Dynamics of a delay differential equation model of hepatitis B virus infection

Transcription

1 Journal of Biological Dynamics Vol. 2, No. 2, April 2008, Dynamics of a delay differential equation model of hepatitis B virus infection Stephen A. Gourley a, Yang Kuang b * and John D. Nagy c a Department of Mathematics, University of Surrey, Guildford, Surrey, UK; b Department of Mathematics, Arizona State University, Tempe, AZ, USA; c Department of Biology, Scottsdale Community College, Scottsdale, AZ, USA (Received 19 July 2007; final version received 03 October 2007 ) We formulate and systematically study the global dynamics of a simple model of hepatitis B virus in terms of delay differential equations. This model has two important and novel features compared to the wellknown basic virus model in the literature. Specifically, it makes use of the more realistic standard incidence function and explicitly incorporates a time delay in virus production. As a result, the infection reproduction number is no longer dependent on the patient liver size (number of initial healthy liver cells). For this model, the existence and the component values of the endemic steady state are explicitly dependent on the time delay. In certain biologically interesting limiting scenarios, a globally attractive endemic equilibrium can exist regardless of the time delay length. Keywords: basic infection reproduction number; mass action; standard incidence; time delay; global stability AMS Subject Classification: 34K20; 92C50; 92D25 1. Introduction Hepatitis B, caused by the hepatitis B virus (HBV), represents an enormous challenge to global public health. Some 2 billion people have been infected with the virus [31], with 5 million new cases each year [14]. The disease s clinical course varies, in adults usually resulting in self-limiting acute hepatitis or, rarely, fatal fulminant disease [4,8,27]. However, chronic hepatitis develops in most individuals infected as children and about 5% of adults. This chronic form often leads to early death from cirrhosis-induced liver failure or hepatocellular carcinoma (HCC), a particularly deadly malignancy [5,8,14]. Some 500,000 to 1.2 million people die each year from complications associated with HBV, with HCC accounting for about a third of these deaths [14]. Currently, some 350 million people worldwide live with chronic HBV infection [31]. It is currently widely accepted that HBV infection is non-cytopathic. Infected hepatocytes are killed not by the virus but by HBV-specific cytotoxic T lymphocytes (CTLs) [8,20]. This mortality *Corresponding author. Kuang@asu.edu ISSN print/issn online 2008 Taylor & Francis DOI: /

