c 2007 Society for Industrial and Applied Mathematics

Transcription

1 SIAM J. APPL. MATH. Vol. 7, No. 3, pp c 27 Society for Inustrial an Applie Mathematics MATHEMATICAL ANALYSIS OF AGE-STRUCTURED HIV- DYNAMICS WITH COMBINATION ANTIRETROVIRAL THERAPY LIBIN RONG, ZHILAN FENG, AND ALAN S. PERELSON Abstract. Various classes of antiretroviral rugs are use to treat HIV infection, an they target ifferent stages of the viral life cycle. Age-structure moels can be employe to stuy the impact of these rugs on viral ynamics. We consier two moels with age-of-infection an combination therapies involving reverse transcriptase, protease, an entry/fusion inhibitors. The reprouctive number R is obtaine, an a etaile stability analysis is provie for each moel. Interestingly, we fin in the age-structure moel a ifferent functional epenence of R on ɛ RT, the efficacy of a reverse transcriptase inhibitor, than that foun previously in nonage-structure moels, which has significant implications in preicting the effects of rug therapy. The influence of rug therapy on the within-host viral fitness an the possible evelopment of rug-resistant strains are also iscusse. Numerical simulations are performe to stuy the ynamical behavior of solutions of the moels, an the effects of ifferent combinations of antiretroviral rugs on viral ynamics are compare. Key wors. human immunoeficiency virus type, antiretroviral therapy, rug resistance, optimal viral fitness, age-structure moel, stability analysis AMS subject classifications. 35L, 5D5, 92C37, 92C5, 92C5 DOI..37/395. Introuction. Since the iscovery of the human immunoeficiency virus type (HIV-) in the early 98s, the isease has sprea in successive waves to most regions aroun the globe. It is reporte that HIV has infecte more than million people, an over a thir of them subsequently ie []. Consierable scientific effort has been evote to the unerstaning of viral pathogenesis, host/virus interactions, immune response to infection, an antiretroviral therapy. Over the last ecae, there has been a great effort in the mathematical moeling of HIV infection an treatment strategies. These moels mainly investigate the ynamics of the target cells an infecte cells, viral prouction an clearance, an the effects of antiretroviral rugs treatment. Perelson et al. [] an Ho et al. [22] use a simple mathematical moel to analyze a set of viral loa ata collecte from infecte patients after the aministration of a protease inhibitor, an the virion clearance rate, the rate of loss of prouctively cells, an the viral prouction rate were estimate. These estimates were minimal estimates since the effects of antiretroviral rugs were assume to be % effective, an cells were assume to prouce new virus immei- Receive by the eitors June 28, 2; accepte for publication (in revise form) January 3, 27; publishe electronically March 2, 27. Portions of this work were performe uner the auspices of the U.S. Department of Energy uner contract DE-AC52-NA2539. The U.S. Government retains a nonexclusive, royalty-free license to publish or reprouce the publishe form of this contribution, or allow others to o so, for U.S. Government purposes. Copyright is owne by SIAM to the extent not limite by these rights. Department of Mathematics, Purue University, West Lafayette, IN 797 (rong@math.purue. eu, zfeng@math.purue.eu). The manuscript was finalize when the first author visite the Theoretical Biology an Biophysics Group, Los Alamos National Laboratory in 2. The research of the secon author was supporte in part by NSF grant DMS-3575 an the James S. McDonnell Founation 2st Century Science Initiative. Theoretical Biology an Biophysics, Los Alamos National Laboratory, MS K7, Los Alamos, NM 8755 (asp@lanl.gov). The research of this author was supporte by NIH grants AI2833 an RR

2 732 LIBIN RONG, ZHILAN FENG, AND ALAN S. PERELSON ately after they were infecte [35, 3]. In orer to characterize the time between the infection of target cells an the prouction of virus particles, an intracellular elay was introuce by Herz et al. [9] in a mathematical moel to analyze the clinical ata. Subsequently, Culshaw an Ruan [3] investigate the effect of the time elay on the stability of the enemical equilibrium in their moel. Criteria were presente to guarantee the asymptotic stability of the infecte steay state inepenent of the time elay. In [35], Nelson, Murray, an Perelson stuie a generalize moel that inclue a iscrete elay an allowe for less than perfect rug effects. The estimation of kinetic parameters unerlying HIV infection was improve by the use of a elay ifferential equation moel. In [3, 32], the authors use a gamma istribution function to escribe a continuous elay between infection an viral prouction an foun no change in the estimate of δ, the eath rate of prouctively infecte cells. However, Nelson an Perelson [3] extene this moel an showe that the constancy of δ was ue to the assumption of % rug effectiveness. When rug effectiveness was less than %, the estimate of δ epene on the elay, i.e., the variance an mean of the assume gamma istribution. Recently, a moel incluing both pharmacokinetics an the intracellular elay has been employe to obtain new estimates of intracellular elay an the antiviral efficacy of ritonavir [7]. Age-structure moels have also been evelope to stuy the epiemiology of HIV. Thieme an Castillo-Chavez [52] kept track of an iniviual s infection age to stuy the effect of infection-age-epenent infectivity on the ynamics of HIV transmission in a homogeneously mixing population. Kirschner an Webb [2] propose a moel that incorporate age structure into the infecte cells to account for the mechanism of AZT (ziovuine) treatment. Recently, for the within-host ynamics of HIV, age-structure moels have receive increasing interest ue to their greater flexibility in moeling viral prouction an mortality of infecte cells [, 3]. Nelson et al. [3] consiere an age-structure moel that allowe for variations in the prouction rate of virus particles an the eath rate of infecte T cells. For a specific form of the viral prouction function an constant eath rate of infecte cells, the authors performe a local stability analysis of the nontrivial equilibrium point. They use numerical simulations to illustrate that the time to reach the peak viral level epene not only on the initial conitions but also on the spee at which viral prouction achieves its maximum value. Base on this age-structure moel, Gilchrist, Coombs, an Perelson [] use the various life history trae-offs between viral prouction an clearance of infecte cells to erive the within-host relative viral fitness. In this article, we evelop two age-structure moels to stuy HIV- infection ynamics. These moels exten the existing age-structure moels [, 2, 3] by incorporating combination therapies to stuy the influence of antiretroviral therapy on the evolution of HIV-. The first moel inclues therapy with a combination of a reverse transcriptase (RT) inhibitor an a protease inhibitor, while the secon moel inclues an entry inhibitor an a protease inhibitor. To account for the fact that reverse transcription takes place in the early stage of infection before an infecte T cell prouces virus particles, we ivie the infecte cells into two subclasses. One subclass represents the cells that have been infecte by the virus but in which reverse transcription has not been complete. The other subclass contains infecte cells that have finishe the reverse transcription process an are capable of proucing new virions. Our stability analysis is performe for a general form of both the viral prouction rate an the mortality rate of infecte cells. The stability of the infectionfree or the infecte steay state is shown to epen on the reprouctive ratio R being

