con-centration is eÿdt

. In order to get this result, it was assumed the drug was completely e€ective and that the virus clearance rate constant, c, was much larger than d, and therefore associated with processes that decayed more quickly and consequently could be ignored after a brief initial pe-riod. It was later shown [7] that if the drug or combination of drugs was less then perfect, the exponential decline of the ®rst phase could be represented by eÿdn_ct

e€ects of drug therapy are not studied. Herz et al. [5] examined the e€ect of including a constant delay in the source term for productively infected T-cells but their analysis was limited to the case where the drug treatment was assumed completely e€ective. Mittler et al. [6] also assumed that there was a delay between initial infection and the start of virus production, a period that vi-rologists call the eclipse phase of the viral life cycle, but rather than using a ®xed delay they assumed that the delay varied between cells according to a gamma distribution. They used this model to examine if the parameters that characterize viral dynamics could be identi®ed from biological data and then compared the parameter estimates obtained in a previous model without delay and those obtained with their delay model [10]. These works extended the ability of models to describe the relevant biological processes and addressed the implications of ignoring the intracellular delays that are part of the viral life cycle. In the models of Herz et al. [5] and Mittler et al. [6], the drug was assumed to completely block viral production. This assumption, we will show, cancels out the e€ect the intracellular delay has on the decay rate of virus pro-ducing T-cells. Hence, while noticing a change in viral dynamics during the eclipse phase, it was assumed the delay had no e€ect during the ®rst phase's exponential decline. This paper's main focus is to examine the change in dynamics when a delay of discrete type is considered and the drug, a protease inhibitor, is assumed less then perfect. We have already shown [6,11] that when ®tting a model to experimental data there are signi®cant changes in the values of the in-fected T-cell loss rate and the viral clearance rate constant if the model includes a delay. Here, we derive a new approximation, eÿdn_pC_s;n_p;d†t

, for the asymptotic ®rst-phase viral decay when a constant intracellular delaysis considered and protease inhibitor treatment is applied. Here the new factor is C_s;n_{p;d† ˆ ‰1‡ …1ÿ}n_p†dsŠÿ1_{, where} n_{p is the e€ectiveness of the protease} inhib-itor. We provide a detailed analytical study of the model equations and examine the e€ects of adding a delay on the stability of solutions, something that was neglected in the analysis of previous delay models (see Table 1).

Table 1

List of variables and parametersa

De®nition Mean/Initial value Source

Variable

Tt_{† Uninfected T-cells 180 cells/mm}3 _[4]

Tt_{† Productively infected T-cells 2 % T-cells [4]}

V_It_{† Infectious virus 13410}3_{virions/ml [4]}

V_NIt_{† Non-infectious virus 0 virions/ml [4]}

n_pt_† Protease inhibitor ecacy (0,1) Variable

Parameter

k T-cell source term 0_ÿ10 cells/mm3 _[30]

d1 Death rate of target T-cells 0:03/day [31]

k Viral infectivity rate 3:4310ÿ5_{ml/virions/day [4]}

d Death rate of an infected T-cell 0:5/day [1]

N Bursting term for viral production after lysis 480 virions/cell [1]

c Clearance rate of virus 3/day [1]

a_{The parameter values used in generating the ®gures shown later in the paper are given and the references from which}

2. General model

The models in this paper only deal with dynamics occurring after drug treatment. The models are not designed to examine the progression from time of infection until drug initiation [12,13]. The study of HIV-1 dynamics has focused on the use of a general set of model equations that keep track of di€erent cell populations and virus. In one case the model has been linear [4], while in most cases the model includes non-linear e€ects [14±17]. In most cases the models, while trivial mathematically, have provided important results when ®tted to experimental data [3,4]. Models have distinguished between infected and non-infected T-cells [1], included other populations of immune cells, such as macrophages [2], looked at variability in certain crucial parameters, such as the infectivity rate,k[18], viral burst size,N, and included drug treatments [7,19,20]. The general form of the non-delay model includes compartments for the densities of uninfected T-cells, T, infected T-cells that are producing virus, T_{, infectious virus,} V_{I, and non-infectious virus} (pro-duced by the action of a protease inhibitor), V_{NI, and is written as}

