Journal of Applied Mathematics

Volume 2012 (2012), Article ID 578561, 25 pages

doi:10.1155/2012/578561

Research Article

Analysis of a HBV Model with Diffusion and Time Delay

Noé Chan Chí, Eric ÁvilaVales, and Gerardo García Almeida

Facultad de Matemáticas, Universidad Autónoma de Yucatán, Anillo Periférico Norte, Tablaje Catastral 13615, Colonia Chuburna Hidalgo Inn, 97203 Mérida, YUC, Mexico

Received 7 March 2012; Accepted 27 August 2012

Academic Editor: Alexander Timokha

Copyright © 2012 Noé Chan Chí 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.

Abstract.

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.

1. Introduction

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. [1]. 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 [2], 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 [3]; therefore, the authors introduce an HBV model with diffusion and delay. Xu and Ma [4] 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).

2. Preliminaries

Consider problem (1.3)–(1.5) and the following definitions.

Definition 2.1. A pair of functions , in are called coupled upper and lower solutions to system (1.3)–(1.5) if , , in and the following differential inequalities hold:

for , and

The following lemma then follows from Theorem 3.4 developed by Redlinger [5].

Lemma 2.2. Let and be a pair of coupled upper and lower solutions for problem (1.3)–(1.5) and suppose that the initial functions () are Hölder continuous in . Then problem (1.3)–(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)–(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 [2], 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.

Theorem 4.2. Let be a solution to problem (1.3)–(1.5), let . If and

,

then

that is, the infected steady state is globally asymptotically stable.

Proof. Let be a solution to problem (1.3)–(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)–(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)–(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)–(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. [6]; consider and the following expressions for , , , , , and

We prove the result by induction so we first show that ,

and ,

For the next step consider the function , with , , and positive, which is monotone increasing. The induction hypothesis is and ,

The last function is increasing and by the induction hypothesis we have

therefore the sequence is nonincreasing. For the sequence we use similar ideas and the behaviour of the sequence just proved. One has

The last function is increasing and by the induction hypothesis we have

therefore the sequence is nondecreasing. The behaviour for the sequences , , , and follows from the nonincreasing sequence and the nondecreasing sequence .

Hence, the limit of each sequence in , , , , , and exists. Denote

We therefore obtain from (4.57) and (4.66) that

Noting that (H1) holds and , it follows that

which together with the previous equation yields . We therefore derive from (4.57) (4.66) that , . Noting that if , by Theorem 3.1, the virus-infected equilibrium is locally asymptotically stable, and if in addition (H1) holds, we conclude that is globally asymptotically stable.

Now we prove the global stability for the disease-free equilibrium

Theorem 4.3. If the disease-free equilibrium of (1.3) is globally asymptotically stable.

Proof. Let be a solution to problem with , . We have , , and for all . Let be a solution to the following problem:

Therefore for , we have

We derive from the first equation that

so we can conclude

Hence, for sufficiently small, there exists a such that for all and . Hence, is a lower solution to the following problem:

Consider as the solution for

Then with according to Lemma 4.1 we have that

By comparison, it follows that

uniformly for . Hence, for sufficiently small, by comparison there is a such that if , , for all and .

As in the proof of Theorem 4.2   is an upper solution for the following problem:

from the above equation we have that

uniformly for . Since this holds for arbitrary sufficiently small, by comparison we conclude that

which together with (4.72) gives

uniformly for . We already have by Theorem 3.1 that the disease-free equilibrium is locally asymptotically stable. And now we have proved that it is also globally asymptotically stable.

5. Numerical Simulations

In this section we illustrate some numerical solutions for systems (1.3). In the numerical simulation display in Figure 1 we illustrate the stability for the disease-free equilibrium according to Theorem 3.1. In this case the basic reproductive number is . In the graphics we see how the level of uninfected cells increases from the initial condition and the number of infected cells and virus in the body goes to zero.

Figure 1: Simulations with parameters , , , , , , , and . .

In Figure 2 consider the case in this case we consider a bigger rate of infection for the cells in the graphics we see how the number of infected cells and viruses increases when the time passes, and when the number of susceptible cells decreases the number of virus also decreases to the value .

Figure 2: Simulations with parameters , , , , , , , and . .

Now in Figure 3 we just show the level of virus in different for different values of the diffusion constant and the delay . We see that a bigger delay increases the time needed for the virus to reach the value , meanwhile a mayor value for the constant just affects the levels of the virus according to the space and does not affect significantly the time needed for the virus to reach . In Figure 4 we consider a lower value for , which is the uninfected cell production rate. In this case we see how the time to reach the value of the endemic equilibrium is lower and again the diffusion rate has no significant effect on the time; its effect is on the level of virus in the system. The delay is what really affect the time to reach the value .

(a)

(b)

(c)

(d)

Figure 3: In this case the parameters are , , , , , and .

(a)  

(b)

(c)

(d)

Figure 4: In this case the parameter are , , , , , and .

Acknowledgments

This paper was supported by UADY under Grant UADY-CA-34 and Mexican SNI under Grants 15284 and 33365.

References

M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boheme, H. C. Thomas, and H. McDade, “Viral dynamics in hepatitis B virus infection,” Proceedings of the National Academy of Sciences of the United States of America, vol. 93, no. 9, pp. 4398–4402, 1996.

X. Tian and R. Xu, “Asymptotic properties of a hepatitis B virus infection model with time delay,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 182340, 21 pages, 2010.

K. Wang and W. Wang, “Propagation of HBV with spatial dependence,” Mathematical Biosciences, vol. 210, no. 1, pp. 78–95, 2007.

R. Xu and Z. Ma, “HBV model with diffusion and time delay,” Journal of Theoretical Biology, vol. 257, pp. 499–509, 2009.

R. Redlinger, “Existence theorems for semilinear parabolic systems with functionals,” Nonlinear Analysis. Theory, Methods & Applications, vol. 8, no. 6, pp. 667–682, 1984.

M. Uh Zapata, E. Avila Vales, and A. G. Estrella, “Asymptotic behavior of a competition-diffusion system with variable coefficients and time delays,” Journal of Applied Mathematics, vol. 2008, Article ID 537284, 17 pages, 2008.