2 Journal of Biological Dynamics 141 is somehow amplified by inflammation responses within the liver, but CTLs appear to be the major mediator of hepatitis B pathogenesis [12]. How HBV enters hepatocytes is still unclear, but the molecular pathogenesis of HBV infection once it occurs is well understood (see reviews [8,20,26,27]). Once inside the cell, the HBV DNA is translocated to the nucleus, where it is stabilized as a circular, episomal chromosome roughly similar to a bacterial plasmid. Following transcription of this episomal body, viral mrnas then return to the cell cytoplasm to be translated into viral structural and non-structural proteins. From these elements, new viral capsids, complete with an RNA genome and polymerase, are constructed. The RNA is then reverse transcribed into the proper viral DNA genome, completing intracellular viral assembly. Once in this stage, the assembly has one of two fates. Either it travels, via the endoplasmic reticulum and Gogli apparatus, to the plasma membrane and subsequently out of the cell as a new, infectious virion, or it delivers its DNA genome back to the nucleus to become yet another episomal viral chromosome. Precise dynamical parameters of HBV infection are difficult to measure since HBV infects only humans and chimps, very few adequate animal models exist, and there are no adequate in vitro models [12]. In addition, as described earlier, the clinical course of disease varies. However, in the acute course, it appears that the HBV population in the liver peaks about 8 weeks after infection, with clinical signs of hepatitis reaching their peak 8 13 weeks later, at least in chimpanzees [20]. During acute infection, newly synthesized virions destined to exit the cell transit the cytoplasm with a half-life of 17 hours to 1.5 days [21]. In the plasma, virion half-life is probably about 4 hours, the often-cited 1 day being most likely an over-estimate [21]. Because HBV is non-cytopathic, clearance requires a massive adaptive immune response [12,27]. Indeed, chronic infection is largely thought to result most often from a suboptimal immune response [27]. Although modern HBV vaccines are largely safe and effective [31], current curative chemotherapies have limited effectiveness against chronic infection. The most commonly used drugs include lamivudine, adefovir dipivoxil and interferon alpha [16,28]. The first two are nucleos(t)ide analogues that inhibit viral reverse transcriptase, while interferon is an immune system modulator. Roughly speaking, all currently available treatments, when used alone, are about equally efficacious, greatly reducing viral burden in the short term [13,28]. However, none are capable of routinely clearing chronic infection [16], and the virus typically evolves resistance to the nucleos(t)ide analogues [17,28]. Until drug resistance is overcome or more effective and inexpensive therapies are introduced, hepatitis B will remain a major threat to health around the world. In an effort to model HBV infection dynamics and its treatment with the reverse transcriptase inhibitor lamivudine, Nowak et al. [22] employed the following phenomenological model (see also [23]): ẋ = λ dx βvx, ẏ = βvx ay, v = ky v, (1) where x, y and v are numbers of uninfected (susceptible) liver cells infected liver cells and free virions, respectively. Uninfected liver cells are assumed to be produced at a constant rate, λ, to maintain tissue homeostasis in the face of hepatocyte turnover, described by the linear term dx, where d is the per-capita death rate. A healthy liver maintains λ/d cells as its homeostatic setpoint. However, during infection, healthy (uninfected) liver cells are assumed to become infected at rate βvx, where β is the mass action rate constant describing the infection process. Infected liver cells are killed by immune cells at rate ay and produce free virions at rate ky, where k is the so-called burst constant. Free virions are cleared by lymphatic and other mechanisms at rate v, where is a constant. In addition to its use for describing the dynamics of HBV, this

3 142 S.A. Gourley et al. model and various extensions of it have been employed to describe within host HIV dynamics [24,25]. Indeed, its generality makes it equally applicable to essentially any virus parasitizing any vertebrate species. The basic infection reproductive number of Equation (1) is R 0 = λβk ad. (2) It is straightforward to show that if R0 < 1, then the disease-free equilibrium is globally stable, and if R0 > 1 an endemic equilibrium exists [15]. Furthermore, R 0 in Equation (2) is proportional to a patient s liver size λ/d, which implies that individuals with larger livers (e.g., adults) are more likely to be infected than individuals with smaller organs (e.g., children). This implication is highly questionable on biological grounds [19]. Indeed, as we show subsequently, this implication is simply an artefact of the use of the mass action incidence function, which is often inappropriate for infection dynamics involving a large number of hosts in this context, cells and when the total number of cells varies. This demonstration is one of the two main objectives of this paper. Our second objective is to study how delay in virus production, described previously, affects HBV infection dynamics. In particular, we address whether or not such a delay can generate the complex dynamics characteristic of other viral infections [1,2] and more general population models [7,11]. 2. The model A typical chronically infected HBV patient has a total serum load of about to virions [21,22]. The average human liver has about an equal number of cells (assuming a liver mass of about 1.5 kg). These large numbers suggest that a more plausible HBV model should employ a standard incidence function, instead of the mass action incidence used in Model (1). Also, the HBV incubation period, which varies from 45 to 180 days, and the delay in viral shedding mentioned previously both suggest that viral production delay may significantly impact infection dynamics and, hence, should be explicitly modeled. Therefore, we propose the following model to accommodate these two biologically motivated modifications: x (t) = λ dx(t) e (t) = de(t) + βv(t)x(t) x(t) + y(t) + e(t), βv(t)x(t) x(t) + y(t) + e(t) βe dτ v(t τ)x(t τ) x(t τ)+ y(t τ)+ e(t τ), y βe dτ v(t τ)x(t τ) (t) = x(t τ)+ y(t τ)+ e(t τ) ay(t), v (t) = ky(t) v(t). (3) In this system, x(t) represents the number of uninfected cells, y(t) represents the number of infected cells, e(t) represents the number of exposed cells (i.e., cells that have acquired the virus but are not yet producing new virions) and v(t) represents the number of free virions. We assume that exposed and uninfected cells have the same death rate, implying the biological assumption that cells currently not shedding viruses do not display viral antigens to CTLs. Exposed cells begin shedding virions after τ units of time, representing the time required to construct, transcribe and translate the episomal viral genome, construct and then release mature virions. Other parameters are the same as in the basic virus model (1). Precise parameter values of the model are very difficult

