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Journal of Theoretical Biology 313 (2012) 127–135 Contents lists available at SciVerse ScienceDirect Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi An HIV infection model based on a vectored immunoprophylaxis experiment$ Xiunan Wang, Wendi Wang n Key Laboratory of Eco-environments in Three Gorges Reservoir Region, School of Mathematics and Statistics, Southwest University, Chongqing 400715, PR China H I G H L I G H T S c c c The introduction of immunoprophylaxis can induce a backward bifurcation. The ignorance of antibodies’ involvement causes the loss of a backward bifurcation. Burst size of virus induces five types of behaviors including bistability. a r t i c l e i n f o abstract Article history: Received 21 June 2012 Received in revised form 18 August 2012 Accepted 20 August 2012 Available online 28 August 2012 A medical experiment published in Nature has shown that humanized mice receiving the vectored immunoprophylaxis can be fully protected from HIV infection. In this paper, a mathematical model is proposed to investigate the viral dynamics under the effect of antibodies in the experiment. It is shown that the introduction of vectored immunoprophylaxis can induce the backward bifurcation and the ignorance of antibodies’ loss due to their involvement with virus may result in the loss of backward bifurcation. By numerical simulations, it is found that the model also exhibits some other complicated dynamical behaviors. A subcritical Hopf bifurcation, a fold bifurcation of equilibria and a limit point bifurcation of limit cycles are detected, which induce five typical patterns of dynamical behaviors including the bistable phenomenon. & 2012 Elsevier Ltd. All rights reserved. Keywords: Viral dynamics Global stability Backward bifurcation Hopf bifurcation Bistability 1. Introduction Since the first case of AIDS was diagnosed on December 1st in 1981 this deadly disease has been pandemic around the world and has caused about 34 million infected and 25 million deaths so far. Although the advent of highly active antiretroviral therapy has dramatically reduced HIV-related morbidity and mortality (Simon and Ho, 2003), the clinical benefits of combination therapy are often challenged with the increasing emergence of drug resistance driven by within-host selective pressure of antiretroviral drugs (Clavel and Hance, 2004). Meanwhile, a latentinfection state can be established in resting memory CD4 þ T cells (Chun et al., 1995, 1997a). These latently infected cells are capable of abrogating CTL recognition because they express very low levels of HIV-1 messenger RNA and proteins (Lambotte et al., 2004). In the presence of antiretroviral drug therapy memory CD4 þ T cells can remain in the resting state for a long time, $ Research supported by the National Science Foundation of PR China (11171276) and by the Ministry of Education of PR China (20100182110003). n Corresponding author. Tel.: þ86 23 68367563; fax: þ 86 23 68252397. E-mail address: [email protected] (W. Wang). 0022-5193/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2012.08.023 providing a hiding place for virus (Wong et al., 1997; Finzi et al., 1997; Chun et al., 1997b). When activated they will start to produce new virus particles (Chun et al., 1998). Thus, a viral level rebound seems to be inevitable once the therapy is stopped and this has become a primary obstacle to the long-term control of HIV in infected individuals. In addition, viral escape from immune responses is indeed an impediment for HIV vaccination (Bailey et al., 2004; Davenport et al., 2008). All these facts have made the eradication of AIDS an elusive goal. Recently, however, a vectored immunoprophylaxis experiment carried out by Balazs et al. (2012) brought new hope to eradicate HIV infection. In the experiment, vector-mediated gene transfer was used to engineer secretion of the existing broadly neutralizing antibodies into the circulation. Vectored immunoprophylaxis (VIP) in mice induced the lifelong expression of these monoclonal antibodies at high concentrations from a single intramuscular injection. They showed that the humanized mice receiving VIP appear to be fully protected from HIV infection, even when challenged intravenously with very high doses of replication-competent virus. Mathematical models have made great contributions to getting insights into HIV infection and dynamics, as well as on how an infection can be managed, reduced or even eradicated. Since the 128 X. Wang, W. Wang / Journal of Theoretical Biology 313 (2012) 127–135 basic three-dimensional viral dynamical model was proposed (Perelson et al., 1993, 1996; Bonhoeffer et al., 1997; Perelson and Nelson, 1999; Nowak and May, 2000) considerable progress has been made in mathematical modeling of dynamics of viral infections. Some researchers have taken into account the effect of immune responses including CTL responses or humoral immunity or both (Wodarz, 2003; Pang et al., 2005; Wang et al., 2006a,b; Iwami et al., 2009). Some other researchers have incorporated the effect of intracellular delays into the model (Herz et al., 1996; Mittler et al., 1999; Nelson et al., 2000; Culshaw and Ruan, 2000; Nelson and Perelson, 2002; Dixit and Perelson, 2004; Li and Shu, 2010; Yan and Wang, 2012). A number of models use time-varying drug concentrations to determine the efficacy of antiviral treatment (Wahl and Nowak, 2000; Huang et al., 2003; Dixit and Perelson, 2004; Wu et al., 2005) and several other models have been proposed