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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 578561, 25 pages
Analysis of a HBV Model with Diffusion and Time Delay
Facultad de Matem####^~^~^~^~^~^####xe1;ticas, Universidad Aut####^~^~^~^~^~^####xf3;noma de Yucat####^~^~^~^~^~^####xe1;n, Anillo Perif####^~^~^~^~^~^####xe9;rico Norte, Tablaje Catastral 13615, Colonia Chuburna Hidalgo Inn, 97203 M####^~^~^~^~^~^####xe9;rida, YUC, Mexico
Received 7 March 2012; Accepted 27 August 2012
Academic Editor: Alexander Timokha
Copyright © 2012 No####^~^~^~^~^~^####xe9; Chan Ch####^~^~^~^~^~^####xed; et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper discussed a hepatitis B virus infection with delay, spatial diffusion, and standard incidence function. The local stability of equilibrium is obtained via characteristic equations. By using comparison arguments, it is proved that if the basic reproduction number is less than unity, the infection-free equilibrium is globally asymptotically stable. If the basic reproductive number is greater than unity, by means of an iteration technique, sufficiently conditions are obtained for the global asymptotic stability of the infected steady state. Numerical simulations are carried out to illustrate our findings.
Human infection with hepatitis B virus (HBV) is a major global health problem. Between 300 and 400 million people are chronically infected worldwide. The virus is contracted through contact with blood or other fluids from the body, which could lead to develop viral persistence in the individual in the absence of strong antibody or some immune depression. Mathematical models have the potential to improve the understanding of the dynamics of this disease; one of the earliest models is referred to as the basic virus infection model, introduced by Nowak et al. . They proposed a basic mathematical model for uninfected susceptible host cells (hepatocytes), , infected host cells, , and free virus particles, , as follows: where hepatocytes are produced at a rate , uninfected cells die at rate and become and infected at rate , infected hepatocytes are produced at rate and die at rate . Free viruses are produced from infected cells at rate and are removed at rate . It is assumed that all parameters are positive constants. Previous models assume that the infectious process is instantaneous; that is, in the very moment that the virus enters an uninfected cell, this one starts to produce virus particles; we know that this is not biologically reasonable. Thus, models with delays have been considered; in , the authors studied the following hepatitis B virus infection model with a time delay: The authors gave results about local and global stability of feasible equilibria.
For HBV infection, susceptible host cells and infected cells are hepatocytes and cannot move under normal conditions, but viruses move freely in liver ; therefore, the authors introduce an HBV model with diffusion and delay. Xu and Ma  considered also a diffusion model with delay but instead of bilinear response of the infection rate, they considered saturation response.
In this work motivated by the work of Xu and Ma, we study the following model: for , , with homogeneous Neumann boundary conditions and initial conditions In the previous problem is a bounded domain in with smooth boundary , denotes the outward normal derivative on .
This paper is ordered as follows. In the next section we present a result about the existence, uniqueness, and positivity. In Section 3 we discuss the local stability of each of the feasible equilibria of system (1.3), by analyzing the corresponding characteristic equations. In Section 4, by using comparison arguments and an iterative technique, we establish sufficient conditions for the global stability of the equilibria of system (1.3). In Section 5 numerical simulations are carried out to illustrate our principal results and we compare the effect of the diffusion and the delay on the system (1.3).
Definition 2.1. A pair of functions , in are called coupled upper and lower solutions to system (1.3)####^~^~^~^~^~^####x2013;(1.5) if , , in and the following differential inequalities hold: for , and
The following lemma then follows from Theorem 3.4 developed by Redlinger .
Lemma 2.2. Let and be a pair of coupled upper and lower solutions for problem (1.3)####^~^~^~^~^~^####x2013;(1.5) and suppose that the initial functions () are H####^~^~^~^~^~^####xf6;lder continuous in . Then problem (1.3)####^~^~^~^~^~^####x2013;(1.5) has exactly one regular solution satisfying in .
It is not hard to see that and are a pair of coupled lower-upper solutions to problem (1.3)####^~^~^~^~^~^####x2013;(1.5), where Hence, , , for , and also, by the maximum principle, if , we have , , for all , .
3. Local Stability
System (1.3) has the equilibrium . Let then system (1.3) has a unique infected steady state ; the previous notation is because the equilibrium involves and we use this as the parameter for the stability analysis, where
Let be the eigenvalues of the operator on with the homogeneous Neumann boundary conditions, and let be the eigenspace corresponding to in .
Let , let be an orthonormal basis of , and let , then Let , , , where and represents any feasible steady state of the system (1.3). The linearization of system (1.3) at is of the form . For each , is invariant under the operator , and is an eigenvalue of the matrix for some , then, there is an eigenvector in .
