The influence of anti-viral drug therapy on the evolution of HIV-1 pathogens

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1 DIMACS Series in Discrete Mathematics an Theoretical Computer Science Volume 7, 26 The influence of anti-viral rug therapy on the evolution of HIV- pathogens Zhilan Feng an Libin Rong Abstract. An age-structure moel is use to stuy the possible impact of rug treatment of infections with the human immunoeficiency virus type (HIV-) on evolution of the pathogen. Inappropriate rug therapy often leas to the evelopment of rug-resistant mutants of the virus. Previous stuies have shown that natural selection within a host favors viruses that maximize their fitness. By emonstrating how rug therapy may influence the within host viral fitness we show that while a higher treatment efficacy reuces the fitness of the rug-sensitive virus, it may provie a stronger force of selection for rug-resistant viruses allowing a wier range of resistant strains to invae.. Introuction Reverse transcriptase inhibitors an protease inhibitors are the two major types of rugs that have been use as inhibitors of HIV- replication in vivo. Mathematical moels of HIV- infection uner the impact of rug treatments have been stuie using orinary an/or elay ifferential equations (see, for example [4, 8, ]). Nelson et al. evelope an age-structure moel of HIV- infection (without rug treatments) an showe that this moel is a generalization of ODE an DDE moels mentione above in the absence of treatments [9]. Such generalize moels are consiere to have greater flexibility that may better represent the unerlying biology of an infection [2]. In [9] the local stability of both the infection-free an the infecte steay states are shown for the case when the viral prouction rate has a special functional form (see Eq. (2.2)). These results are applie to a similar age-structure moel in [2] to stuy the effect of life history parameters of HIV- on maximizing within host viral fitness uner various assumptions on trae-offs between the virion prouction rate an other parameters. Drug treatments are again not inclue in this moel. In this chapter we generalize the moel in [9] by incorporating the effect of two classes of anti-hiv rugs which help to reuce the HIV replication at two ifferent stages of the cell infection. One class is the reverse transcriptase inhibitor which Key wors an phrases. HIV-, Combination therapy, Drug-resistance, Optimal viral fitness, Age-structure moel, Stability analysis. This work is supporte in part by NSF grant DMS an by James S. McDonnell Founation 2 st Century Science Initiative. 26 c 26 American Mathematical Society

2 262 ZHILAN FENG AND LIBIN RONG blocks the translation of viral RNA into DNA so that CD4 cells cannot prouce virion particles (the cells then become uninfecte). The other class is the protease inhibitor which targets the HIV protease enzyme to prevent new copies of HIV from being mae (so that some of the virion particles will remain non-infectious). Stability results are provie for a general form of the viral prouction rate, an the stability of the infection-free or the infecte steay state is shown to epen on the reprouctive ratio R being 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 virion prouction rate for which R is maximize. Recent clinical stuies have suggeste that prolonge treatment with a single anti-hiv rug may be responsible for the emergence of resistant virus [3, 4, 5, 6,, 2]. The impact of rug treatments on the ynamics of resistant stains of pathogens has been stuie using age-inepenent mathematical moels (see, for example, [, 4, 3]). We show that if the 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. This chapter is organize as follows: In Section 2, we formulate a mathematical moel for HIV- infection which generalizes the age-structure moel propose in [9] by incorporating both types of rug treatments. Section 3 is evote to the analysis of our moel incluing the existence an stability of both the infectionfree an the infection steay states. In Section 4, we erive a criterion for invasion by resistant strains an explore how rug treatments may affect the optimal viral fitness of resistant strains. Section 5 iscusses the results. 2. Moel formulation In [9] the following age-structure moel of HIV infection is propose: T (t) = s T kv T, t (2.) 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, with appropriate initial conitions. Here, T (t) enotes the population of uninfecte target T cells at time t, T (a, t) enotes the ensity 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 population 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 an infectious virus, k is the rate at which an uninfecte cell becomes infecte by one infectious virus, an p(a) is the virion prouction rate by an infecte cell of age a.

