The Effects of Latent Infection on the Dynamics of HIV
aa r X i v : . [ m a t h . D S ] D ec The Effects of Latent Infection on the Dynamics of HIV
Stephen Pankavich Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO 80401
Abstract
One way in which the human immunodeficiency virus (HIV-1) replicates within a host is by infecting activatedCD4+ T-cells, which then produce additional copies of the virus. Even with the introduction of antiretroviraldrug therapy, which has been very successful over the past decade, a large obstacle to the complete eradicationof the virus is the presence of viral reservoirs in the form of latently infected CD4+ T-cells. We consider amodel of HIV infection that describes T-cell and viral interactions, as well as, the production and activationof latently infected T-cells. Upon determining equilibrium states of the latent cell model, the local and globalasymptotic behavior of solutions is examined, and the basic reproduction number of the system is computedto be strictly less than that of the corresponding three-component model, which omits the effects of latentinfection. In particular, this implies that a wider variety of parameter values will lead to viral eradication as t → ∞ due to the appearance of latent CD4+ T-cells. With this realization we discuss possible alternativenotions for eradication and persistence of infection other than traditional tools. These results are furtherillustrated by a number of numerical simulations. Keywords:
HIV-1, Mathematical model, Latently infected T-cells, Antiretroviral therapy, Globalasymptotic stability
1. Introduction
The majority of cells infected with the human immunodeficiency virus (HIV-1) are activated CD4+T-cells. Once infected, these cells produce additional copies of virus, thereby prolonging the infection.Upon detecting such an infection, the immune system mounts a complex adaptive response, controllingthe virus population to a limited extent. Further control is available in the form of antiretroviral drugs,such as Reverse Transcriptase Inhibitors (RTIs) and Protease Inhibitors (PIs) [17]. If such drugs are takenwith sufficient frequency, the virus population is largely limited and remains below the level of detection [6].However, antiretroviral therapy (ART) cannot fully eradicate the virus, as viral rebound occurs once therapyis interrupted [1, 12] A number of factors have been proposed to explain this viral rebound. Most notably,it has been suggested that HIV lay dormant within a number of reservoirs. Primary among these reservoirsare latently infected CD4+ T-cells [2]. Though latently infected T-cells exist in the body with a muchlower frequency than susceptible CD4+ T-cells, the reservoir appears to decay very slowly, with a half-lifemeasured between 6 and 48 months [26]. Although infected, these cells do not produce new virions untilactivated, thus potentially providing a longer-lived hiding place where the virus may evade control by eitherthe immune system or antiretroviral treatment [2]. Consequently, in this paper, we analyze a mathematicalmodel that includes latent infection and examine the control of infection by ART. We also assume that suchlatent T-cells exist significantly longer than productively infected CD4+ T-cells. Ultimately, we will showthat a mathematical analysis of the most basic latent model demonstrates that the inclusion of such cellsincreases the likelihood for viral clearance under the traditional approach of analyzing the basic reproductionnumber and the associated stability of equilibria. While this will seem intuitive from a modeling perspective,
Email address: [email protected] (Stephen Pankavich) The author was supported in part by NSF grants DMS-0908413 and DMS-1211667
Preprint submitted to Elsevier October 5, 2018 s described later, it also appears contradictory to the widely-held notion that latently infected T-cells arean important mechanism for the inability of ART to eradicate an established infection. What this reallyimplies is that standard mathematical tools are insufficient to realistically describe the dynamics of HIVwhen latent cells are considered. Instead, one must focus on the rate of decay of the infection, which issignificantly slowed by the latent T-cell population.A number of authors have studied the biological aspects of mathematical models concerning HIV dynamicsthat consider latently infected cells. Notably, Callaway and Perelson [4] studied low-level viremia, Rong andPerelson [28] modeled viral blips and showed that a latent reservoir could produce viral transients whenactivated by infection, while Sedaghat et al. [31] employed a simple model for the dynamics of the latentreservoir to show that its stability was unlikely to stem from ongoing replication during ART. In each ofthese studies, a reduced or linearized mathematical analysis was performed, but the nonlinear behavior ofthe associated model was not fully elucidated. In the current study, we describe latently infected cells usinga separate compartment, as did these authors, by assuming that a proportion of newly-infected cells becomelatently infected upon contact with the virus, but that they are not productively infected until they leavethe latent state, which occurs at a rate α proportionate to the strength of the latent cell population. Wenote that the effects of viral mutation, which may continuously change model and parameter values, andthe possible spatial dependence of parameters are ignored. Using this model, we study the influence of thelatent reservoir on the persistence of HIV infection and viral rebound. Our results provide a new perspectiveon the methods of mathematical and stability analysis for viral and latent reservoir persistence.The paper proceeds as follows. In the next section, we will review some known results concerning thestandard three-component model of HIV dynamics. In Section 3, we introduce an additional populationrepresenting latently infected CD4+ T-cells, and study the effects that these cells have on the structure andbehavior of the long-time dynamics of the model. In Section 4, we discuss the ramifications of our resultsand, in particular, the need to construct more precise notions of viral eradication and persistence. The fifthsection contains proofs of the theorems contained within previous sections. In the final section, we concludewith a discussion of our results.