4 Journal of Biological Dynamics 143 Table 1. Parameter description, value and origin for Model (3). Parameter Description Value References λ Liver cell production rate /day/ml [3,19] d Liver cell death rate /day/ml [19] β Maximum infection rate /day/ml [19] τ Time delay for virion production 1 2 days [21] a Death rate of infected liver cells /day [3,19] k Virion production rate /day [3,19] Virion death rate 0.67/day [3,19] to obtain since HBV infects only human and chimps and there are no good in vitro models. Some typical parameter values from the literature are presented in Table 1. The term in Model (3) representing the transfer rate between exposed and infected classes can be rigorously derived using an age-structured modeling approach, as follows. Let Y(t,α)denotes the density of cells at time t that were infected α time units before t (i.e., cells of disease age α). Then, we should write y (t) = Y(t,τ) ay(t), because Y(t,τ) is the rate at which cells move from the exposed to the infected class, since it takes τ time units for an infection to mature in a given cell. We need to find Y(t,τ). Since the per-capita death rate for the exposed class is constant (d), it is appropriate to assume that Y(t,α) satisfies the McKendrick von Foerster age-structured model, Y (t, α) t subject to the condition + Y (t, α) α = dy(t,α), 0 <α<τ, (4) βv(t)x(t) Y(t,0) = x(t) + y(t) + e(t), (5) because Y(t,0) is the rate at which new infections arise. Define Y ξ (α) = Y(α+ ξ,α). Using Equation (4), we see that Y ξ (α) satisfies dy ξ (α)/dα = dy ξ (α). Thus, Setting α = τ and ξ = t τ yields Y ξ (α) = Y ξ (0)e dα = Y(ξ,0)e dα = βe dα v(ξ)x(ξ) x(ξ) + y(ξ) + e(ξ). βe dτ v(t τ)x(t τ) Y(t,τ) = x(t τ)+ y(t τ)+ e(t τ), completing the derivation. Strictly speaking, the above derivation holds only for t>τ, but we shall assume that the model Equations (3) hold for all t>0with standard initial data on the interval [ τ,0]. 3. Basic properties of the full model Note that the differential equation for e(t) has the implicit solution t e d(t s) v(s)x(s) e(t) = β ds, (6) t τ x(s) + y(s) + e(s) which is the accumulated total of exposed cells.