3 THE INFLUENCE OF DRUG THERAPY ON THE EVOLUTION OF HIV 263 One of the special forms of p(a) consiere in Moel (2.) is (2.2) p(a) = { p ( e θ(a a)) if a a, else where θ is a constant which etermines how quickly p(a) reaches the saturation level, p, an a is the maximum age at which reverse transcription takes place. We provie stability results in this chapter for an arbitrary function p(a) which allows the possibility that an infecte cell may start proucing viruses before the age a (i.e., in some infecte cells the reverse transcription may occur earlier than a ) an the possibility that p(a) may not be a monotone function of a (e.g., it may have a peak prouction rate at some intermeiate age). To account for the fact that an infecte cell oes not prouce any virus before the reverse transcription has taken place we introuce a function β(a) which escribes the proportion of infecte cells of age a that are not yet actively prouctive. Assume that β(a) L [, ) is a non-increasing function with the following properties: (2.3) β(a), β() =. Then, β(a) can be use to ivie the class of infecte cells, T (a, t), into two subclasses, TpreRT (a, t) an T postrt (a, t) which are efine by: (2.4) T prert (a, t) = β(a)t (a, t), T postrt (a, t) = ( β(a))t (a, t). TpreRT (a, t) represents the ensity of cells that have been infecte by an HIV virion but reverse transcription has not been complete at infection age a, an an RT inhibitor coul revert it back to uninfecte class (because RT fails to occur) or reuce the probability that a prert cell progresses to the postrt state. TpostRT (a, t) represents the ensity of infecte cells that have progresse to the phase at infection age a, an the presence of a protease inhibitor coul affect the rate at which new infectious virion particles are prouce. Let r rt an r p enote the efficacy of the treatment therapy with reverse transcriptase inhibitors an protease inhibitors, respectively ( r rt, r p ), an let η enote the maximal (age-inepenent) per capita rate at which prert cells of become uninfecte. Then the rate at which prert cells become uninfecte is given by r rt ηt prert (a, t)a, an new infectious virion particles are prouce at the rate ( r p )p(a)t postrt (a, t)a.

4 264 ZHILAN FENG AND LIBIN RONG Incorporating these rug treatments in the equations for T, T an V in Moel (2.) we have: T (t) = s T kv T + t r rt ηt prert (a, t)a, t T (a, t) + a T (a, t) = δ(a)t (a, t) r rt ηtprert (a, t), t V (t) = ( r p )p(a)t postrt (a, t)a cv. Thus, using the relation (2.4) we moify Moel (2.) to get the following moel: T (t) = s T kv T + t r rt ηβ(a)t (a, t)a, (2.5) t T (a, t) + a T (a, t) = δ(a)t (a, t) r rt ηβ(a)t (a, t), t V (t) = ( r p )( β(a))p(a)t (a, t)a cv, T (, t) = kv T, T () = T >, T (a, ) = T (a), V () = V >. As mentione earlier, we allow the virion prouction rate p(a) to be an arbitrary function (e.g., it oes not have to be a monotone function). p(a) an δ(a) are assume to be boune. System (2.5) can be reformulate as a system of Volterra integral equations. To simplify expressions we introuce the following notation: K (a) = e R a (δ(τ)+r rtηβ(τ))τ (2.6) K (a) = r rt ηβ(a)k (a) (a) = ( r p )( 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 (a) gives the probability that an infecte cell becomes noninfecte at age a given that the cell has not ie at age a. The probability that an infecte cell has not ie at age a is (2.7) e R a δ(τ)τ. The probability that an infecte cell is still in the prert stage an has not been treate at age a by an RT inhibitor is e Ra r rtηβ(τ)τ an hence the probability