to investigate the effects of nonperfect adherence to drug regimens (Wahl and Nowak, 2000; Phillips et al., 2001; Huang et al., 2003; Ferguson et al., 2005; Smith, 2006; Wu et al., 2006; Rong et al., 2007a). Some more complex models have been employed to study the question of drug resistance or viral evolution (Kirschner and Webb, 1996; Nelson et al., 2004; Gilchrist et al., 2004; Iwasa et al., 2005; Iwami et al., 2006; Rong et al., 2007a,b). In previous papers, most of the models exhibit only forward bifurcation. However, during the last two decades, backward bifurcations have been observed in a fraction of epidemic models (Liu et al., 1986; Hadeler and van den Driessche, 1997; Dushoff et al., 1998; van den Driessche and Watmough, 2000; Arino et al., 2003; Martcheva and Thieme, 2003; Inaba and Sekine, 2004; Wang, 2006; Blayneh et al., 2010; Buonomo and Lacitignola, 2011). In recent years, backward bifurcations are also found in viral dynamical models (Shu and Wang, 2012; Qesmi et al., 2010). When a backward bifurcation occurs it is insufficient to reduce the basic reproduction number below 1 to eradicate the infection. In this case, the basic reproduction number must be reduced further to a threshold value which is less than 1 in order to guarantee the eradication. Therefore, identifying the occurrence of a backward bifurcation has important implications in studying viral dynamics. The purpose of the present paper is to develop a mathematical model to depict the viral dynamics under the effects of antibodies in the experiment (Balazs et al., 2012). We provide theoretical analysis and implement numerical simulations to investigate the antibody-based protection against HIV infection by vectored immunoprophylaxis. We find that the recruitment of antibody can lead to the occurrence of backward bifurcation and patterns of bistability, which may be helpful for the selection of treatment strategies in the future. The paper is organized as follows. In the next section, we state the model in consideration and the hypotheses on which it is formulated. The existence and stability of equilibria and the backward bifurcation are analyzed in Section 3. Some new and complicated dynamical behaviors are given by detailed numerical simulations in Section 4 and a general discussion concludes the paper in Section 5. 2. Model formulation We start from the following system which was proposed by Bonhoeffer et al. (1997) and Nowak and May (2000), and is the basis for mathematical studies of HIV dynamics dT ¼ ldTbTV, dt n dT ¼ bTVdT n , dt dV ¼ N dT n cV, dt ð1Þ where the state variables TðtÞ,T n ðtÞ and V(t) represent the concentrations of healthy CD4 þ T cells, infected cells and free immunodeficiency virus (HIV) at time t, respectively, l is the recruitment rate of uninfected CD4 þ T cells, and d represents the natural death rate of CD4 þ T cells. The term bTV describes the incidence of HIV infection of uninfected CD4 þ T cells, where b is the infection coefficient. Infected cells die at a rate d. Furthermore, N represents the total number of free virus particles released by each productively infected cell over its lifespan, and c is the clearance rate of virus particles. The humoral immunity, which is mediated by B lymphocytes expressing antigen specific receptors, is a very important kind of adaptive immune responses in HIV infection process. Kajiwara and Sasaki (2004) and Murase et al. (2005) introduced the effect of humoral immunity to system (1) to obtain the following system of differential equations: dT ¼ ldTbTV, dt n dT ¼ bTVdT n , dt dV ¼ NdT n cVpAV, dt dA ¼ aAVbA, dt ð2Þ where A represents the concentration of antibodies in humoral immune responses. The term pAV depicts the loss rate of virus under attack of antibodies and p is the killing rate. Since the humoral immunity is stimulated by virus in HIV infection, the production rate of antibodies is supposed to be aAV. Moreover, the natural clearance rate of antibodies is bA. Considering that the vectored immunoprophylaxis (VIP) in mice can induce the lifelong expression of monoclonal antibodies at high concentrations from a single intramuscular injection, we suppose that neutralizing antibodies are produced at a constant rate m after the injection. Observe that graph (d) of Fig. 1 in Balazs et al. (2012) depicts the comparison of the experiments in which, under HIV challenge, most mice expressing luciferase (i.e., those mice without receiving VIP) show the dramatic loss of CD4 cells whereas mice expressing b12 antibody (i.e., those mice receiving VIP) show no CD4 cell depletion. This indicates that without vectored immunoprophylaxis the antibody response induced by HIV in mice is so weak that the level of CD4 cells drops dramatically so that it cannot protect mice from infection at all, and the full protection from HIV infection in mice results from the effect of immunity induced by the vectored immunoprophylaxis. Thus, it is reasonable to consider the case where the immunity by vectored immunoprophylaxis is much stronger than the humoral immunity induced by HIV, and so we can neglect the term aAV in (2). Furthermore, since we care mainly about the effects of antibodies induced by vectored immunolprophylaxis, we ignore also the immune responses from the CTL immunity induced by infected cells. Note that antibodies, which are involved in the immune response to virus, can be removed because antibodies bind to specific antigen, forming precipitation or cell groups and then swallowed by phagocytes. Thus, we incorporate the loss