2. The Three-component model
In general, the modeling of HIV dynamics in vivo is complicated by the appearance of spatial inhomo-geneities, which can arise from various reservoirs, such as those occurring within lymphatic tissues [19, 25].Even when such inhomogeneities are ignored, however, these systems are often described to a sufficient degreeby systems of ordinary differential equations that include no spatial dependence. We begin by consideringa three-component model for the evolution of within-host HIV, that does not include spatial fluctuationsor effects due to long-lived infected or latently infected cells. This model has been widely-accepted as adescriptive representation for the basic dynamics of HIV [3, 24, 35]. It represents the populations of threecomponents in a fixed volume at a given time t : T ( t ), the number of CD4+ T-cells that are susceptible toHIV-1 infection, I ( t ) the number of infected T-cells that are actively producing virus particles, and V ( t ) thenumber of free virions. These quantities approximately satisfy the system of ordinary differential equations(1) dTdt = λ − d T T − kT VdIdt = kT V − d I IdVdt = N d I I − d V V. Here λ is the recruitment rate of susceptible T-cells and d T is their mortality rate. The constant k representsthe rate of infection, which is included within a bilinear mass action term, while d I is the death rate ofproductively infected cells and d V is the clearance rate of free virus. The parameter N is the burst size, i.e.the total number of virions produced by an infected cell during its life span.2
20 40 60 80 100 120 140 160 180 2004.555.56 l og T ( t ) time (days)0 20 40 60 80 100 120 140 160 180 200123456 l og I ( t ) time (days)0 20 40 60 80 100 120 140 160 180 2003456 l og V ( t ) time (days) Figure 1: A representative solution of (1) with parameter values stated in Table 1. The initial T-cellpopulation is T (0) = 4 × , while the initial viral load is V (0) = 10 , and I (0) = 0. In this example, thesystem tends to the endemic equilibrium as t → ∞ because R = 2 . T, I, V ), namelya non-infective equilibrium E NI : (cid:18) λd T , , (cid:19) and an infective or endemic equilibrium E I : (cid:18) λd T R , d T d V kN d I ( R − , d T k ( R − (cid:19) where R = λkNd V d T . The stability properties of these steady states are also well-known and depend only upon the single parameter R , called the basic reproduction number. In particular, one can study the linearized analogue of (1) andprove the local asymptotic stability of E NI if R ≤ E I if R > t → ∞ . Additionally,the global asymptotic stability of these equilibria is known. In [20] it was shown that initial populationsare irrelevant in determining the long term dynamics of the solution. More specifically, if R ≤
1, thenfor any initial population of uninfected cells, infected cells, and virions the solutions of (1) tend to E NI as t → ∞ . Contrastingly, if R > E I . Figure 1 displays arepresentative graph of solutions for which R > . Inclusion of Latently-infected cells Though (1) describes the basic mechanisms which account for the spread of HIV, it lacks the abilityto describe the latent stage of a specific subpopulation of infected T-cells. Many studies [7, 9, 10] havedetermined that upon infection and transcription of viral RNA into cell DNA, a fraction of CD4+ T-cellsfail to actively produce virus until they are activated, possibly years after their initial infection. Suchcells may possess a much longer lifespan than their counterparts, and are termed latently infected. Uponactivation, latently infected cells do become actively productive, and hence begin to increase the viral loadthrough viral replication. A basic model of latent cell activation was initially developed to examine cellpopulations that contribute to the viral decline that occurs after administration of antiretroviral therapy[23]. However, within [23] and other articles by related authors [24, 28, 29], the mathematical analysis of themodel is performed under a number of limiting assumptions, including a constant background populationof susceptible T-cells and perfect efficacy of anti-retroviral drugs. Thus, we focus on rigorously proving theresulting nonlinear dynamics without these assumptions.As for (1) we consider a model describing T-cells that may be susceptible or infected. In addition, welet L ( t ) represent the new population of latently infected T-cells that cannot produce virions at time t butbegin to do so once they are activated by recall antigens. With this addition, the previously describedthree-component model now contains four components and is given by(2) dTdt = λ − d T T − kT VdIdt = (1 − p ) kT V + αL − d I IdLdt = pkT V − αL − d L LdVdt = N d I I − d V V. Here, p ∈ (0 ,
1) is the proportion of infections that lead to the production of a latently infected T-cell, ratherthan a productively infected T-cell, and α is the rate at which latently infected cells transition to becomeactively productive. Additionally, d L is the rate at which latent cells are cleared from the system. Figure2 displays a representative graph of solutions to (2) for which R >
1. We note that the oscillations of
T, I ,and V seem quite damped in comparison to those of Figure 1. In this section and the previous one, we have adopted parameter values from other studies. A few ofthe parameters possess generally agreed upon values, including λ , d T , d I , and d V . However, it should benoted that λ and d T are typically estimated for healthy individuals, and thus may not be reliable estimatesto describe values within HIV patients, especially for those who experience impaired thymic function [18].Obviously, there are many parameters, and these are summarized within Table 1, along with descriptions ofthe variables, their associated units, and references from which parameter values stem.The parameter that displays the most uncertainty within the literature is the viral infectivity k whichfluctuates by an order of magnitude from a value of 2 . × − ml/day [24, 28] to 2 . × − ml/day [34].The value we utilize here is at the low end of this range and stems from [24]. Biologically relevant values ofthe in vivo burst size N are also somewhat uncertain. Estimates based on counting HIV-1 RNA moleculesin an infected cell vary between hundreds and thousands [15, 16, 28], and estimates based on viral productionhave been as high as 5 × [5, 11]. Here, we choose N = 2000 HIV-1 RNA/cell as reported in [16].Parameters that stem specifically from (2) are generally not well-known. In particular, the fraction of newviral infections resulting in latency α varies from study to study, but based on previous work [4, 18], we use α = 0 .
01 per day. Similarly, the removal rate of latently infected cells, d L , has been discussed as anywherefrom 10 − per day [28] to 0 .
24 per day [34]. Hence, we chose a value with this range, namely d L = 4 × − per day, as reported within [14]. The proportion p of cells which are categorized as latent upon becoming4
50 100 150 20055.25.45.65.8 l og T ( t ) time (days) 0 50 100 150 200123456 l og I ( t ) time (days)0 50 100 150 200123456 l og L ( t ) time (days) 0 50 100 150 20044.555.56 l og V ( t ) time (days) Figure 2: A representative solution of (2) with parameters from Table 1. The initial values are T (0) = 4 × , V (0) = 10 , and I (0) = L (0) = 0. In this example, the system tends to the endemic equilibrium as t → ∞ because R L = 2 . . × − in [4] to 0 . T (0) = 4 × , I (0) = 0, L (0) = 0, and V (0) = 10 . Next, we analyze properties of solutions to (2) so as to compare their dynamicsand large time behavior with solutions of (1). As a first step, we can say with certainty that biologically reasonable values of the parameters give riseto positive populations assuming that at some earlier point (perhaps at the initial time) the populationspossessed positive values.
Theorem 3.1.
Assume all constants in (2) are nonnegative and the initial values T (0) , I (0) , L (0) , and V (0) are positive. Then, the solutions of (2), namely T ( t ) , I ( t ) , L ( t ) , and V ( t ) exist, are unique, and remainbounded on the interval [0 , t ∗ ] for any t ∗ > . Additionally, each function remains positive for any t ≥ . Of course, the requirement of initial positivity is not completely necessary since we may translate orrescale the time variable to alter the initial time. Hence, what is necessary for the theorem to hold is that allpopulations must attain positive values at some time. This result provides some general validation for themodel since it implies that negative population values cannot occur if one begins with biologically reasonable(i.e., positive) values.Next, we proceed as for (1) and investigate the possible equilibrium states of (2) and their stability5able 1: Variable and parameter values for (1) and (2).Variable Units Description Value Reference T ( t ) cells ml − Susceptible CD4+ T-cells – – I ( t ) cells ml − Actively Infected CD4+ T-cells – – L ( t ) cells ml − Latently Infected CD4+ T-cells – – V ( t ) virions ml − Infectious virions – – λ ml − day − Production rate of CD4+ T-cells 10 [4] d T day − Death rate of susceptible T cells 0 .