5 THE INFLUENCE OF DRUG THERAPY ON THE EVOLUTION OF HIV 265 that the cell becomes noninfecte at age a ue to an RT inhibitor is (2.8) ( e R R a r rtηβ(τ))τ ) a = r rt ηβ(a)e rrtηβ(τ))τ. t K (a) is the prouct of the two probabilities given by (2.7) an (2.8). (a) is a prouct of the the age-specific survival probability of an infecte cell an the rate, ( r p )( β(a))p(a), at which infectious virion particles are prouce by an actively prouctive cell of age a. Thus, the integral of (a) over all ages, i.e., = ( r p )( β(a))p(a)k (a)a gives the total amount of infectious virion particles prouce by one infecte cell in its lifespan. This is an important quantity which will be use later to efine the viral fitness. For mathematical convenience we introuce the new variable, B(t), to escribe the rate at which an uninfecte T cell becomes infecte at time t, (2.9) B(t) = kv (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.) T (a, t) = T K (a) (a t) for a t. K (a t) Substituting (2.) into the T an V equations in (2.5): (2.) where (2.2) T (t) = s T B(t) + t t V (t) = F (t) = F 2 (t) = t t K (a)b(t a)a + F (t), (a)b(t a)a cv + F 2 (t), r rt ηβ(a)t K (a) (a t) K (a t) a, ( r p )( β(a))p(a)t K (a) (a t) K (a t) a. Clearly, F (t) as t. Integrating the T equation in (2.) an changing the orer of integration: (2.3) T (t) = T e t + = + u e (t u) [s B(u) ] B(u τ)k (τ)τ + F (u) u [ ] e (t u) (s B(u)) + B(u)H (t u) u + F (t),

6 266 ZHILAN FENG AND LIBIN RONG where (2.4) H (t) = e t e τ K (τ)τ, F (t) = T e t + e (t u) F (u)u. For the erivation of (2.3) we have use the following fact: (2.5) = = = = u e (t u) B(u τ)k (τ)τu u e (t u) B(α)K (u α)αu B(α) α e (t u) K (u α)uα (t α) B(α)e (t α) e σ K (σ)σα B(α)H (t α)α. Similarly, by integrating the V equation in (2.) we get [ u ] V (t) = V e ct + e c(t u) B(u τ) (τ)τ + F 2 (u) u where (2.6) = B(u)H 2 (t u)u + F 2 (t), H 2 (t) = e ct e cτ (τ)τ, F 2 (t) = T e ct + e c(t u) F2 (u)u. Equations (2.3) an (2.5), with B(t) replace by kv (t)t (t), form a system of Volterra integral equations which is equivalent to the original system (2.5). Hence, for the iscussion of existence an uniqueness of the solutions we only nee to consier the following system [ ] T (t) = e (t u) (s kv (u)t (u)) + kv (u)t (u)h (t u) u (2.7) +F (t), V (t) = kv (u)t (u)h 2 (t u)u + F 2 (t), where H i an F i (i =, 2) are given in (2.4) an (2.6).

7 THE INFLUENCE OF DRUG THERAPY ON THE EVOLUTION OF HIV Moel analysis In this section we provie analytic results on the existence of positive solutions as well as possible steay states an their stability. 3.. Existence of positive solutions. Let x(t) = (T (t), V (t)). System (2.7) can be written in the form x(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 following 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 T ). Obviously, f C([, ); R 2 ), g C(R 2, R 2 ), an κ L loc ([, ); R2 2 ). Theorem. in Gripenberg et al. (99), Section 2., now provies us with a continuous solution efine on a maximal interval such that the solution goes to infinity if this maximal interval is finite. To see that all solutions will remain non-negative for positive initial ata, we use the following system (see (2.9) an (2.)) which is also equivalent to System (2.5): T (t) = s T B(t) + t K (a)b(t a)a + F (t), (3.) t V (t) = (a)b(t a)a cv + F 2 (t), B(t) = kv (t)t (t), where F i is given in Eq. (2.2) an F i (t) >, lim t Fi (t) = for i =, 2. Suppose that there exists a t > such that T ( t) = an T (t), V (t) > for t < t. Then B( t) = kv ( t)t ( t) =, B(t) = kv (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 (t) an B(t) for all t an for all positive initial ata Steay states an their stability. We use System (3.) for our stability analysis. Accoring to [7], any equilibrium of System (3.), if it exists, must be