3 ANALYSIS OF AGE-STRUCTURED HIV- MODELS 733 smaller or greater than. The formulation of this reprouctive ratio also provies an appropriate measure for the within-host viral fitness, which can be use to explore the optimal viral prouction rate for which R is maximize. We also iscuss the possible influence of treatment for rug-sensitive strains of HIV- on the evelopment of rug-resistant strains of the pathogen. Clinical stuies have suggeste that prolonge treatment with a single antiretroviral rug may be associate with the emergence of resistant virus [2, 2, 27, 28, 39]. The impact of rug treatment on the ynamics of resistant stains of pathogens has been stuie using age-inepenent mathematical moels (see, for example, [, 2, 55]). We show that if viral prouction is linke to resistance, then higher treatment efficacy with antiretroviral agents (such as protease inhibitors) may lea to the establishment of multiple viral strains with a wier range of resistance levels. The organization of the remaining part is as follows. In section 2, we formulate a mathematical moel for HIV- infection that generalizes the age-structure moel propose in [3] by incorporating an RT inhibitor an a protease inhibitor. Section 3 is evote to the analysis of our moel, incluing the existence an stability of both the infection-free an the infecte steay states. In section, another moel incluing therapy with a new class of rugs, fusion/entry inhibitors, is evelope. Stability properties of the steay states are also obtaine in this section. In section 5, we erive a criterion for invasion by rug-resistant strains an explore how rug treatment may affect the optimal viral fitness of resistant strains. Some numerical simulations are presente in section to illustrate/exten our analytical results. We also compare the treatment effects of these two combination antiretroviral therapies. Section 7 contains concluing remarks. 2. The moel with RT an protease inhibitors. HIV infection begins by the attachment of a virus to a CD + cell. Insie the cell, the HIV- enzyme RT makes a DNA copy of the virus s RNA genome. During this process, if an RT inhibitor is present, then the viral genome will not be copie into DNA, an therefore the host cell will not prouce new virus. When the virus replicates, its DNA is rea out to prouce viral proteins. A large polyprotein is mae, an a viral protease is neee to cut the long polypeptie chain into iniviual components that are neee to prouce infectious virus particles. If the HIV- protease is inhibite, the newly prouce virus will be noninfectious. From the above escription of the HIV life cycle an the roles of various inhibitors, it is clear that the infection age of an infecte cell can be important for the stuy of HIV ynamics uner the influence of antiretroviral rug treatment. In [3] the following age-structure moel of HIV infection (without rug treatment) was propose: (2.) T (t) =s T kv T, t t T (a, t)+ a T (a, t) = δ(a)t (a, t), t V (t) = p(a)t (a, t)a cv, T (,t)=kv T, where T (t) enotes the concentration of uninfecte target T cells at time t, T (a, t)

4 73 LIBIN RONG, ZHILAN FENG, AND ALAN S. PERELSON enotes the concentration of infecte T cells of infection age a (i.e., the time that has elapse since an HIV virion has penetrate the cell) at time t, an V (t) enotes the concentration of infectious virus at t. s is the recruitment rate of healthy T cells, is the per capita eath rate of uninfecte cells, δ(a) is the age-epenent per capita eath rate of infecte cells, c is the clearance rate of virions, k is the rate at which an uninfecte cell becomes infecte by an infectious virus, an p(a) is the viral prouction rate of an infecte cell with age a. The functional forms of the viral prouction kernel, p(a), an the eath rate of infecte cells, δ(a), nee to be etermine experimentally [2, 3]. In [3], the authors choose the following function for the prouction rate: (2.2) p(a) = { p ( e θ(a a)) if a a, else, where θ etermines how quickly p(a) reaches the saturation level p, an a is the age at which reverse transcription is complete. To incorporate the two types of treatments mentione above, we ivie the class of infecte cells, T (a, t), into two subclasses: TpreRT (a, t) an T postrt (a, t). T prert (a, t) represents the ensity of cells that have been infecte by an HIV virion but in which reverse transcription has not been complete at infection age a. An RT inhibitor coul allow a prert cell to revert back to an uninfecte cell (because if reverse transcription fails to complete, cellular nucleases will egrae the HIV RNA that entere the cell) or reuce the probability that a prert cell progresses to the postrt state [9]. TpostRT (a, t) represents the ensity of infecte cells that have progresse to the postrt phase at infection age a. The ensities of the prert an postrt cells are relate by a function β(a) ( β(a) ) that escribes the proportion of infecte cells that have not complete reverse transcription, i.e., (2.3) T prert (a, t) =β(a)t (a, t), T postrt (a, t) =( β(a))t (a, t). We assume that β(a) L [, ) is a nonincreasing function with the following properties: β(a) ; β()=; β(a) =fora a ; β (a) a.e. Let ɛ RT an ɛ PI enote the efficacy of the therapy with RT inhibitors an protease inhibitors, respectively ( ɛ RT, ɛ PI < ). The efficacy is scale such that zero represents complete ineffectiveness an unity represents % effectiveness. To stuy the effect of protease inhibitor, we ivie the newly prouce virus particles into two classes: infectious virions with concentration V I (t) an noninfectious virions with concentration V NI (t). New infectious virus particles are prouce at the rate ( ɛ PI )p(a)tpostrt (a, t)a. Let η(ɛ RT ) enote the rate at which prert cells revert to the uninfecte state ue to the failure of reverse transcription. The rate at which prert cells of all ages become uninfecte is then given by η(ɛ RT )TpreRT (a, t)a. The reversion rate η(ɛ RT ) is an increasing function of rug efficacy ɛ RT. In the absence of rug therapy, we assume there are no infecte cells going back to the uninfecte class, i.e., η() =. As the limit case, when RT inhibitors are % effective (ɛ RT ), η(ɛ RT ) shoul be very large. We shall iscuss the functional form of η(ɛ RT ) more in the simulation section. Our analytical results are obtaine for a general reversion rate function.