01 [21] d I day − Death rate of actively infected T cells 1 [22] d V day − Clearance rate of virions 23 [27] k ml day − Rate of infection of susceptible cells 2 . × − [24] N - Burst rate of actively infected T-cells 2000 [16] d L day − Death rate of latent cells 4 × − [14] α day − Activation rate of latent cells 0 .
01 [4] p - Proportion of latent infection 0 . ǫ RT - Efficacy of RT inhibitor varies – ǫ P I - Efficacy of protease inhibitor varies –properties. We find steady states by solving the nonlinear system of algebraic equations(3) λ − d T T − kT V − p ) kT V + αL − d I I pkT V − αL − d L L N d I I − d V V. for the unknown constants T, I, L , and V in terms of the parameters λ, k, p , α, N, d T , d I , d L , and d V . Thisis a nontrivial task, but eventually we find the existence of exactly two steady states. We begin by solvingfor the nonlinear interaction term in the first equation so that kT V = λ − d T T. With this, we may solve for L , I , and hence V in terms of T alone. From the third equation, pkT V = ( d L + α ) L and thus L = pd L + α ( λ − d T T ) . Next, in the second equation, we find (1 − p ) kT V = − αL + d I I and thus I = 1 d I (cid:18) − p + αpd L + α (cid:19) ( λ − d T T ) . The last equation yields V in terms of I , whence T , so that V = N d I d V I = Nd V (cid:18) − p + αpd L + α (cid:19) ( λ − d T T ) . Finally, we may use the representation of V in terms of T within the first equation and solve a simplequadratic in T to determine the possible steady state values. With this, the first equation becomes0 = λ − d T T − kNd V (cid:18) − p + αpd L + α (cid:19) ( λ − d T T ) T T = λd T and T = λd T · R L where(4) R L = kN λd T d V · (1 − p ) d L + αd L + α . Continuing in this manner, we obtain two corresponding values for
I, L , and V . To summarize, we find twoequilibria, which we write in the form ( T, I, L, V ) as E NI : (cid:18) λd T , , , (cid:19) E I : (cid:18) λd T R L , d T d V kN d I ( R L − , pλR L ( d L + α ) ( R L − , d T k ( R L − (cid:19) . As before, we denote the non-infective equilibrium by E NI and the infective equilibrium by E I . Noticethat the limiting values of T, I , and V for the infective state are of the same form as those of (1), with R L replacing the role of R . Additionally, we see that if R L = 1, then the equilibria coincide, and if R L <
1, thenthe endemic equilibrium corresponds to negative values which, in view of Theorem 3.1, cannot be obtainedfrom biologically relevant initial data.By studying the linearized version of the system, we may examine the local stability of these equilibriaand find that their behavior mimics that of (1).
Theorem 3.2. If R L ≤ , then the non-infective equilibrium is locally asymptotically stable. If R L > thenthe non-infective equilibrium is an unstable saddle point, and the endemic equilibrium is locally asymptoticallystable. Therefore, if R L ≤ E NI , then theywill tend to E NI as t → ∞ . Contrastingly, if R L > E I ,they will tend to E I in the long run. Theorem 3.2 also emphasizes the crucial feature that equilibria are notstable simultaneously, that is, bistability of E NI and E I does not occur. Furthermore, it expresses that thequalitative behavior of system (2) changes exactly when R L transitions from less than one to greater thanone, and hence a bifurcation occurs at R L = 1.The final theorem of the section demonstrates the stronger result that initial values of these populationshave no effect on their long term ( t → ∞ ) limiting values. Theorem 3.3. If R L ≤ , then the non-infective equilibrium is globally asymptotically stable. If R L > ,then the endemic equilibrium is globally asymptotically stable. This analysis reveals one very important fact about the overall system: the end states of populations areonly dependent on the value of R L , and not any other parameter or initial value. If R L is greater than one,then the system tends to E I , an end state with a non-zero population of infected cells and virions, but if R L is less than one, then the final equilibrium is E NI , which contains neither virions nor infected T-cells.The most important feature of these results is the explicit formula for R L , which can be related exactlyto the basic reproductive number of the three-component model (1). In order to investigate the differencesbetween the two reproductive ratios, we define the quantity(5) Q := R L R = (1 − p ) d L + αd L + α . Notice that Q depends only upon the three new parameters included within (2), namely the activationratio α , proportion of cells which become latent upon infection p , and the death rate of latent cells d L .7dditionally, if the proportion p of infections which produce latently infected T-cells is identically zero, then R L = R . However, since we consider p ∈ (0 ,
1) we find
Q < d L + αd L + α = 1and the relationship R L < R follows directly. Thus, the reproduction number of the latent cell model (2) is strictly less than that ofthe standard three-component model (1). Therefore, the stability of the the non-infective state is enhancedby the inclusion of the latently-infected cell population. Namely, there are more values of λ, d T , d I , d V , k and N which correspond to R L ≤ R ≤
1. From a modeling standpoint, this result is somewhatintuitive. Because (2) assumes that a fraction of newly infected cells become latently infected and the lattercan only activate (becoming actively productive) or die, the average number of infected cells generated bythe introduction of a single infected cell into a susceptible system is decreased in comparison to a modelwithout latently infected cells, namely (1). Hence, one should expect that the basic reproduction number,representing this average number of infected cells, does in fact decrease. Another consequence of this resultsis that there exist a number of parameter values for which R > R L ≤
1, and in such cases the solutionsof (1) tend to E NI while those of (2) tend to E I as t → ∞ . Clearly, the converse ( R ≤ R L >
1) isnot possible by the above inequality. In fact, we may rewrite their ratio Q as Q = 1 − pd L d L + α so that the difference between R and R L is greatest for large values of p and d L , but small values of α .With the representative parameter values given in Table 1, we see that R L ≈ .
978 and R ≈ . . Hence, the change in system behavior caused by the difference between the reproduction numbers appearssomewhat negligible, as both values are significantly larger than their respective bifurcation points. Addi-tionally, Q = 0 .
97 in this case, so that the relative difference between R and R L is merely R − R L R = 1 − Q = 3% . Exactly quantifying this relative change, however, is difficult since many of the parameter values of Table1, in particular k, p, α , and d L , are not well-established, and hence this percentage could be much larger orperhaps even smaller. For example, if we utilize the smallest value of α = 3 × − , stemming from [24],and the largest values of p = 0 . d L = 0 .
