Modeling within-host HIV-1 dynamics and the evolution of drug resistance: Trade-offs between viral enzyme function and drug susceptibility

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1 Journal of Theoretical Biology 247 (27) Moeling within-host HIV- ynamics an the evolution of rug resistance: Trae-offs between viral enzyme function an rug susceptibility Libin Rong a, Michael A. Gilchrist b, Zhilan Feng a, Alan S. Perelson c, a Department of Mathematics, Purue University, West Lafayette, IN 4797, USA b Department of Ecology an Evolutionary Biology, University of Tennessee, Knoxville, TN 37996, USA c Theoretical Biology an Biophysics, Los Alamos National Laboratory, MS K7, Los Alamos, NM 87545, USA Receive 2 December 26; receive in revise form April 27; accepte 6 April 27 Available online 9 April 27 Abstract There are many biological steps between viral infection of CD4 þ T cells an the prouction of HIV- virions. Here we incorporate an eclipse phase, representing the stage in which infecte T cells have not starte to prouce new virus, into a simple HIV- moel. Moel calculations suggest that the quicker infecte T cells progress from the eclipse stage to the prouctively infecte stage, the more likely that a viral strain will persist. Long-term treatment effectiveness of antiretroviral rugs is often hinere by the frequent emergence of rug resistant virus uring therapy. We link rug resistance to both the rate of progression of the eclipse phase an the rate of viral prouction of the resistant strain, an explore how the resistant strain coul evolve to maximize its within-host viral fitness. We obtaine the optimal progression rate an the optimal viral prouction rate, which maximize the fitness of a rug resistant strain in the presence of rugs. We show that the winow of opportunity for invasion of rug resistant strains is wiene for a higher level of rug efficacy provie that the treatment is not potent enough to eraicate both the sensitive an resistant virus. Publishe by Elsevier Lt. Keywors: HIV-; Drug resistance; Viral fitness; Mathematical moel; Viral ynamics. Introuction Mathematical moels have proven valuable in the unerstaning of human immunoeficiency virus type (HIV-) ynamics, isease progression an antiretroviral responses (see reviews in Nowak an May (2), Perelson (22),Callaway an Perelson (22), Perelson an Nelson (999, 22)). Many important insights into the host pathogen interaction in HIV- infection have been erive from mathematical moeling an analyses of changes in the level of HIV- RNA in plasma when antiretroviral rugs are aministere to perturb the equilibrium between viral prouction an viral clearance in infecte iniviuals (Ho et al., 995; Perelson et al., 996, 997; Wei et al., 995). Corresponing author. Tel.: ; fax: aress: asp@lanl.gov (A.S. Perelson). In a basic HIV moel that has been frequently use to escribe virus infection, there are three variables: uninfecte CD4 þ T cells, prouctively infecte T cells, an free virus (Nowak an May, 2; Perelson et al., 996). In this moel, infecte cells are assume to prouce new virions immeiately after target cells are infecte by a free virus. However, there are many biological processes between viral infection an subsequent prouction within a cell. For example, after viral entry into the host cell, the viral RNA genome is reverse transcribe into a complementary DNA sequence by the enzyme reverse transcriptase (RT). The DNA copy of the viral genome is then importe into the nucleus an integrate into the genome of the lymphocyte. When the infecte lymphocyte is activate, the viral genome is transcribe back into RNA. These RNAs are translate into proteins that require a viral protease to cleave them into active forms. Finally, the mature proteins assemble with the viral RNA to prouce new virus particles /$ - see front matter Publishe by Elsevier Lt. oi:.6/j.jtbi

2 L. Rong et al. / Journal of Theoretical Biology 247 (27) that bu from the cell. The portion of the viral life cycle before prouction of virions is calle the eclipse phase. Several mathematical moels have been evelope that either introuce a constant (iscrete) elay (Culshaw an Ruan, 2; Dixit an Perelson, 24; Herz et al., 996; Nelson et al., 2) to enote the eclipse phase, or assume that the time elay is approximate by some istribution functions (e.g., a gamma istribution) (Mittler et al., 999; Nelson an Perelson, 22). The introuction of a time elay in moels of HIV- primary infection to analyze the viral loa ecay uner antiretroviral therapy has refine the estimates of important kinetic parameters, such as the viral clearance rate an the mortality rate of prouctively infecte cells (Nelson et al., 2; Nelson an Perelson, 22). Some more complex moels, incluing age-structure moels, have been employe to stuy virus ynamics (Nelson et al., 24) an the influence of rug therapy on the evolution of HIV- (Kirschner an Webb, 996; Rong et al., 27a). It shoul be note that the above-mentione agestructure moels essentially treat the transition of a cell from the uninfecte state to the prouctively infecte state as a eterministic process by taking into account the time elay that occurs between various steps in the virus life cycle within a target cell. In contrast, in this stuy we incorporate an eclipse stage to escribe the stage of an infecte cell between viral attachment an generation of new virus. The present stage-structure moel implicitly treats the progression of an infecte cell from the initial infection to subsequent reprouction as an exponentially istribute process. We have chosen to aopt the stagestructure approach because it allows us to explore mechanistically biological trae-offs between protein functions an rug resistance while avoiing the complications of time elay moels. The avent of highly active antiretroviral therapy (HAART) has been an important breakthrough in HIV- treatment, resulting in a great reuction in the morbiity an mortality associate with HIV infection (Simon an Ho, 23). However, the clinical benefits of combination therapy are often compromise by the frequent emergence of rug resistance riven by the within-host selective pressure of antiretroviral rugs (Clavel an Hance, 24). In aition, the persistence of viral reservoirs, incluing latently infecte resting memory CD4 þ T cells that show minimal ecay even in patients on HAART up to many years (Chun et al., 997; Finzi et al., 997; Wong et al., 997; Zhang et al., 999), has been a major obstacle to the long-term control or eraication of HIV- in infecte iniviuals. Drug resistance results from mutations that emerge in the viral proteins targete by antiretroviral agents. Most of our knowlege regaring resistance comes from the genotypic analysis of virus isolates from patients receiving prolonge rug treatment (Larer, 996). Important insights into the mechanisms unerlying the evolution of rug resistant viral strains have also been erive from mathematical moeling of virus ynamics an antiretroviral responses (Bonhoeffer an Nowak, 997; Kirschner an Webb, 997; Nowak et al., 997; Ribeiro an Bonhoeffer, 2; Ribeiro et al., 998; Stilianakis et al., 997). Both eterministic an stochastic moeling approaches suggest that treatment failure is mostly likely ue to the preexistence of rug resistant strains before the initiation of therapy rather than the generation of resistant virus uring the course of treatment (Bonhoeffer an Nowak, 997; Ribeiro an Bonhoeffer, 2). The evolution of HIV resistance is associate with selective pressures exerte by rug treatments that are not potent enough to completely suppress the viral replication. The longer the rug efficacy remains in the intermeiate