dT ducing T-cells, c is the viral clearance rate constant, and n_{p represents the drug ecacy of a} protease inhibitor. The system (1) models the dynamics of virus levels in the plasma during drug treatment. The term 1_ÿn_{p†represents the level of leakiness of a protease inhibitor, a drug that} inhibits the cleaving of viral polyproteins and renders newly produced virions non-infectious. Thus, if n_{pˆ1, the protease inhibitor is 100% e€ective and no infectious virus particles are} produced. Analysis of (1) has shown that allowing T to vary dynamically has not yielded sig-ni®cantly di€erent behavior in the total viral output, V _ˆV_I‡V_{NI, over a period of a week or so} [21,22]. Because of this, we will analyze a model in which Tis held constant at the value T_0.

2.1. Approximation to dominant eigenvalue

The eigenvalues of (1) determine the rate of viral decline, after therapy is initiated. Assuming that the system is in steady state before therapy is initiated so thatc_ˆNkT_{0 and that}T _ˆT_{0, then} the eigenvalues of the ®nal three equations of (1) can be expressed as [22]

k1;2 ˆ ÿ

k1 _ÿdn_{p: …3†}

This shows when therapy is less then perfect the ®rst phase slope of viral decline is not simply the death rate,d, of productively infected T-cells. Previously, the value ofdhas been estimated from experimental data, assuming a completely e€ective drug [3,4]. Relaxing this assumption of a perfect drug will change the estimate ofd. However, as we show below, when the drug ecacy is not 100%, delays between the time of cell infection and viral production will also in¯uence the estimate of d because k1 now becomes k1 _ÿdn_pC_s;n_{p;d† where} C_s;n_{p;d† is a new factor} in-troduced by the delay.

3. Intracellular delay model

In HIV-1 infection, the virus life cycle plays a crucial role in disease progression. The binding of a viral particle to a receptor on a target cell initiates a cascade of events that can ultimately lead to the target cell becoming productively infected, i.e. producing new virus. The previous model (1) assumed this process to occur instantaneously. In other words, it is assumed that as soon as virus contacts a target cell the cell begins producing virus. However, in reality there is a time delay between initial viral entry into a cell and subsequent viral production. Previous work has shown that this delay needs to be taken into account to accurately determine the half life of free virus from drug perturbation experiments, but if the drug is assumed to be completely ecacious, the delay does not a€ect the estimated rate of decay of viral producing T-cells [5,6,10]. Here we show that the delay does a€ect the estimated value for the infected T-cell loss rate when one assumes the drug is not completely e€ective.

We incorporate the intracellular delay in the model given by (1) by assuming that the gener-ation of virus producing cells at timetis due the infection of target cells at timet_{ÿs, wheresis a} constant. As before, we assume uninfected T-cells remain constant, i.e., T _ˆT_{0. The equations} describing this model are then

dT dt ˆ

kT₀V_It_ÿs†eÿms_ÿdT_; dV_I

dt ˆ …1ÿnp†NdT

_ÿcV

I; …4†

dV_NI

dt ˆnpNdT

_ÿcV

NI;

where the termV_It_{ÿs†allows for the time delay between contact and virus production and e}ÿms

3.1. Analysis of delay model

The model (4) in vector±matrix notation is

dyt_†

The initial conditions, on the interval 06t_{6s are de®ned as}

yt_{† ˆ}gt_†; 6_†

Using the results in [23] it is easy to show that a system of the form (5), with initial conditions (6), has a unique solution which is continuous for tP0 and which satis®es (5) for t_>s.

3.1.1. Characteristic equation

Time delays when used in population dynamic models have been shown to create ¯uctuations in population size due to the generation of instabilities [24,25]. Here, we will show that the ei-genvalues of the delay model remain in the left half plane for all values of the delay. With this being the case the dominant eigenvalue will then be the negative eigenvalue whose real part is closest to zero.

Introducing a delay of discrete type in a system of di€erential equations changes the structure of the solution as determined by the characteristic equation. In the case when no delay is present a linear system ofNdi€erential equations will haveNsolutions and a characteristic polynomial of degree N. When a delay is included the characteristic equation is no longer a polynomial but rather takes the form (see [26])

H_{k† ˆ}G_{k† ‡}K_k†eÿks_{ˆ0; …7†} whereG_{k† and} K_{k† are polynomials and}

P_{k† ˆ}G_{k† ‡}K_{k† ˆ0; …8†}

[27]) that ass_!0,nof the roots of (7) tend towards the roots ofP_{k† ˆ0, while the remainder of} the roots have real parts tending to large negative values. Hence ass_!0 the stability properties of the equilibrium points in the non-delay model (1) are recovered.

