Analysis of a HBV Model with Diffusion and Time Delay

This paper discussed a hepatitis B virus infection with delay, spatial di ﬀ usion, 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, su ﬃ ciently conditions are obtained for the global asymptotic stability of the infected steady state. Numerical simulations are carried out to illustrate our ﬁndings.


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 , u, infected host cells, w, and free virus particles, v, as follows: where hepatocytes are produced at a rate s, uninfected cells die at rate μ and become and infected at rate βu t v t , infected hepatocytes are produced at rate βu t v t and die at rate aw t . Free viruses are produced from infected cells at rate kw t and are removed at rate dv t . 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: v t ky t − uv t .

1.2
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: ∂w ∂t x, t ∈ Ω × −τ, 0 .

2.2
The following lemma then follows from Theorem 3.4 developed by Redlinger It is not hard to see that 0 0, 0, 0 and K K 1 , K 2 , K 3 are a pair of coupled lowerupper solutions to problem 1.3 -1.5 , where

Local Stability
System 1.3 has the equilibrium E 1 L/d, 0, 0 . Let R 0 βke −mτ /ap > 1 then system 1.3 has a unique infected steady state E 2 u * τ , w * τ , v * τ ; the previous notation is because the equilibrium involves τ and we use this as the parameter for the stability analysis, where

3.1
Let 0 μ 1 < μ 2 < · · · be the eigenvalues of the operator −Δ on Ω with the homogeneous Neumann boundary conditions, and let E μ i be the eigenspace corresponding to and E * u, w, v represents any feasible steady state of the system 1.3 . The linearization of system 1.3 at E * is of the form Z t LZ. For each i ≥ 1, X i is invariant under the operator L, and λ is an eigenvalue of the matrix −μ i D J E * J τE * for some i ≥ 1, then, there is an eigenvector in X i . The characteristic equation on the equilibrium E 1 is where a 0 a p μ I D , The characteristic equation has the negative root λ −d. All other roots of 3.4 are given by the transcendental equation if R 0 > 1, note that for λ real and i 1 in this case μ 1 0 , Hence, 3.7 has a positive root. Therefore, there is a characteristic root λ with positive real part in the spectrum of L. Accordingly, if R 0 > 1, the disease-free steady state E 1 λ/d, 0, 0 is unstable.
If R 0 < 1, when τ 0 the coefficients of 3.7 are a 1 and a 0 b 0 0 , and under the hypothesis R 0 < 1 the coefficients are positive and according to the criterion of Routh-Hurwitz, the equilibrium E 1 λ/d, 0, 0 is locally asymptotically stable.

3.9
Squaring and adding the above equations and taking z ω 2 we obtain the last inequality is true because R 0 < 1. Therefore there is no positive root z ω 2 of 3.10 .
In conclusion if R 0 < 1 the equilibrium E 1 λ/d, 0, 0 is locally asymptotically stable. The characteristic equation of system 1.3 at the endemic equilibrium E 2 u * , w * , v * is of the form when τ 0 becomes Note that a 2 0 b 2 0 > 0; adding a 0 0 b 0 0 and replacing u * 0 and w * 0 we obtain

3.15
By the Routh-Hurwitz criteria, all roots have negative real parts if R 0 > 1. For the case τ > 0 we look for solutions λ iω ω > 0 for 3.12 , separating real and imaginary parts, it follows that

3.16
Squaring and adding the two equations, we derive that implying that 3.17 has no positive roots z ω 2 .

Global Stability
We will discuss in this section the global stability of the infected steady state and the diseasefree 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: and according to 2 , for system 4.1 , one has the following.
Lemma 4.1. If kβe −mτ > ap, then the positive equilibrium A * u * 1 , u * 2 is globally stable. If kβe −mτ < ap, then the equilibrium A 0 0, 0 is globally stable. Now we stablish and prove our result about global stability.
that is, the infected steady state E * is globally asymptotically stable.