8 268 ZHILAN FENG AND LIBIN RONG a constant solution of the limiting system: T (t) = s T B(t) + t K (a)b(t a)a, (3.2) t V (t) = (a)b(t a)a cv, B(t) = kv (t)t (t). System (3.2) has two constant solutions: the infection-free steay state (3.3) Ē = ( T, V, B) = ( s,, ), an the infecte steay state (3.4) T = c k, E = (T, V, B ) V = sk c kc( K ), where B = kt V with K an given in (2.6). Notice that β(a) = for a a an K < r rt ηβ(a)e R a r rtηβ(s)s a R ( ) a = e r rtηβ(s)s a a = e R r rtηβ(s)s. Thus, V > if an only if sk c >, or R >, where (3.5) R = sk c. Clearly, the infecte steay state (3.4) is feasible if an only if R >. We can interpret R by noticing that s/ is the cell ensity in the absence of infection; k an c are the cell infection an viral clearance rates, respectively; an gives the infectious virion particles prouce by one infecte cell uring its entire life. Therefore, R is the reprouctive ratio of the virus uner the impact of rug treatments. We now consier the stability of steay states. Let us first consier the infection-free steay state Ē. The following result suggests that the population sizes of virus an infecte cells will go to zero as t if the reprouctive ratio is less than. Result Ē is locally asymptotically stable if R <, an it is unstable if R >. This stability result can be verifie as the follows. Using System (3.2) we get the characteristic equation at the steay state Ē (3.6) et λ ks/ ˆK (λ) c λ ˆK2 (λ) =, ks/

9 THE INFLUENCE OF DRUG THERAPY ON THE EVOLUTION OF HIV 269 where λ is an eigenvalue an ˆK i (λ) enotes the Laplace transform of K i (a), i.e., (3.7) ˆKi (λ) = K i (a)e λa a, i =, 2. The characteristic equation (3.6) can be written as (3.8) (λ + ) [ λ + c sk ˆ (λ) ] =. One negative root of Eq. equation (3.8) is λ = an all other roots are given by the (3.9) λ + c = sk ˆ (λ), which can be rewritten as (3.) λ c + = R ˆ (λ). Notice that ˆ (λ) for all complex roots λ with non-negative real part (i.e., Rλ ). Hence, the moulus of the right-han sie of Eq. (3.) is less than provie that R <. Since the moulus of the left han sie of Eq. (3.) is always greater than if Rλ, we conclue that all roots of (3.9) have negative real part if R <. It follows that Ē is locally asymptotically stable when R <. For the case of R >, let (3.) f(λ) = λ c + R ˆ (λ). Clearly, any real root of f(λ) = is also a root of (3.9). Recognizing that (3.2) f() = R <, lim f(λ) =, λ we know that f(λ) = has at least one positive root λ > which is a positive eigenvalue of the characteristic equation (3.8). This shows that the infection-free steay state is unstable when R >. 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. Result 2 The infecte steay state E is locally asymptotically stable if R >. For the verification of Result 2 we now look at the characteristic equation at the steay state E : et kv λ kt ˆK (λ) c λ ˆK2 (λ) = kv kt or equivalently [ (λ + sk ck c( K ) ][ λ + c c ˆK2 (λ) ] sk c[ = (λ + c) ˆK (λ) c ˆ (λ)]. c( K ) Using the notation R = sk /c we can rewrite the above equation as [ ( K )λ + (R K ) ][ λ + c c ˆ (λ)] [ = (R ) (λ + c) ˆK (λ) c ˆ (λ)]