5 ANALYSIS OF AGE-STRUCTURED HIV- MODELS 735 Incorporating these rugs into the equations for T, T, an V in moel (2.), we have t T (t) =s T kv IT + η(ɛ RT )TpreRT (a, t)a, t T (a, t)+ a T (a, t) = δ(a)t (a, t) η(ɛ RT )TpreRT (a, t)a, (2.) t V I(t) = ( ɛ PI )p(a)tpostrt (a, t)a cv I, t V NI(t) = ɛ PI p(a)tpostrt (a, t)a cv NI, T (,t)=kv I T. Notice that the variable V NI oes not appear in equations for other variables. Thus, we can ignore the V NI equation when stuying the ynamics of infection. Using the relation (2.3), we have the following system: (2.5) t T (t) =s T kv IT + η(ɛ RT )β(a)t (a, t)a, t T (a, t)+ a T (a, t) = δ(a)t (a, t) η(ɛ RT )β(a)t (a, t), t V I(t) = ( ɛ PI )( β(a))p(a)t (a, t)a cv I, T (,t)=kv I T. In our analysis, we allow the viral prouction rate p(a) to be an arbitrary function that is boune (e.g., it oes not have to be a monotone function). δ(a) is also assume to be a boune function. Since we are intereste in the effect of combination therapy on virus ynamics, we assume that the patients are initially at steay state an the combination of rugs is aministere at time. We choose the initial conitions to be T () = T, V I () = V I, V NI () =, an T (a, ) = T (a), where T an V I are the steay state levels of target cells an infectious virions, respectively. T (a) is the age istribution of infecte cells at the initial time t =, an T (a)a represents the steay state level of infecte cells before the onset of rug therapy. System (2.5) can be reformulate as a system of Volterra integral equations. To simplify expressions, we introuce the following notations: (2.) K (a) =e a (δ(s)+η(ɛ RT )β(s))s, K (a) =η(ɛ RT )β(a)k (a), K 2 (a) =( ɛ PI )( β(a))p(a)k (a), K i = K i (a)a, i =, 2. K (a) is the probability of an infecte cell remaining infecte at age a, hereafter the age-specific survival probability of an infecte cell. K 2 (a) is the prouct of the agespecific survival probability of an infecte cell an the rate at which infectious virus particles are prouce by an infecte cell of age a. Thus, the integral of K 2 (a) over all ages, i.e., K 2 = ( ɛ PI)( β(a))p(a)k (a)a, gives the total number of infectious virus particles prouce by one infecte cell over its lifespan. For convenience, we call K 2 the infectious virus burst size.

6 73 LIBIN RONG, ZHILAN FENG, AND ALAN S. PERELSON For mathematical convenience, we introuce a new variable, B(t), to escribe the rate at which an uninfecte T cell becomes infecte at time t, (2.7) B(t) =kv I (t)t (t). Integrating the T equation in system (2.5) along the characteristic lines, t a = constant, we get the following formula: B(t a)k (a) for a<t, (2.8) T (a, t) = T K (a) (a t) for a t. K (a t) Substituting (2.8) into the T an V I equations in (2.5), (2.9) where (2.) t T (t) =s T B(t)+ K (a)b(t a)a + t F (t), t t V I(t) = K 2 (a)b(t a)a cv I + F 2 (t), F (t) = F 2 (t) = t t η(ɛ RT )β(a)t K (a) (a t) K (a t) a, ( ɛ PI )( β(a))p(a)t K (a) (a t) K (a t) a. Clearly, Fi (t) ast, i =, 2. Integrating the T equation in (2.9) an changing the orer of integration, we have t u ] T (t) =T e t + e [s (t u) B(u)+ B(u τ)k (τ)τ + F (u) u (2.) t [ ] = e (t u) (s B(u)) + B(u)H (t u) u + F (t), where (2.2) t H (t) =e t e τ K (τ)τ, F (t) =T e t + t e (t u) F (u)u. Similarly, by integrating the V I equation in (2.9), we get t [ u ] V I (t) =V I e ct + e c(t u) B(u τ)k 2 (τ)τ + F 2 (u) u (2.3) t = B(u)H 2 (t u)u + F 2 (t), where (2.) t H 2 (t) =e ct e cτ K 2 (τ)τ, F 2 (t) =T e ct + t e c(t u) F2 (u)u. Equations (2.) an (2.3), with B(t) replace by kv I (t)t (t), form a system of Volterra integral equations that are equivalent to the original system (2.5). Hence,

7 ANALYSIS OF AGE-STRUCTURED HIV- MODELS 737 for etermining the existence an uniqueness of the solutions we nee only consier the following system: t [ T (t) = e (t u) (s kv I (u)t (u)) + kv I (u)t (u)h (t u) ] u + F (t), (2.5) t V I (t) = kv I (u)t (u)h 2 (t u)u + F 2 (t), where H i an F i (i =, 2) are given in (2.2) an (2.). 3. Analysis of the system (2.5). In this section, we provie analytic results on the existence of positive solutions as well as possible steay states an their stability for the system (2.5) or the equivalent system (2.5). 3.. Existence of positive solutions. Let x(t) = (T (t),v I (t)), where enotes the transpose of the vector. System (2.5) can be written in the form x(t) = t κ(t u)g(x(u))u + f(t), where f(t) =(F (t),f 2 (t)) is a continuous function from [, ) to[, ) 2, κ is the 2 2 matrix with entries being locally integrable functions on [, ), ( ) se t H κ(t) = (t) e t, H 2 (t) an g is efine by g(x) =(,kv I T ). Obviously, f C([, ); R 2 ), g C(R 2, R 2 ), an κ L loc ([, ); R2 2 ). Theorem. in Gripenberg, Lonen, an Staffans [7, section 2.], shows that a continuous solution exists on a maximal interval such that the solution goes to infinity if this maximal interval is finite. To see that all solutions will remain nonnegative for positive initial ata, we use the following system (see (2.7) an (2.9)) that is also equivalent to system (2.5): (3.) t T (t) =s T B(t)+ K (a)b(t a)a + t F (t), t t V I(t) = K 2 (a)b(t a)a cv I + F 2 (t), B(t) =kv I (t)t (t), where F i is given in (2.) an F i (t) >, lim t Fi (t) =fori =, 2. Suppose that there exists a t > such that T ( t) =ant (t),v I (t) > for t< t. Then B( t) =kv I ( t)t ( t) =,B(t) =kv I (t)t (t) > for t< t, an thus from the T equation in (3.) we have t T ( t) =s+ t K (a)b( t a)a+ F ( t) >. Hence, T (t) for all t. Similarly, we can show that V I (t) an B(t) for all t an for all positive initial ata Steay states an their stability. We use the system (3.) for our stability analysis. Accoring to [3], any equilibrium of system (3.), if it exists, must be a constant solution of the following limiting system: T (t) =s T (t) B(t)+ K (a)b(t a)a, t (3.2) t V I(t) = K 2 (a)b(t a)a cv I, B(t) =kv I (t)t (t).