The characteristic function of (4), determined from the Laplace transform [23], is

H_{1…k;s† …k}2_{‡ …d‡}c_†k‡dc_ÿgeÿks_†…k‡c_{†; …9†} whereg_ˆ1_ÿn_p†dNkT_0eÿm_s

. Further, if we assume that the system is in quasi-steady state before drug therapy, i.e.,c_ˆNkT_{0, and that the death during transition from infected to productively} infected is negligible, i.e., m_{1, we can then simplifyg to}

g_ˆ1_ÿn_p†dc_{: …10†}

From (9) one eigenvalue is equal to_ÿc _{and hence always negative. In what follows we determine} the other eigenvalues from the solutions of

H_{k;s† ˆk}2_{‡ …d‡}c_†k‡dc_ÿgeÿks_{ˆ0: …11†} The following results show that the eigenvalues will remain in the left half plane and show the location of the largest eigenvalue. Setk_ˆl_‡im wherel;m_2Rand substitute into (11) to get

H_{l;m;s† ˆ ‰l}2_ÿm2_{‡ …d‡}c_†l‡dc_ÿgeÿls_{cosmsŠ ‡i‰2lm‡m…d‡}c_{† ‡ge}ÿls_{sinmsŠ; …12†} whereH_l;0;s† H_{l† ˆl}2_{‡ …d‡}c_†l‡dc_ÿgeÿls_.

Lemma 1. IfH_{l†>0then there is nom2Rsuch that}H_{l‡im† ˆ0.Moreover,if}H_{l† ˆ0then} H_{l‡im† ˆ0 if and only ifmˆ0.}

Proof (by contradiction). Assume thatH_{l‡im† ˆ0. Then}H_{l‡im†can be rewritten in terms of} its real and imaginary parts as

l2_ÿm2_‡ld_‡c_{† ‡d}c_ˆgeÿls_{cosms and 2lm‡m…d‡}c_{† ˆ ÿge}ÿls_sinms:

Taking the squares of the absolute values of both equations and adding together gives

‰m2_‡l_‡c_†2_Š‰m2_{‡ …l‡d†}2_{Š ˆg}2_eÿ2ls which implies that

…l_‡c_†2_l‡d†2_ÿg2_eÿ2ls_{ˆ ÿm}4_ÿm2_‰…l‡c_†2_{‡ …l‡d†}2_Š: The left-hand side of this equation equals

H_l†‰…l‡c_{†…l‡d† ‡ge}ÿls_{Š ˆ}H_l†‰H_{l† ‡2ge}ÿls_Š;

which is always positive given H_{l†>0. The right-hand side is always negative leading to a} contradiction. Therefore, H_{l‡im† 6ˆ0 given} H_{l†>0. The second part of the proof is easily} seen.

In the remainder of this section we will show that there exists one real eigenvalue, calledl^, on the interval_‰ÿc_{;0† (see Fig. 1) and that for certain parameter values all other eigenvalues, either} real or complex, have real parts at least an order of magnitude smaller thenl_^. (This is similar to showing the approximation ofk1 ÿdnp was valid givencd.)

Theorem 1. For each sP0 there is exactly one l^s_{† 2 ‰}min_ÿc_{;ÿd†;1† such that} H_{l^† ˆ0.}

Moreover, whenH_{l†>0 forl>l^ thenl^ is the unique eigenvalue with the largest real part.}

Proof.

H_{l† ˆl}2_{‡ …d‡}c_†l‡dc_ÿgeÿls _and H0_{l† ˆ2l‡ …d‡}c_{† ‡gse}ÿls_:

For l_{2 ‰ÿ}c_;ÿdŠ, H_{l†<0 and for l}P ÿ …c_‡d†=2,H0_{l†>0. Hence}H_{l† is a monotonic} in-creasing function on the interval _‰ÿ…c‡₂d†;_1†. Therefore, there exists exactly one unique l^ such thatH_{l^† ˆ0 forl^2 ‰min…ÿ}c_{;ÿd†;1†. Now forl>l^,}H_{l†>0 and by Lemma 1 cannot be the} real part of any eigenvalue. Thus, l^is the largest real eigenvalue.