10 27 ZHILAN FENG AND LIBIN RONG or as (3.3) where ( + λ )( ) A(λ + ) + c ˆK (λ) = ˆ (λ) A ( λ + ), (3.4) A = K (R ). The possibility of a negative real root of Eq. (3.3) can be exclue as follows. Suppose λ. Then (3.5) ˆK (λ) ˆK () = K <. It follows that A > an ( + λ )( ) A(λ + ) + c ˆK (λ) From (3.3) we have (3.6) ˆ (λ) >. A(λ + ). However, since λ, ˆK2 (λ) ˆ () =, which contraicts with (3.6). Thus, Equation (3.3) has no non-negative real roots. We can show that Eq. (3.2) or Eq. (3.3) has no complex roots λ with nonnegative real part. Suppose not, then Eq. (3.2) has a root λ = x +iy with x, y >. If R, from (3.2) we have ( (3.7) (λ + ) λ + c c ˆ (λ) ) =. i) If x > then using a similar argument as in Result we know that λ = x + iy cannot be a root. ii) If x = then (3.7) has one negative root an other roots that are etermine by the equation or (3.8) + y c i = + λ c = ˆ (λ) (a) cos(ya)a (a) sin(ya)a i. Comparison of the real parts of both sies yiels that cos(ya) =. Thus, sin(ya) = which implies that (3.8) oes not hol. Therefore, (3.2) has no roots with nonnegative real part when R. For general 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 an close to. That is, the characteristic equation (3.3) has a pure imaginary root, say, iy, with y >. Replacing λ in (3.3) with iy we see that the moulus of the left-han sie of (3.3) satisfies (3.9) LHS > Ayi + A + ˆK (iy) = A + ( K (a) cos(ya)a + i Ay + K (a) sin(ya)a).

11 THE INFLUENCE OF DRUG THERAPY ON THE EVOLUTION OF HIV 27 We claim that K (a) sin(ya)a. In fact, notice that (3.2) K (a) sin(ya)a = a K (a) sin(ya)a. Notice also that K () = r rt η an K (a) = r rt η[β (a)k (a) + β(a)k (a)] almost everywhere on [, ). Integrating a K (a) sin(ya)a by parts, a K (a) sin(ya)a = r rtη y y K (a ) cos(ya ) + y r rtη y y K (a ) cos(ya ) + y = y K (a )( cos(ya )). a a K (a) cos(ya)a K (a)a Thus K (a) sin(ya)a. We also observe that K (a) cos(ya)a K >. It follows from (3.9) that (3.2) LHS > A + iy. On the other han, the moulus of the right-han sie of (3.3) satisfies (3.22) RHS A + iy. This leas to a contraiction. We conclue that the characteristic equation (3.3) has no roots with nonnegative real part. Therefore, Result 2 hols. It is obvious that the threshol conition R > an the magnitue of R play a key role in the maintenance of the virus. In the following section we use these results to emonstrate how rug efficacy may affect the invasion of rug resistant strains. 4. Influence of rug therapy on viral fitness In this section we focus on the issue concerning the impact of rug treatments on evolution of pathogens. In particular we consier the evelopment of mutant strains meiate by rug therapy. Suppose that the rug-sensitive strain of HIV- infection is at the infecte steay state E = (T, V, B ) (see (3.4)), an that a small number of rug resistant virus have been introuce into the population. We erive an invasion criterion for a resistant strain by using a heuristic argument as is one in [2]. Denote the reprouctive ratio of the sensitive strain by R s (which is the same R as efine in (3.5)). From results in Section 2 we know that R s > an that the population size of uninfecte cells is T = s/(r s ) (see (3.4)). Let K (a) enote the age-specific survival probability of a T cell infecte with the resistant strain (an equivalent quantity for the sensitive strain is given in (2.6)), r rt an r p enote the efficacy of the two types of rugs for the resistant strain, an p(a) enote the virion prouction rate of the resistant strain. For ease of illustration we assume that all other parameters are the same for both strains. We erive the invasion criterion for the case when both types of rugs are inclue. This criterion will be applie to ifferent scenarios of chemotherapy such as a single-rug therapy (e.g., r p > an r rt = ) or combination therapy (i.e., r p > an r rt > ).