rate qAV of antibody from the effect of antibodies’ involvement with virus. Taking all of the above into considerations, we obtain the following model: dT ¼ ldTbTV, dt n dT ¼ bTVdT n , dt dV ¼ NdT n cVpAV, dt dA ¼ mbAqAV: dt ð3Þ X. Wang, W. Wang / Journal of Theoretical Biology 313 (2012) 127–135 The basic reproduction number for system (3) is R0 ¼ b LaSalle–Lyapunov theorem, every nonnegative solution of (3) approaches E0 as t-1. & l cþp m d N: Now let us consider positive equilibria of (3). An infection equilibrium E1 ðT 1 ,T n1 ,V 1 ,A1 Þ of system (3) satisfies 8 ldT 1 bT 1 V 1 ¼ 0, > > > > < bT 1 V 1 dT n ¼ 0, 1 b Set Rn0 ¼ blN cd , Rn0A ¼ a Nblcd : cbb n 129 n Then R0 is the basic reproduction number of (2) and R0A is the antibody immune reproductive number of (2) (see Yan and Wang, 2012). It is shown in Yan and Wang (2012) that the dynamical behaviors of system (2) are completely determined by Rn0 and Rn0A . Specifically, the infection-free equilibrium is globally asymptotically stable if Rn0 r1; an immune-free equilibrium is globally stable if Rn0 41 and Rn0A r1, and a coexistence equilibrium of healthy CD4 þ T cells, infected cells, virus and antibodies is globally stable if Rn0 41 and Rn0A 41. However, we will show that the incorporation of vectored immunoprophylaxis and the loss of antibodies due to involvement with virus induces rich dynamical behaviors in (3), including the backward bifurcation, the subcritical Hopf bifurcation, and the bistable phenomena. These findings may be helpful for designing the strategies of vectored immunoprophylaxis. N dT n1 cV 1 pA1 V 1 ¼ 0, > > > > : mbA1 qA V 1 ¼ 0: 1 It follows that T1 ¼ l d þ bV 1 , T n1 ¼ blV 1 m , A1 ¼ , dðd þ bV 1 Þ bþ qV 1 where V1 is a positive solution of the following equation: m2 V 21 þ m1 V 1 þ m0 ¼ 0, ð4Þ in which m2 ¼ cqb, m1 ¼ cdq þcbb þ pmb dqðcb þ pmÞ R0 , b m0 ¼ dðbc þ pmÞð1R0 Þ: 3. Equilibria and backward bifurcation There is always an infection-free equilibrium E0 ¼ ðl=d,0,0, m=bÞ in (3). Evaluating the Jacobian matrix of system (3) at E0 gives 2 3 0 d 0 bl d 6 7 bl 6 0 d 0 7 6 7 d JðE0 Þ ¼ 6 7: 6 0 Nd c pm 0 7 b 4 5 b 0 0 qbm Its characteristic equation is pm dðbc þpmÞ ð1R0 Þ ¼ 0: ðo þ dÞðo þ bÞ o2 þ d þc þ oþ b b If R0 o1, then all the solutions of the above characteristic equation have negative real parts. If R0 41, the above characteristic equation has one positive root. Thus, by the Routh–Hurwitz criterion, we have the following theorem: Theorem 1. The infection-free equilibrium E0 is locally asymptotically stable if R0 o 1 and is unstable if R0 41. Next we analyze the global stability of the infection-free equilibrium E0 by constructing a Lyapunov function. Theorem 2. The infection-free equilibrium E0 is globally asymptotically stable if Rn0 o 1, i.e., R0 obc=ðbc þ pmÞ. Proof. We show below that E0 attracts the nonnegative solutions of (3). From the first equation of system (3), we have dT=dt r ldT: It follows that for any given E 40, there is some t0 such that TðtÞ r l=d þ E,t r t 0 . Define a Lyapunov function: L ¼ NT n þ V: For positive equilibria to exist, the solutions of Eq. (4) must be positive. Note that m2 4 0 and m0 has the same sign as 1R0 . It follows that (4) admits a unique positive solution if R0 4 1, which means that (3) admits a unique positive equilibrium whenever R0 41. To consider the case of R0 r 1, we set D ¼ m21 4m2 m0 . When R0 o1, it is easy to see that (4) admits two positive solutions if m1 o 0 and D 4 0, ð5Þ admits a unique positive solution if m1 o 0 and D ¼ 0, ð6Þ and there is no positive solution in (4) if m1 o 0 and D o 0: ð7Þ Note that m1 o 0 is equivalent to R0 4 ðpmb þ cdq þ cbbÞb :¼ R01 : dqðcb þ pmÞ ð8Þ Since we aim to find positive equilibria of (3) when R0 o 1, we need to ensure R01 o1 according to (8). R01 o 1 equals to the following inequality: ðpmb þ cdq þcbbÞb o 1: dqðcb þ pmÞ ð9Þ Solving b from (9) we get the following condition: bo dqpm :¼ b1 : bðcb þpmÞ ð10Þ By direct calculations we obtain 2 b D ¼ a2 R20 þ a1 R0 þa0 , ð11Þ where 2 a2 ¼ d q2 ðcb þ pmÞ2 , Since Rn0 ¼ Nbl=ðcdÞ o 1, we can select E small enough such that Nbðl=d þ EÞc o 0. Thus, dL l ¼ N bTVcV pAV r Nb þ E c V r 0, dt d a1 ¼ 2bdðcb þ pmÞqðcbbcdqpmbÞ, and dL=dt ¼ 0 if and only if V ¼0. The largest compact invariant set in fðT,T n ,V,AÞ : dL=dt ¼ 0g is the singleton fE0 g. Therefore, by the If D0 ¼ a21 4a2 a0 , by direct calculations we see that D0 o 0 if b 4 dq=b. This case is not interesting because b1 o dq=b. When 2 a0 ¼ b ðcbb þ pmbcdqÞ2 : 130 X. Wang, W. Wang / Journal of Theoretical Biology 313 (2012) 127–135 b o b1 , we have D0 4 0. Note that (10) implies 2 2 2 b D9R0 ¼ R01 ¼ 4cbb ðpmbb þ cbb dqpmÞ o0, ð12Þ and 2 2 b D9R0 ¼ 1 ¼ ðcbb þ pmbbdqpmÞ2 4 0: ð13Þ Since a2 4 0, it follows from (12) and (13) that the equation of D ¼ 0 in R0 admits a root Rc0 in the interval ðR01 ,1Þ where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 þ a21 4a2 a0 Rc0 ¼ ð14Þ 2a2 dynamical behaviors of system (3). By fixing q ¼ 104 , which is higher than q ¼ 106 for the case of forward bifurcation in Fig. 1, and choosing R0 as a bifurcation parameter, we observe a backward bifurcation, which occurs through the branch point ðBPÞ at R0 ¼ 1 (see Fig. 2). In addition, we find a Hopf bifurcation point ðHÞ at R0 ¼ 0.8243 and a fold bifurcation point ðLPÞ at R0 ¼ 0.6111 (see Fig. 2). At the Hopf point ðHÞ, the values of the four variables T, Tn, 11000 10000 9000 such that D ¼ 0 when R0 ¼ Rc0 , D 40 when Rc0 oR0 o1, and D o 0 when R0 oRc0 . Consequently, we can state the following results for positive equilibria of (3). Theorem 3 indicates that system (3) exhibits multiple infection equilibria via a backward bifurcation when (10) holds. This has two biological implications. First, the introduction of the vectored immunoprophylaxis induces the backward bifurcations. In contrast, model (2) with only humoral immunity exhibits the global stability of an equilibrium (Yan and Wang, 2012). Second, since (10) cannot be satisfied when q¼ 0, it follows that the ignorance of antibodies’ loss due to the involvement with virus may result in the loss of backward bifurcation, and thus give wrong predictions of HIV evolutions. 