The characteristic equation on the equilibrium is where The characteristic equation has the negative root . All other roots of (3.4) are given by the transcendental equation Let if , note that for real and (in this case ), Hence, (3.7) has a positive root. Therefore, there is a characteristic root with positive real part in the spectrum of . Accordingly, if , the disease-free steady state is unstable.
If , when the coefficients of (3.7) are and , and under the hypothesis the coefficients are positive and according to the criterion of Routh-Hurwitz, the equilibrium is locally asymptotically stable.
For if is a solution of (3.6), separating in real and imaginary parts, we obtain Squaring and adding the above equations and taking we obtain where the last inequality is true because . Therefore there is no positive root of (3.10). In conclusion if the equilibrium is locally asymptotically stable.
The characteristic equation of system (1.3) at the endemic equilibrium is of the form where when becomes Note that ; adding and replacing and we obtain By the Routh-Hurwitz criteria, all roots have negative real parts if .
For the case we look for solutions for (3.12), separating real and imaginary parts, it follows that Squaring and adding the two equations, we derive that where implying that (3.17) has no positive roots .
Theorem 3.1. If the disease-free equilibrium is locally asymptotically stable; if it is unstable and the endemic equilibrium exists and it is locally asymptotically stable.
4. Global Stability
We will discuss in this section the global stability of the infected steady state and the disease-free equilibrium. The technique of proof is to use comparison arguments and to successively modify the coupled lower-upper solutions pairs.
Consider the following delay system: with initial conditions System (4.1) always have the trivial equilibrium . If , then system (4.1) has a unique positive equilibrium where and according to , for system (4.1), one has the following.
Lemma 4.1. If , then the positive equilibrium is globally stable.
If , then the equilibrium is globally stable.
Now we stablish and prove our result about global stability.
Proof. Let be a solution to problem (1.3)####^~^~^~^~^~^####x2013;(1.5), let . We have , , and for all , . Denote First we look for upper solutions for the system (1.3). Let be a solution for the following problem: We note that the solution of this system is an upper solution of system (1.3)####^~^~^~^~^~^####x2013;(1.5). For , we have From the first equation of (4.6) Hence, by comparison, for all sufficiently small, there exists such that if since is arbitrary and sufficiently small we can conclude that Now consider the problem related with the second and third equations of (4.6) Consider the solution for Note that is an upper solution for system (4.11), and using the assumption that , by Lemma 4.1, it follows from (4.12) that Hence, for all sufficiently small, by comparison there exists a such that if where Since is arbitrary and sufficiently small, we conclude that Now for lower solutions, let be the solution for the following problem: Note that the solution of (4.17) is a lower solution to (1.3)####^~^~^~^~^~^####x2013;(1.5). For all sufficiently small, from the first equation of (4.17) and (4.16) it follows By comparing the above equation with the following problem: we obtain so , , and . Hence, for all sufficiently small, there is a such that if , where Since is arbitrary sufficiently small, by comparison we conclude that Now consider the following problem related with the second and third equations of (4.17): Now let us consider the solution for the problem and according to Lemma 4.1 Hence, for all sufficiently small, by comparison there exists a such that if where Since is arbitrary and sufficiently small, we conclude that Now we look for the closest upper and lower solutions. Let be a solution for the problem For all sufficiently small it follows form the first equation of (4.30) and the inequalities (4.27) and (4.14) that Let be the solution for the following problem: it follows that By comparison we have that , , and . Hence, for all sufficiently small, by comparison, there is a such that if where Since (4.34) is valid for arbitrary and sufficiently small, by comparison we conclude that Now consider the following problem related with the second and third equations of (4.30): Let be the positive solution to the following problem: Then by Lemma 4.1 and the previous system we have Hence for all sufficiently small, by comparison there is a such that if where Since is arbitrary and sufficiently small, we conclude that Let be a solution for the following problem: Then and are a pair of coupled lower and upper solutions to system (1.3)####^~^~^~^~^~^####x2013;(1.5). Hence we have that for , For all sufficiently small, it follow from the first equation of (4.43), and the inequalities By comparison we have that , , and where is the solution to problem which has satisfies Hence for all sufficiently small, by comparison, there is a such that if with Since this holds true for arbitrary sufficiently small, by comparison we conclude that Now consider the following problem: Let be the positive solution for the following problem: By Lemma 4.1 it follows hence, for all sufficiently small, by comparison there exists a such that if , where Since is arbitrary and sufficiently small, we conclude that continuing this process, we derive six sequences , , , , , and such that, for , It is readily seen that The sequences , , and are nonincreasing and the sequences , , and are nondecreasing.
To prove the monotonicity of and , we follow the ideas of Uh Zapata et al. ; consider and the following expressions for , , , , , and We prove the result by induction so we first show that , and ,