5 144 S.A. Gourley et al. The initial data for System (3) has the form x(s) = x 0 (s) 0, y(s) = y 0 (s) 0, v(s) = v 0 (s) 0, s [ τ,0], 0 e ds v 0 (s)x 0 (s) e 0 (0) = β x 0 (s) + y 0 (s) + e 0 (s) ds, (7) τ where x 0 (s), y 0 (s) and v 0 (s) are prescribed initial functions. The initial data for the variable e has to be related to that for the other variables such as to satisfy Equation (7), which is effectively an integral constraint on the function e 0 :[ τ,0] R. If this constraint is not satisfied, positivity of solutions may not be preserved, though the effect of an incorrect e 0 on the dynamics would only be transient. Stage-structured population models commonly have such a constraint on their initial data. PROPOSITION 3.1 Each component of the solution of System (3), subject to Equation (7), remains non-negative for all t>0. Proof If x were to lose its non-negativity, there would have to be a time at which x = 0 with x 0. However, this is clearly impossible given the equation for x in System (3). In fact, x(t) > 0 for all t>0. Non-negativity of y, v and e will be shown next, using the method of steps. On the interval t (0,τ], y ay, so that y(t) 0 on this interval by a standard comparison argument. This in turn implies that v v on (0,τ], so that v(t) 0 on this interval. Now suppose that e(t) loses its non-negativity at some time t (0,τ]. Then, e(t ) = 0 and, since x(t) is strictly positive, Expression (6) yields that v(t) 0on[t τ,t ]. The v in Equation (3) then implies that y(t) 0on[t τ,t ]. Now, the only way that y and v can be identically zero on an interval of length τ, is if they remain identically zero for all subsequent time, as the y and v equations of System (3) show. Thus, for non-trivial solutions, e(t) > 0on(0,τ]. These arguments can now be repeated to deduce non-negativity of y, v and e on the interval t (τ, 2τ] and then on successive intervals t (nτ, (n + 1)τ], n = 2, 3,...to include all positive times. Next, we present a result on the global stability of the virus-free equilibrium (x,e,y,v)= (λ/d, 0, 0, 0) of Equation (3). A linearization at the virus-free equilibrium and a study of its stability properties furnishes us with an expression for the basic infection reproduction number R 0 of Equation (3), which turns out to be given by R 0 = βke dτ a. (8) This linearized analysis is effectively part of the proof of our next theorem, in which we use a comparison argument to show that the y and v components of solutions of Equation (3) are bounded by solutions of the linearization of the y and v equations. Observe that R 0 is no longer proportional to the disease-free equilibrium value of x(= λ/d). THEOREM 3.1 If R 0 < 1, where R 0 is given by Equation (8), then the virus-free equilibrium (x, e, y, v) = (λ/d, 0, 0, 0) of Equation (3) is globally asymptotically stable for initial data satisfying Equation (7). Proof The proof proceeds via a comparison argument. It is sufficient to show that (y(t), v(t)) (0, 0), for then it is clear from the x and e equations in Equation (3) that e(t) 0 and x(t) λ/d.

6 Journal of Biological Dynamics 145 From positivity of solutions, it is clear that y and v satisfy the coupled differential inequality y (t) βe dτ v(t τ) ay(t), v (t) = ky(t) v(t). (9) This differential inequality system has certain significant properties: the right-hand side of the first is increasing as a function of the delayed variable v(t τ), and the right-hand side of the second increases with respect to y(t). These properties give the system a quasi-monotone structure and Theorem on page 78 of [29] is applicable, assuring us that (y(t), v(t)) is bounded by the solution of the corresponding system of differential equations obtained by replacing by = in Equation (9) and subject to the initial data for y and v in Equation (7). This solution we shall also denote by (y(t), v(t)), and of course it suffices to show that it tends to zero as t. Being linear, the corresponding system of differential equations associated with Equation (9) can be analyzed via its characteristic equation. Corresponding to trial solutions of the form exp(σ t), we obtain the characteristic equation (σ + a)(σ + ) = βke dτ e στ, and Theorem on page 92 of [29] assures us that it is sufficient to consider only the real roots of the characteristic equation (any complex roots would have smaller real part than the largest real root). A simple graphical argument then demonstrates that if R 0 < 1 then its real roots are all negative, so that (y(t), v(t)) (0, 0) as t. This completes the proof. 4. Dynamics of a simplified model In this section, we consider a simplification of the full model (3). This simplification follows from two observations. First, variable v (virus particles) evolves on much faster time scale than the liver cells do, so a quasi-steady state assumption can be applied to v; i.e., to a good approximation, v = ky/. Second, the time delay associated with virus production is on the order of a day or two [21], much shorter than the life expectancy of a typical hepatocyte, which is 6 12 months or more [27]. This makes e much smaller than x and y. Hence, e can be omitted from the denominators of the infection term. Under these assumptions, System (3) can be replaced by the simpler system x (t) = λ dx(t) e (t) = de(t) + βky(t)x(t) (x(t) + y(t)), y (t) = βke dτ y(t τ)x(t τ) (x(t τ)+ y(t τ)) βky(t)x(t) (x(t) + y(t)) βke dτ y(t τ)x(t τ) (x(t τ)+ y(t τ)), ay(t). (10) Note that in this simpler model the x and y equations do not involve e and form a closed subsystem of two equations. Mechanistically, Model (10) is similar to the cell-to-cell model of HIV virus infection studied in [6]. The basic infection reproduction number R 0 for this simplified model is the same as that of Equation (3). It can be shown, using a similar technique to that used in the proof of Theorem 3.2, that solutions of this simpler system again approach the virus-free state if R 0 < 1, with R 0 given by Equation (8). Our next result concerns persistence of the simplified system (10). We say System (10) is persistent for solutions with non-negative initial conditions if there is a positive constant M