7000 Virus Theorem 3. System (3) has a unique infection equilibrium whenever R0 4 1, and has two infection equilibria when Rc0 o R0 o1 and (10) holds. Moreover, there exists a unique infection equilibrium of multiplicity 2 in (3) when (10) holds and R0 ¼ Rc0 o 1. 8000 6000 5000 4000 3000 2000 1000 0 0.6 0.7 0.8 0.9 1 1.1 1.2 2.2 x 104 2 1.8 1.6 1.4 V In this section, we implement numerical simulations to testify the theoretical results above and explore more patterns of dynamical behaviors of model (3). The parameter values are chosen from literatures (Zack et al., 1990; Perelson et al., 1996; Pawelek et al., 2012; Rong et al., 2007b; Stafford et al., 2000) by fixing l ¼ 50 cells ml day 1, d ¼0.008 day 1, b ¼ 5 107 ml 1 virion day 1, d ¼ 0:8 day , c¼3 day 1. The estimate of burst size, N, varies from 100 to a few thousands (Haase et al., 1996; Hlavacek et al., 1999) and possibly could be significantly larger (Chen et al., 2007). The value of m is mainly determined by the dosage of vectored immunoprophylaxis and the values of the other three parameters p, b and q may vary with various kinds of antibodies. We set m ¼ 12 cells ml day 1, p ¼ 5 103 ml virion day 1 and b¼0.02 day 1 in all the following simulations (see also Table 1). A forward bifurcation is shown in Fig. 1 where q ¼ 106 and R0 is a bifurcation parameter. With the help of the MatCont package (Dhooge et al., 2003, 2008) we obtain some more complicated 1.2 1 H 0.8 0.6 0.4 0.2 0 0.4 LP BP 0.5 0.6 0.7 0.8 R0 0.9 1 Parameter Value Description Reference l 50 cells ml day 1 0.008 day 1 5 10 7 ml virion day 1 0.8 day 1 Varied 3 day 1 Varied Varied Varied Varied Recruitment rate of uninfected cells Death rate of uninfected cells Infection rate of target cells by virus Death rate of infected cells Burst size of virus Clearance rate of free virus Killing rate of antibody Recruitment rate of antibody Clearance rate of antibody Involvement rate of antibody Stafford et al. (2000) Stafford et al. (2000) Stafford et al. (2000) Zack et al. (1990) See text Perelson et al. (1996) See text See text See text See text N c p m b q 1.1 1.2 Fig. 2. Graph of V component in equilibria versus R0 by fixing l ¼ 50,d ¼ 0:008, b ¼ 5 107 , d ¼ 0:8,c ¼ 3,p ¼ 5 103 , m ¼ 12,b ¼ 0:02,q ¼ 104 , where H denotes a Hopf bifurcation point, LP is a fold bifurcation point and BP represents a branch point. Table 1 Parameter definitions and values used in numerical simulations. b d 1.4 Fig. 1. The diagram of a forward bifurcation. The parameters are l ¼ 50,d ¼ 0:008, b ¼ 5 107 , d ¼ 0:8,c ¼ 3,p ¼ 5 103 , m ¼ 12,b ¼ 0:02,q ¼ 106 . 4. Numerical simulations d 1.3 R0 X. Wang, W. Wang / Journal of Theoretical Biology 313 (2012) 127–135 V and A are 3866:5166,23:8348,9863:0727 and 11.9248, respectively. At the fold bifurcation point ðLPÞ, the values of T,T n ,V and A are 5689:0464,5:6095,1577:6382 and 67.5053, respectively. At the branch point ðBPÞ, the values of these four variables are 6250:0000,0:0000,0:0000 and 600.0000, respectively. The first Lyapunov coefficient at the Hopf point is l1 ¼ 3:2165 104 , which means that the Hopf bifurcation is subcritical and the periodic orbits are born unstable. The fold bifurcation and the backward bifurcation lead to the existence of multiple infection equilibria for 0:6111oR0 o1. Note that the qualitative difference between Figs. 1 and 2 results from the fact that the value of q in Fig. 1 is smaller than it in Fig. 2. This indicates that a stronger loss 1600 1400 1200 Period 1000 800 600 400 200 LPC 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 R0 Fig. 3. Bifurcation curve of period of cycles versus R0 by fixing l ¼ 50,d ¼ 0:008, b ¼ 5 107 , d ¼ 0:8,c ¼ 3,p ¼ 5 103 , m ¼ 12,b ¼ 0:02,q ¼ 104 , where LPC represents the limit point bifurcation of cycles. 131 of antibodies’ involvement with virus can lead to a backward bifurcation, which supports the result of Theorem 3. Using the detected Hopf point in Fig. 2 to start a limit cycle continuation we find that the periodic orbits are initially unstable. A limit point of cycles (LPC) is detected at R0 ¼1.8214 and stability is gained at this moment. At this limit point (LPC), the period of the cycle is 125.6932. As the value of R0 decreases, the stability is preserved but the period appears to be infinity as the value of R0 tends to 1.0000 from right (see Fig. 3), which suggests that the stable limit cycle turns to a homoclinic orbit as R0 approaches 1.0000. From Figs. 2 and 3 we conclude that if we increase R0 from 0 continuously, we can obtain the five typical patterns of dynamical behaviors. These typical patterns of dynamical behaviors are listed briefly in Table 2. As an example, we choose a value of N within each interval corresponding to each pattern to illustrate the five dynamical behaviors of system (3). The other parameters are fixed as the same as above: l ¼ 50, d ¼ 0:008, b ¼ 5 107 , d ¼ 0:8,c ¼ 3,p ¼ 5 103 , m ¼ 12,b ¼ 0:02, q ¼ 104 . Firstly, if 