range, the greater the possibility that rug resistant virus variants will arise uring therapy (Mugavero an Hicks, 24). Nonetheless, the conitions of mutant selection are very complex in treate patients ue to time-epenent intracellular rug concentrations in vivo (Dixit an Perelson, 24; Huang et al., 23) an spatial heterogeneity (Kepler an Perelson, 998). The management of such patients requires a careful unerstaning of the mechanistic evolution of HIV- variants uring treatment. The evolution of resistant strains in the presence of rugs is thought to epen on inherent trae-offs that exist between the proper functioning of HIV s RT an protease enzymes an their reuce susceptibility to antiretroviral regimens in their mutate forms. Inirect evience for such trae-offs is foun in the observation that there is a reuction in replication capacity for rug resistant virus variants in the absence of rug therapy (Clavel et al., 2; Nijhuis et al., 2). These trae-offs not only help explain that even after rug resistance arises viral loa often remains partially suppresse below pre-therapy levels but also coul be potentially exploite in orer to better manage the evolution of rug resistance within a patient. The main purpose of this stuy is to evelop a mathematical framework that can be use to formalize an examine simple hypotheses about the life-history trae-offs that allow rug-resistant viral strains within a patient to persist in the presence of rug therapy. We incorporate the eclipse phase of viral replication into a mathematical moel to characterize the stage uring which infecte CD4 þ T cells have not yet starte to prouce new virus. The inclusion of the progression of infecte cells from this eclipse phase to the prouctive stage enables us to capture more variability in HIV ynamics. We observe that the strain of virus with a faster progression rate essentially has a quicker process of reverse transcription of RNA into DNA an integration of the DNA into the chromosome, which gives rise to an increase chance for that viral strain to persist. More importantly, our approach allows us to link rug resistance to RT inhibitors to the progression of the eclipse phase an ientify the optimal evolutionary strategy for the rug resistant strain uner some simple assumptions. It is wiely believe that most HIV rug resistance mutations affect highly conserve amino aci

3 86 L. Rong et al. / Journal of Theoretical Biology 247 (27) resiues that are thought to be important for optimal enzyme functions, an thus for the full replicative potential of virus (Clavel et al., 2). Consequently, we assume that in the absence of rug therapy the wil-type strain will evolve to replicate as fast as possible an prouce as many new virions as possible. Thus, a viral strain with a slower progression rate, which is operating suboptimally, will possibly have a higher level of resistance to antiretroviral rugs, creating a trae-off between the progression rate an the rug efficacy of RT inhibitors. In aition, there are trae-offs between the viral prouction rate an the clearance rate of prouctively infecte cells (De Paepe an Taei, 26), an between the viral prouction rate an the rug efficacy of protease inhibitors (see the last section for more iscussions). We will investigate how these traeoffs may affect the fitness of rug-resistant viral strains in the presence of rugs at ifferent concentration levels. The optimal progression rate an the optimal viral prouction rate are erive by maximizing the viral fitness of rugresistant strains. An invasion criterion of resistant strains is also obtaine in the presence of rug therapy. Both analytical results an numerical simulations suggest that with a more effective rug treatment (yet not potent enough to eraicate the virus), a wier range of rugresistant strains will be able to invae in response to the selective pressure of rugs. 2. Moel formulation A basic mathematical moel has been wiely aopte to escribe the virus ynamics of HIV- infection in vivo (see Perelson et al. (996) an reviews in Nowak an May (2), Perelson (22), Perelson an Nelson (999)). Important features of the interaction between virus particles an cells have been etermine by fitting the moel to experimental ata. In this paper, we exten the basic moel by incluing a class of infecte cells that are not yet proucing virus an two viral strains to stuy the evolution of rug resistant strains. 2.. Inclusion of cells in the eclipse phase After a virus enters a target CD4 þ T cell, there are a number of biological events before the prouction of new virions: reverse transcription from viral RNA to DNA, integration of the DNA copy into the DNA of the infecte cell (the integrate viral DNA is calle the provirus), transcription of the provirus an translation to generate viral polypepties, cleavage of polypepties by the HIV protease, assembly an buing of new virus. Perelson et al. (993) examine a moel for the interaction of HIV with CD4 þ T cells that consiers a class of infecte T cells, which contain the provirus but are not proucing virus. In this work, we begin with a moification of the moel in Perelson et al. (993), an then incorporate antiretroviral effects to stuy the evolution of rug resistance. As suggeste in Zack et al. (99), when a virus enters a resting CD4 þ T cell, the viral RNA may not be completely reverse transcribe into DNA. If the cell is activate shortly following infection, reverse transcription can procee to completion. However, the unintegrate virus harbore in resting cells may ecay with time an partial DNA transcripts are labile an egrae quickly (Zack et al., 992). Hence a proportion of resting infecte cells can revert to the uninfecte state before the viral genome is integrate into the genome of the lymphocyte (Essunger an Perelson, 994). To moel these events, we inclue a class of infecte cells in the eclipse stage of viral replication, i.e., the stage between the initial infection an subsequent viral prouction. Thus, a portion of infecte cells in the eclipse phase can revert to the uninfecte class. Let TðtÞ, T E ðtþ, T ðtþ an VðtÞ enote the concentrations of uninfecte CD4 þ T cells, infecte cells in the eclipse stage, prouctively infecte cells, an free virus particles at time t, respectively. The moel can be escribe by the following equations: t TðtÞ ¼l T kvt þ bt E, t T E ðtþ ¼kVT ðb þ f þ EÞT E, t T ðtþ ¼fT E T, t VðtÞ ¼pT cv, () where l is the recruitment rate of uninfecte T cells, is the per capita eath rate of uninfecte cells, k is the rate constant at which uninfecte cells get infecte by free virus. is the per capita eath rate of prouctively infecte cells, p is the viral prouction rate of an infecte cell, an c is the clearance rate of free virus. Cells in the eclipse phase revert to the uninfecte T class at a constant rate b. In aition, they may alternatively progress to the prouctively infecte class T at the rate f, or ie at the rate E. Note that our moel assumes that the expecte resience time of a cell in the eclipse phase is exponentially istribute, an the parameter f is etermine, in part, by the activity of RT. For example, if reverse transcription is quick, then f will be large an the infecte cells in the eclipse phase will progress to the prouctively infecte state with a high probability, i.e., f=ðb þ f þ E Þ. As with the basic HIV moel, there are two possible steay states of moel (). One steay state is the infectionfree steay state, the other is the infecte steay state. If we efine klpf R ¼ ðb þ f þ E Þc, (2) then it can be shown in Appenix A that the infecte steay state exists if an only if R 4. In fact, R can be written as the prouct of klp=ðcþ an f=ðb þ f þ E Þ.