8 738 LIBIN RONG, ZHILAN FENG, AND ALAN S. PERELSON We mention that the introuction of the variable B(t) is just for mathematical convenience. If we substitute kv I (t)t (t) for B(t) in the first two equations of (3.2), then we will obtain the same stability results. System (3.2) has two constant solutions, the infection-free steay state Ē = ( T, V I, B) =(s/,, ), an the infecte steay state E =(T,V I,B ), where (3.3) T = c kk 2, V I = skk 2 c kc( K ), B = kt V I, with K an K 2 given in (2.). Notice that K is less than. Thus, V > ifan only if skk 2 c >, or R >, where (3.) R = skk 2 c. Clearly, the infecte steay state (3.3) is feasible if an only if R >. Notice that s/ is the cell ensity in the absence of infection, an k an c are the cell infection an viral clearance rate, respectively. Recall that K 2, the infectious virus burst size, gives the number of infectious virus particles prouce by one infecte cell over its lifespan. Therefore, R gives the reprouctive ratio of the virus uner the impact of rugs. We now consier the stability of steay states. Let us first consier the infectionfree steay state Ē. The following result suggests that the population sizes of virus an infecte cells will go to zero if the reprouctive ratio is less than. Theorem. The noninfecte steay state Ē is locally asymptotically stable (l.a.s) if R <, an it is unstable if R >. Proof. The Jacobian matrix of (3.2) at the steay state Ē is J = λ ks/ ˆK (λ) c λ ˆK2 (λ), ks/ where λ is an eigenvalue an ˆK i (λ) enotes the Laplace transform of K i (a), i.e., ˆK i (λ) = K i (a)e λa a, i =, 2. The corresponing characteristic equation is ( (λ + ) λ + c sk ) (3.5) ˆK 2 (λ) =. One negative root of equation (3.5) is λ =, an all other roots are given by the equation (3.) λ + c = sk ˆK 2 (λ), which can be rewritten as (3.7) λ c +=R ˆK 2 (λ). K 2 Notice that ˆK 2 (λ) K 2 for all complex roots λ with nonnegative real parts (i.e., Rλ ). Hence, the moulus of the right-han sie of (3.7) is less than, provie that R <. Since the moulus of the left-han sie of (3.7) is always greater than

9 ANALYSIS OF AGE-STRUCTURED HIV- MODELS 739 or equal to if Rλ, we conclue that all roots of (3.) have negative real parts if R <. It follows that Ē is l.a.s. when R <. In the case of R >, let ψ(λ) = λ c + R ˆK 2(λ) K 2. Thus, any real roots of ψ(λ) = are also roots of (3.). Recognizing that ψ()= R < an lim λ ψ(λ) =, we know that ψ(λ) = has at least one positive root λ >, which is a positive eigenvalue of the characteristic equation (3.5). This shows that the infection-free steay state is unstable when R >. The following theorem eals with the global stability of the noninfecte steay state Ē. Theorem 2. For R <, the noninfecte steay state Ē is a global attractor, i.e., lim t (T (t),v I (t),b(t))=(s/,, ). In orer to prove Theorem 2, we nee the following lemma, in which the following notations are use: ϕ = lim inf t ϕ(t), ϕ = lim sup t ϕ(t), where ϕ is a realvalue function on [, ). Lemma (see [5]). Let ϕ: [, ) R be boune an continuously ifferentiable. Then there exist sequences s n,t n as n such that ϕ(s n ) ϕ, ϕ (s n ) an ϕ(t n ) ϕ, ϕ (t n ). Proof of Theorem 2. It is ifficult to apply Lemma to the T equation of (2.5) irectly. We introuce a new variable, W (t) =T (t) +T (t), where T (t) enotes the total number of infecte cells at t. Notice from the T equation in (2.5) that T satisfies the equation T t = kv I T [ δ(a)+η(ɛrt )β(a) ] T (a, t)a. Then we get W t = s (W T ) δ(a)t (a, t)a = s W (δ(a) )T (a, t)a s W. The last inequality hols because of the fact that δ(a) (i.e., the eath rate of infecte cells δ(a) is equal to the natural eath rate plus an extra eath rate ue to the infection). By Lemma, we can choose a sequence t n such that W (t n ) W, W (t n ). From W t s W,wehaveW s/. Rewrite the V I equation in (2.5) as V I (t) = t kv I(t u)t (t u)h 2 (u)u+f 2 (t). We use Lemma to choose a sequence s n such that V I (s n ) VI as n. Taking supremum limit on both sies of the above V I equation for t = s n, we have VI kvi T H 2 (u)u. Noticing that T W s/ an that H 2 (u)u = K 2 /c, we get VI ksk 2 VI /(c) =R VI. Since R <, we see that VI =. Thus, V I (t) ast. It also follows that B(t) since B(t) =kv I (t)t (t) an T W s/. We use Lemma again to choose a sequence s n such that T (s n ) T an T (s n ). Using the T equation in (3.2) we get T s/. But T W s/. This shows that T (t) s/ as t, which finishes the proof of Theorem 2. Next, we consier the stability of the infecte steay state E. As note earlier, this steay state exists if an only if R >. The following result suggests that the virus population will be establishe if the reprouctive ratio is greater than. Theorem 3. The infecte steay state E is l.a.s if R >. Proof. The Jacobian at the steay state E is kvi λ kt ˆK (λ) J = c λ ˆK2 (λ). kvi kt Using the notation R = skk 2 /c, the corresponing characteristic equation can be