4.5
First we look for upper solutions for the system 1.3 . Let u 1 x, t , w 1 x, t , v 1 x, t be a solution for the following problem:

4.6
We note that the solution of this system is an upper solution of system 1.3 -1.5 . For t > 0, From the first equation of 4.6 Hence, by comparison, for all > 0 sufficiently small, there exists since is arbitrary and sufficiently small we can conclude that

4.10
Now consider the problem related with the second and third equations of 4.6

4.11
Consider the solution foru

4.12
Note that u 1 t , u 2 t is an upper solution for system 4.11 , and using the assumption that R 0 > 1, by Lemma 4.1, it follows from 4.12 that

4.13
Hence, for all > 0 sufficiently small, by comparison there exists a t 2 > t 1 such that if t > t 2 Since > 0 is arbitrary and sufficiently small, we conclude that

4.16
Journal of Applied Mathematics 11 Now for lower solutions, let u 1 x, t , w 1 x, t , v 1 x, t be the solution for the following problem:

4.17
Note that the solution of 4.17 is a lower solution to 1.3 -1.5 . For all > 0 sufficiently small, from the first equation of 4.17 and 4.16 it follows By comparing the above equation with the following problem: Since > 0 is arbitrary sufficiently small, by comparison we conclude that Now consider the following problem related with the second and third equations of 4.17 :

4.24
Now let us consider the solution for the probleṁ

4.25
and according to Lemma 4.1

4.26
Hence, for all > 0 sufficiently small, by comparison there exists a t 4 > t 3 such that if t > t 4 min x∈Ω w 1 x, t > N w 1 − , min

4.28
Since > 0 is arbitrary and sufficiently small, we conclude that

4.29
Now we look for the closest upper and lower solutions. Let u 2 , w 2 , v 2 be a solution for the problem

4.30
For all > 0 sufficiently small it follows form the first equation of 4.30 and the inequalities 4.27 and 4.14 that Let ω 2 1 x, t be the solution for the following problem:

4.35
Since 4.34 is valid for > 0 arbitrary and sufficiently small, by comparison we conclude that Now consider the following problem related with the second and third equations of 4.30 :

4.37
Let u 2 t , u 3 t be the positive solution to the following problem:

4.41
Since > 0 is arbitrary and sufficiently small, we conclude that Let u 2 , w 2 , v 2 be a solution for the following problem:

4.43
Then u 2 , w 2 , v 2 and u 2 , w 2 , v 2 are a pair of coupled lower and upper solutions to system 1.3 -1.5 . Hence we have that for t ≥ t 6 , x ∈ Ω

Journal of Applied Mathematics
For all > 0 sufficiently small, it follow from the first equation of 4.43 , and the inequalities

4.45
By comparison we have that u 2 x, t ≥ v 2 1 x, t , t > t 6 , and x ∈ Ω where v 2 1 is the solution to problem

4.47
Hence for all > 0 sufficiently small, by comparison, there is a t 7 > t 6 such that if t > t 7

4.49
Since this holds true for arbitrary > 0 sufficiently small, by comparison we conclude that Now consider the following problem: Proof. Let u x, t , w x, t , v x, t be a solution to problem with φ i x, 0 / 0, i 1, 2, 3 . We have u x, t > 0, w x, t > 0, and v x, t > 0 for all x ∈ Ω. Let u 1 x, t , w 1 x, t , v 1 x, t be a solution to the following problem:

4.69
Therefore for t > 0, x ∈ Ω we have We derive from the first equation that Hence, for > 0 sufficiently small, there exists a t 1 such that u 1 x, t ≤ L/d for all x ∈ Ω and t ≥ t 1 . Hence, w x, t , v x, t is a lower solution to the following problem: Consider u 2 t , u 3 t as the solution foṙ

4.74
Then with R 0 < 1 according to Lemma   uniformly for x ∈ Ω. Hence, for > 0 sufficiently small, by comparison there is a t 2 ≥ t 1 such that if t ≥ t 2 , w x, t < , v x, t < for all x ∈ Ω and t ≥ t 2 .
As in the proof of Theorem 4.2 u x, t is an upper solution for the following problem: uniformly for x ∈ Ω. Since this holds for arbitrary > 0 sufficiently small, by comparison we conclude that lim inf uniformly for x ∈ Ω. We already have by Theorem 3.1 that the disease-free equilibrium E 1 is locally asymptotically stable. And now we have proved that it is also globally asymptotically stable.
Journal of Applied Mathematics 25

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 R 0 0.039426. 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.
In Figure 2 consider the case R 0 > 1 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 v * . Now in Figure 3 we just show the level of virus in different for different values of the diffusion constant D and the delay τ. We see that a bigger delay increases the time needed for the virus to reach the value v * , meanwhile a mayor value for the constant D just affects the levels of the virus according to the space and does not affect significantly the time needed for the virus to reach v * . In Figure 4 we consider a lower value for λ, which is the uninfected cell production rate. In this case we see how the time to reach the value v * 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 v * .