4 L. Rong et al. / Journal of Theoretical Biology 247 (27) Obviously, klp=ðcþ is the basic reprouctive ratio of the stanar moel without the eclipse phase. f=ðb þ f þ E Þ is the probability that an infecte T cell survives the eclipse phase. Therefore, R in (2) efines the basic reprouctive ratio for moel (). It is further shown that R etermines whether the virus population ies out or persists. The infection-free steay state Ē is locally asymptotically stable (l.a.s.) if R o an unstable if R 4. The infecte steay state ~E is l.a.s. whenever it exists, i.e., when R 4. It is clear that R efine in (2) is an increasing function of the progression rate f (a larger value of f correspons to quicker reverse transcription) an a ecreasing function of the mortality rate E. Thus, with all else equal, we expect that the viral strain that can complete reverse transcription more quickly is more likely to lea to a more severe infection (e.g., viral persistence at a higher infection level). This is supporte by numerical simulations (Figs. an 2). In Fig., R is plotte as either a function of f or a function of E.InFig. (a), E ¼ :7 ay (or ln 2= E ¼ ay (Zack et al., 99)) is fixe. It shows that R 4 for f4:23 ay, in which case the viral loa will converge to the infecte steay state, an that R o for fo:23 ay, in which case the virus population will ie out (the infection-free steay state). In Fig. (b), f ¼ : ay (or =f ¼ :9 ays (Perelson et al., 996)) is fixe. Other parameter values are chosen from the literatures: k ¼ 2:4 8 ml ay (Perelson et al., 993); l ¼ 4 ml ay (Dixit an Perelson, 24); ¼ : ay (Mohri et al., 998); c ¼ 23 ay (Ramratnam et al., 999); ¼ ay (Markowitz et al., 23). The viral prouction rate p can be written as N, where N (burst size) is the total number of virus particles release by a prouctively infecte cell over its lifespan (Perelson et al., 996). The estimate of burst size varies from to a few thousans (Haase et al., 996; Hockett et al., 999) an possibly coul be significantly larger (Yuan Chen et al., submitte for publication). Here, as an example, we choose N ¼ 4. Thus, p ¼ 4 ay. Because only a small fraction of cells in the eclipse phase will revert to the uninfecte state (Essunger an Perelson, 994), we assume that b ¼ : ay. Fig. 2 emonstrates the ynamic behavior of the viral loa for ifferent progression rate f or mortality rate E. We observe that there is a viral peak followe by an oscillatory approach to a set-point value. As f increases, the time neee to reach the peak viral loa is shortene, while the amplitue of the peak an the subsequent setpoint value are increase (Fig. 2(a)). We observe similar behaviors as the mortality rate E ecreases (Fig. 2(b)). The steay state of the viral loa is presente as either a function of f ( E ¼ :7ay is fixe, see Fig. 2(c)) or a function of E (f ¼ : ay is fixe, see Fig. 2()). These results show that the viral strain that has a larger progression rate f or a smaller mortality rate E will have a higher viral steay state level, an thus is more likely to inuce faster isease progression The moel with two strains To stuy the invasion of rug-resistant mutant variants into an environment in which the wil-type strain is alreay establishe, we incorporate both rug-resistant an rugsensitive strains in the moel () an get the following twostrain moel: t TðtÞ ¼l T k sv s T k r V r T þ b s T Es þ b rt Er, t T Es ðtþ ¼k sv s T ðb s þ f s þ Es ÞT Es, t T s ðtþ ¼f st Es st s, t V sðtþ ¼p s T s c sv s, t T Er ðtþ ¼k rv r T ðb r þ f r þ Er ÞT Er, R R φ δ E Fig.. (a) Plot of the basic reprouctive ratio R in (2) as a function of the progression rate f for a fixe mortality rate of expose cells E ¼ :7 ay. (b) Plot of R as a function of E for a fixe value of f ¼ : ay. Other parameter values are given in the text.

5 88 ARTICLE IN PRESS L. Rong et al. / Journal of Theoretical Biology 247 (27) δ E =.7 φ =. log RNA copies /ml 5-5 φ =.8 φ =. log RNA copies /ml 5-5 δ E =.6 δ E =.2 log RNA copies /ml ays δ E = φ log RNA copies /ml ays φ = δ E Fig. 2. Time plots for the virus ynamics of moel () for ifferent f or for ifferent E : (a) f ¼ :8or: ay, E ¼ :7 ay ; (b) E ¼ :2or:6 ay, f ¼ : ay. The values of other parameters are the same as those in Fig.. The initial values for T, T E, T an V are 6 ml (Perelson et al., 993),,, an 6 ml (Staffor et al., 2), respectively. The steay state of the viral loa is plotte as a function of f or E in (c) an (). When fo:23, the virus population ies out. t T r ðtþ ¼f rt Er rt r, t V rðtþ ¼p r T r c rv r, (3) where the subscripts s an r represent the rug sensitive an resistant strains, respectively. For each strain, we obtain the corresponing reprouctive ratio, which is given by k i lp R i ¼ i f i ; i ¼ s; r. (4) ðb i þ f i þ Ei Þc i i Let ~E s enote the steay state in which only the rugsensitive strain is present an ~E r enote the steay state in which only the rug-resistant strain is present. We prove in Appenix B that each steay state is biologically feasible if an only if the reprouctive ratio for the corresponing strain is greater than. Furthermore, if R s 4R r 4, then ~E s is l.a.s. an ~E r is unstable. If R r 4R s, then ~E s is unstable an ~E r is l.a.s. Therefore, the resistant strain cannot invae the sensitive strain if R r or s.ifr r 4 maxðr s ; Þ then the resistant strain is able to invae an out-compete the sensitive strain. We will apply this result to etermine the criterion for invasion an to examine how the resistant virus may evolve to optimize its fitness in the presence of antiretroviral treatment The moel with rug therapy an resistance We moify moel (3) by incorporating combination antiretroviral therapy. Currently, a combination of reverse transcriptase inhibitors (RTIs) an protease inhibitors (PIs) is commonly use in the treatment of HIV infection. RTIs interfere with the process of reverse transcription an prevent the infection of new target cells. PIs prevent infecte cells from proucing new infectious virus particles (Nowak an May, 2). To incorporate these rug effects into our moel, we efine RTI an PI to be the efficacies of RTIs an PIs for the wil-type strain, respectively. We efine these constants relative to the impact of the rugs on the most susceptible genetic variants of RT an protease. As a result, i ¼ (i ¼ RTI or PI) implies that the inhibitor is completely ineffective against wil-type virus, while i ¼ implies that the inhibitor is % effective against them. Note that in reality % effectiveness may not be clinically feasible ue to problems with rug elivery or absorption. When PI is say.7, this implies that 7% of the wiltype virus particles prouce are non-infectious ue to the