7 146 S.A. Gourley et al. such that lim sup max{x(t), e(t), y(t)} <M, t and if lim inf min{x(t), e(t), y(t)} > 0. t We shall show below that System (10) is persistent, in the above sense, if R 0 > 1. This result will be needed when we prove global stability of the endemic equilibrium for R 0 > 1. While the result can be established by applying an existing generic result [30], we prefer to provide below a direct and simpler proof. LEMMA 4.1 For non-negative initial conditions in which y 0 (s) 0 on [ τ,0], System (10) is persistent if R 0 > 1, with R 0 given by Equation (8). Proof For System (10), non-negativity of solution components can be established with an argument similar to the proof of Proposition 3.1. Moreover, refinements to the techniques used there show that, if y 0 (s) 0 then y(t) > 0 for some time in [0,τ] and that y(t) > 0 for all subsequent times. By translating time if necessary, we may therefore assume that all components of the solution of Equation (10) satisfy initial data that is strictly positive for all s [ τ,0]. The components x(t), e(t) and y(t) each remain strictly positive for all subsequent times. Note that d(x + e + y) = λ dx de ay λ (x + e + y)max(a, d), dt and that d(x + e + y) λ (x + e + y)min(a, d). dt Thus, λ lim inf(x(t) + e(t) + y(t)) t max(a, d) and lim sup(x(t) + e(t) + y(t)) t In particular, it follows that both x(t) and y(t) are bounded from above for all t. Next, we note that from the x equation in Equation (10) and positivity of solutions, λ min(a, d). lim sup x(t) λ t δ. We now claim that, for solutions of Equation (10), lim inf t x(t) > 0. Suppose this is false, then there exists a sequence of times t j such that x(t j ) 0 and x (t j ) 0asj. Noting that y(t)/(x(t) + y(t)) remains bounded by 1, if we evaluate the x equation in Equation (10) at t j and let j we obtain a contradiction. In summary, we have shown that x(t) remains bounded from above, and bounded away from zero. Choose δ>0sufficiently small so that δ< min y 0(s), s [ τ,0] which is possible since, as explained earlier, we can assume without loss of generality that min s [ τ,0] y 0 (s) > 0. By shrinking δ if necessary, assume that δ satisfies the additional smallness constraint that, for all t, ( ) x(t τ) R 0 1 > 0, (11) x(t τ)+ δ which is possible because x(t)is uniformly bounded from above and away from zero, and because R 0 > 1.We now claim that y(t), which starts above δ, can never get below δ throughout the model s

8 Journal of Biological Dynamics 147 evolution. Indeed, if this were not the case there would exist t > 0 such that y(t ) = δ, y (t ) 0 and y(t)>δfor all t<t. Evaluating the y equation in Equation (10) at time t, and using that yx/(x + y) increases with respect to y, we obtain y (t ) = βke dτ y(t τ)x(t τ) (x(t τ)+ y(t τ)) aδ βke dτ δx(t τ) (x(t τ)+ δ) aδ ( ( x(t ) ) τ) = aδ R 0 1 x(t τ)+ δ > 0 by Equation (11). This contradicts y (t ) 0. We thus have shown that y(t) is bounded away from zero. Since t e d(t s) ky(s)x(s) e(t) = β t τ (x(s) + y(s)) ds, we see that e(t) is also bounded away from zero. This completes the proof of persistence for System (10). We next establish the local asymptotic stability of the unique endemic equilibrium (x, e,y ) of Model (10) for R 0 > 1. THEOREM 4.2 If R 0 > 1, with R 0 given by Equation (8), then System (10) possesses a unique equilibrium (x, e,y ), with x, e,y > 0. Moreover, this equilibrium is locally asymptotically stable. Proof From the equations that determine the equilibria, it is straightforward to prove the existence of the unique endemic equilibrium for R 0 > 1. For the stability, it is of course sufficient to consider only the first and third equations in Equation (10). After a fair amount of algebra, and using the equations satisfied by the equilibrium components and Expression (8), we can show that the linearization of these equations about (x,y ) has non-trivial solutions with exp(σ t) dependence if and only if σ satisfies ( σ 2 + a + d + βk ) ) 2 (1 1R0 σ + ad + aβk ( 1 1 R 0 ) 2 = a R 0 (σ + d)e στ. (12) We need to show that all roots of Equation (12) satisfy Reσ <0. This is easily seen to be the case when τ = 0 with R 0 > 1, since Equation (12) can be rearranged into a quadratic equation all of whose coefficients have the same sign, and so the question is whether a pair of complex conjugate roots could cross the imaginary axis as τ is varied. We will show that this cannot happen. Indeed, suppose that a pair of purely imaginary roots σ =±iω, with ω real and positive, has been found for Equation (12). Inserting σ = iω into Equation (12), taking the complex conjugate and

