MODELING THE DRUG THERAPY FOR HIV INFECTION

Transcription

1 Journal of Biological Systems, Vol. 17, No. 2 (9) c World Scientific Publishing Company MODELING THE DRUG THERAPY FOR HIV INFECTION P. K. SRIVASTAVA,M.BANERJEE and PEEYUSH CHANDRA Department of Mathematics and Statistics Indian Institute of Technology, Kanpur Kanpur 816, India pksri@iitk.ac.in malayb@iitk.ac.in peeyush@iitk.ac.in Received 7 July 8 Accepted 25 November 8 A mathematical model for the effect of Reverse Transcriptase (RT) Inhibitor on the dynamics of HIV is proposed and analyzed. Further, with help of numerical simulations, the relation between efficacy of administered drug, the total number of virus particles emitted from the infected cell and the transition period is also discussed. Keywords: HIV; Stability; CD4 + T Cells; Drug Therapy. 1. Introduction Mathematical modelling has proved its importance in understanding the dynamics of many biological processes e.g. epidemiology, ecology, virology etc. 1 3 It has helped to improve our understanding for diseases like HIV. Modelling the level of viremia, when anti-retroviral drugs are administered, has provided insights into the hostpathogenesis interaction of HIV with CD4 + T cells. 4 8 HIV is a retrovirus and its main target is CD4 + T cells. The process of infection completes in the following steps: at the first stage, HIV attaches itself to the CD4 + T cell and then inserts its genetic material, which is in form of RNA, into the host cell. This process is called fusion. Now in the host cell, viral RNA is reverse transcribed into DNA using the protein reverse transcriptase and then this DNA is integrated with host cell genome by integrase.atthismomentitiscalledprovirus. When the cell is activated, it starts producing the viral RNA which is then cleaved into proteins using protease to become infectious virus. All these proteins are assembled into a capsid and then viral particles bud out of the cell and mature to infect other CD4 + T cells. 9 The phase of virus life cycle before the production of virus is called as eclipse phase. 1 Corresponding author. 213

2 214 Srivastava et al. It may be pointed out here that CD4 + T cells are important constituent of human immune system and are primarily attacked by HIV. The earlier models of HIV primary infection have mainly considered the interaction between virus population and CD4 + T cells 6,11 through three classes of populations: virus, uninfected and infected CD4 + T cells. It was assumed in these models that after infection, CD4 + T cells immediately become actively infected and start producing virus. Perelson et al. 12 considered an additional class of CD4 + T cells, namely latently infected cells. They considered that virus infected CD4 + T cells first become latently infected and then proceed to a class of productively infected CD4 + T cells, which then produce virus. Some researchers took account of this fact with the help of delay differential equation models. 13,14 A number of mathematical models have been proposed to understand the effect of drug therapy on viremia Drugs e.g., fusion inhibitors, reverse transcriptase (RT) inhibitors and protease inhibitors have been developed so as to attack on different phases of viral life cycle during infection. To get better results a combination of these drugs is used. The eclipse phase is an important phase of virus life cycle on which the drug therapy primarily depends. In this paper we propose and analyze a primary infection model with reverse transcriptase (RT) inhibitor. We study the dynamics of the uninfected CD4 + T cells, the infected CD4 + T cells and the virus population where the development of infected CD4 + T cells is through the interaction between uninfected CD4 + T cells and the virus. The interaction is taken to be of mass action type. The infected class of CD4 + T cells is further subdivided into two subclasses to account for the fact that the reverse transcription takes place in early stage of infection i.e. before infected cell start producing virus. One of these subclass is of the infected cells in which the reverse transcription is not completed and second class is of those infected cells in which reverse transcription is completed. We call the first class as pre-rt class and second class as post-rt class. Further, as suggested in Zack et al., 19, when a virus enters a resting CD4 + T cell, the viral RNA may not be completely reverse transcribed into DNA and the un-integrated virus may decay with time and partial DNA transcripts are labile and degrade quickly. Hence a proportion of infected cells will revert back to uninfected class. In previous models of drug therapy using RT inhibitor, it has been considered that the drug actually affects the interaction-infection rate constant k. 6 These models are based on the assumption that if η be the efficacy of RT inhibitor then the actual infection rate in presence of drug will be (1 η)k. But it is important to note that reverse transcriptase inhibitor inhibits the reverse transcription which takes place only after the virus has entered the host cell i.e. the process of attachment is over, so drug actually does not directly affect k. To overcome this situation the age structured model was proposed by Rong et al. 21 In view of this we divide the class of infected CD4 + T cells into pre-rt and post-rt class. Now, the RT inhibitor will not allow the reverse transcription and hence the the infected cells in pre-rt class will revert back to uninfected class. But since, drug may not be 1% effective