6 L. Rong et al. / Journal of Theoretical Biology 247 (27) action of the protease inhibitor. This population of virions has previously been enote V NI (Perelson et al., 996). The remaining 3% of particles are assume not to be affecte by the PI an contain the same population of virions as in an untreate patient. Although this population has a mixture of infectious an non-infectious virions, it has been previously enote V I (Perelson et al., 996) an for simplicity calle the infectious population. A more precise efinition woul call V I the virions not mae noninfectious by the protease inhibitor. In the moel below we will follow the rug sensitive an rug resistant forms of the V I population only, an enote them V s an V r, respectively. The equations for the rug sensitive an resistant virion populations corresponing to V NI will be ignore as they can be ecouple from the system (see (5)). To moel the reuce susceptibility of rug-resistant virus variants to antiretroviral agents, we assume that the rug efficacies of RTIs an PIs for the resistant strain are reuce by factors s RTI an s PI, respectively. s RTI an s PI are between an. Therefore, RTI s RTI an PI s PI are the rug efficacies of RTIs an PIs for the resistant strain. s i ¼ (i ¼ RTI or PI) correspons to the completely rugsensitive strain while s i ¼ correspons to the completely rug-resistant strain. In orer to focus on the role of the progression rate of cells in the eclipse phase, f, an the viral prouction rate, p, on the evolution of rug-resistant virus, we assume that all other parameters for the two strains in moel (3) are equal, i.e., k i ¼ k; b i ¼ b; Ei ¼ E ; c i ¼ c; where i ¼ s; r. The cost of rug resistance will be iscusse later. The moifie moel incluing antiretroviral rugs an resistance can be escribe by the following equations: t TðtÞ ¼l T kv st kv r T þ bt Es þ bt Er, t T Es ðtþ ¼kV st ðb þ f s ð RTI Þþ E ÞT Es, t T s ðtþ ¼f sð RTI ÞT Es st s, t V sðtþ ¼p s ð PI ÞT s cv s, t T Er ðtþ ¼kV rt ðb þ f r ð RTI s RTI ðf r ÞÞ þ E ÞT Er, t T r ðtþ ¼f rð RTI s RTI ðf r ÞÞT Er rðp r ÞT r, t V rðtþ ¼p r ð PI s PI ðp r ÞÞT r cv r. (5) In the above moel, the progression rate of cells in the eclipse class to the prouctively infecte state is reuce ue to the effect of RTIs an the viral prouction rate of infectious virus is reuce ue to the effect of HIV protease inhibitors. As note above, the equations for the noninfectious particles generate by the PI ecouple from the system. These equations are t V NI s ðtþ ¼p s PI T NI s cv s, t V NI r ðtþ ¼p r PI s PI ðp r ÞT NI r cv r. The total rug sensitive an rug resistant populations are then V s þ V NI s an V r þ V NI r, respectively. Because HIV resistance is usually associate with changes of highly conserve amino aci resiues that are believe to be essential for the optimal enzyme function, rug-resistant variants isplay some extent of resistanceassociate loss of viral fitness in the absence of therapy (Clavel et al., 2; Coffin, 995). We incorporate this feature in our moel by assuming a reuce progression rate f an a reuce viral prouction rate p for the resistant strain. Base on the arguments given previously, we assume that infecte cells of the wil-type strain, the most susceptible strain to rug therapy, have the maximal progression rate, f s, an the maximal viral prouction rate, p s. Therefore, for all resistant strains we have f r of s an p r op s. We further assume that the resistance factor, s RTI,isan increasing function of f r. The justification for this assumption is the following. f r is mainly etermine by the activity of RT, an the more resistant a strain is to an RTI (a smaller s RTI ) the more likely that the RT of that strain functions poorly (a smaller f r ). Thus, we assume that s RTI is an increasing function of f r. Similarly, we assume that s PI is an increasing function of the viral prouction rate p r, which reflects the fact that the more rug-resistant a strain is to a protease inhibitor, the more poorly its protease functions an hence the lower the capacity to prouce new infectious virus (Clavel et al., 2; Zennou et al., 998). As suggeste in Coombs et al. (23), Gilchrist et al. (24), the mortality rate of prouctively infecte cells is also an increasing function of the viral prouction rate. This is because the loss of cell resources utilize to prouce virus may impair cell functions. In aition, cell-meiate immune responses are likely to rapily kill cells expressing more viral proteins. To summarize, in moel (5) we have assume that s RTI ðf r Þ, s PI ðp r Þ an r ðp r Þ are all increasing functions, with s RTI ðf s Þ¼s PI ðp s Þ¼an r ðp r Þ¼ s when p r ¼ p s, i.e., r ðp s Þ¼ s. 3. Results In this section, we use moel (5) an the results in previous sections to investigate the evolution of rugresistant strains in the presence of antiretroviral treatment. Specifically, we stuy how the resistant virus evolves to maximize its fitness, an erive the range of rug efficacy in

7 8 ARTICLE IN PRESS L. Rong et al. / Journal of Theoretical Biology 247 (27) which the rug-resistant strain will be able to invae an out-compete the wil-type strain. 3.. Optimal f r an p r that maximize viral fitness Within-host viral fitness has receive increasing interest ue to its potential clinical implications for viral loa, rug resistance, an isease progression (see reviews in Clavel et al. (2), Nijhuis et al. (2), Quinones-Mateu an Arts (2)). The term fitness is commonly use in clinical settings to escribe the ability of a virus to effectively replicate in a particular environment. Due to the fact that rug-resistant virus is less susceptible to antiretroviral regimens, the mutant variant is more fit than the wil-type virus in the presence of rug, although resistance mutations may ecrease the intrinsic capacity of the virus to replicate. In practice, it still remains unclear which assay is most appropriate to measure the fitness of HIV- isolates, an many stuies have been performe to test ifferent hypotheses that exten the efinition of relative fitness (reviewe in Quinones-Mateu an Arts (2)). The basic reprouctive ratio is a commonly use measure of the absolute fitness of a virus within a host (Gilchrist et al., 24). In this section we examine the effect of antiretroviral treatment on the HIV- fitness of resistant virus by analyzing the reprouctive ratio in the presence of therapy. The reprouctive ratio of the resistant strain (in the presence of therapy) for moel (5), enote by R r, is given by a function of f r an p r (see (4)): R r ðf r ; p r Þ¼ kl c F ðf r ÞF 2 ðp r Þ, (6) where f F ðf