0 oR0 r 0:6111, i.e., 0 oN r 1173, that is, the reproduction number (or equivalently, the burst size) is relatively small, then the infection-free equilibrium is the unique steady state and is globally asymptotically stable. Fig. 4 illustrates this pattern where the burst size is chosen as N ¼1000 (R0 ¼0.5208). Secondly, if 0:6111 oR0 r 0:8243, i.e., 1173 oN r 1582, that is, the burst size becomes a little larger, then the stable infectionfree equilibrium coexists with two unstable positive equilibria. For this pattern, we choose N ¼1400 (R0 ¼0.7292) to obtain Fig. 5, in which a numerical solution of system (3) tends also to the infection-free equilibrium. Thirdly, if 0:8243o R0 o1:0000, i.e., 1582 o N o1920, then there is a stable infection-free equilibrium, an unstable positive Table 2 Five typical patterns of dynamical behaviors of system (3). Pattern Range of R0 (Range of N) Steady states of system (3) 1 0 o R0 r 0:6111 ð0 o N r 1173Þ 0:6111 o R0 r 0:8243 ð1173 o N r 1582Þ 0:8243 o R0 o 1:0000 ð1582 o N o 1920Þ 1:0000 o R0 r 1:8214 ð1920 o N r 3497Þ R0 41:8214 ðN 4 3497Þ A stable infection-free equilibrium 2 3 4 5 4 A stable infection-free equilibrium and two unstable positive equilibria A stable infection-free equilibrium, an unstable positive equilibrium, a stable positive equilibrium and an unstable limit cycle An unstable infection-free equilibrium, a stable positive equilibrium, an unstable limit cycle and a stable limit cycle An unstable infection-free equilibrium and a stable positive equilibrium x 104 700 3.5 600 3 500 Antibody Virus 2.5 2 1.5 400 300 1 200 0.5 100 0 0 0 20 40 60 80 100 120 140 160 180 200 Time 0 100 200 300 400 500 600 700 800 900 1000 Time Fig. 4. A numerical solution of system (3) tends to the infection-free equilibrium as time tends to infinity, where N ¼ 1000,R0 ¼ 0:5208. The initial values are Tð0Þ ¼ 8000,T n ð0Þ ¼ 0,V ð0Þ ¼ 2000,Að0Þ ¼ 0. (a) Time series of V. (b) Time series of A. 132 X. Wang, W. Wang / Journal of Theoretical Biology 313 (2012) 127–135 2.5 x 104 700 600 2 Antibody Virus 500 1.5 1 400 300 200 0.5 100 0 0 0 0 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 900 1000 Time Time Fig. 5. A numerical solution of system (3) tends to the infection-free equilibrium as time tends to infinity, where N ¼ 1400,R0 ¼ 0:7292. The initial values are Tð0Þ ¼ 4400,T n ð0Þ ¼ 9,Vð0Þ ¼ 7000,Að0Þ ¼ 0. (a) Time series of V. (b) Time series of A. 4.5 x 104 700 4 600 3.5 500 Antibody Virus 3 2.5 2 1.5 400 300 200 1 100 0.5 0 0 0 50 100 150 200 250 300 350 400 450 500 0 100 200 Time 300 400 500 600 700 800 Time Fig. 6. A numerical solution of system (3) tends to the infection-free equilibrium as time tends to infinity, where N ¼ 1700,R0 ¼ 0:8854. The initial values are Tð0Þ ¼ 4080,T n ð0Þ ¼ 30,Vð0Þ ¼ 11,800,Að0Þ ¼ 0. (a) Time series of V. (b) Time series of A. 1.24 x 104 1.23 15 1.22 Antibody Virus 1.21 1.2 1.19 1.18 1.17 5 0 27.5 1.16 1.15 1.14 10 0 500 1000 1500 2000 2500 3000 3500 4000 Time 27 26.5 Inf ect ed Tc ells 26 25.5 3575 3580 3585 3590 ted T c Uninfe 3595 3600 cells Fig. 7. A numerical solution of system (3) tends to the positive equilibrium as time tends to infinity, where N ¼ 1700,R0 ¼ 0:8854. The initial values are Tð0Þ ¼ 3580,T n ð0Þ ¼ 27,V ð0Þ ¼ 11,870,Að0Þ ¼ 0. (a) Time series of V. (b) An orbit in the TT n A space. equilibrium, a stable positive equilibrium and an unstable limit cycle which is born from the subcritical Hopf bifurcation. The limit cycle encloses the stable positive equilibrium and it contracts as R0 increases in the interval. Hence, the bistability occurs, where a stable disease-free equilibrium coexists with a stable positive equilibrium. For this pattern, the burst size is chosen to be N ¼ 1700 (R0 ¼0.8854). As depicted in Figs. 6 and 7, the solution with the initial condition Tð0Þ ¼ 4080,T n ð0Þ ¼ 30,Vð0Þ ¼ 11 800,Að0Þ ¼ 0 tends to the infection-free equilibrium while the solution with the initial condition Tð0Þ ¼ 3580,T n ð0Þ ¼ 27,Vð0Þ ¼ 11870,Að0Þ ¼ 0 tends to the stable positive equilibrium inside the unstable limit cycle. Fourthly, if 1:0000 o R0 r1:8214, i.e., 1920o N r3497, that is, the burst size is larger than that in the third case, then another type of bistability occurs, where a stable positive equilibrium coexists with a stable limit cycle. Specifically, the infection-free equilibrium becomes unstable and it coexists with a stable positive equilibrium and two limit cycles, where the smaller limit X. Wang, W. Wang / Journal of Theoretical Biology 313 (2012) 127–135 cycle is unstable and the larger one is stable. The smaller limit cycle encloses the positive equilibrium and firstly contracts and then enlarges as R0 increases in the interval. The larger limit cycle encloses the smaller limit cycle and contracts as R0 increases in the interval. These two limit cycles collide when R0 reaches 1.8214. In this case, the ultimate infection state in vivo depends on the initial infection state. If the initial state lies in the basin of attraction for the stable positive equilibrium, then the infection state will tend to this positive equilibrium at last. If the initial state locates in the basin of attraction for the stable limit cycle, then stable periodic oscillations will occur as time tends to infinity. For this pattern, we choose N ¼3400 (R0 ¼1.7708). In Fig. 8, the solution with the initial condition Tð0Þ ¼ 1770, T n ð0Þ ¼ 120,Vð0Þ ¼ 40,392,Að0Þ ¼ 0 tends to the positive equilibrium. In Fig. 9, the solution with the initial condition Tð0Þ ¼ 6000,T n ð0Þ ¼ 0,Vð0Þ ¼ 106 ,Að0Þ ¼ 0 tends to