3 Modeling the Drug Therapy for HIV Infection 215 hence only a part of infected cells in pre-rt class will revert back to uninfected class and the remaining will progress to complete reverse transcription and become productively infected and then produce virus. 2. Model Formulation We develop a mathematical model for primary infection with RT inhibitor under the above mentioned assumptions. We consider three populations of CD4 + T cells: (1) T represents density of susceptible CD4 + T cells, (2) T1 represents density of infected CD4 + T cells before reverse transcription (i.e. those infected cells which are in pre-rt class), and (3) T represents density of infected CD4 + T cells in which reverse transcription is completed (post-rt class) and they are capable of producing virus. V is virus density. After infection, infected cells progress to pre-rt class T1 and then they leave pre-rt class at a rate α to productively infected (post- RT) class. These infected cells are capable of producing virus particles. In view of the above discussion, we consider that due to presence of RT inhibitor a fraction of cells ηαt1 in pre-rt class reverts back to uninfected class and remaining (1 η)αt 1 proceeds to post-rt class and become productively infected, where <η<1is the efficacy of RT inhibitor. Hence we have the following model: dt dt = s kv T µt +(ηα + b)t 1, (2.1) dt1 = kv T (µ 1 + α + b)t1 dt, (2.2) dt =(1 η)αt1 δt, (2.3) dt dv dt = NδT cv, (2.4) with T () = T,T1 () =, T () =, V () = V. Here s is the inflow rate of CD4 + T cells and µ its natural death rate. The parameter k is interaction-infection rate of CD4 + T cells (here by infection we mean the attachment and fusion of virus with cell) and µ 1 is death rate of infected cells. α is the transition rate from pre-rt infected CD4 + T cells class to productively infected class (post-rt). b is the reverting rate of infected cells to uninfected class due to non-completion of reverse transcription. 19, The δ represents death rate of actively infected cells and includes the possibility of death by bursting of infected T cells, c is the clearance rate of virus and N is the total number of viral particles produced by an infected cell. The above system has the following two steady states: (1) E 1 =( T = s µ,,, ), and (2) E 2 =(T,T, I,V ).

4 216 Srivastava et al. The components of E 2 are given as T = (µ 1 + α + b)c Nαk(1 η), T s µt 1 = µ 1 + α(1 η), T = α(1 η) T 1 and V = Nδ δ c T. (2.5) Feasible existence of E 2 is ensured whenever T >T. Remark 2.1. From the existence condition for E 2, i.e. T > T, we may find the critical value for the drug efficacy: η crit =1 µc(µ 1 + α + b). (2.6) Nαks Whenever η<η crit both E 1 and E 2 coexist and when η>η crit, the infection is cleared and only uninfected steady state E 1 will exist. Boundedness of solution: Equations (2.1) and (2.2) of the system give d dt (T + T 1 )=s µt µ 1 T1 (1 η)αt1 s µ m (T + T1 ), where µ m =min(µ, µ 1 ). Hence lim sup t (T + T1 ) s µ m. Without loss of generality, we can assume that lim sup t T s µ m and lim sup t T1 s µ m.nowusing the bound for T1 in Eq. (2.3), we get the bound for T and using this in (2.4), we get the following positively invariant set { Γ= (T,T1,T,V) R 4 : T,T1 s }, T Φ, V Ψ µ m with respect to the system (2.1) (2.4), where Φ = αs(1 η) µ mδ and Ψ = Nαs(1 η) µ mc. 3. Stability Analysis In this section we shall establish stability results for steady states E 1 and E 2. Proposition 3.1. The non-infected steady state E 1 is locally stable if and only if T T. (3.1) Proof. The linearized matrix evaluated at E 1 is given as, µ ηα+ b k s µ (µ M 1 = 1 + α + b) k s µ (1 η)α δ. Nδ c