r Þ¼ r ð RTI s RTI ðf r ÞÞ, b þ f r ð RTI s RTI ðf r ÞÞ þ E F 2 ðp r Þ¼ p rð PI s PI ðp r ÞÞ ð7þ r ðp r Þ an s RTI ðf r Þ, s PI ðp r Þ, an r ðp r Þ are increasing functions as mentione previously. Using the formulas (6) an (7) we can fin the optimal f r an p r that maximize the reprouctive ratio R rðf r ; p r Þ. Because we assume that f r an p r are inepenent, we can maximize F ðf r Þ an F 2 ðp r Þ iniviually. When specific forms of the functions s RTI ðf r Þ, s PI ðp r Þ, an r ðp r Þ are given we are able to obtain explicit formulas for f r an p r. Before we iscuss some particular forms of these functions, we present the following result in terms of general functions, which provies some convenient criteria for fining the optimal f r an p r. The proof can be foun in Appenix C. Proposition. (i) R r is maximize at f r 2ð; f sþ if there exists a unique value f r satisfying RTI s RTI ðf r Þ f r RTIs RTI ðf r Þ¼; of r of s, (8) an s RTI ðf r ÞX. (9) (ii) R r is maximize at p r 2ð; p sþ if there exists a unique value p r satisfying ½ PI s PI ðp r Þ PIp r s PI ðp r ÞŠ rðp r Þ ¼ p r ½ PIs PI ðp r ÞŠ r ðp r Þ an ðþ s PI ðp r ÞX; r ðp r ÞX. () Proposition suggests that if s RTI ðf r Þ, s PI ðp r Þ an r ðp r Þ are concave up functions, then Eqs. (8) an () etermine the optimal progression rate f r an the optimal viral prouction rate p r, respectively. It shoul be note that the concave up property is not require in Proposition, as the four conitions, (8) (), involve only f r an p r. We now consier some specific forms of the increasing functions s RTI ðf r Þ, s PI ðp r Þ an r ðp r Þ. Noticing that f r pf s an s RTI ðf s Þ¼, we choose s RTI ðf r Þ to be a simple power function s RTI ðf r Þ¼ f a r, (2) f s where ax is a constant, ensuring that the secon erivative is nonnegative. From Proposition we get the optimal progression rate f r ¼ f pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a s= RTI ð þ aþ (see (8)). Therefore, we can establish the following result: Result. Let s RTI ðf r Þ be given in (2). Then (i) if o RTI o þa, then the optimal progression rate f r is f s ; an (ii) if þa o RTIo, then the optimal progression rate is an pintermeiate ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi value within ð; f s Þ; i.e., f r ¼ a f s = RTI ð þ aþof s. This result suggests that when the rug efficacy RTI is low, the best strategy for a resistant strain to achieve the maximal viral fitness is unchange from the non-treatment scenario, i.e., infecte cells in the eclipse phase nee to progress to the prouctively infecte state as soon as possible. When the rug efficacy is high, the optimal viral fitness is achieve at an intermeiate value f r ¼ f s=ð2 RTI Þ (in the case of a ¼ ), instea of the maximal progression rate f s. To examine the optimal prouction rate p r, we assume that the rug resistance factor s PI ðp r Þ is a linear function of p r, an that the eath rate of prouctively infecte cells of the resistant strain follows a non-linear relationship between the cell eath an viral prouction as examine in Coombs et al. (23); i.e., s PI ðp r Þ¼ p r an r ðp p r Þ¼ b 2 pr þ m, (3) s 2 p s

8 L. Rong et al. / Journal of Theoretical Biology 247 (27) where p r pp s, b is a constant an m is a fixe backgroun mortality rate. Since we require that the function value r ðpþ evaluate at p ¼ p s (the wil-type strain) is exactly the constant s, b can be chosen to be 2ð s mþ. Then, using () we obtain the following result for the optimal prouction rate p r. Result 2. Let s PI ðp r Þ an r ðp r Þ be given by (3). Then the optimal prouction rate p r etermine by Eq. () is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p r ¼ 4m 2 2 PI þ 2mb 2m PI p b s. (4) Moreover, if s o2m, then ~ PI ¼ s 2m provies a threshol such that (i) the optimal prouction rate of resistant virus is p s if o PI o~ PI ; an (ii) the optimal prouction rate of resistant virus is an intermeiate value given by (4) if ~ PI o PI o. The formula (4) allows us to stuy the effect of the backgroun mortality rate, m, on the viral fitness. As m!, the optimal prouction rate p r!. A straightforwar calculation shows that F 2 ðp r Þ!þ. This implies that slow prouction is the best strategy for long-live infecte cells. This result is consistent with the observation in Coombs et al. (23). These results are emonstrate in Fig. 3. Fig. 3(a) illustrates the optimal progression rate f r for the special case a ¼ in (2). f r is plotte as a function of the rug efficacy of RTIs, RTI. Fig. 3(b) plots the optimal prouction rate p r as a function of the rug efficacy of protease inhibitors, PI. In these graphs, m is chosen to be the same as E, an the values for other parameters are the same as those in Fig.. Fig. 3(c) an () plot the reprouctive ratios for the rug-resistant strain using the optimal values f r an p r (as shown in the upper panel) an the wil-type strain. The flat surface is constant, the upper surface is for the reprouctive ratio of the rug-resistant strain (R r, r for resistant strain), an the lower surface is for that of the wil-type strain (R s, s for sensitive strain). We choose ifferent backgroun mortality rates of infecte cells. For example, in Fig. 3(c) m ¼ E an hence b ¼ 2ð s E Þ, an in Fig. 3() m ¼ an hence b ¼ 2ð s Þ. In both cases, the reprouctive ratio of the resistant strain (R r ) is always greater than or equal to that of the sensitive strain (R s ). We observe that for a large backgroun mortality rate m (for example, m is equal to the eath rate of infecte cells in the eclipse phase), R r becomes less than one as rug Optimal progression rate ( φ s ) /2 Optimal prouction rate ( p s ) /2 /2 Drug efficacy ε RTI /3 2/3 Drug efficacy ε PI 2 R (ε RTI, ε PI ) ε PI.5.5 ε RTI R (ε RTI, ε PI ) 5 ε PI.5.5 ε RTI Fig. 3. (a) Plot of the optimal progression rate f r as a function of the rug efficacy of RTIs. (b) Plot of the optimal prouction rate p r as a function of the rug efficacy of PIs. (c) an () are plots of the reprouctive ratios of the rug-resistant strain using the optimal f r an p r (as shown in (a) an (b)) an the wil-type strain. In (c) we assume m ¼ E ¼ :7 ay ; in () m ¼ ¼ : ay. The other parameters are: k ¼ 2:4 8 ml ay, l ¼ 4 ml ay, ¼ : ay, c ¼ 23 ay, b ¼ : ay, E ¼ :7 ay, s ¼ ay, f s ¼ :25 ay, p s ¼ 4 ay. The flat surface is constant, the upper surface is for the reprouctive ratio of the rug-resistant strain (R r ), an the lower surface is for that of the wil-type strain (R s ). We observe that in both cases, R r is always greater than or equal to R s. In (c), R r becomes less than for a high level of rug efficacy, while in () R r is always greater than.