12 272 ZHILAN FENG AND LIBIN RONG Since T is the amount of available uninfecte T cells, a typical resistant virus can infect kt /c cells in its lifespan. Each of these infecte cells can prouce a total of N r = ( r p )( β(a)) p(a) K (a)a virion particles uring its whole life time (burst size). Thus the reprouctive ratio of the resistant strain (when the sensitive strain is at its positive equilibrium), R r, is R r = kt ( r p )( β(a)) p(a) c K (a)a, an the invasion criterion is R r >. Substituting s/(r s ) for T we obtain the conition for the resistant strain to invae the sensitive strain: (4.) R r > R s where the quantity (4.2) R r = s k c ( r p )( β(a)) p(a) K (a)a actually 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 state). We use the quantity R r as a measure of fitness of a resistant virus. Eq. (4.) shows that natural selection within a host favors viruses that maximize its reprouctive ratio. We can also efine a relative viral fitness (cf. [2]) by iviing the reprouctive ratio by the factor (s/)(k/c) in both R s an R r since they are assume equal for both sensitive an resistant strains. Next, for the calculation of the optimal reprouctive ratio we consier the case when the viral prouction rates for both strains have the form given in (2.2). That is, { ( p e (4.3) p(a) = θ(a a)) if a a, else where p is the saturation level for the resistant strain. Accoringly, we choose β(a) to be (see (2.3)) {, a < a, (4.4) β(a) =, a a. The eath rate of cells is assume to be the same for both strains with form { δ, a < a (4.5) δ(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 actively reprouctive cells. Drug resistance is incorporate by assuming that the efficacy of chemotherapy for the resistant strain is lower than that for the sensitive strain by a factor σ, σ, i.e., (4.6) r rt = σ rt r rt, r p = σ p r p. For ease of emonstration we assume in this chapter that σ rt = σ p = σ which coul be relaxe in later stuies. σ = correspons to the completely resistant strain while σ = correspons to the completely sensitive strain. Other strains have an intermeiate value < σ <. To incorporate the cost that resistant strains pay

13 THE INFLUENCE OF DRUG THERAPY ON THE EVOLUTION OF HIV 273 for the evelopment of resistance we consier a trae-off between rug resistance an virion prouction rate p(a). Various functional forms for the cost can be use. Herein, we consier two types of costs by which the saturation level p is reuce accoring to the following formulas (4.7) Type I : p(a) = σp ( e θ(a a ) ), ( ) (4.8) 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 Type I cost an illustrate that the qualitative properties of the two types of costs are similar. To make the calculation transparent we rewrite the reprouctive ratio R r using (4.4) (4.7) as (4.9) R r = s k ( ) σrp σp [ e θ(a a)] e (δ+σrrtη)a e (δ+µ)(a a) a c a = ( σr p ) σe (δ +σr rt η)a D, where D = s k c p ( e θu) e (δ +µ)u u is a quantity that is inepenent of σ, r rt, r p, an a. Similarly we can rewrite R s in the form (4.) R s = ( r p )e (δ+rrtη)a D. From Eqs (4.9) an (4.) we get the following relationship between R r an R s : (4.) R r = σ( σr p)e r rtη( σ)a r p R s. Consier R r = R r (σ) as a function of σ. A resistant strain with resistance σ can invae the sensitive strain if R r (σ) > R s, which (from the fact that R r () = R s ) is possible only if Rr(σ) σ < for some σ (, ). Notice that < if σ < σ < σ + R r (4.2) = if σ = σ, σ + σ > else, where 2r p + a r rt η ± 4rp 2 + a 2 r2 rtη 2 σ ± =, 2a r p r rt η if r p > an r rt >. Since 4rp 2 + a 2 r2 rtη 2 = (2r p + a r rt η) 2 4a r p r rt η 2r p + a r rt η