24 [24], then a simple computation shows that Q = 0 . − Q between 1% and 5%.Regardless of the quantified distinction between the reproductive ratios, it seems somewhat counterintu-itive that R L < R , especially since so many authors [1, 2, 6, 7, 9, 10, 18, 19, 23, 28, 32] have detailed the largedegree to which latent reservoirs contribute to the increased persistence of HIV infection via viral reboundafter treatment with ART. Hence, the result of the mathematical analysis, namely that the effects of latentinfection reduce the basic reproductive ratio, seems to contradict this theory. However, as we previouslystated, the alterations in the mathematical model explain this effect. Additionally, the reproductive ratio isbut one parameter, and so it seems unlikely that this particular metric will completely determine the realisticbehavior of the system. In fact, a more detailed analysis of the behavior of solutions over the timescalesof biological relevance, rather than considering only the limiting behavior as t → ∞ , will demonstrate theshortcomings of the basic reproductive ratio. We illustrate this using the effects of antiretroviral therapyand some associated computational results within the next section.8 . Antiretroviral Therapy In order to further contrast these two models and the effects of the latent cell population, we will introduceadditional parameters to represent the application of antiretroviral therapy. The inclusion of ART will allowus to determine the range of drug efficacies that distinguish between the limiting dynamics of (1) and (2)and further elucidate the differing behaviors of the two models.Two classes of antiretroviral drugs are often used to reduce the viral load and limit the infected T-cellpopulation. One class is known as Reverse Transcriptase Inhibitors (RTIs), which can block the infectionof target T-cells by infectious virions. The other category is Protease Inhibitors (PIs), which prevent HIV-1protease from cleaving the HIV polyprotein into functional units, thereby causing infected cells to produceimmature virus particles that are non-infectious. In this way, RTIs serve to reduce the rate of infection ofactivated CD4+ T-cells, whereas PIs decrease the number of new infectious virions that are produced. Bothdrugs thus diminish the propagation of the virus [17, 30]. While we expect that latently infected cells mayabsorb PIs and that such cells, when activated, will produce noninfectious virus, we will instead assume thatPIs have no effect on the proportion of cells that are latently infected. This is in line with some experimentalfindings, that suggest that antiretroviral drugs do not effectively block replication of virus from the latentreservoir [8]. Hence, in our model, susceptible T-cells may be inhibited with either RTIs, or PIs, or theymay become infected. Infected cells may be inhibited with PIs, and cells inhibited with one drug may beinhibited with the other. In the presence of these two inhibitors, the model equations (2) are modified tobecome:(6) dTdt = λ − d T T − k (1 − ǫ RT ) T V I dIdt = (1 − p ) k (1 − ǫ RT ) T V I + αL − d I IdLdt = pk (1 − ǫ RT ) T V I − αL − d L IdV I dt = N (1 − ǫ P I ) d I I − d V V I . where ǫ RT , ǫ P I ∈ [0 ,
1] are the efficacies of RTIs and PIs, and V I represents the population of infectiousvirions. We may include the number of non-infectious virions V NI , with the total viral load V = V I + V NI ,but V NI decouples from the remaining equations, and hence plays no role in the evolution of the system.To compute the steady states and basic reproduction number for (6), we may reproduce the analysis of(2), but clearly the new terms are introduced only where the parameters k and N appear. Thus, we needonly replace k with k (1 − ǫ RT ) and N with N (1 − ǫ P I ). The new basic reproduction number then becomes(7) R ǫL = kN (1 − ǫ ) λd T d V · (1 − p ) d L + αd L + α where we define the quantity ǫ = ǫ RT + ǫ P I − ǫ RT ǫ P I , so that1 − ǫ = (1 − ǫ RT )(1 − ǫ P I ) . Since (6) is identical to (2) with the minor change in parameter values described above, Theorems 3.2 and3.3 hold for (6) with the corresponding value R ǫL instead of R L . Writing the corresponding viral steady statefor (6) we find V = d T R L k (1 − ǫ P I ) − d T k (1 − ǫ RT )and we notice that its partial derivative ∂V∂ǫ RT = − d T k (1 − ǫ RT )
100 200 300 400 500 600 700 800 900 1000−15−10−505 l og V ( t ) time (days) V V Latent l og V ( t ) time (days) Figure 3: Viral loads for (6) and (8) with ǫ = 0 . ǫ yields R ǫ = 1 .
003 and R ǫL = 0 . t ∈ [0 , ǫ RT ≈
1. Thus, V is sensitive to small changes in ǫ RT , and this sensitivity increaseswith the efficacy of the RTI. Hence, this model does not realistically describe the persistence of low-levelviremia in patients on reverse transcriptase inhibitors, as previously addressed within [28]. However, we notethat ∂V∂ǫ P I = − d T R L k which is constant for all values of ǫ P I , and does not possess the same sensitivity. Thus, to simplify theanalysis, we will assume throughout that only PIs are used, and hence ǫ = ǫ P I while ǫ RT = 0.Upon incorporating the use of antiretroviral drugs into (1), the system becomes(8) dTdt = λ − d T T − k (1 − ǫ RT ) T V I dIdt = k (1 − ǫ RT ) T V I − d I IdV I dt = N (1 − ǫ P I ) d I I − d V V I . with associated basic reproduction number(9) R ǫ = kN (1 − ǫ ) λd T d V . As for (6), the stability results for (1) contained in Section 2 hold for (8) by replacing R with R ǫ . Comparingthe two values R ǫ and R ǫL , we see that their ratio is again Q given by (5), so that R ǫL = QR ǫ . As before,since Q < R ǫL < R ǫ and even with the incorporation of ART, the latent cell population decreases R ǫL ≈ . − ǫ ) and R ǫ ≈ . − ǫ ) . Hence, in order for the non-infective state to be realized in (8), we must have ǫ > . ǫ that is needed to reach the same non-infective state in (6), namely ǫ > . ǫ which yield R ǫ > R ǫL < V CM ( t ) and its latent model analogue by V Latent ( t )as in the figure. In this case, we choose ǫ = 0 .
519 and find R ǫ = 1 .