10 7 LIBIN RONG, ZHILAN FENG, AND ALAN S. PERELSON written as (3.8) ( )( ( K )λ + (R K ) ( = (R ) ) λ + c c ˆK 2(λ) K 2 (λ + c) ˆK (λ) c ˆK 2(λ) K 2 ), or ( + λ )( ) A(λ + )+ c ˆK (λ) = ˆK 2 (λ) (3.9) A ( λ + ), K 2 where A =( K )/((R )). We can exclue the possibility of a nonnegative real root of (3.9) as follows. Suppose λ. Then ˆK (λ) ˆK () = K <. It follows that A>an ( + λ c )(A(λ + )+ ˆK (λ)) >A(λ + ). Hence, (3.9) yiels ˆK 2 (λ)/k 2 >. However, since λ, we have ˆK 2 (λ) ˆK 2 () = K 2, which leas to a contraiction. Thus, (3.9) has no nonnegative real roots. In the next step, we will exclue the possibility that (3.9) has a complex root λ with a nonnegative real part. We prove this by contraiction. Suppose that λ = x + iy is a root with x an y >. From (3.8), we have (3.) ( (λ + ) λ + c c ˆK 2 (λ) K 2 ) as R. It follows from a similar argument as in Theorem that λ = x + iy cannot be a root if x >. Now we let x = an y >. In this case, (3.) has a negative root, an all other roots are etermine by the equation ( + λ c )= ˆK 2(λ) or (3.) + y c i = K 2 (a) cos(ya)a K 2 K 2 K 2 (a) sin(ya)a K 2 i. Comparison of the real parts of both sies yiels cos(ya) =. Thus, sin(ya) =, which implies that (3.) cannot hol. Therefore, (3.8) has no roots with nonnegative real parts when R. By the continuous epenence of roots of the characteristic equation on R,we know that the curve etermine by the roots must cross the imaginary axis as R ecreases close to. That is, the characteristic equation (3.8) or (3.9) has a pure imaginary root, say, iy, with y>. Replacing λ in (3.9) with iy, we see that the moulus of the left-han sie of (3.9) satisfies ( (3.2) LHS > A + K (a) cos(ya)a + i Ay + K (a) sin(ya)a). We claim that K (a) sin(ya)a. In fact, notice that K (a) sin(ya)a = a K (a) sin(ya)a, where a is the age at which reverse transcription is complete. Notice also that K () = η(ɛ RT ) an K (a) =η(ɛ RT )[β (a)k (a) +β(a)k (a)] a.e. on [, ). Integrating a K (a) sin(ya)a by parts, we get a K (a) sin(ya)a = η(ɛ RT ) y y K (a ) cos(ya )+ a K y (a) cos(ya)a η(ɛ RT ) y y K (a ) cos(ya )+ a K y (a)a = y K (a )( cos(ya )).

11 ANALYSIS OF AGE-STRUCTURED HIV- MODELS 7 Thus, we have K (a) sin(ya)a. We also observe that K (a) cos(ya)a K >. It follows from (3.2) that LHS >A + iy. On the other han, the moulus of the right-han sie of (3.9) satisfies RHS A + iy. This leas to a contraiction. We conclue that the characteristic equation (3.9) has no roots with nonnegative real parts. Therefore, Theorem 3 is prove.. The moel with entry an protease inhibitors. Since the iscovery of RT inhibitors an protease inhibitors, significant progress in rug evelopment has been mae. Recently, a new class of rugs, entry/fusion inhibitors, has been introuce [, 8]. These compouns can block the fusion of the viral envelope to the target cell membrane an interfere with continue infection. They became available with the FDA approval of enfuvirtie (Fuzeon) in 23. In this section, we evelop an age-structure moel that takes into account the effects of both entry inhibitors an protease inhibitors. The moel can be escribe by the following equations: t T (t) =s T ( ɛ EI)kV I T, t T (a, t)+ a T (a, t) = δ(a)t (a, t), (.) t V I(t) = ( ɛ PI )( β(a))p(a)t (a, t)a cv I, t V NI(t) = ɛ PI ( β(a))p(a)t (a, t)a cv NI, T (,t)=( ɛ EI )kv I T, where ɛ EI represents the efficacy of the entry inhibitor. The other parameters an variables have the same meaning as in the moel (2.). We remark that the moel in [3] is a special case of our moel (.) when ɛ EI = ɛ PI = β(a) =. Our result applies to a general form of the viral prouction rate p(a) an the eath rate δ(a). The existence an uniqueness of (nonnegative) solutions for the system (.) can be prove in a similar way as for the system (2.). Here we present only the stability analysis. The following notations are use throughout the rest of this section: K 3 (a) =e a δ(s)s, K (a) =( ɛ PI )( β(a))p(a)k 3 (a), K = The following limiting system is use to erive stability results: T (t) =s T (t) Y (t), t (.2) t V I(t) = K (a)y (t a)a cv I, Y (t) =( ɛ EI )kv I (t)t (t), K (a)a. where the variable Y (t) is introuce for mathematical convenience. System (.2) has two constant solutions (steay states): the noninfecte steay state Ē =( T, V I, Ȳ )=(s/,, ), an the infecte steay state E =(T,VI,Y ), where T = c, VI k( ɛ EI )K = sk( ɛ EI)K c, Y =( ɛ EI )kt VI. kc( ɛ EI )

12 72 LIBIN RONG, ZHILAN FENG, AND ALAN S. PERELSON Clearly, VI > if an only if R 2 >, where R 2 = sk( ɛ EI )K /(c) isthe reprouctive ratio for moel (.). Hence, E exists if an only if R 2 >. The stability results are given in the following theorem. It can be prove similarly by previous arguments. Here we omit the proof ue to the space limit. Theorem. (a) The noninfecte steay state Ē is a global attractor if R 2 < ; an it is unstable if R 2 >. (b) When R 2 >, the infecte steay state E is l.a.s. Results obtaine in this section an in the previous section will be use in the next section to explore the impact of rug treatment on the evolution of HIV-. 5. Influence of rug therapy on the invasion of resistant strains. In the previous sections, we have shown that a virus population can establish itself if an only if its reprouctive ratio excees. Consier an environment in which the rugsensitive strain of HIV- infection is at the infecte steay state E =(T,VI,B ) (see (3.3)), an a small number of rug-resistant virions has been introuce into the virus population. Denote the reprouctive ratio of the sensitive strain by R s, which is the same as R efine in (3.). We can rewrite the population size of uninfecte cells in terms of R s, i.e., T = s/(r s ). Assume that R s is greater than. Let ɛ RT an ɛ PI enote the efficacies of the two types of rugs for the resistant strain, respectively, an let p(a) enote the viral prouction rate of the resistant strain. We can efine the corresponing K (a) as the age-specific survival probability of T cells infecte with the resistant strain (an equivalent quantity for the sensitive strain is given in (2.)). For ease of illustration, we assume that all other parameters are the same for both strains. We erive an invasion criterion for a resistant strain by using a heuristic argument, as is one in []. This criterion will be applie to ifferent scenarios of antiretroviral therapy, such as single-rug therapy (e.g., ɛ PI > an ɛ RT = ) or combination therapy (i.e., ɛ PI > an ɛ RT > ). Notice that /c is the average lifespan of a free virus. Thus a single resistant virus can infect on average kt /c cells in its whole life. Each of these infecte cells can prouce a total of N r = ( ɛ PI )( β(a)) p(a) K (a)a infectious rug-resistant virus particles uring its lifespan (burst size). Thus, the reprouctive ratio of the resistant strain at the resient equilibrium ensity T is R r = kt ( ɛ PI )( β(a)) p(a) c K (a)a, an the invasion criterion is R r >. Substituting s/(r s ) for T, we obtain that the conition for the resistant strain to invae the sensitive strain is R r > R s, where the quantity (5.) R r = s k c ( ɛ PI )( β(a)) p(a) K (a)a represents the reprouctive ratio of the resistant strain when the equilibrium ensity of uninfecte cells is s/ (which is the value of T at the infection-free steay state). Viral fitness is often use to escribe the relative replication competence of a virus in a given environment. R r can be regare as a goo measure of the fitness of a resistant virus. Thus the inequality R r > R s implies that natural selection within a host favors the virus strain that maximizes its reprouctive ratio.