5 Modeling the Drug Therapy for HIV Infection 217 Hence the characteristic equation for M 1 is (λ + µ)(λ 3 + Pλ 2 + Qλ + R) =, (3.2) where P = µ 1 + α + b + δ + c >, Q = (µ 1 + α + b)(δ + c) +cδ > and R = cδ(µ 1 + α + b) Nδα(1 η) ks µ. Since one eigenvalue is negative, λ = µ, the local stability of E 1 demands the negative real parts of all roots of the equation λ 3 + Pλ 2 + Qλ + R =.Now,itcan be easily seen that PQ R>. Hence for Routh-Hurwitz criterion to be satisfied, we need R>, which gives c(µ 1 + α + b) Nα(1 η) ks > 1 i.e. T <T. µ Therefore the non-infected steady state E 1 will be locally asymptotically stable if T < T.Also,for T = T, the characteristic equation has a zero root which is simple and other roots have negative real parts. Hence E 1 will be locally stable when T = T. Proposition 3.2. The infected steady state E 2, whenever it exists, is locally asymptotically stable provided the following condition is satisfied: where C A 2 D>, (3.3) A = µ + kv + µ 1 + α + b + δ + c, B =(c + δ)(α + µ 1 + µ + kv + b)+cδ + µ(µ 1 + α + b)+kv (µ 1 +(1 η)α), C = cδ(µ + kv )+(c + δ)(µµ 1 + µα + µb + µ 1 kv +(1 η)αkv ), D = cδkv (µ 1 + α(1 η)), and =AB C. Proof. The Jacobian matrix evaluated at E 2 is given by µ kv ηα+ b kt M 2 = kv (µ 1 + α + b) kt (1 η)α δ. (3.4) Nδ c The characteristic equation of M 2 then becomes λ 4 + Aλ 3 + Bλ 2 + Cλ + D =, (3.5) Clearly A >, D >, and >. Therefore using Routh Hurwitz criterion, all the roots of characteristic equation (3.5) will have negative real part if condition (3.3) is satisfied.

6 218 Srivastava et al. Proposition 3.3. The non-infected steady state E 1 is globally asymptotically stable if T T. Proof. Define a Lyapunov function L of system (2.1) (2.4) as follows: L = α(1 η) (µ 1 + α + b) T 1 + T + 1 V. (3.6) N Its derivative along a solution of system (2.1) (2.4), ( α(1 η)kt L = (µ 1 + α + b) c ) V (3.7) N It is clear from (3.7) that for T T, L ast s µ.furthermore,ifm is the set of solutions of the system where L =, then the Lyapunov Lasalle Theorem 22 implies that all paths in Γ approach the largest positively invariant subset of the set M. Here, M is the set where V =. On the boundary of Γ where V =,we have T =,I =,andt = s µt.sot s µ as t. Hence all solution paths in Γ approach the non-infected steady state E 1 when T T. Remark 3.1. Whenever T >T, one root of characteristic equation (3.2) becomes positive and hence E 1 becomes unstable. Using the same Lyapunov function as in (3.6), in this case we shall get L > and using the results in 23,24 we conclude that the system (2.1) (2.4) is persistent. Details are omitted here. 4. Numerical Simulation We integrated the system (2.1) (2.4) numerically in MATLAB for the following set of parametric values 12 : s =1mm 3 day 1, k =.24 mm 3 day 1, δ =.26 day 1 and c =2.4 day 1.Wetookµ =.1 day 125 and µ 1 =.15 (since death rate of cells with viral particle will be slightly higher than those of uninfected cells) and N = 1 (N varies from 1 to some thousands in literature 26,27 ). The drug efficacy η varies and we simulated for different values of η (η =.6,.7,.8,.9) keeping α =.4 day 1 and b =.5 day 1 28, and with initial condition T () = mm 3, T () = 1 mm 3, I() = 1 mm 3, V () = 1 mm 3.Itmay be pointed out that for η =.9 weobtain T <T while for η =.6 to.8, T >T. The numerical simulation shows that system approaches to the infection free steady state for η =.9 and to infected steady state for other values of η considered [cf. Propositions 2 and 3]. As η increases from.6 to.8 the viral level decreases and in case when η =.9 it approaches to zero. The results are plotted in Fig. 1. It can be easily seen that the level of CD4 + T cells increases with increase in η. Since the transition rate α from pre-rt infected class to post-rt infected class is not clear in literature we simulated the system for α =.3 to.5, keeping all other parameters same as above and η =.6, and see that variation in α does not have

7 Modeling the Drug Therapy for HIV Infection 219 T(t) T 1 * (t) T * (t) V(t) η=.6 η=.7 η=.8 η=.9 Fig. 1. Solution trajectories of system (2.1) (2.4) for different values of η keeping α =.4and b =.5. Remaining all other parameters are same as in Sec T(t) 35 T 1 * (t) T * (t) V(t) α=.3 α=.4 α=. 5 5 Fig. 2. Solution trajectories of system (2.1) (2.4) for different values of α keeping η =.6and b =.5. Remaining all other parameters are same as in Sec. 4. much difference in viral level, though increase in α increases the viral level (see Fig. 2). Further, since only small fraction of infected cells will revert back due to incompletion of reverse transcription, 28 we expect the reverting rate b to be small. We simulated numerically for b = to.1 again for same set of parameters as above