003 and R ǫL = 0 . V CM ( t ) → d T k ( R ǫ − ≈ × as t → ∞ , while V Latent ( t ) → t → ∞ . However, one can distinctly see from Figure 3 that the early decayrate of V CM is much greater than that of V Latent .Notice that the effects of latent infection do not influence the viral load for the first thirty days of treatmentas V CM ( t ) and V Latent ( t ) follow the same approximate trajectory during this time period. However, oncethe latently infected T-cell population grows sufficiently large, the effects are tremendous. Throughoutthe first three years of continuous treatment, V CM diminishes greatly, past 10 − in fact, while V Latent remains O (1) even up to day 1000. Certainly this seems strange as R ǫ > R L < V CM rebound and tend to a positive equilibrium, whilethose of V Latent continue their slow, steady decline to eradication. Unfortunately, these events occur nearlyfifteen years after the introduction of ART and well outside the timescale of biological relevance. Hence, itappears that the values of R ǫ and R ǫL alone cannot provide sufficient information to account for the realisticbiological dynamics of the model due to the change in timescales and decay rates introduced by the latent cellpopulation. A better estimate of the behavior would certainly be given by the rates of decay to eradication,but precise estimates on these quantities are more difficult to obtain analytically. Instead, we examine aslightly different metric of viral persistence or clearance.The feature of viral clearance that one must capture here is not just the decay of the viral load, but asufficiently rapid speed of decay so as to be realized within a time period of biological relevance. Hence, weconsider a specific value of the virus population to represent clearance, and proceed to study the minimumarrival time of the viral load to that value. In this vein, we define the functions P n ( r ) = inf (cid:26) t > (cid:18) V CM ( t ) (cid:19) ≤ − n for R ǫ = r (cid:27) and Q n ( r ) = inf (cid:26) t > (cid:18) V Latent ( t ) (cid:19) ≤ − n for R ǫL = r (cid:27) . We note that either of these functions may become infinite if the population of virions fails to reach theprescribed value for any positive time. For example, P (2) = Q (2) = ∞ , since neither viral load obtains avalue as small as 10 − for a corresponding reproductive ratio of 2, while P (0 . ≈
25 and Q (0 . ≈ P n ( r ) and Q n ( r ) will depend uponthe initial population values that are chosen. Within the present study, however, we will continue to utilizethe initial populations of previous sections to serve as a representative example.11 P ( da ys ) R ε Q ( da ys ) R L ε Figure 4: Comparison of P ( R ǫ ) and Q ( R ǫL )We first select the value of 10 − copies per ml for our definition of viral eradication and study theassociated times to eradication provided by the functions P and Q . Namely, what we are assuming is thatonce the viral population is suitably dilute - in this case less that 10 − copies per ml - then the infectionhas been cleared and no rebound can occur. Figure 4 provides a comparison of P and Q for differingvalues of R ǫ and R ǫL , respectively. Though their general shapes are quite similar, the associated time periodsdiffer dramatically. Typical values of P range from 20 to 100 days, while the majority of values of Q range between 1000 and 3000 days. As can be seen in Figure 4, even if the efficacy of the RT inhibitor, ǫ ,approaches 100%, and thus R ǫ approaches zero, V CM requires around 15 days to reach a value of 10 − . Inthis same situation, R ǫL approaches zero, but V Latent requires nearly 500 to 1000 days to reach a value of10 − . Thus, even for values of R ǫL which are significantly less than one, we see that it would require nearlythree years for the viral load to reach this threshold due to the influence of latent infection. In addition, wesee that V CM will reach values of 10 − even if R ǫ >
1, and this will occur within 100 days, almost ten timesfaster than it would take V Latent to reach the same value for a constant drug efficacy around 90%.Considering that the biological detection threshold is around 50 viral copies per ml [4, 12], one possibilityis that the 10 − threshold above has been chosen too small in Figure 4 to effectively serve as a realisticmeasure of eradication. Hence, we consider P and Q and perform a similar analysis. Figure 5 containsthese simulations and displays a decrease in the time necessary to reach the defined threshold of 10 − .However, even for a 70% constant drug efficacy, in which case R ǫL = 0 .
6, we see that approximately one yearof continuous ART would still be required to reach a viral load of 0 . P remain around 60 for (8) even if R ǫ is near 1 .
5, which exceeds the bifurcation point by nearly50%. Thus, if we define viral clearance as a decay in the viral load to 0 . R ǫ to be less than 1 . R ǫL to be less than 0 . R ǫ < . R ǫL < .
2, and we see that thelatent T-cell population does, in fact, extend the period of time during which viremia persists, even thoughthe behavior as t → ∞ , as given by Theorems 3.2 and 3.3, may provide seemingly contrary information.From this, the biological influence of latent infection becomes clear - the time needed to decrease the12 P ( da ys ) R ε Q ( da ys ) R L ε Figure 5: Comparison of P ( R ǫ ) and Q ( R ǫL )viral load to values from which rebound is unlikely or unable to occur is increased by a factor of ten ortwenty. Hence, the value of the basic reproduction number alone does not represent a proper definition forviral persistence or eradication, and the functions provided above P n ( r ) and Q n ( r ), for well-chosen values of n , possess the information required to better determine the behavior of the infection. Further analysis canbe performed for smaller (and negative) values of n , but the results discussed above are typical. In the nextsection, we prove Theorems 3.1, 3.2, and 3.3 regarding the qualitative behavior of the latent infection model.
5. Proofs of main theorems
With the analysis concluded, we finally prove the main results of the previous sections. In what follows, C will be used to denote a positive, but arbitrary constant which may change from line to line. First, weprove the existence, uniqueness, and positivity of solutions. Proof (Theorem 3.1).
While one may prove that a certain positive set remains invariant under the flow(as in [13]), this requires assumptions which bound the initial data from above. In our proof, we utilize acontinuity argument instead and do not assume any upper bounds on initial data. Using the Picard-Lindelofftheorem and the quadratic nature of the equation, the local-in-time existence of a unique, C solution followsimmediately. Hence, we will concentrate on proving positivity of solutions as long as they remain continuous,and this property will yield bounds on the growth of solutions. From the bounds obtained below, then, itfollows that the solution exists globally and is both unique and continuously differentiable for all t > T ∗ = sup { t ≥ T ( s ) , I ( s ) , L ( s ) , V ( s ) > , for all s ∈ [0 , t ] } . Since each initial condition is nonnegative and the solution is continuous, there must be an interval on whichthe solution remains positive, and we see that T ∗ >
0. Then on the interval [0 , T ∗ ] we estimate each term.Lower bounds on I, L , and V instantly follow since the decay terms are linear. More specifically, we find dIdt = (1 − p ) kT V + αL − d I I ≥ − d I I I ( t ) ≥ I (0) e − d I t > t ∈ [0 , T ∗ ]. Similarly, for the latent T-cell population dLdt = pkT V − ( d L + α ) L ≥ − ( d L + α ) L and thus L ( t ) ≥ L (0) e − ( d L + α ) t > t ∈ [0 , T ∗ ]. The positivity of the virion population follows in the same manner since dVdt = N d I I − d V V ≥ − d V V and thus V ( t ) ≥ L (0) e − d V t > t ∈ [0 , T ∗ ]. The positivity of T requires extra effort since it decreases due to the nonlinearity. We firstconstruct an upper bound on dTdt as dTdt = λ − µT − kT V ≤ λ and thus T ( t ) ≤ T (0) + λt ≤ C (1 + t ) . Next, we sum the equations for
I, L , and V , and by positivity of these functions, obtain upper bounds oneach one. Using the upper bound on T ( t ), we find ddt ( I + L + V ) = kT V + ( N − d I I − d L L − d V V ≤ C (1 + t ) ( I + L + V ) . By Gronwall’s Inequality, we have I ( t ) + L ( t ) + V ( t ) ≤ Ce t for t ∈ [0 , T ∗ ]. Since I ( t ) and L ( t ) are positive on this interval, the same upper bound follows on V ( t ) alone.With this, we can now obtain a lower bound on T . We find dTdt = λ − d T T − kT V ≥ − d T T − kT V ≥ − C (1 + e t ) T or stated equivalently dTdt + C (1 + e t ) T ≥ . It follows that ddt (cid:16) T ( t ) e C R t (1+ e τ ) dτ (cid:17) ≥ T ( t ) ≥ T (0) e − C R t (1+ e τ ) dτ > t ∈ [0 , T ∗ ]. Finally, if T ∗ < ∞ , then all functions are strictly positiveat time T ∗ , contradicting its definition as the supremum of such values. Hence, we find T ∗ = ∞ and theresult follows.Next, we prove the local stability results. Proof (Theorem 3.2).