the stable limit cycle. The phenomenon of bistability is depicted in Fig. 10. The blue and the green orbits, whose initial values lie in the basin of attraction for the stable limit cycle, both oscillate to the stable limit cycle. The magenta orbit, whose initial value locates in the basin of attraction for the stable positive equilibrium, tends to the stable positive equilibrium also in an oscillatory form. Fifthly, if R0 4 1:8214, i.e., N 43497, that is, the burst size is large enough, then the unstable infection-free equilibrium coexists 10 16 133 x 105 14 12 Virus 10 8 6 4 2 0 0 1000 2000 3000 4000 Uninfected T cells 5000 6000 Fig. 10. Graph of bistability (a stable positive equilibrium and a stable limit cycle) on the TV-plane. A blue orbit, with initial values Tð0Þ ¼ 6000,T n ð0Þ ¼ 0, Vð0Þ ¼ 106 ,Að0Þ ¼ 0, tends to the stable limit cycle from outside. A green orbit, with initial values Tð0Þ ¼ 3770,T n ð0Þ ¼ 120,V ð0Þ ¼ 40,392,Að0Þ ¼ 0, tends to the stable limit cycle from inside. A magenta orbit, with initial values Tð0Þ ¼ 2770, T n ð0Þ ¼ 120,Vð0Þ ¼ 40,392,Að0Þ ¼ 0, tends to the positive equilibrium inside the cycle, where N ¼ 3400,R0 ¼ 1:7708. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.) x 104 9 8 Antibody 7 Virus 6 5 4 10 5 0 150 2 1 2000 1900 100 Inf ec ted Tc 3 1800 ells 0 100 200 300 400 500 600 700 1700 50 ed T 1600 1500 0 800 ect ninf s cell U Time Fig. 8. A numerical solution of system (3) tends to the positive equilibrium as time tends to infinity, where N ¼ 3400, R0 ¼1.7708. The initial values are Tð0Þ ¼ 1770,T n ð0Þ ¼ 120,V ð0Þ ¼ 40,392,Að0Þ ¼ 0. (a) Time series of V. (b) An orbit in the TT n A space. 16 x 105 14 600 Antibody 12 10 8 400 200 6 0 2000 4 2 0 0 500 1000 Time 1500 1500 1000 Infe cted 500 T ce lls 0 0 4000 2000 6000 8000 ells dTc cte ninfe U Fig. 9. A numerical solution of system (3) tends to a stable limit cycle as time tends to infinity. N ¼ 3400, R0 ¼1.7708. The initial values are Tð0Þ ¼ 6000,T n ð0Þ ¼ 0,V ð0Þ ¼ 106 ,Að0Þ ¼ 0. (a) Time series of V. (b) An orbit in the TT n A space. 134 X. Wang, W. Wang / Journal of Theoretical Biology 313 (2012) 127–135 8 x 105 400 6 300 Antibody 7 Virus 5 4 3 2 1 0 50 100 150 200 250 300 350 400 450 500 550 600 200 100 0 2500 2000 1500 Infe 1000 cte 500 dT cell s 0 0 2000 4000 ected Uninf 8000 6000 T cell s Time Fig. 11. A numerical solution of system (3) tends to the positive equilibrium as time tends to infinity, where N ¼ 5000, R0 ¼ 2.6042. The initial values are Tð0Þ ¼ 6000,T n ð0Þ ¼ 0, Vð0Þ ¼ 106 , Að0Þ ¼ 0. (a) Time series of V. (b) An orbit in the TT n A space. with a stable positive equilibrium, but there is no limit cycle. All positive orbits tend to the positive equilibrium. For this pattern, the burst size is chosen as N¼5000 (R0 ¼2.6042). As depicted in Fig. 11, the solution tends to the positive equilibrium in this case. Acknowledgments We are very grateful to anonymous reviewers for their valuable comments and suggestions. 5. Concluding remarks In this paper, we have developed a mathematical model based on a vectored immunoprophylaxis experiment published in Nature. By combining theoretical analysis and numerical simulations we have found that the model can exhibit some new and complicated dynamical behaviors. We have shown that the infection-free equilibrium is globally asymptotically stable when the basic reproduction number is smaller than a threshold. In this case, the mouse can be completely protected from the infection. We have also proved that the introduction of the vectored immunoprophylaxis can induce the backward bifurcation and the ignorance of antibodies’ loss due to their involvement with virus may result in the loss of backward bifurcation. This means that driving the basic reproduction number below 1 is not enough to eradicate the infection. Another interesting result is that the model can exhibit five different patterns of dynamical behaviors if we choose the burst size N as a bifurcation parameter. When the burst size is relatively small, the infection-free equilibrium is globally asymptotically stable. When the burst size is a little larger, the stable infectionfree equilibrium coexists with two unstable positive equilibria. In these two cases, the infection could be thoroughly eradicated. But when the burst size is larger than a critical value, there is a stable infection-free equilibrium, an unstable positive equilibrium, a stable positive equilibrium and an unstable limit cycle. In this case, whether the infection can be eradicated depends on the initial state. When the burst size is larger than the second critical value, the unstable infection-free equilibrium coexists with a stable positive equilibrium, a small unstable limit cycle and a large stable limit cycle. Thus, the bistability occurs in this case. If the burst size is large enough, the model admits a unique stable positive equilibrium. In the last two cases, the infection persists in vivo. The mathematical analysis and numerical simulations of the model proposed in this paper provide insights into the underlying viral infection mechanisms under vectored immunoprophylaxis. It is interesting to consider the synthetical effects of immunoprophylaxis and cell immunity. We leave this as a future work. References Arino, J., McCluskey, C.C., van den Driessche, P., 2003. Global results for an epidemic model with vaccination that exhibits backward bifurcation. SIAM J. Appl. Math. 64, 260–276. Bailey, J., Blankson, J.N., Wind-Rotolo, M., Siliciano, R.F., 2004. Mechanisms of HIV1 escape from immune responses and antiretroviral drugs. Curr. Opin. Immunol. 