9 82 ARTICLE IN PRESS L. Rong et al. / Journal of Theoretical Biology 247 (27) efficacy increases although it is always greater than or equal to R s. Thus, in this case both strains of virus will be eraicate for a high rug efficacy (Fig. 3(c)). However, if the backgroun mortality rate is very small (for example, m is equal to the eath rate of uninfecte T cells), then the threshol value of PI corresponing to (4) is ~ PI ¼ s 2, which is less than zero. Hence, the optimal prouction rate p r is always given by the intermeiate value etermine by (4). In this case, simulation results show that R r is always greater than both R s an (Fig. 3()), an accoring to the result given in Section 2.2, the rug-resistant strain that evolves with the optimal f r an p r will always be able to invae an out-compete the wil-type strain in the presence of rug therapy Invasion criterion In the previous section, we have shown that if the rugresistant strain continuously evolves to aopt the optimal f r an p r that maximize its viral fitness, then the resistant strain will always be expecte to emerge an out-compete the establishe wil-type strain, provie that the antiretroviral treatment is not potent enough to eraicate both strains. Now a natural question arises: if the optimal viral fitness is not achieve, is it possible that the rug-resistant strain can still invae the population of the wil-type virus? If yes, what is the invasion criterion? Below we attempt to aress these questions using moel (5). To erive the conition uner which a rug-resistant strain (with parameters f r an p r, f r of s an p r op s ) can invae the sensitive-strain in the presence of rug therapy, we assume that the population of wil-type virus is at the infecte steay state. Recall that the infecte steay state exists only if the reprouctive ratio of the wil-type strain is greater than. From Section 2.2, the rug-resistant strain will be able to invae the wil-type strain if the following conition is satisfie: R r 4R s. (5) The reprouctive ratio for the rug-resistant strain in the presence of therapy is given by (see (4), (6) an (7)) R r ¼ k l c F ðf r ÞF 2 ðp r Þ; of r of s ; op r op s (6) an the reprouctive ratio for the wil-type strain is R s ¼ k l c F ðf s ÞF 2 ðp s Þ, (7) where the functions F an F 2 are given in (7)). Using the criterion (5) an formulas (6) (7), we can establish the following result. The proof is given in Appenix D. Result 3. (i) When both rug efficacies, RTI an PI, are low then the resistant strain cannot invae the sensitive strain. (ii) If the rug efficacies are above certain threshol values then invasion is possible by a resistant strain for which the progression rate f r an the viral prouction rate p r are in some given ranges. Clearly, the invasion ranges efine by (37) an (38) in Appenix D epen on the rug efficacies RTI an PI.In fact, such ranges increase with increasing RTI an PI (see Fig. 4). Also, if the backgroun eath rate m is much smaller than s, then from the formula (38) we can see that R r 4R s for almost all values of p r such that p r op s. In Fig. 4, the reprouctive ratios R s an R r are plotte either as a function of f r (Figs. 4(a) an (b)) or as a function of p r (Figs. 4(c) an ()) for ifferent values of RTI or PI. For example, Figs. 4(a) an (b) are for RTI ¼ :4 an RTI ¼ :5, respectively, for fixe values of PI ¼ an a ¼ 3 (see Eq. (2)). We observe that the range in which R r 4R s is bigger for a larger value of RTI, suggesting that for a more effective rug therapy, the resistant strain can invae the sensitive strain at a smaller progression rate f r. Figs. 4(c) an () are for PI ¼ :5 an PI ¼ :6, respectively, for a fixe value of RTI ¼. We have assume that the backgroun mortality rate m is equal to E, hence b ¼ 2ð s E Þ. We observe again that the range in which R r 4R s is bigger for a larger value of PI. Therefore, for a higher protease inhibitor rug efficacy, the resistant strain can invae the sensitive strain at a smaller prouction rate p r. 4. Discussion an conclusion Avances in the evelopment of potent combination antiretroviral therapy have ramatically reuce HIVrelate morbiity an mortality in the evelope worl. However, increasing emergence of resistance to antiretroviral rugs coul challenge this achievement. The rapi evelopment of rug resistant HIV variants is ue to the high turnover of HIV approximately billion new virus particles are prouce per ay in the average mi-stage HIV-infecte untreate patient (Perelson et al., 996) an the exceptionally high error rate of HIV reverse transcriptase (RT). This leas to a high mutation rate an constant prouction of new viral strains, even in the absence of rug therapy. Unerstaning the evolution of viral resistance uring therapy has far-reaching implications in preicting treatment outcomes an esigning treatment strategies employe in clinical practice. In this work, we have evelope a mathematical moel to explore the initial constraints that may shape the evolution of viral resistance to antiretroviral rugs. We focuse on the interactions between two classes of rugs (reverse transcriptase inhibitors (RTIs) an protease inhibitors) an the enzymes they target, an the trae-offs that are likely to result from such interactions. For RT an its inhibitor we assume that there is a trae-off between the efficiency of RT an its susceptibility to the inhibitor. Our rationale was as follows: within-patient selection shoul favor the virus that maximizes its burst size N, the total number of virions mae by an infecte cell uring its lifetime (Gilchrist et al., 24). The burst size is a function of the lifespan of the infecte cell, with longer living cells

10 L. Rong et al. / Journal of Theoretical Biology 247 (27) Reprouctive ratio R s R r Reprouctive ratio R s R r.5.5 φ r.5.5 φ r 2.5 Reprouctive ratio.5.5 R s R r Reprouctive ratio.5 R s R r p r p r Fig. 4. Plots of R s an R r for ifferent values of RTI or PI. The long ashe line is for R s, an the soli curve is for R r : (a) RTI ¼ :4; (b) RTI ¼ :5 ( PI ¼ an a ¼ 3 for both (a) an (b)); (c) PI ¼ :5; () PI ¼ :6 ( RTI ¼ for both (c) an ()). It is shown that the range in which R r 4R s becomes bigger for larger values of RTI or PI. potentially able to make more virions. Due to the mortality rate of an infecte cell the contribution of virion prouction to N is effectively iscounte as the infecte cell ages. In aition, viral mrna is susceptible to attack by host nucleases once it enters the cell. As a result, withinhost selection will inherently favor the virus with an RT that can rapily reverse transcribe the virus genome an integrate it into the host s genome. Because we expect these forms of RT to be favore by within-host selection we also expect them to be the most susceptible to inhibition by rugs esigne to interfere with their activity. Along the same line of reasoning, other forms of RT that have low activity levels are expecte to have low frequencies within the host, maintaine primarily by rift an mutation. However, the