We proceed by linearizing the system and using the Routh-Hurwitz criterion todetermine conditions under which the linear system possesses only negative eigenvalues. Then, as a conse-quence of the Hartman-Grobman Theorem, the local behavior of the linearized system is equivalent to thatof the nonlinear system. 14irst, we compute the Jacobian evaluated at the non-infective equilibrium E NI = ( λd T , , , J ( E NI ) = − d T − kλd T − d I α k (1 − p ) λd T − d L − α kpλd T N d I − d V From this, we compute the associated characteristic polynomial for eigenvalues η J − η I )= ( d T + η ) (cid:20) ( d I + η )( d L + α + η )( d V + η ) − αN Kλpd I d T − N Kλ (1 − p ) d I d T (cid:18) d L + α − p + η (cid:19)(cid:21) . After expanding the terms and ordering by powers of η , this equation ultimately simplifies to(10) η + A η + A η + A = 0where A = d V + d I + d L + αA = d I d V + ( d L + α )( d I + d V ) − (1 − p ) λN kd I d T A = ( d L + α ) d I d V − λN kd I d T ((1 − p ) d L + α ) . The Routh-Hurwitz criterion requires A , A , A > A A − A >
0. Clearly, A >
0, and afterrewriting A in terms of R L , we find A = ( d L + α ) d I d V (1 − R L ) . Thus, in order for all of the eigenvalues of the system to be negative, it is necessary that R L <
1. Similarly,we rewrite A as A = ( d L + α )( d I + d V ) + d I d V (cid:20) − R L (1 − p )( d L + α )(1 − p ) d L + α (cid:21) . Using the inequality(11) (1 − p )( d L + α )(1 − p ) d L + α = 1 − pα (1 − p ) d L + α < , and the previous condition R L <
1, we find A > A > d I d V (1 − R L ), and clearly A > d L + α . Therefore, we find A A > d I d V ( d L + α )(1 − R L ) = A and the Routh-Hurwitz criteria are satisfied. Thus, R L < E NI follows. Conversely, if R L >
1, thenthe linearized system possesses at least one positive eigenvalue, and the equilibrium is unstable.The analysis for E I is similar to that of E NI . For notational purposes, we label the equilibrium as( ¯ T , ¯ I, ¯ L, ¯ V ). Linearizing (2) about E I , we find the Jacobian J ( E I ) = − ( d T + k ¯ V ) 0 0 − k ¯ T (1 − p ) k ¯ V − d I α k (1 − p ) ¯ Tpk ¯ V − d L − α kp ¯ T N d I − d V d T R L + η )( d I + η )( d L + α + η )( d V + η ) − N Kλ (1 − p ) d I d T R L (cid:18) d L + α − p + η (cid:19) ( d T + η ) . After expanding terms and simplifying, we arrive at the quartic polynomial η + A η + A η + A η + A = 0where A = d T R L + d V + d I + d L + αA = d T R L ( d L + α + d I + d V ) + ( d L + α )( d I + d V ) + d I d V − (1 − p ) λN kd I d T R L A = d T R L ( d L + α )( d I + d V ) + d T R L d I d V + ( d L + α ) d I d V − λN kd I d T R L ((1 − p ) d T + (1 − p ) d L + α ) A = d T R L ( d L + α ) d I d V − λN kd I R L ((1 − p ) d L + α )As before, the Routh-Hurwitz criterion requires all coefficients to be positive, as well as, A A − A > A ( A A − A ) − A A >
0. As for the E NI analysis, the positivity of A follows directly from thepositivity of the coefficients, and after rewriting A , we find A = d T ( d L + α ) d I d V ( R L − . Hence, it is necessary that R L > A as A = d T R L ( d L + α )( d I + d V ) + d T R L d I d V + ( d L + α ) d I d V − (cid:20) d T d I d V (1 − p )( d L + α )(1 − p ) d L + α + d I d V ( d L + α ) (cid:21) > d T R L ( d L + α )( d I + d V ) + d T d I d V ( R L − > . In this inequality we have canceled the third term with the last term and utilized the inequality (11) tobound the fourth term. The only nonpositive term in A can be rewritten as − (1 − p ) λN kd I d T R L = − d I d V (1 − p )( d L + α )(1 − p ) d L + α > − d I d V , and hence A > d T R L ( d L + α + d I + d V ) + ( d L + α )( d I + d V ) > . By the definition of A , we have A > d I + d V and using the above inequality for A , we find A A > ( d I + d V ) · [ d T R L ( d L + α + d I + d V ) + ( d L + α )( d I + d V )] > ( d I + d V ) · d T R L ( d L + α ) + d V · d T R L d I + d V · ( d L + α ) d I > A . Finally, we verify the last inequality, namely A ( A A − A ) − A A >
0. After a long calculation, wefind A A − A > ( d L + α ) (cid:20) d T R L ( d T R L + 2 d I + 2 d V + d L + α )+( d L + α )( d I + 2 d V ) + ( d I + d V ) (cid:21) . A A − A > d I (cid:20) ( d L + α ) + ( d L + α ) d I + d T R L ( d L + α ) (cid:21) . In addition, we see from a previous computation that A > d T d V R L ( d L + α ) + d T d I d V ( R L − . Hence, we obtain a lower bound for A ( A A − A ) by multiplying the first term in the inequality for A bythe right side of (12) and the second term of the A inequality by the previous lower bound for A A − A .This results in A ( A A − A ) > d T d V R L ( d L + α ) · d I (cid:20) ( d L + α ) + ( d L + α ) d I + d T R L ( d L + α ) (cid:21) + d T d I d V ( R L − d L + α ) (cid:20) d T R L ( d T R L + 2 d I + 2 d V + d L + α )+( d L + α )( d I + 2 d V ) + ( d I + d V ) (cid:21) > d T d I d V ( R L − d L + α ) · ( d T R L + d V + d I + d L + α ) = A A With this, all of the criteria have been satisfied and E I is stable if R L >
1. Conversely, if R L <
1, thenthe Jacobian possesses at least one positive eigenvalue, and the endemic state is unstable. Finally, the localbehavior of the system for R = 1 is implied by the result of Theorem 3.3.Lastly, we include a proof of the previously stated global stability theorem. Proof (Theorem 3.3).