Moreover, by direct computation,H_{0† ˆ}n_pdc_{>0. Thus, by Lemma 1,l^<0. In other words,}

^

ls_{† 2 ‰}min_ÿc_{;ÿd†;0†, and further assuming} c_{d, l^…s† 2 ‰ÿ}c_{;0†. Thus, there is only one real} root in the interval,_‰ÿc_{;0†, To complete this analysis we must show that this interval contains no} complex eigenvalues. If there exists complex eigenvalues in _‰ÿc_{;0† then we would be unable to} claim that the real eigenvalue,l_^, is the slowest varying, or dominant, eigenvalue since the complex eigenvalues would have real parts of the same order of magnitude asl^. Hence we would be unable to isolate a single dominant term.

3.2. Numerical analysis of location of complex roots

By de®nition if there exists a pair of complex eigenvalues in _‰ÿc_{;0† then given l2 ‰ÿ}c_;0†, H_{l‡im† ˆ0 for some m2R. We can write the eigenvalue equation}H_{k† in terms of two real} equations de®ned as

l2

‡ …d_‡c_†l‡dc_ÿm2 _ˆgeÿls_{cosms; …13a†}

2lm_‡md_‡c_{† ˆ ÿge}ÿls_{sinms: …13b†}

We can rewrite (13a) as

H_{l† ÿm}2_ˆgeÿls_{cosms; …14†}

whereH _ˆl2_‡c_‡d†l‡dc_{. It is easily seen that on the intervall2 ‰ÿ}c_;ÿdŠ,H_{l†60. We can} also de®ne some constraints to validate numerical results.

3.2.1. Constraint I

Given l_{2 ‰ÿ}c_{;ÿdŠ, the left-hand side of (14) is always less then or equal to zero. For} 0<ms<p=2 the right-hand side of (14) is positive and hence there can be no solution to the equation. Subsequently, for 0<ms<p=2 and l_{2 ‰ÿ}d;0_†, (13b) cannot be satis®ed. In general, for (13a) and (13b) to have complex roots, the following has to hold:

maxH_ÿm2_†Pmin…geÿlscosms†: …15†

Forl_{2 ‰ÿ}c_{;ÿdŠ, min…ge}ÿls_{cosms† ˆ ÿge}c_s

and maxH_ÿm2_{† ˆ ÿm}2_{. Thus, (15) implies that for a}

root to exist,m2_6gecs_{. Since there is no solution if 0<ms<p=2,p=2s<m<}p_gec_{s is a necessary}

condition for a non-zero root. Hence, if 

gec_s

p _{<p=2s there can be no non-zero solution.}

Rear-ranging, we ®nd that if

0<spec_s

< p

2p_g

then (13a) and (13b) has no solution form_6ˆ0.

3.2.2. Constraint II

As pointed out to us by D. Calloway, Cornell University, the derivative of (13a) with respect to

lis 1=m times the left-hand side of (13b). Therefore,

o

o_l…ge

ÿls_cosms

† ˆ ÿ1_mgeÿls_sinms;

and hence a condition for the system (13a) and (13b) to have a solution is

ms_ˆtanms: 16_†

This result gives an in®nite number of solutions due to the periodicity of the tangent function. Solving this equation numerically, we ®ndms_ˆ0,4:4934,7:7235, . . . But, from (14), we can write

m2<gec_s

which gives a bound on m necessary for (13a) and (13b) to have a solution. For example, with

d_ˆ0:5,c_{ˆ3 and a drug ecacy} n_{pˆ0:7,gˆ0:45. Hence for s<1:5, m<6:36 and hence the} solutions with msP7:7235 can be excluded. Thus the constraints provide a ®nite range of m in which to search for solutions, simplifying the following numerical calculations.