9 148 S.A. Gourley et al. eliminating exp( iωτ) in the usual way, we find that ω must satisfy ( ω 4 + a + d + βk ( 1 1 ) ) 2 2 ( 2 ad + aβk ) ) 2 (1 1R0 a2 ω 2 R 0 ( + ad + aβk ) ) 2 2 (1 1R0 a2 d 2 = 0. (13) R 2 0 Since R 0 > 1, it is easy to see that the term independent of ω and the coefficient of ω 2 are both positive. Therefore, it is not possible to find a real value of ω from Equation (13). Next, we present a theorem giving sufficient conditions for global stability of the endemic steady state of System (10). THEOREM 4.3 Suppose that R 0 > 1, with R 0 given by Equation (8), and that the equations λ = du + βk ( ) (R0 1)uv, λ = dv + βk ( ) (R0 1)uv (14) (R 0 1)v + u (R 0 1)u + v have no solution with u, v > 0 other than (u, v) = (x,x ). Then, the unique equilibrium (x, e,y ), x, e,y > 0, of System (10) is globally asymptotically stable for all non-negative initial data such that y 0 (s) 0, s [ τ,0]. Proof As noted in the proof of Lemma 4.1, the assumption that y 0 (s) 0, together with R 0 > 1, assures that y(t) is bounded away from zero after perhaps a short initial transient (certainly for all t τ), and therefore y(t) cannot approach zero. We noted earlier that lim sup t x(t) λ/δ. This provides an asymptotic upper bound x 1 := λ/δ which shall form the starting point for a sequence of asymptotic upper and lower bounds in which the transition from the nth to the (n + 1)th stage is as follows. Suppose we have an upper bound x n for lim sup t x(t), i.e., lim sup x(t) x n. t Let ɛ>0be arbitrary. There exists T>0such that, for all t T, x(t) x n + ɛ. Using this estimate in the y equation of Equation (10), and the fact that the expression xy/(x + y) increases with respect to both x and y, we find that for t T + τ, y (t) βke dτ y(t τ)(x n + ɛ) ay(t). (15) (x n + ɛ + y(t τ)) The right-hand side of Equation (15) is increasing with respect to y(t τ) and, therefore, by Theorem on page 78 of [29], y(t)is bounded above by the solution ỹ(t)of the corresponding differential equation, which in terms of R 0 is ( ) ỹ R0 ỹ(t τ)(x n + ɛ) (t) = a x n + ɛ +ỹ(t τ) ỹ(t), (16) such that ỹ(s) = y 0 (s), s [ τ,0]. The dynamics of solutions of Equation (16) are well known. Theorem on page 159 of [11], or Proposition on page 90 of [29], assures us that ỹ(t) R 2 0