As in [20] for the case of (1), we will prove the global stability using a Lyapunovfunction. We will denote the non-infective equilibrium by ( T , , , T ( t ) − T − T ln (cid:16) T ( t ) T (cid:17) vanishes when evaluated at T ( t ) = T and is nonnegative as long as T ( t ) > U ( t ) = ((1 − p ) d L + α ) (cid:20) T ( t ) − T − T ln (cid:18) T ( t ) T (cid:19)(cid:21) + ( d L + α ) (cid:20) I ( t ) + 1 N V ( t ) (cid:21) + αL ( t ) . Notice that U is nonnegative, and U is identically zero if and only if it is evaluated at the non-infectiveequilibrium point. We compute the derivative along trajectories and find dUdt = ((1 − p ) d L + α ) (cid:18) − T T (cid:19) [ λ − d T T − kT V ]+ ( d L + α ) (cid:20) (1 − p ) kT V + αL − d I I + 1 N ( N d I I − d V V ) (cid:21) + α [ pkT V − ( α + d L ) L ]The I, L , and
T V terms all cancel and after using the definition of T , we are left with dUdt = ((1 − p ) d L + α ) ( λ − d T T ) (cid:18) − λd T T (cid:19) + (cid:20) ((1 − p ) d L + α ) kT ∗ − ( d L + α ) d V N (cid:21) V = − (1 − p ) d L + αd T T ( λ − d T T ) + ( d L + α ) d V N ( R L − V. R L ≤
1, we see that dUdt ≤ T, I, L , and V , andthe global asymptotic stability follows by LaSalle’s Invariance Principle.Turning to the endemic equilibrium, none of the end values are zero, so we denote this steady state by( T ∗ , I ∗ , L ∗ , V ∗ ) and define U ( t ) = ((1 − p ) d L + α ) (cid:20) T ( t ) − T ∗ − T ∗ ln (cid:18) T ( t ) T ∗ (cid:19)(cid:21) + ( d L + α ) (cid:20) I ( t ) − I ∗ − I ∗ ln (cid:18) I ( t ) I ∗ (cid:19) + 1 N (cid:18) V ( t ) − V ∗ − V ∗ ln (cid:18) V ( t ) V ∗ (cid:19)(cid:19)(cid:21) + α (cid:20) L ( t ) − L ∗ − L ∗ ln (cid:18) L ( t ) L ∗ (cid:19)(cid:21) . As before, this function is nonnegative and identically zero only when evaluated at the endemic equilibrium.Computing the derivative along trajectories yields dUdt = ((1 − p ) d L + α ) (cid:18) − T ∗ T (cid:19) [ λ − d T T − kT V ]+ ( d L + α ) (cid:20)(cid:18) − I ∗ I (cid:19) ((1 − p ) kT V + αL − d I I )+ 1 N (cid:18) − V ∗ V (cid:19) ( N d I I − d V V ) (cid:21) + α (cid:18) − L ∗ L (cid:19) [ pkT V − ( α + d L ) L ]= ((1 − p ) d L + α ) [ λ − d T T − kT V ]+ ( d L + α ) (cid:20) (1 − p ) kT V + αL − d I I + (cid:18) d I I − d V N V (cid:19)(cid:21) + α [ pkT V − ( α + d L ) L ] − ((1 − p ) d L + α ) (cid:20) λT ∗ T − d T T ∗ − kT ∗ V (cid:21) − ( d L + α ) (cid:20) (1 − p ) kT V I ∗ I + αLI ∗ I − d I I ∗ + d I IV ∗ V − d V V ∗ N (cid:21) + α (cid:20) pkT V L ∗ L − ( α + d L ) L ∗ (cid:21) . Nicely, the
I, L, V , and
T V terms all vanish and what remains is dUdt = ((1 − p ) d L + α ) (cid:20) λ − d T T + d T T ∗ − λT ∗ T (cid:21) + ( d L + α ) (cid:20) − (1 − p ) k T V I ∗ I − α LI ∗ I + d I I ∗ − d I IV ∗ V + d V V ∗ N + αL ∗ − αpkd L + α T V L ∗ L (cid:21) =: I + II.