3.2.3. Numerical calculations

Using the above results, we examine the possible existence of complex eigenvalues numerically. We restrict our attention to the casec_{ˆ3. Discretizingl,m,s, and}n_{pwith steps sizes of 0:01 and} using Matlab we ®nd forl_{2 ‰ÿ}3;0_†, andn_{p2 …0:7;0:9†there exists no solution of (13a) and (13b)} with m_6ˆ0 for determined ranges of s. As an example, with d_ˆ0:5, c_{ˆ3 and} n_pˆ0:8, nu-merically we ®nd no solutions of (13a) and (13b) when s<0:825. This agrees surprisingly well with constraint I, which implies no solutions givens<0:828. If we use a higher ecacy of drug treatment, say n_{pˆ0:95 then numerically we can show there are no solutions with s<1:18 and} using constraint I, we ®nd no solutions fors<1:105. Varyingdbetween0:25;0:75_† adjusts the upper bound ons. For example, withd_ˆ0:35,c_{ˆ3 and} n_{p ˆ0:95, (13a) and (13b) is not} sat-is®ed with l>_ÿ3 givens<1:15. These numerical results compared with the above constraints forsandmprovide sucient evidence to support the conclusion that there exists only one unique root which is real and dominant while also providing sucient reasoning for neglecting, to ®rst approximation, all eigenvalues of the delay model except the unique real eigenvalue.

The value ofc may be higher than c_{ˆ3 [10,28,29]. If this is so, then a high drug ecacy is} required to guarantee that there will not be any complex eigenvalues. For example, if c_{ˆ5 and}

s_ˆ1 then only with a drug ecacy of 0:92 or higher, will there be one unique real eigenvalue. Ifc

is much higher, then our results show the existence of complex eigenvalues but the order of magnitude of their real parts is still dominated by the single negative and real eigenvalue. Therefore, it should still be possible to describe the viral decline seen in patients after drug treatment to a good approximation by a single exponential but the shoulder region no longer depends onc alone but a combination ofc and the complex eigenvalues.

4. Delay e€ects on dominant eigenvalue

With the results of the previous section we can now present the main ®nding of this work: an asymptotic representation for the dominant eigenvalue and how the delay e€ects its value (see Fig. 1). Since we have shown that forl_{2 …ÿ}3;0_†there are no complex roots for a de®ned s, we can let l^ be the unique root of H_l;0;s† H_{l^;s† ˆl^}2_{‡ …d‡}c_†l^‡dc_ÿgeÿls^ _{ˆ0 with the} largest real part. Expanding eÿ^ls_{in a Taylor series gives}

eÿls^

1_ÿls^ _‡l^ 2_s2

2 ‡0…l^

3_s3

†; 18_†

notice from (22), that assincreases,l^decreases, hence_jls^ _jremains suciently small. Substituting

The square root term can be approximated by the ®rst two terms of its binomial expansion, since for the parameter values of interest

Using this and taking the positive term we get

^

l _ÿ dcÿg

d_‡c_‡gs: …21†

Assumingc_{d[2,10,29], and using the de®nition of g (10), we ®nd}

^

is the correction introduced by incorporating a delay. Notice the rate of decay,l^, is proportional to the product of the rate constant for death of virus producing T-cells,d, and the ecacy of the protease inhibitor,n_{p, but now this product is reduced by the delay since assgets bigger the value} ofl_^gets smaller.

We illustrate the e€ects of delay in Figs. 2 and 3. Whenn_{p6ˆ1, the delay reduces the long term} rate of decline of the viral load and hence any estimate ofdthat is made which does not take the delay into account will be ¯awed. The e€ect, however, is not large. For example, ifs_ˆ1 day, a typical value, andn_{p ˆ0:8 then the correction due to the delay}

C_s;n_{p;d† ˆ} 1 1_‡0:2d:

Fig. 2. The top plot shows the numerical solution for viral decay when a perfect drug is used. Notice the slopes of the decay with and without a delay are parallel. The bottom two graphs compare the viral decay with and without a delay assuming a less then perfect drug; withn_{pˆ0:8 in the left panel and}n_{pˆ0:5 in the right panel. Both cases show the} change in the slope is a€ected by the delay. The model parameters, besides the ones noted, are given in Table 1.