16, 470–476. Balazs, A.B., Chen, J., Hong, C.M., Rao, D.S., Yang, L., Baltimore, D., 2012. Antibodybased protection against HIV infection by vectored immunoprophylaxis. Nature 481, 81–84. Blayneh, K.W., Gumel, A.B., Lenhart, S., Clayton, T., 2010. Backward bifurcation and optimal control in transmission dynamics of West Nile Virus. Bull. Math. Biol. 72, 1006–1028. Bonhoeffer, S., May, R.M., Shaw, G.M., Nowak, M.A., 1997. Virus dynamics and drug therapy. Proc. Natl. Acad. Sci. U.S.A. 94, 6971–6976. Buonomo, B., Lacitignola, D., 2011. On the backward bifurcation of a vaccination model with nonlinear incidence. Nonlinear Anal. Model. 16, 30–46. Chen, H.Y., Mascio, M.D., Perelson, A.S., Ho, D.D., Zhang, L., 2007. Determination of virus burst size in vivo using a single-cycle SIV in rhesus macaques. Proc. Natl. Acad. Sci. U.S.A. 104, 19079–19084. Chun, T.W., Carruth, L., Finzi, D., Shen, X., DiGiuseppe, J.A., Taylor, H., Hermankova, M., Chadwick, K., Margolick, J., Quinn, T.C., Kuo, Y.H., Brookmeyer, R., Zeiger, M.A., Barditch-Crovo, P., Siliciano, R.F., 1997a. Quantification of latent tissue reservoirs and total body viral load in HIV-1 infection. Nat. Rev. Microbiol. 387, 183–188. Chun, T.W., Engel, D., Mizell, S.B., Ehler, L.A., Fauci, A.S., 1998. Induction of HIV-1 replication in latently infected CD4 þ T cells using a combination of cytokines. J. Exp. Med 188, 83–91. Chun, T.W., Finzi, D., Margolick, J., Chadwick, K., Schwartz, D., Siliciano, R.F., 1995. In vivo fate of HIV-1-infected T cells: quantitative analysis of the transition to stable latency. Nat. Med. 1, 1284–1290. Chun, T.W., Stuyver, L., Mizell, S.B., Ehler, L.A., Mican, J.A., Baseler, M., Lloyd, A.L., Nowak, M.A., Fauci, A.S., 1997b. Presence of an inducible HIV-1 latent reservoir during highly active antiretroviral therapy. Proc. Natl. Acad. Sci. U.S.A. 94, 13193–13197. Clavel, F., Hance, A.J., 2004. HIV drug resistance. N. Engl. J. Med. 350, 1023–1035. Culshaw, R.V., Ruan, S.G., 2000. A delay-differential equation model of HIV infection of CD4þ T cells. Math. Biosci. 165, 27–39. Davenport, M.P., Loh, L., Petravic, J., Kent, S.J., 2008. Rates of HIV immune escape and reversion: implications for vaccination. Trends Microbiol. 16, 561–566. Dhooge, A., Govaerts, W., Kuznetsov, Y.A., 2003. Matcont: A Matlab package for numerical bifurcation analysis of odes. ACM Trans. Math. Software 29, 141–164. Dhooge, A., Govaerts, W., Kuznetsov, Y.A., Meijer, H.G.E., Sautois, B., 2008. New features of the software matcont for bifurcation analysis of dynamical systems. Math. Comput. Model. Dyn. Syst. 14, 145–175. X. Wang, W. Wang / Journal of Theoretical Biology 313 (2012) 127–135 Dixit, N.M., Perelson, A.S., 2004. Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokinetics and intracellular delay. J. Theor. Biol. 226, 95–109. van den Driessche, P., Watmough, J., 2000. A simple SIS epidemic model with a backward bifurcation. J. Math. Biol. 40, 525–540. Dushoff, J., Huang, W., Castillo-Chavez, C., 1998. Backward bifurcations and catastrophe in simple models of fatal diseases. J. Math. Biol. 36, 227–248. Ferguson, N.M., Donnelly, C.A., Hooper, J., Ghani, A.C., Fraser, C., Bartley, L.M., Rode, R.A., Vernazza, P., Lapins, D., Mayer, S.L., Anderson, R.M., 2005. Adherence to antiretroviral therapy and its impact on clinical outcome in HIV-infected patients. J. R. Soc. Interface 2, 349–363. Finzi, D., Hermankova, M., Pierson, T., Carruth, L.M., Buck, C., Chaisson, R.E., Quinn, T.C., Chadwick, K., Margolick, J., Brookmeyer, R., Gallant, J., Maarkowitz, M., Ho, D.D., Richman, D.D., Siliciano, R.F., 1997. Identification of a reservoir for HIV-1 in patients on highly active antiretroviral therapy. Science 278, 1295–1300. Gilchrist, M.A., Coombs, D., Perelson, A.S., 2004. Optimizing within-host viral fitness: infected cell lifespan and virion production rate. J. Theor. Biol. 229, 281–288. Haase, A.T., Henry, K., Zupancic, M., Sedgewick, G., Faust, R.A., Melroe, H., Cavert, W., Gebhard, K., Staskus, K., Zhang, Z.Q., Dailey, P.J., Balfour, H.H.J., Erice, A., Perelson, A.S., 1996. Quantitative image analysis of HIV-1 infection in lymphoid tissue. Science 274, 985–989. Hadeler, K.P., van den Driessche, P., 1997. Backward bifurcation in epidemic control. Math. Biosci. 146, 15–35. Herz, V., Bonhoeffer, S., Anderson, R., May, R., Nowak, M., 1996. Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay. Proc. Natl. Acad. Sci. U.S.A. 93, 7247–7251. Hlavacek, W.S., Wofsy, C., Perelson, A.S., 1999. Dissociation of HIV-1 from follicular dendritic cells during HAART: mathematical analysis. Proc. Natl. Acad. Sci. U.S.A. 96, 14681–14686. Huang, Y., Rosenkranz, S.L., Wu, H., 2003. Modeling HIV dynamics and antiviral response with consideration of time-varying drug exposures, adherence and phenotypic sensitivity. Math. Biosci. 184, 165–186. Inaba, H., Sekine, H., 2004. A mathematical model for chagas disease with infection-age-dependent infectivity. Math. Biosci. 190, 39–69. Iwami, S., Miura, T., Nakaoka, S., Takeuchi, Y., 2009. Immune impairment in HIV infection: existence of risky and immunodeficiency thresholds. J. Theor. Biol 260, 490–501. Iwami, S., Nakaoka, S., Takeuchi, Y., 2006. Frequency dependence and viral diversity imply chaos in an HIV model. Phys D: Nonlinear Phenom. 