very genetic changes that confer low activity levels to these RT variants are also likely to confer some resistance to the rugs esigne to target RT with high activity levels. As a result we posit that there is likely a simple trae-off between RT activity an susceptibility to RT inhibitors. The HIV protease also plays a critical role in the virus life cycle by converting a viral polypeptie into mature an functional viral proteins necessary for viral infectivity. Because mutations associate with the emergence of rug resistance to protease inhibitors moify some key viral proteins (Barrie et al., 996; Winslow et al., 995), the virus force to evelop resistance uner rug pressure is thought to have a substantial impairment in its replicative capacity (Clavel an Hance, 24) even though some aitional mutations can compensate for this impaire viral replication potential (Nijhuis et al., 2). We thus expect that there is a trae-off between the efficacy of protease inhibitors an the viral prouction rate for the rugresistant virus variants selecte uring therapy. Once a cell begins actively proucing virions it becomes highly susceptible to attack by the patient s immune response an viral cytopathic effects. Viral cytopathicity an cell-meiate immune responses are assume to epen on the rate of viral prouction. If the mortality rate of infecte cells is a concave up function with respect to the viral prouction rate, then the optimal viral prouction rate is likely to be at some intermeiate level below its physiological maximum (Gilchrist et al., 24). Uner such conitions, an intermeiate prouction rate will maximize the within-patient viral fitness by maximizing the burst size N. This is consistent with our finings when rug resistance to antiretroviral regimens is consiere in the moel. It shoul be mentione that our moel assumes that the viral prouction rate is time inepenent. When the prouction rate is allowe to vary with time uring infection, the optimal prouction scheule to maximize the burst size is still to prouce virus at a constant rate

11 84 ARTICLE IN PRESS L. Rong et al. / Journal of Theoretical Biology 247 (27) (Coombs et al., 23). More results on the optimal viral prouction scheule from the perspective of virus can be foun in Coombs et al. (23). Taken together, the moel evelope here allows us to investigate the fitness of ifferent HIV variants taking into account the trae-offs between the progression of infecte cells in the eclipse phase an resistance to RT inhibitors, between viral prouction an cell mortality, an between viral prouction an resistance to protease inhibitors. The moel preicts that when the rug efficacy is not high enough to exert sufficient selective pressure (the threshol values in our example are RTI ¼ :5 an PI ¼ s 2m :3), the resistant strain will be unable to invae the establishe sensitive strain. For a more effective rug therapy (but not potent enough to eraicate both the wil-type an resistant strains), a wier range of resistant virus variants can invae an out-compete the rugsensitive strain. In the present moel, the efficacies of antiretroviral rugs are assume to be constant. However, this assumption may not be realistic because rug concentrations in the bloo an in cells continuously vary ue to rug absorption, istribution an metabolism. There are some existing moels that use time-varying rug concentrations to etermine the efficacy of antiviral treatment (Dixit an Perelson, 24; Huang et al., 23; Wahl an Nowak, 2; Wu et al., 25). The pharmacokinetic moel evelope by Dixit an Perelson (24) was also employe to etermine rug efficacies for both the sensitive an resistant strains (Rong et al., 27b). They showe that using the average rug efficacy can still give a goo preiction of the longterm outcome of therapy although the viral loa isplays frequent oscillations when the time-varying rug efficacy is employe. Another important factor that affects rug efficacy is patients aherence to prescribe regimen protocols. In fact, non-aherence an non-persistence with antiretroviral therapy is the major reason most iniviuals fail to benefit from their treatments (Becker et al., 22). A number of mathematical moels have been evelope to stuy the effects of non-perfect aherence to rug regimens (Ferguson et al., 25; Huang et al., 23; Phillips et al., 2; Rong et al., 27b; Smith, 26; Wahl an Nowak, 2; Wu et al., 26). An overview can be foun in Heffernan an Wahl (25). Careful moeling of rug pharmacokinetics an more realistic aherence patterns can provie an important tool in the stuy of the kinetics of evolutionary aaptation of HIV to rug therapy an ultimately may improve our ability to evelop proceures to efeat this ealy virus. Acknowlegments Portions of this work were performe uner the auspices of the U.S. Department of Energy uner contract DE- AC52-6NA This work was supporte by NSF grant DMS an James S. McDonnell Founation 2st Century Science Initiative (ZF), an NIH grants AI28433 an RR6555 (ASP). The manuscript was finalize when LR visite the Theoretical Biology an Biophysics Group, Los Alamos National Laboratory in 26. The authors thank three anonymous referees for their constructive comments that improve this manuscript. Appenix A. Stability of steay states of moel () The infection-free steay state of moel () is Ē ¼ð T; T E ; T ; VÞ ¼ l ; ; ;. (8) The infecte steay state is ~E ¼ð~T; ~T E ; ~T ; ~VÞ, where ~T ¼ ðb þ f þ EÞc kpf ; ~T E ¼ klpf ðb þ f þ EÞc, kpfðf þ E Þ ~T ¼ f T ~ E ; ~V ¼ pf T c ~ E. ð9þ Using (2), ~T E can be rewritten as ~T E ¼ ðb þ f þ EÞc ðr Þ. kpfðf þ E Þ Therefore, the infecte steay state exists if an only if R 4. Let ^E ¼ð^T; ^T E ; ^T ; ^VÞ enote a steay state of moel (). Then the characteristic equation at ^E is k ^V z b k ^T k ^V ðb þ f þ E Þ z k ^T f z ¼, p c z (2) where z is an eigenvalue. Eq. (2) can be simplifie to ½ðz þ þ k ^VÞðz þ b þ f þ E Þ k ^VbŠðz þ cþðz þ Þ ¼ðz þ Þfpk ^T. ð2þ (i) Let R o. Evaluating (2) at the infection-free steay state Ē, we get ðz þ Þðz þ b þ f þ E Þðz þ cþðz þ Þ ¼ðz þ Þfpk l. Clearly, there is one negative eigenvalue, an other eigenvalues are etermine by ðz þ b þ f þ E Þðz þ cþðz þ Þ ¼fpk l, which can be rewritten as (see (2)) ðz þ b þ f þ E Þðz þ cþðz þ Þ ¼R ðb þ f þ E Þc. (22) If z has a nonnegative real part, then the moulus of the left-han sie of (22) satisfies jðz þ b þ f þ E Þðz þ cþðz þ ÞjXðb þ f þ E Þc, (23) which leas to a contraiction in (22) since R o. Therefore, all the eigenvalues have negative real parts, an hence Ē is l.a.s.