5. Conclusions

The main goal of this paper was to show that the viral decay seen in HIV-1 infected patients put under antiretroviral therapy depends not only on the lifespan of viral producing cells,d, but also the ecacy of the therapy,n_{p and the length of the intracellular delay, s, that measures the time} between viral infection of a cell and the time the cell begins releasing virus. Here we have focused on the e€ects of protease inhibitor therapy because detailed data have been published about the kinetics of viral load decline and used to estimate bothcand d[1]. In [1] a model with no delay and n_{pˆ1 was used to estimate c and d, and the claim was made that the estimates of both of} these parameters were minimal estimates. Our results support the claim that the estimate ofdwas minimal since the inclusion of drug ecacy and delay both reduce the slope of the viral decay curve. Hence estimates ofd made by examining the slope, as was in essence done in [1], under-estimate the true value ofd. Recent work by Mittler et al. [6,10] using a model with delay argue that the published estimates ofc were also minimal as claimed. Previous work of Herz et al. [5] and Mittler et al. [6] have shown that an intracellular delay can e€ect the length of the `shoulder region' observed in Fig. 3, i.e., the time between initiation of therapy and the observed decay in plasma virus. But in both of these theoretical models, no changes were predicted during the ex-ponential decline of the viral load. We have shown that this is due to assuming the protease in-hibitor to be completely e€ective. When the ecacy,n_{pˆ1, the e€ects of the delay cancel out. The} above results imply that the inclusion of a constant delay and allowing the protease inhibitor to be less then perfect changes the estimates for the productively infected T-cell loss rate,d, and the viral decay rate,c.

Including a delay also changes the estimated lifespan of a productively infected cell. In previous work without delays the average lifespan of a productively infected cell was calculated as 1=d. Here we have taken into consideration that there are two phases in the life of cell from time of infection to death. First, there is an eclipse phase in which the virus enters the cell, uses reverse transcriptase to copy its RNA genome into DNA, and proceeds through a number of steps in its life cycle before new virus particles are produced. The length of the eclipse phase is represented by the delays. Following the eclipse phase, the cell produces virus. In the current model the average lifespan of a virus producing cell is 1=d. Thus the total average lifespan of cell from infection to death iss_‡1=d. The precise value ofsis not known but current estimates suggest values of 1±1.5 days [1,6,10]. Thus, to a ®rst approximation T-cells infected with HIV-1 might live an average of 2±3 days rather than the average of 1±2 days previously reported [1,3,4]. Currently, work is in progress examining delays in drug activation and modeling the ecacy of the drug as a dy-namically varying parameter. Also, there is a need to explore the e€ects of di€erent types of delays such as distributed delays or multiple discrete delays; some of which is already in progress. The length of the delay is not directly measurable in vivo. Thus, additional in vitro data would prove valuable in expanding our knowledge of the delay characteristics.

Acknowledgements

the auspices of the US Department of Energy and supported by NIH grant RR06555 (ASP) and an Institute for Mathematics and Its Applications Postdoctoral Fellowship (PWN) with funds provided by the National Science Foundation.

References

[1] A.S. Perelson, A.U. Neumann, M. Markowitz, J.M. Leonard, D.D. Ho, HIV-1 dynamics in vivo: Virion clearancerate infected cell life-span and viral generation time, Science 271 (1996) 1582.

[2] A.S. Perelson, P. Essunger, Y. Cao, M. Vesanen, A. Hurley, K. Saksela, M. Markowitz, D.D. Ho, Decay characteristics of HIV-1-infected compartments during combination therapy, Nature 387 (1997) 188.

[3] X. Wei, S.K. Ghosh, M.E. Taylor, V.A. Johnson, E.A. Emini, P. Deutsch, J.D. Lifson, S. Bonhoe€er, M.A. Nowak, B.H. Hahn, S.S. Saag, G.M. Shaw, Viral dynamics in human immunode®ciency virus type 1 infection, Nature 373 (1995) 117.

[4] D.D. Ho, A.U. Neumann, A.S. Perelson, W. Chen, J.M. Leonard, M. Markowitz, Rapid turnover of plasma virions and CD4 lymphocytesin HIV-1 infection, Nature 373 (1995) 123.

[5] V.M. Herz, S. Bonhoe€er, R.M. Anderson, R.M. May, M.A. Nowak, Viral dynamics in vivo: Limitations onestimations on intracellular delay and virus decay, Proc. Nat. Acad. Sci. USA 93 (1996) 7247.

[6] J.E. Mittler, B. Sulzer, A.U. Neumann, A.S. Perelson, In¯uence of delayed virus production on viral dynamics in HIV-1 infected patients, Math. Biosci. 152 (1998) 143.