13 ANALYSIS OF AGE-STRUCTURED HIV- MODELS 73 In orer to calculate the reprouctive ratio, we consier the case when the viral prouction rate for the resistant strain has the form given in (2.2). That is, (5.2) p(a) = { p ( e θ(a a)) if a a, else, where p is the saturation level for prouction of the resistant strain. Accoringly, we choose β(a) tobe (5.3) β(a) = {, a<a,, a a. The eath rate of cells is assume to be the same for both strains with the form (5.) δ(a) = { δ, a<a, δ + μ, a a, where δ an μ are positive constants with δ representing a backgroun eath rate of cells an μ representing an extra eath rate for prouctively infecte cells ue to either viral cytopathicity or cell-meiate immune responses. Drug resistance is incorporate by assuming that the efficacy of antiretroviral therapy for the resistant strain is lower than that for the rug sensitive strain by a factor between an, i.e., ɛ RT = σ RT ɛ RT, ɛ PI = σ PI ɛ PI. For ease of emonstration, we assume that σ RT = σ PI = σ. σ = correspons to the completely resistant strain, while σ = correspons to the completely sensitive strain. Other strains have an intermeiate value <σ<. Many rug-resistant HIV variants isplay some extent of resistance-associate loss of fitness as the resistant viral strains propagate at a reuce rate when compare to sensitive strains [2]. Therefore, there is a traeoff between rug resistance an viral prouction rate p(a). We choose two types of functional forms for the cost by which the saturation level p is reuce in resistant strains, using the following formulas: (5.5) (5.) Type I : p(a) =σp ( e θ(a a)), Type II : p(a) =e φ( σ ) p ( e θ(a a)), where φ is a measure for the level of cost. We provie analytic results for the Type I cost an illustrate that the qualitative properties of the two types of costs are similar. Using (5.2) (5.5), we have the following relationship between R r an R s (see [3]): (5.7) R r = σ( σɛ PI)e η(ɛ RT )( σ)a ɛ PI R s. We consier R r = R r (σ) as a function of σ. A rug-resistant strain with resistance σ can invae the sensitive strain if R r (σ) > R s. Obviously, it is not easy to raw conclusions from this conition. We first erive some analytic unerstaning for a simpler case in which only single-rug therapy with a protease inhibitor is consiere, i.e., ɛ PI > an ɛ RT =. The case of combine therapy will be explore numerically. (a) Single-rug therapy. In this case, since ɛ PI > an ɛ RT =, (5.7) simplifies to R r (σ) = σ( σɛ PI) ɛ PI R s. It is easy to check that in orer to have R r (σ) R s for some σ (, ) it is necessary that ɛ PI > 2. In fact, there exists a maximum

14 7 LIBIN RONG, ZHILAN FENG, AND ALAN S. PERELSON R s 8 R r (a) ε PI =..5 σ min σ R s, 8 R r max R r (b) ε PI = σ R r (c) ε PI =.7 R r () ε PI =.8 R r max R s R r max R s. σ min.5 σ opt.8 σ. σ min.5 σ opt.8 σ Fig.. Plots of the reprouctive ratio R r of a resistant strain vs. the resistance σ for ifferent treatment efficacy ɛ PI (ɛ RT is chosen to be ). In (a) an (b), it is shown that R r < R s for all σ<. Therefore, no resistant strains can invae. In (c) an (), resistant strains with resistance σ in (σ min, ) can invae. The optimal resistance is σ opt at which R r reaches its maximum R r max. level of resistance (corresponing to the smallest value of σ), σ min = ɛ PI ɛ PI <, such that R r (σ) > R s if an only if σ min <σ< (see Figure ). Clearly, if ɛ PI < 2, then σ min >, an hence R r < R s for all σ. This inicates that when the rug efficacy is very low, the sensitive strain is favore. The intuitive reason for this is that if the cost of resistance is high, one woul not expect resistance when there is little selection pressure from the rugs. Other nonresistant strains woul outcompete it uner these conitions. Resistant strains can increase in frequency only when the selection pressure (rug efficacy) is high. We can also etermine an optimal resistance, σ opt, which maximizes the reprouctive ratio. In fact, we can easily check that R r (σ) has only one critical point in the interval (σ min, ), σ = 2ɛ PI, at which Rr(σ) σ = (see Figure ). Hence, σ opt =/(2ɛ PI ). We summarize the following results for the case of single-rug therapy. Recall that a resistant strain with resistance σ can invae the sensitive strain if an only if R r (σ) > R s. (i) There exists a threshol rug efficacy ɛ PI (ɛ PI =/2 for Type I cost) below which no resistant strains can invae (see Figure (a) (b)). Analytically, this is ue to the fact that σ min when ɛ PI <ɛ PI. Hence, R r(σ) < R s for all σ<. (ii) When the rug efficacy is above the threshol ɛ PI, there is a range of resistance levels for which the resistant strains are able to invae. This is because, analytically, σ min < when ɛ PI >ɛ PI, an R r(σ) > R s for all σ in (σ min, ). (iii) When σ min <, the range of invasion strains, (σ min, ), increases with the rug efficacy ɛ PI. The optimal resistance, σ opt, ecreases with the rug efficacy ɛ PI (a more resistant strain correspons to a smaller σ value; see Figure (c) ()).