10 Journal of Biological Dynamics 149 approaches one of the equilibria of Equation (16). As noted earlier, y(t) cannot approach zero, and therefore neither can ỹ(t). Therefore, ỹ(t) approaches a non-zero equilibrium y ɛ n satisfying R 0 (x n + ɛ) x n + ɛ + y ɛ = 1. n Since R 0 > 1, existence of y ɛ n > 0 is guaranteed. Taking limits as t in the inequality y(t) ỹ(t) then gives us y ɛ n as an upper bound for lim sup t y(t), and we shrink ɛ to zero to obtain lim sup y(t) y n, (17) t where y n = (R 0 1)x n. (18) Another ɛ argument involving the use of Equation (17) in the x equation of System (10) yields a differential inequality for x, from which we can conclude that where x n satisfies the equation lim inf t x(t) x n, (19) λ = dx n + βky nx n (y n + x n ). (20) Yet another comparison argument involving the use of Equation (19) in the y equation of Equation (10) yields that lim inf y(t) y, (21) t n where y n = (R 0 1)x n. (22) Finally, the use of Equation (21) in the x equation of Equation (10) gives us where x n+1 satisfies lim sup x(t) x n+1, t λ = dx n+1 + βky x n n+1 (y n + x n+1 ). (23) Using Equations (18) and (22), we see that the nth asymptotic upper bound x n for x(t) generates the nth lower bound x n via the equation λ = dx n + βk ( (R0 1)x n x n (R 0 1)x n + x n and x n then generates the next upper bound x n+1 via the equation λ = dx n+1 + βk ( (R0 1)x n x n+1 (R 0 1)x n + x n+1 Our claim is that, for each positive integer n, ), (24) ). (25) x n x n+1 x x n+1 x n, (26) which we shall now prove by induction. Note that if Equation (26) holds, then Equations (18) and (22) assure us that a similar property holds for the variable y. Assume that Equation (26) holds

11 150 S.A. Gourley et al. for a particular integer n. We shall prove that it also holds for the next integer as well, and this involves showing that x x n+2 x n+1 and x n+1 x n+2 x. The above inequalities must be proved in the order shown. We prove just the first of them: x x n+2 x n+1 ; this inequality is then used in the proof of x n+1 x n+2 x, which is similar. Now, x n+2 satisfies the equation g(x) = 0, where g(x) = λ dx βk ( (R0 1)x n+1 x (R 0 1)x n+1 + x This function g(x) is monotone decreasing. Therefore, to show that x x n+2 x n+1 it is enough to show that g(x ) 0 and g(x n+1 ) 0. But g(x n+1 ) = λ dx n+1 βk λ dx n+1 βk = 0 by Equation (25), ( (R0 1)x n+1 x n+1 (R 0 1)x n+1 + x n+1 ( ) (R0 1)x n x n+1 (R 0 1)x n + x n+1 ) ). since x n x n+1 and g(x ) = λ dx βk λ dx βk ( (R0 1)x n+1 x (R 0 1)x n+1 + x ( ) (R 0 1)x 2 (R 0 1)x + x ) since x x n+1 = 0 since x is an equilibrium component. From Equation (26), it follows that there exist the limits x = lim n x n, x = lim n x n, and x x, x x. To show that lim t x(t) = x, it suffices to show that x = x = x. The result for y(t) follows similarly, and that for e(t) follows from the asymptotically autonomous limiting form of the e equation in Equation (10) using arguments that are straightforward to justify (e.g., [18]). Letting n in Equations (24) and (25) gives and λ = dx + βk λ = dx + βk ( ) (R0 1)xx (R 0 1)x + x (27) ( ) (R0 1)xx. (28) (R 0 1)x + x By hypothesis, it follows from Equations (27) and (28) that (x, x) = (x,x ). In the next result, we provide some explicit conditions that are sufficient for global convergence to the endemic state.