14 274 ZHILAN FENG AND LIBIN RONG we know that σ ±. Clearly R r (σ) R s for all σ (, ) if σ, i.e., if r p an r rt satisfy 2r p + a r rt η 4rp 2 + a 2 r2 rtη 2 (4.3). 2a r p r rt η Obviously the conition (4.3) is not easy to use to raw conclusions. Let us first erive some analytic unerstaning for a simpler case in which only a single-rug therapy with a protease inhibitor is consiere, i.e., r p > an r rt =. The case of combine therapy will be explore numerically. Single-rug therapy. In this case, since r p > an r rt =, Eq. (4.) simplifies to (4.4) R r (σ) = σ( σr p) r p R s. It is easy to check that in orer to have R r (σ) R s for some σ (, ) it is necessary that r p > 2, in which case R r (σ) σ <, for 2r p < σ <. The above inequalities suggest that there exists a maximum level of resistance, σ max, such that (4.5) R r (σ) > R s if an only if σ max < σ <. (Remark: A higher resistance level correspons to a smaller value of σ.) That is, strains with resistance σ < σ max cannot invae. We can etermine σ max by noticing that it must be a solution of the equation R r (σ) = R s, from Eq. (4.4) we know that σ max satisfies (4.6) Two solutions of (4.6) are σ( σr p ) r p =. Since the conition r p > 2 σ =, σ 2 = r p r p. guarantees that rp r p <, we have an (4.7) σ max = σ 2 = r p r p <, if r p > 2, R r (σ) > R s, σ max < σ <, R r (σ) < R s, σ < σ max, R(σ) = R s, σ =, σ max (see Figure ). Clearly, if r p < 2 then σ max > 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

15 THE INFLUENCE OF DRUG THERAPY ON THE EVOLUTION OF HIV 275 Rr a rp.4 Rr b rp.5 Rs Rs, Rr max Σmax Σ.5 Rr c rp.7 Rr rp Rr max Rs 4 Rr max 4 Rs. Σmax.5 Σopt.8 Σ. Σmax.5 Σopt.8 Figure. Plots of the reprouctive ratio R r vs. the resistant level /σ for ifferent treatment efficacy r p (r 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 () it is shown that resistant strains with resistance σ in (σ max, ) can invae. The optimal resistance is σ opt at which R r reaches its maximum R r max. expect resistance when there is little selection pressure from the rugs. Other nonresistant strains woul outcompete it uner these conitions. Resistant strains can only increase in frequency 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 max, ), σ = 2r p, at which Rr(σ) σ =. Hence, (4.8) σ opt = 2r p (see Figure ). We summarize the following results for the case of a 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.. There exists a threshol rug efficacy r p (r p = /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 max when r p < r p. Hence R r (σ) < R s for all σ <. 2. When the rug efficacy is above the threshol r p there is a range of resistance levels for which the resistant strains are able to invae. This is because, analytically, σ max < when r p > r p, an R r (σ) > R s for all σ in (σ max, ).

16 276 ZHILAN FENG AND LIBIN RONG R r 7 6 r rt.,r r Σ r rt.,r s r rt.3,r r Σ r rt.3,r s r rt.5,r r Σ r rt.5,r s Σ Figure 2. Plots of the reprouctive ratio R r vs. resistance σ for r nt =. (soli), r rt =.3 (long ashe), r rt =.5 (short ashe). The value of r p is fixe at r p =.6 for which invasion is possible in the absence of the an RT rug (i.e., if r rt = ). For each given r 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 two corresponing curves. 3. When σ max <, the range of invasion strains, (σ max, ), increase with the rug efficacy r p. The optimal resistance, σ opt, ecrease with the rug efficacy r p (a more resistant strain correspons to a smaller σ value, see Figure (c) ()). This increasing property is also clear from the formulas σ max = ( r p )/r p an σ opt = /(2r p ). 4. As the rug efficacy increases, the optimal viral fitness, R r (σ opt ), ecreases (see Figure (c) ()). We can also see this from the formula R r max = R r (σ opt ) = D/(4r p ) which is a ecreasing function of r p. Combination therapy. We now consier the case of combination therapy, i.e., r p > an r rt >. For simplicity we choose η =. From (4.9) an (4.) we have (4.9) R r = σ( σr p)e r rt( σ)a r p R s. Again, consier R r = R r (σ) as a function of σ. Then R r (σ) > R s if an only if σ satisfies the inequality (4.2) σ( σr p )e rrt( σ)a r p >. To explore the role of r rt we fix r p (e.g., r p =.7 in Figure 2). Eq. (4.2) cannot be solve analytically for σ. However, plots of R r (σ) for ifferent values of r rt suggest that, as r rt increases, the range for R r (σ) > R s also increases (see Figure 2). Figure 3 illustrates the joine effect of r rt an r p on the reprouctive ratios R s