223, 222–228. Iwasa, Y., Franziska, M., Nowak, M.A., 2005. Virus evolution with patients increases pathogenicity. J. Theor. Biol. 232, 17–26. Kajiwara, T., Sasaki, T., 2004. A note on the stability analysis of pathogen-immune interaction dynamics. Discrete Cont. Dyn. B 4, 615–622. Kirschner, D.E., Webb, G.F., 1996. A model for treatment strategy in the chemotherapy of AIDS. Bull. Math. Biol. 58, 367–390. Lambotte, O., Chaix, M.L., Gubler, B., Nasreddine, N., Wallon, C., Goujard, C., Rouzioux, C., Taoufik, Y., Delfraissy, J.F., 2004. The lymphocyte HIV reservoir in patients on long-term HAART is a memory of virus evolution. AIDS 18, 1147–1158. Li, M.Y., Shu, H., 2010. Global dynamics of an in-host viral model with intracellular delay. Bull. Math. Biol. 72, 1492–1505. Liu, W.M., Levin, S.A., Iwasa, Y., 1986. Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23, 187–204. Martcheva, M., Thieme, H.R., 2003. Progression age enhanced backward bifurcation in an epidemic model with superinfection. J. Math. Biol. 46, 385–424. Mittler, J.E., Markowitz, M., Ho, D.D., Perelson, A.S., 1999. Refined estimates for HIV-1 clearance rate and intracellular delay. AIDS 13, 1415–1417. Murase, A., Sasaki, T., Kajiwara, T., 2005. Stability analysis of pathogen–immune interaction dynamics. J. Math. Biol. 51, 247–267. Nelson, P.W., Gilchrist, M.A., Coombs, D., Hyman, J.M., Perelson, A.S., 2004. An agestructured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells. Math. Biosci. Eng. 1, 267–288. 135 Nelson, P.W., Murray, J.D., Perelson, A.S., 2000. A model of HIV-1 pathogenesis that includes an intracellular delay. Math. Biosci. 163, 201–215. Nelson, P.W., Perelson, A.S., 2002. Mathematical analysis of delay differential equation models of HIV-1 infection. Math. Biosci. 179, 73–94. Nowak, M.A., May, R.M., 2000. Virus Dynamics: Mathematical Principles of Immunology and Virology. Oxford University Press, Oxford. Pang, H., Wang, W., Wang, K., 2005. Global properties of virus dynamics model with immune response. J. Southwest China Norm. Univ. (Nat. Sci.) 30, 796–799. Pawelek, K.A., Liu, S., Pahlevani, F., Rong, L., 2012. A model of HIV-1 infection with two time delays: mathematical analysis and comparison with patient data. Math. Biosci. 235, 98–109. Perelson, A.S., Kirschner, D.E., Boer, R.D., 1993. Dynamics of HIV infection of CD4 þ T cells. Math. Biosci. 114, 81–125. Perelson, A.S., Nelson, P.W., 1999. Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev. 41, 3–44. Perelson, A.S., Neumann, A.U., Markowitz, M., Leonard, J.M., Ho, D.D., 1996. HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science 271, 1582–1586. Phillips, A.N., Youle, M., Johnson, M., Loveday, C., 2001. Use of a stochastic model to develop understanding of the impact of different patterns of antiretroviral drug use on resistance development. AIDS 15, 2211–2220. Qesmi, R., Wu, J., Wu, J., Heffernan, J.M., 2010. Influence of backward bifurcation in a model of hepatitis B and C viruses. Math. Biosci. 224, 118–125. Rong, L., Feng, Z., Perelson, A.S., 2007a. Emergence of HIV-1 drug resistance during antiretroviral treatment. Bull. Math. Biol. 69, 2027–2060. Rong, L., Gilchrist, M.A., Feng, Z., Perelson, A.S., 2007b. Modeling within-host HIV-1 dynamics and the evolution of drug resistance: trade-offs between viral enzyme function and drug susceptibility. J. Theor. Biol. 247, 804–818. Shu, H., Wang, L., 2012. Role of CD4 þ T-cell proliferation in HIV infection under antiretroviral therapy. J. Math. Anal. Appl. 394, 529–544. Simon, V., Ho, D.D., 2003. HIV-1 dynamics in vivo: implications for therapy. Nat. Rev. Microbiol. 1, 181–190. Smith, R.J., 2006. Adherence to antiretroviral HIV drugs: how many doses can you miss before resistance emerges?. Proc. R. Soc. B 273, 617–624. Stafford, M.A., Corey, L., Cao, Y., Daar, E.S., Ho, D.D., Perelson, A.S., 2000. Modeling plasma virus concentration during primary HIV infection. J. Theor. Biol. 203, 285–301. Wahl, L.M., Nowak, M.A., 2000. Adherence and drug resistance: predictions for therapy outcome. Proc. R. Soc. Lond. B 267, 835–843. Wang, K., Wang, W., Liu, X., 2006a. Global stability in a viral infection model with lytic and nonlytic immune responses. Comput. Math. Appl. 51, 1593–1610. Wang, K., Wang, W., Liu, X., 2006b. Viral infection model with periodic lytic immune response. Chaos Solutions Fractals 28, 90–99. Wang, W., 2006. Backward bifurcation of an epidemic model with treatment. Math. Biosci. 201, 58–71. Wodarz, D., 2003. Hepatitis C virus dynamics and pathology: the role of CTL and antibody responses. J. Gen. Virol. 84, 1743–1750. Wong, J.K., Hezareh, M., Gunthard, H.F., Havlir, D.V., Ignacio, C.C., Spina, C.A., Richman, D.D., 1997. Recovery of replication-competent HIV despite prolonged suppression of plasma viremia. Science 278, 1291–1295. Wu, H., Huang, Y., Acosta, E.P., Park, J.G., Yu, S., Rosenkranz, S.L., Kuritzkes, D.R., Eron, J.J., Perelson, A.S., Gerber, J.G., 2006. Pharmacodynamics of antiretroviral agents in HIV-1 infected patients: using viral dynamic models that incorporate drug susceptibility and adherence. J. Pharmacokinet. Pharmacodyn. 33, 399–419. Wu, H., Huang, Y., Acosta, E.P., Rosenkranz, S.L., Kuritzkes, D.R., Eron, J.J., Perelson, A.S., Gerber, J.G., 2005. Modeling long-term HIV dynamics and antiretroviral response: effects of drug potency, pharmacokinetics, adherence, and drug resistance. J. Acquir. Immune Defic. Syndr. 39, 272–283. Yan, Y., Wang, W., 2012. Global stability of a five-dimensional model with immune responses and delay. Discrete Cont. Dyn. B 17, 401–416. Zack, J.A., Arrigo, S.J., Weitsman, S.R., Go, A.S., Haislip, A., Chen, I.S., 1990. HIV-1 entry into quiescent primary lymphocytes: molecular analysis reveals a labile latent viral structure. Cell 61, 213–222.