15 ANALYSIS OF AGE-STRUCTURED HIV- MODELS 75 R r 7 ε RT =.,R r (σ) ε RT =.,R s ε RT =.3,R r (σ) ε RT =.3,R s ε RT =.5,R r (σ) ε RT =.5,R s σ Fig. 2. Plots of the reprouctive ratio R r vs. resistance σ for ɛ RT =. (soli), ɛ RT =.3 (long ashe), ɛ RT =.5 (short ashe). The value of ɛ PI is fixe at ɛ PI =. for which invasion is possible in the absence of the an RT rug (i.e., if ɛ RT =). For each given ɛ RT, the values of σ for which R r(σ) > R s give the range for resistance invasion, which is the range between the two intersection points of the R r curve an the R s horizontal line. This increasing property is also clear from the formulas σ min =( ɛ PI )/ɛ PI an σ opt =/(2ɛ PI ). (iv) As the rug efficacy increases, the optimal viral fitness, R r (σ opt ), ecreases (see Figure (c) ()). (b) Combination therapy. We now consier the case of combination therapy, i.e., ɛ PI > an ɛ RT >. Again, we consier R r = R r (σ) in (5.7) as a function of σ. Then R r (σ) > R s if an only if σ satisfies the inequality (5.8) σ( σɛ PI )e η(ɛ RT )( σ)a ɛ PI >. To explore the role of ɛ RT,wefixɛ PI (e.g., ɛ PI =.in Figure 2). Because the numerical simulations appear qualitatively similar for ifferent increasing reversion rate functions, we choose η(ɛ RT ) = ɛ RT for simplicity here. We will iscuss the selection of the function η(ɛ RT ) in the next section. Equation (5.8) cannot be solve analytically for σ. However, plots of R r (σ) for ifferent values of ɛ RT suggest that, as ɛ RT increases, the range for R r (σ) > R s also increases (see Figure 2). Figure 3 illustrates the joint effect of ɛ RT an ɛ PI on the reprouctive ratios R s an R r. From the contour plot (see Figure 3(c)), we see that when the rug efficacy is low (the region in the lower-left corner in which R s > R r > ) the resistant strain cannot invae. Neither strain can survive when the rug efficacy is high (the top-right region in which R s < an R r < ). In the mile region, the invasion of resistant strains is possible as R r > R s.

16 7 LIBIN RONG, ZHILAN FENG, AND ALAN S. PERELSON (a) (b) 3 R s,r r RT PI.25 R s RT PI (c) R s <<R r R s <R r < PI. <R s <R r.2 <R r <R s RT Fig. 3. Plots of the reprouctive ratios R s an R r as functions of ɛ RT an ɛ PI. Three surfaces are plotte in (a): R r(ɛ RT,ɛ PI ) (the top surface near the origin), R s(ɛ RT,ɛ PI ) (mile surface), an the constant (the bottom surface). The intersection of the top two surfaces is the curve on which R r = R s. In (b), two surfaces, R s(ɛ RT,ɛ PI ) an the constant, are plotte to show the curve on which R s =. (c) is a contour plot of the surfaces R r(ɛ RT,ɛ PI ) an R s(ɛ RT,ɛ PI ). Figure shows that when the Type II cost is use, the qualitative property of the reprouctive ratio R r as a function of σ is very similar to that when the Type I cost is use. For example, the function R r (σ) amits a unique σ min an a unique σ opt for sufficiently small values of φ.. Numerical results. In this section, we provie numerical simulations to confirm an/or exten our analytical results. Backwar Euler an the linearize finite ifference metho are use to iscretize the ODE an PDE, respectively, an the integral is evaluate using Simpson s rule. For all simulations, we choose the viral prouction rate p(a) as (2.2) an β(a) as (5.3) with a =.25 ays [2]. The eath rate of infecte cells δ(a) is assume to be constant δ = ay [29], an the virion clearance rate is set to our best estimate c =23ay [5]. The other moel parameters are chosen as follows [8]: s = ml ay, =. ay, k =2. 8 ml ay, an the burst size is N = 25. The reversion rate function, η(ɛ RT ), remains to be etermine. We know η()=, an when ɛ RT, η(ɛ RT ) shoul be sufficiently large such that all the prert cells will revert back to the uninfecte class. In our simulation, we assume the reversion rate function takes the following form: η(ɛ RT )= ρln( ɛ RT ), where the constant

17 ANALYSIS OF AGE-STRUCTURED HIV- MODELS 77 R r 5 φ=.5 φ=.9 φ=2.5 Type I cost R s σ Fig.. Plots of the reprouctive ratio R r vs. resistance σ when Type II cost is consiere. The value of φ measures the cost of resistance. Invasion is possible for σ in the range between the two intersection points at which R r = R s. It also shows that invasion is impossible if the cost is too high (e.g., φ =2.5). ρ controls the steepness of the function. From the stanar moel in which there are only short-live infecte cells (see []), the viral level will be theoretically suppresse to be below the limit of viral etection (5 RNA copies ml in the bloo) in.2 ays if RT inhibitors are assume to be % effective (we assume the same parameters as above an choose the initial viral loa to be ml ). In our moel (2.), uner the same initial conitions an parameters, if we choose ρ = 2 ay, then the viral loa can reach the same limit in.2 ays when the rug efficacy of RT inhibitors is very close to. Therefore, we will use the value ρ = 2 ay in our simulation to stuy the RT inhibitor s effects on the ynamics of viral loa. The abilities of RT inhibitors with ifferent ρ to suppress the viral loa will be iscusse later. Figures 5 an show numerical simulations of the first moel (2.) an the secon moel (.), respectively. For the calculations unerlying Figure 5, the maximum age of infecte cells a max is chosen to be ays [3]. In (2.2), we choose p =.2 3 an θ = to guarantee the burst size is 25 [8]. To see the influence of antiretroviral rug therapy on the viral ynamics, we choose the initial conitions to be the steay states of the stanar moel [] in the absence of rug treatment. We use T ()= ml [2] an V ()= ml [5] in the stanar moel to get the following steay state values: T = ml, T =.75 3 ml, V = ml, which are use as the initial values of our moels (2.) an (.). The value for the efficacy of the protease inhibitor is fixe at ɛ PI =.5. Figures 5(a) (b) an (c) () are for ifferent values of ɛ RT that increase from ɛ RT =.2 (Figure 5(a) (b)) to ɛ RT =.5 (Figure 5(c) ()). We observe that, when ɛ RT is increase, the infection In reality, the time to reach this limit is much longer, probably ue to the existence of long-live infecte cells an latently infecte cells [, ].