17 THE INFLUENCE OF DRUG THERAPY ON THE EVOLUTION OF HIV 277 a b 3 R s, R r r rt r p.25 3 R 2 s.25.5 r rt r p c R s R r R s R r rp.4 R s R r.2 R r R s r rt Figure 3. Plots of the reprouctive ratios R s an R r as functions of r rt an r p. Three surfaces are plotte in Figure 3(a): R r (r rt, r p ) (the top surface near the origin), R s (r rt, r p ) (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 Figure 3(b) two surfaces, R s (r rt, r p ) an the constant, are plotte to show the curve on which R s =. Figure 3(c) is a contour plot of the surfaces R r (r rt, r p ) an R s (r rt, r p ). 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 are possible as R r > R s. Figure 4 shows that when Type II cost is use the qualitative property of the reprouctive ratio R r as a function of σ is very similar to those when Type I cost is use. For example, the function R r (σ) amits a unique σ max an a unique σ opt for sufficiently small values of φ. 5. Discussion We have formulate an age-structure moel for HIV- infection with rug treatments to stuy the impact of chemotherapy on the emergence of resistant HIV- strains. We have exhibite the reprouctive ratio of the rug sensitive strain R s,

18 278 ZHILAN FENG AND LIBIN RONG Rr 6 5 Φ.5 Φ.9 Φ 2.5 Type I cost Rs Σ Figure 4. Plots of the reprouctive ratio R r vs. resistance σ when Type II cost is consiere. The value of φ measures the level of cost. 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). an emonstrate the asymptotical stability of the infection-free steay state Ē if R s < an the infecte steay state E if R s >. We have consiere two types of rug treatments with reverse transcriptase inhibitors an protease inhibitors an the possible evelopment of rug resistance. The cost for resistant strains is assume to be a reuce viral prouction rate. We have calculate the reprouctive ratio for the resistant strain R r (σ) with resistance σ, an provie a criterion for the potential invasion of resistant strains, R r (σ) > R s, in an environment where the wil strain is alreay establishe. We argue that natural selection within a host favors viruses that maximize the reprouctive ratio which is consistent with earlier finings (see, for example, [2]). Consequently, we show that natural selection shoul favor viral strains that have an intermeiate level of resistance, an the optimal resistance (σ opt ) ecreases with increasing rug efficacy (see Figure an Figure 2). Mathematically, increasing the values of r p an r rt results in: (i) a reuction in the reprouctive ratio R s of the rug sensitive strain (see Figure 4) an hence a reuction in the equilibrium level of infection (see T an V in Eq. (3.4)); (ii) a ecrease in the optimal viral fitness R r max of the resistant strain an a ecrease in the optimal resistance σ opt (see Figure an Figure 3); an (iii) an increase in the range σ max < σ < of resistance (that is, σ max ecreases with both r p an r rt, see Figure an Figure 2). These are strains that has a high fitness (i.e., R r (σ) > R s ) are hence are able to invae a host population. On the other han, if r p an r rt are small such that σ max is greater than, then R r (σ) < R s for all resistance σ. These strains will not be maintaine in a population. Acknowlegments We woul like to thank M. Gilchrist an P. Nelson for very useful suggestions which improve this manuscript. We also woul like to thank D. Coombs for helpful iscussions.

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