8 2 Srivastava et al T(t) T 1 * (t) 4 25 T * (t) V(t) b= b=.25 b=.5 b=.75 b=.1 Fig. 3. Solution trajectories of system (2.1) (2.4) for different values of b keeping α =.4 and η =.6. Remaining all other parameters are same as in Sec. 4. keeping η =.6 andα =.4. The results are plotted in Fig. 3. It can be easily seen that increase in b shows increase in CD4 + T cells and decrease in viral level. 5. Results and Discussion In this paper we proposed and analyzed a primary infection model of HIV with CD4 + T cells during therapy. It was considered that only RT inhibitor was given, and accordingly we developed the drug therapy model. The proposed model is different from the existing models in literature and accounts for the events that occur during infection and drug therapy. We subdivided the infected cells class into two subclasses: pre-rt and post-rt. The cells in pre-rt proceed to post-rt with completion of reverse transcription. It is argued that the RT inhibitor will prevent the infected cells in pre-rt class from proceeding to the post-rt class. If the efficacy of drug is not 1% then a fraction of infected cells in pre-rt class will revert back to uninfected class and remaining will proceed to post-rt class. We analyzed the stability of the equilibrium points. If condition (3.1) is satisfied then the infection will be cleared and the non-infected steady state E 1 will be globally stable. It was shown using numerical simulations that for the given set of

9 Modeling the Drug Therapy for HIV Infection η α N Fig. 4. Graph of η crit with various magnitudes of α and N, within the range.2 α.8 and 5 N η crit.9.85 η crit N α Fig. 5. (Top panel) Plot for critical magnitude of drug efficacy (η crit ) against the number of virus particles emitted from an infected cell (N) whenα =.4. (Bottom panel:) Plot for critical magnitude of drug efficacy (η crit ) against the transition rate for T cells, from pre-rt infected to post-rt infected class, α when (N = 1).

10 222 Srivastava et al. parameters, when the drug efficacy was 6%, the infected steady state is stable, and when we assume the drug efficacy to be 8%, the level of uninfected cell increases but infection still persists. When the efficacy was increased to 9% the infection was cleared. In literature the value of N varies from 1 to a few thousands. 26,27 We simulated η crit numerically for.2 α.8 and 5 N 25. It can be easily seen from (2.6) that η crit is an increasing function of both N and α (Fig. 4). We plotted η crit for N in Fig. 5 (top) and found that larger the value of N, we require higher the efficacy of the drug. Also we plotted η for α and found from graph that increase in α requires high efficacy of drugs (Fig. 5). 1/α is actually the transition period for CD4 + T cells from pre-rt class to post-rt class. Hence if the transition period is lower we require the efficacy of drug to be higher. Also, it may be noted that for single drug we need it to be highly effective. Therefore for better results of drug therapy we need a combination of therapy. In future we shall work on the effect of combination of drugs. Further, we hope that the model will stimulate experimenters to pose relevant questions that ultimately will help to unravel the nature of the biological mechanism itself. Acknowledgments The work of one of authors [P. K. S.] is supported by Council of Scientific and Industrial Research, India. The authors are thankful to the reviewers of this paper for their valuable suggestions and critical comments. References 1. Kot M, Elements of Mathematical Ecology, Cambridge University Press, Cambridge, Brauer F, Castillo-Chavez C, Mathmatical Model in Population Biology and Epidemiology, Springer, New York, Nowak MA, May RM, Virus Dynamics, Oxford University Press, UK,. 4. Ho DD, Neumann AU, Perelson AS, Chen W, Leonard JM, Markowitz M, Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection, Nature 373: , Perelson AS, Neumann AU, Markowitz M, Leonard JM, Ho DD, HIV-1 dynamics in vivo: virion clearance rate, infected cell life span, and viral generation time, Science 271: , Perelson AS, Nelson PW, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev. 41:3 44, Wei X, Ghosh SK, Taylor ME, Johnson VA, Emini EA, Deutsch P, Lifson JD, Bonhoeffer S, Nowak MA, Hahn BH, Saag MS, Shaw GM, Viral dynamics in human immunodeficiency-virus type-1 infection, Nature 373: , Bonhoeffer S, May RM, Shaw GM, Nowak M, Virus dynamics and drug therapy, Proc Nat Acad Sci USA 94: , Sierra S, Kupfer B, Kaiser R, Basics of virology of HIV-1 and its replication, JClin Virol 34: , 5.

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