For I , we factor out a d T T ∗ term and use the form of T ∗ to find I = ( d L + α ) d T T ∗ (cid:20) R L + 1 − TT ∗ − R L T ∗ T (cid:21) = ( d L + α ) d T T ∗ (cid:20) − TT ∗ − T ∗ T + ( R L − (cid:18) − T ∗ T (cid:19)(cid:21) = ( d L + α ) d T T ∗ (cid:20) − TT ∗ − T ∗ T (cid:21) + ( d L + α ) d T T ∗ ( R L − (cid:18) − T ∗ T (cid:19) II , we factor an L ∗ term and use the identities T ∗ V ∗ = d L + αkp L ∗ and N d I I ∗ = d V V ∗ to find II = ( d L + α ) L ∗ (cid:20) α + d I I ∗ L ∗ + d V V ∗ N L ∗ − (1 − p ) k T V I ∗ L ∗ I − d I I ∗ L ∗ IV ∗ I ∗ V − αpkd L + α T VL − α LI ∗ L ∗ I (cid:21) = ( d L + α ) L ∗ (cid:20) α + 2((1 − p ) d L + α ) p − (1 − p )( d L + α ) p T V I ∗ T ∗ V ∗ I − (1 − p ) d L + αp IV ∗ I ∗ V − α T V L ∗ T ∗ V ∗ L − α LI ∗ L ∗ I (cid:21) = ( d L + α ) L ∗ p (cid:20) ((1 − p ) d L + α ) (cid:18) − IV ∗ I ∗ V (cid:19) − (1 − p )( d L + α ) T V I ∗ T ∗ V ∗ I + αp (cid:18) − T V L ∗ T ∗ V ∗ L − LI ∗ L ∗ I (cid:19)(cid:21) Thus, combining the rearrangements of I and II , we find dUdt = ( d L + α ) d T T ∗ (cid:20) − TT ∗ − T ∗ T (cid:21) + ( d L + α ) d T T ∗ ( R L − (cid:18) − T ∗ T (cid:19) + ( d L + α ) L ∗ p (cid:20) ((1 − p ) d L + α ) (cid:18) − IV ∗ I ∗ V (cid:19) − (1 − p )( d L + α ) T V I ∗ T ∗ V ∗ I + αp (cid:18) − T V L ∗ T ∗ V ∗ L − LI ∗ L ∗ I (cid:19)(cid:21) The second term simplifies to combine with those in the third term since( d L + α ) d T T ∗ ( R L −
1) = ( d L + α ) L ∗ ((1 − p ) d L + α ) p and therefore the expression becomes dUdt = ( d L + α ) d T T ∗ (cid:20) − TT ∗ − T ∗ T (cid:21) + ( d L + α ) L ∗ p (cid:20) ((1 − p ) d L + α ) (cid:18) − T ∗ T − IV ∗ I ∗ V (cid:19) − (1 − p )( d L + α ) T V I ∗ T ∗ V ∗ I + αp (cid:18) − T V L ∗ T ∗ V ∗ L − LI ∗ L ∗ I (cid:19)(cid:21) Since (1 − p )( d L + α ) = (1 − p ) d L + α − αp , we add and subtract αp within the first term of the second lineand place the extra components in the terms on the third line to arrive at dUdt = ( d L + α ) d T T ∗ (cid:20) − TT ∗ − T ∗ T (cid:21) + ( d L + α ) L ∗ p (cid:20) (1 − p )( d L + α ) (cid:18) − T ∗ T − T V I ∗ T ∗ V ∗ I − IV ∗ I ∗ V (cid:19) + αp (cid:18) − T ∗ T − T V L ∗ T ∗ V ∗ L − LI ∗ L ∗ I − IV ∗ I ∗ V (cid:19)(cid:21) (cid:18) TT ∗ + T ∗ T (cid:19) ≥ r TT ∗ · T ∗ T = 113 (cid:18) T ∗ T + T V I ∗ T ∗ V ∗ I + IV ∗ I ∗ V (cid:19) ≥ r T ∗ T · T V I ∗ T ∗ V ∗ I · IV ∗ I ∗ V = 114 (cid:18) T ∗ T + T V L ∗ T ∗ V ∗ L + LI ∗ L ∗ I + IV ∗ I ∗ V (cid:19) ≥ r T ∗ T · T V L ∗ T ∗ V ∗ L · LI ∗ L ∗ I · IV ∗ I ∗ V = 1 . Hence, dUdt ≤ T, I, L , and V . As in the non-infective case, the conclusion thenfollows directly from LaSalle’s Invariance Principle.
6. Discussion
In order to realistically describe and predict the effects of latent HIV infection, models of HIV-1 dynamicsand the associated mathematical tools must be capable of explaining the rich set of dynamics inherent withintheir formulation. We have explored the steady states and asymptotic behavior of the basic three-componentmodel and its well-known variant which includes the effects of latent infection. A rigorous analysis of the largetime behavior of these systems displays a reduction in the basic reproduction number due to the appearanceof the latently infected T-cell population, and at first glance seems contradictory to the known difficultiesof eradicating the latent reservoir with antiretroviral therapy. After undertaking a more detailed analysishere, we find that even though the inclusion of latent T-cells allows for a wider range of parameter values toinduce viral eradication as t → ∞ , the rate at which this decay occurs under ART is retarded so significantlythat, in the majority of cases, the decay could only occur outside time periods of biological relevance. Thisanalysis highlights two major points. First and foremost, the latent cell population drastically extends thelifespan of infection. This can be seen from the rates of decay displayed within Section 4 by the functions P and Q . However, since this property cannot be detected at the level of the basic reproduction number,a second major point becomes clear. The standard tools of computing equilibrium states and the differingconditions under which a system may tend to these states as t → ∞ is clearly insufficient to describe, or moreimportantly predict, the realistic dynamics that these equations model. Hence, a more refined analysis whichinvestigates not only the end states, but the rate of propagation to a supposed equilibrium value within aspecified time period, is clearly needed to describe the propagation of HIV, at least when considering theeffects of latent infection.Of course, our study is not all-inclusive. In attempting to address the question of latently infected cellreservoirs, we have ignored other potential reservoirs of HIV, such as those occurring within the brain,testicles, and dendritic cells [6]. The extent of viral replication in compartments other than resting CD4+ T-cells in patients receiving antiretroviral therapy for extended periods of time has yet to be fully delineated.One may also adapt the model to account for other viral reservoirs and incorporate the immune systemresponse to a viral load. In addition, we assumed the use of antiretroviral therapy that included only PIs.Certainly, the effects of RTIs could also be included, though the picture becomes slightly more complex, andthe results are similar. One can also study effects arising from a number of additional aspects including1. A secondary infective population, such as macrophages [13]2. Pharmacological delays due to drug activation3. The residual effects of decaying drug efficacy or periodic ART schedules4. Spatial effects, such as those characterized by diffusion models and multiple compartment models5. Uncertainty arising from the measurement of parameter values or fluctuations across populations ofindividuals in the form of random coefficients or stochastic differential equations6. Successive mutation of HIV virions 20hat being said, the effects of the latent cell population on viral behavior have been clearly documentedwithin the current study, and it is greatly expected that even when additional mechanisms are incorporatedwithin the model, the basic reproduction number will not serve as a descriptive parameter alone since it onlydescribes the global asymptotic behavior of populations. Future work must examine the aforementionedissues within the context of latent infection using the exponential decay functions, P and Q , as a refinementof the mathematical analysis detailing the long time behavior of the model.In conclusion, the dynamics of models that consider latent infection are so complex, even when spatialfluctuations are ignored, that a single parameter, in this case R L or R ǫL , cannot possibly dictate the realisticbehavior of the corresponding populations. Instead, one must consider a number of factors including thetime of validity inherent within the model, the average time periods underlying treatment, and the rates ofdecay associated with the trend to equilbrium.
7. Acknowledgements
This work is supported by the National Science Foundation under awards DMS-0908413 and DMS-1211667. We also thank Prof. Mrinal Raghupathi (USNA) and ENS Peter Roemer (USN) for helpfulcomments and enthusiasm.
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00 200 300 400 500 600 700 800 9 M ( da ys ) R ε M ( da ys ) R L ε P ( da ys ) R ε Q ( da ys ) R L ε