12 L. Rong et al. / Journal of Theoretical Biology 247 (27) When R 4, we efine f ðzþ ¼ðz þ b þ f þ E Þðz þ cþðz þ Þ R ðb þ f þ E Þc. It is clear that f ðþo an f ðzþ!when z!. By the continuity we know there exists at least one positive root. Hence, the equilibrium point Ē is unstable if R 4. (ii) Let R 4. Substituting the infecte steay state ~E for ^E in the characteristic equation (2), we have ½ðz þ þ k ~VÞðz þ f þ E Þþðz þ ÞbŠðz þ cþðz þ Þ ¼ðz þ Þðb þ f þ E Þc. ð24þ Obviously, (24) oes not have a nonnegative real solution. From (9) an (2), we can write ~V in terms of the basic reprouctive ratio in the form ~V ¼ b þ f þ E f þ E k ðr Þ. Now we want to prove that (24) oes not have any complex root z with a nonnegative real part. Suppose, by contraiction, that z ¼ x þ iy with xx, y4 is a root of (24). When R!, Eq. (24) reuces to ðz þ Þðz þ b þ f þ E Þðz þ cþðz þ Þ ¼ðz þ Þðb þ f þ E Þc. ð25þ Using the same arguments as in part (i), we can show that (25) oes not have any root with a nonnegative real part. By the continuous epenence of roots of the characteristic equation on R, we know that the curve of the roots must cross the imaginary axis as R ecreases sufficiently close to. That is, the characteristic equation (24) has a pure imaginary root, say, iy, where y 4. From (24), we have ½ð þ k ~V þ iy Þðf þ E þ iy Þþð þ iy ÞbŠðc þ iy Þð þ iy Þ ¼ð þ iy Þðb þ f þ E Þc. ð26þ We now claim that the following inequality hols: jð þ k ~V þ iy Þðf þ E þ iy Þþð þ iy Þbj 4j þ iy jðb þ f þ E Þ. In fact, after straightforwar computations, we have jð þ k ~V þ iy Þðf þ E þ iy Þþð þ iy Þbj 2 j þ iy j 2 ðb þ f þ E Þ 2 ¼ y 4 þð þ k ~VÞ 2 y 2 þðf þ E Þ 2 k ~Vð2 þ k ~VÞ þ 2bk ~Vðy 2 þðf þ E ÞÞ 4. Thus, (27) hols. It follows that ð27þ jð þ k ~V þ iy Þðf þ E þ iy Þþð þ iy Þbjjc þ iy jj þ iy j 4j þ iy jðb þ f þ E Þc. This contraicts (26). Therefore, we conclue that the characteristic equation (24) oes not have any root with a nonnegative real part. Thus, the infecte steay state ~E is l.a.s whenever it exists. Appenix B. Steay states an stability of moel (3) Assume that ~E s ¼ð~T s ; ~T Es ; ~T s ; ~V s ; ; ; Þ an ~E r ¼ ð ~T r ; ; ; ; ~T Er ; ~T r ; ~V r Þ.Wehave ~T i ¼ ðb i þ f i þ Ei Þc i i ; ~T Ei k i p i f ¼ k ilp i f i ðb i þ f i þ Ei Þc i i, i k i p i f i ðf i þ Ei Þ ~T i ¼ f i ~T Ei ; ~V i ¼ p if i ~T Ei ; i ¼ s; r. ð28þ i c i i Obviously, each steay state exists if an only if the corresponing reprouctive ratio is greater than. If E ¼ðT; T Es ; T s ; V s ; T Er ; T r ; V r Þ enotes a coexistence steay state (i.e., V s a an V r a, hence both strains are present), then T satisfies T ¼ l R s ¼ l R r. Therefore, E exists only if R r ¼ R s. The Jacobian matrix at ~E s is J ¼ G, H where s k s ~V s b s k s ~T s G ¼ H ¼ k s ~V s ðb s þ f s þ Es Þ k s ~T s f s s p s c s ðb r þ f r þ Er Þ k r ~T s f r r p r c r C A, (29) C A (3) an enotes a 4 3 matrix that oes not affect the proof. Notice that the characteristic equation of G is exactly the same equation (2) with the subscript s ae. From Appenix A an R s 4, all eigenvalues of G have negative real parts. Thus, the stability of ~E s is completely etermine by the eigenvalues of H. Suppose z is an eigenvalue of H, then z satisfies ½z þðb r þ f r þ Er ÞŠðz þ r Þðz þ c r Þ¼k r p r f r ~T s. (3) If we efine R r ¼ k r p r f r ~T s, (32) ðb r þ f r þ Er Þc r r then (3) can be rewritten as ½z þðb r þ f r þ Er ÞŠðz þ r Þðz þ c r Þ ¼ R r ðb r þ f r þ Er Þc r r. ð33þ We remark that R r represents the effective reprouctive ratio for the rug-resistant strain (i.e., the reprouctive ratio when the sensitive strain is at its infecte steay state). If R r 4, then the resistant strain will be able to invae the establishe wil-type strain.

13 86 ARTICLE IN PRESS L. Rong et al. / Journal of Theoretical Biology 247 (27) Using the same arguments as in Appenix A, we have that ~E s is l.a.s. if R r o an it is unstable if R r 4. Notice from (28) an (4) that ~T s ¼ l. R s Substituting this for ~T s in (32) an using (4), we obtain R r ¼ R r. R s Thus, R r 4 if an only if R r4r s an R r o if an only if R r or s. It follows that ~E s is l.a.s. if R r or s, an it is unstable if R r 4R s. From the mathematical symmetry of the two strains we can use the same arguments for the stability analysis of ~E r, an show that ~E r is l.a.s. if R r 4R s an unstable if R r or s. Appenix C. Proof of Proposition (i)wewanttofinf r that maximizes F ðf r Þ (see Eq. (7)). Let f ðf r Þ¼f r ð RTI s RTI ðf r ÞÞ. (34) Then F ðf r Þ is maximize if an only if f ðf r Þ is maximize. Notice that (8) hols if an only if f r is a critical point of f on ð; f s Þ. Since RTI s RTI ðf r Þo, we have f ðf r Þ¼ 2 RTIs RTI ðf r Þ RTIf r s RTI ðf r Þ ¼ 2ð RTIs RTI ðf r ÞÞ f RTI f r s RTI ðf r Þ r o RTI f r s RTI ðf r Þ. Hence, f ðf r Þo ifs RTI ðf r ÞX. It follows that f, an hence R r, assumes its maximum at f r if (8) an (9) hol. (ii) If p r satisfies () then we can easily verify that p r is a critical point of F 2 ðp r Þ; i.e., F 2 ðp r Þ¼. The secon erivative of F 2 ðp r Þ at p r is 2 ðp r Þ¼½ 2 PIs PI ðp r Þ PIp r s PI ðp r ÞŠ rðp r Þ p r ½ PIs PI ðp r ÞŠ r ðp r Þ 2 r ðp r Þ. F It is easy to verify that F 2 ðp r Þo ifs PI ðp r (35) Þ an r ðp r Þ are both nonnegative, which implies that F 2 ðp r Þ has a maximum at p r. Therefore, R r is maximize at p r ¼ p r. This finishes the proof of Proposition. Appenix D. Proof of Result 3 We prove this result using the specific functional forms for s RTI ðf r Þ, s PI ðp r Þ, an r ðp r Þ given by (2) (in the case of a ¼ ) an (3). (i) The invasion conition (5) is equivalent to (see (6) an (7)) F ðf r ÞF 2 ðp r Þ4F ðf s ÞF 2 ðp s Þ. (36) From the analysis in Section 3., we know that for a low level of rug efficacy RTI (e.g., o RTI o=2 when a ¼, see Result ), the maximum of F ðf r Þ can only occur at f r ¼ f s. Thus, F ðf r ÞoF ðf s Þ for f r of s. Similarly, from Result 2 we know that the maximum of F 2 ðp r Þ can only occur at p r ¼ p s if o PI o~ PI, where ~ PI ¼ s 2m. Thus, F 2 ðp r ÞoF 2 ðp s Þ for p r op s. Therefore, the invasion conition (36) oes not hol for any rug efficacies with o RTI o=2 ano PI o s 2m. (ii) When =2o RTI o, solving the inequality F ðf r Þ4F ðf s Þ for f r,wehave f r f r RTI f s f b þ f r f 4 s ð RTI Þ, r b þ f RTI þ s ð RTI Þþ E E f s which is equivalent to f r ð RTI f r =f s Þ4f s ð RTI Þ, or RTI f 2 r f sf r þ f 2 s ð RTIÞo. From the above inequality (an noticing that RTI 4=2), we have f s of r of s. (37) RTI When ~ PI o PI o, solving the inequality F 2 ðp r Þ4F 2 ðp s Þ for p r gives p r p r PI p s b p 2 4 p sð PI Þ, r s þ m 2 p s which can be rewritten as ½2 s PI þ bð PI ÞŠp 2 r 2 sp s p r þ 2mð PI Þp 2 s o. Noticing that b ¼ 2ð s mþ, we can solve the above inequality an obtain mð PI Þ s mð PI Þ p sop r op s. (38) Since PI 4~ PI ¼ s 2m, which guarantees that mð PI Þ s mð PI Þ o, we know that (38) efines an interval on which F 2 ðp r Þ4F 2 ðp s Þ. Therefore, for ðf r ; p r Þ in the regions efine by (37) an (38) the invasion conition (36), or equivalently (5), hols. This finishes the proof of Result 3. References Barrie, K.A., Perez, E.E., Lamers, S.L., Farmerie, W.G., Dunn, B.M., Sleasman, J.W., Gooenow, M.M., 996. Natural variation in HIV- protease, Gag p7 an p6, an protease cleavage sites within gag/pol polyproteins: amino aci substitutions in the absence of protease inhibitors in mothers an chilren infecte by human immunoeficiency virus type. Virology 29,

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