12 Journal of Biological Dynamics 151 COROLLARY 4.1 Suppose that R 0 > 1, with R 0 given by Equation (8), and that either R 0 2 or d βk(r 0 1)(2 R 0 ) R0 2. (29) Then, the unique equilibrium (x, e,y ), x, e,y > 0, of Equation (10) is globally asymptotically stable for all non-negative initial data such that y 0 (s) 0, s [ τ,0]. Proof We will show that either of the Conditions (29) ensures that Equation (14) have no solutions with u, v > 0 other than (u, v) = (x,x ), so that Theorem 4.3 applies. Let F (u, v) = du + βk ( ) (R0 1)uv, G(u, v) = dv + βk ( ) (R0 1)uv. (30) (R 0 1)v + u (R 0 1)u + v If the equations F (u, v) = λ, G(u, v) = λ have a solution in which u = v then, by subtraction, βk(r 0 1)(2 R 0 )uv d = ((R 0 1)u + v)((r 0 1)v + u). This is clearly impossible if R 0 2. So suppose that 1 <R 0 < 2. Now, for u, v > 0 with u = v, ((R 0 1)u + v)((r 0 1)v + u) = (R 0 1)(u 2 + v 2 ) + (R 0 1) 2 uv + uv > R 2 0 uv. Hence d< βk(r 0 1)(2 R 0 ) R0 2, which contradicts the second of Equation (29). Remark 4.1 Conditions that are both necessary and sufficient for global convergence to the endemic equilibrium are difficult to find. For the particular strategy being adopted here, one needs conditions for the two curves in the (u, v) plane defined by the equations F (u, v) = λ, G(u, v) = λ, with F and G given by Equation (30), to have no intersection points in the open first quadrant other than the point (u, v) = (x,x ). These two curves are in fact the mirror image of each other in the line v = u. The curve F (u, v) = λ has a vertical asymptote and meets the u-axis at u = λ/d. Simple graphical arguments suggest (but do not prove conclusively) that if the slope of the curve defined by F (u, v) = λ is less than or equal to 1 at(u, v) = (x,x ) then there would be no other intersections and, if not, that additional intersections definitely do arise. Based on this criterion, we are led to the inequality R 2 0 d + βk(r 0 1) 2 βk(r 0 1). (31) This is precisely the second inequality of Equation (29). Of course, it may be the case that solutions still converge globally to the endemic state even when Equation (29) is not satisfied, but at least the above observation indicates that the methods being used here cannot be extended further. 5. Discussion In this paper, we formulated a plausible model of HBV infection that extends an established model by incorporating a biologically described virus production delay and employing a more

13 152 S.A. Gourley et al. Figure 1. A solution of System (3) with initial condition set as (x(s), e(s), y(s), v(s)) = ( , , , ), s [ 2, 0]. The parameters are τ = 2, λ = , d = 0.011, β = , k = 150, a = 0.1, = In this case, the basic infection reproduction number R 0 is realistic, standard incidence function. The model presented here has more biologically realistic features than its predecessor; in particular, the basic infection reproduction number in our model is independent of the initial liver size λ/d when the virus production delay is ignored (when τ = 0). While our Model (3) is biologically more realistic than Model (1), the analysis of the local and global stability properties of the endemic equilibrium in Model (3) is mathematically intractable. However, our numerical solution results (for a typical example, see Figure 1) strongly suggest that Model (3) produces only steady-state asymptotic dynamics. Fortunately, certain biological properties of HBV allow us to reduce Model (3) to a more amenable system, namely Model (10). Indeed, we show that this simpler model generates only stable steady-state dynamics. While this result essentially agrees with biological realities, it contrasts dramatically with the complex dynamics that can be generated from a similarly structured delayed predator-prey model [7] with different prey production terms. In the delayed predatorprey model of Gourley and Kuang [7], the prey is assumed to grow according to the logistic growth process. Therefore, a time delay in predator recruitment interacts with the logistic process in a delicate fashion. In the model studied here, the time delay is apparently harmless when interacting with a growth function of the form λ dx, a form which represents simple homeostatic tissue regeneration. Importantly, if one replaces the homeostatic regrowth function, λ dx, with a logistic growth function in Equation (3), then the model makes a key biological prediction. Certain susceptibleinfective models with standard incidence functions studied by Hwang and Kuang [9,10] can produce complete extinction of prey. In the context of Model (3), this observation, if preserved, implies that, for logistic hepatocyte growth, all liver cells can eventually become infected. This is generally thought to be the case in chronic HBV infection [27], and therefore the next logical step in this research will be the incorporation of such a logistic term.

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