[7] L.M. Wein, R.M. D'Amato, A.S. Perelson, Mathematical considerations of antiretroviral therapy aimed at HIV-1 eradication or maintenance of low viral loads, J. Theoret. Biol. 192 (1998) 81.

[8] Z. Grossman, M. Feinberg, V. Kuznetsov, D. Dimitrov, W. Paul, HIV infection: how e€ective is drug combinationtreatment? Immunol. Today 19 (1988) 528.

[9] J. Tam, Delay e€ect in a model for virus replication, IMA J. Math. Appl. Med. Biol. 16 (1999) 29.

[10] J.E. Mittler, M. Markowitz, D.D. Ho, A.S. Perelson, Re®ned estimates for HIV-1 clearance rate and intracellular delay, AIDS 13 (1999) 1415.

[11] P.W. Nelson, J. Mittler, A.S. Perelson, E€ect of the eclipse phase of the viral life cycle on estimation of HIV viral dynamic parameters, submitted.

[12] H.J. Bremermann, Mechanism of HIV persistence: implications for vaccines and therapy, J. AIDS 9 (1995) 459. [13] N.I. Stilianakis, C.A.B. Boucher, M.D. DeJong, R. VanLeeuwen, R. Schuurman, R.J. DeBoer, Clinical data sets on human immunode®ciency virus type 1 reverse transcriptase resistant mutants explained by a mathematical model, J. Virol. 71 (1997) 161.

[14] S. Bonhoe€er, R.M. May, G.M. Shaw, M.A. Nowak, Virus dynamics and drug therapy, Proc. Nat. Acad. Sci. USA 94 (1997) 6971.

[15] P. Essunger, A.S. Perelson, Modeling HIV infection of CD4+ T-Cell subpopulations, J. Theoret. Biol. 170 (1994) 367.

[16] D.E. Kirschner, R. Mehr, A.S. Pereson, The role of the thymus in pediatric HIV-1 infection, J. AIDS 18 (1998) 95. [17] M.A. Nowak, R.M. Anderson, M.C. Boerlijist, S. Bonhoe€er, R.M. May, A.J. McMichael, HIV-1 evolution and

disease progression, Science 274 (1996) 1008.

[18] N.I. Stilianakis, K. Dietz, D. Schenzle, Analysis of a model for the pathogenesis of AIDS, Math. Biosci. 145 (1997) 27.

[19] M.A. Nowak, S. Bonhoe€er, G.M. Shaw, R.M. May, Anti-viral drug treatment: dynamics of resistance in free virus and infected cell populations, J. Theoret. Biol. 184 (1997) 205.

[20] D.E. Kirschner, G.F. Webb, Understanding drug resistance for monotherapy treatment of HIV infection, Bull. Math. Biol. 59 (1997) 763.

[21] P.W. Nelson, Mathematical models of HIV pathogenesis and immunology, PhD thesis, University of Washington, pp. 1±150, 1998.

[24] D.M. Pratt, Analysis of population development inDaphniaat di€erent temperatures, Biol. Bull. 85 (1943) 116. [25] C.E. Taylor, R.R. Sokal, Oscillations in house¯y population sizes due to time lags, Ecology 57 (1976) 1060. [26] N. MacDonald, Time Lags in Biological Models, Springer, Berlin, 1970.

[27] L.E. El'sgol'ts, S.B. Norkin, An Introduction to the Theory and Application of Di€erential Equations with Deviating Arguments, Academic Press, New York, 1973.

[28] T. Igarashi, C. Brown, A. Azadegan, N. Haigwood, D. Dimitrov, M.A. Martin, R. Shibata, Human immunode-®ciency virus type 1 neutralizing antibodies accelerate clearance of cell-free virions from blood plasma, Nature Med. 5 (1999) 211.

[29] B. Ramratnam, S. Bonhoe€er, J. Binley, A. Hurley, L. Zhang, J.E. Mittler, M. Markowitz, J.M. Moore, A.S. Perelson, D.D. Ho, Rapid production and clearance of HIV-1 and hepatitis C virus assessed by large volume plasma apheresis, Lancet 354 (1999) 1782.

[30] D.E. Kirschner, G.F. Webb, A model for treatment strategy in the chemotherapy of AIDS, Bull. Math. Biol. 58 (1996) 367.