18 78 LIBIN RONG, ZHILAN FENG, AND ALAN S. PERELSON x 5 (a) 8 (b) Uninfecte cells T /ml ays log HIV RNA copies /ml 2 total virions infectious virions noninfectious virions ays Uninfecte cells T /ml x (c) log HIV RNA copies /ml 5 5 () total virions infectious virions noninfectious virions ays ays Fig. 5. Simulation of moel (2.) with ɛ PI =.5. The upper panel: ɛ RT =.2; the lower panel: ɛ RT =.5. The other parameters for each panel are the same: s = ml ay, =. ay, c =23ay, k =2. 8 ml ay, δ =ay, p =.2 3 ay, θ =, T = ml, V I = ml, V NI =, T =.75 3 ml (see text for escription). The reprouctive numbers of the upper an lower panel are. an.9223, respectively. The upper panel shows that the virus population stabilizes at a steay state an uninfecte T cell concentration remains at 8 μl, an the lower panel shows that the virus ies out an the T cell count reaches μl. level at which the system stabilizes is ecrease as expecte. When ɛ RT is greater than a threshol value (ɛ RT =.; see also Figure 8(c)), the virus population will ie out. Figure shows a similar qualitative behavior of the viral loa, although the efficacy of entry inhibitors has a ifferent threshol value, ɛ EI =.23 (Figure 8(c)). The virus population persists when ɛ EI <.23 an ies out when ɛ EI >.23. This is consistent with our analytic results, as the calculation of the reprouctive ratio for this set of parameters shows that R 2 > when ɛ EI <.23 an R 2 < when ɛ EI >.23. The ifferent behaviors of the moels shown in Figures 5 an inicate that the entry inhibitor appears more effective than the RT inhibitor uner given conitions. However, this comparison of effectiveness epens heavily on the choice of parameter ρ. Ifρ is increase to 5, then the RT inhibitors can suppress viral loa more effectively than entry inhibitors (see more iscussion in Figure 8). Figure 7 emonstrates how the viral loa can be affecte by the virion prouction rate p(a). Each rug efficacy has a fixe value: ɛ EI =.2, ɛ PI =.. We compare

19 ANALYSIS OF AGE-STRUCTURED HIV- MODELS 79 x 5 (a) 8 (b) Uninfecte cells T /ml ays log HIV RNA copies /ml 2 2 total virions infectious virions noninfectious virions 5 5 ays Uninfecte cells T /ml x (c) log HIV RNA copies /ml 5 5 () total virions infectious virions noninfectious virions ays ays Fig.. Simulation of moel (.) with ɛ PI =.5. The upper panel: ɛ EI =.2; the lower panel: ɛ EI =.5. The other parameters are the same as those in Figure 5. The reprouctive numbers of the upper an lower panel are.35 an.522, respectively. The upper panel shows that the virus population stabilizes at a lower steay state than in Figure 5(b) (the graphs o not show this clearly, but the numerical values show the ifference) an the uninfecte T cell concentration remains more than 9μl. The lower panel shows that the virus ies out an the T cell count reaches μl. This implies that the entry inhibitor appears more effective than the RT inhibitor in the given conitions. two sets of parameters p =.2 3,θ= (Figure 7(a) (b)) an p =3.53 3, θ = (Figure 7(c) ()) in the viral prouction function (2.2), which generate the same burst size, N = 25 [8]. However, the viral prouction rate in Figure 7(a) (b) ramps up more slowly to the saturation level than in Figure 7(c) (). We observe that there is not much ifference in the T cell ynamics, the viral peak, the time neee to reach the peak level, an the steay state viral loa, although the nair of the viral loa in panel () is less than that of panel (b). This implies that varying the viral prouction function oes not play an important role, at least in the long-term virus ynamics, given the same burst size. Comparing Figure 7(a) (b) with Figure (a) (b), we observe that when the rug treatment becomes more effective (ɛ PI increases from. to.5, ɛ EI =.2), the amplitue of the viral peak an the steay state viral loa are ecrease. However, it takes longer for the viral loa to reach its peak level when the rug efficacy is higher. A possible explanation for this phenomenon is the following. Because a more effective

20 75 LIBIN RONG, ZHILAN FENG, AND ALAN S. PERELSON x 5 (a) 7 (b) Uninfecte cells T /ml log HIV RNA copies /ml total virions infectious virions noninfectious virions ays 2 8 ays x 5 (c) 8 () Uninfecte cells T /ml ays log HIV RNA copies /ml 2 total virions infectious virions noninfectious virions ays Fig. 7. Simulation of moel (.) with ɛ EI =.2,ɛ PI =.. The upper panel: p =.2 3,θ =; the lower panel: p =3.53 3,θ =. (The burst size of each panel is the same: N = 25.) The other parameters are the same as those in Figure 5. The viral prouction of the lower panel ramps up more quickly to the saturation level than that of the upper panel. There is almost no ifference in the viral peak, the time to reach the peak level, an the steay state viral loa. This shows that the viral prouction function oes not play an important role in the long-term viral ynamics given the same burst size. rug treatment (assuming that it is not potent enough to eliminate the virus) can suppress the virus more substantially, the nair that the viral loa can reach is much lower than when the treatment is more effective. Thus the time for the viral loa to reach its peak level is prolonge. In Figure 8, we compare the effects of two combination therapies on reucing the viral loa. With the choice of p(a) an β(a) given in (2.2) an (5.3), we have the following reprouctive numbers: (.) R = e aη(ɛ RT ) M, R 2 =( ɛ EI )M, skθ where M = cδ(θ+δ) ( ɛ PI)p e δa. Let V () I an V (2) I states of moels an 2, respectively. Then V () I enote the viral steay = (R2 ) k( ɛ EI ), where K = η(ɛ RT ) δ+η(ɛ RT ) ( e (δ+η(ɛ RT ))a ). If we assume the reversion rate takes the = (R ) k( K, V (2) ) I

21 ANALYSIS OF AGE-STRUCTURED HIV- MODELS 75 Reprouctive ratio (a) ε RT ε EI Drug efficacy Infectious virus RNA copies /ml x 5 (b) ε RT ε EI Drug efficacy Reprouctive ratio (c) ε RT ε EI Drug efficacy Infectious virus RNA copies /ml x 5 () ε RT ε EI Drug efficacy Reprouctive ratio (e) ε RT ε EI Drug efficacy Infectious virus RNA copies /ml x 5 (f) ε RT ε EI Drug efficacy Fig. 8. Comparison of the two combination therapies with fixe protease inhibitor rug efficacy ɛ PI =.5. The other parameters are the same as those in Figure 5. Left column: reprouctive numbers R an R 2 as the function of ɛ RT an ɛ EI, respectively. If ɛ EI >.23, then R 2 <, an hence virus will ie out. Right column: steay state V I of moels (2.) an (.) as the function of ɛ RT an ɛ EI, respectively. The upper panel: ρ =, the threshol for R < is ɛ RT >.5; the mile panel: ρ =2, the threshol for ɛ RT is.; the bottom panel: ρ =5, the threshol for ɛ RT is.9. Forasmallρ (ρ < ), the entry inhibitors appear more effective than the RT inhibitors; for a large ρ (ρ >), we have the contrary result.