GLOBAL STABILITY OF AUTONOMOUS AND NONAUTONOMOUS HEPATITIS B VIRUS MODELS IN PATCHY ENVIRONMENT∗

Autonomous and nonautonomous hepatitis B virus infection models in patchy environment are investigated respectively to illustrate the influences of population migration and almost periodicity for infection rate on the spread of hepatitis B virus. The basic reproduction number is determined and asymptotic stabilities of disease-free and endemic equilibria are established in case of autonomous system. Moreover, in the nonautonomous system case, existence and global attractivity of almost periodic solution for this system are studied. Finally, feasibility of main theoretical results is showed with the aid of numerical examples for model with two patches.


Introduction
As a contagious disease triggered by hepatitis B virus (HBV), hepatitis B acutely threatens global public health. According to the latest hepatitis B research report, HBV affects approximately 292 million individuals in 2016, which represents a global prevalence of 3.9% [22]. It is clear that the treatment and prevention for hepatitis B are effective in China and the infection rate is steadily declining by comparing data on HBV infections from then and now. However, there are still nearly 89 million HBV carriers in China, about one third of the world's total, which are the world's largest [22]. It is thus important to monitor HBV infection patterns and predict trends over time.
For the past few years, several mathematics models have been established to analyse the dynamic behaviors of HBV transmission [4, 9, 16, 17, 27-29, 32, 36], especially for the global stabilities of their equilibria. Due to similar main routes of transmission for HBV, HCV (hepatitis C virus) and HIV (human immunodeficiency virus) which include sexual contact, blood transmission and mother-to-child transmission, HBV transmission model is also suitable for describing the spread of HCV and HIV. Dai etc [4] formulated an HBV transmission model to investigate the spread of HBV in mainland China and kinetics of this system. Based on therapy of chronic hepatitis B, Huan etc [9] included antivirus treatment in an HBV transmission model, studied the stability of its equilibria. Khan etc [16] considered a transmission model of HBV by taking into account media coverage, analyzed the stability results for the model and presented the optimal control treatment problem with suggested controls. Khan etc [17] incorporated acute-infected and chronicinfected classes into hepatitis B epidemic model and developed the optimal control strategy of HBV transmission. Wang and Tian [32] introduced a CTLs immune response in a time delay HBV infection model and showed that basic reproduction number and basic immune reproduction number determine the asymptotic stabilities of equilibria. Zhang and Zhang [36] formulated a model for HBV to describe how newborn vaccination and treatment influence HBV prevention and got the result the basic reproduction number, as a critical value, determines the stability and persistence of hepatitis B in this model.
Communicable diseases can easily spread between different countries (or regions). For instance, the first case of SARS was found in Guangdong, China in 2002, however, the cumulative cases involved 32 countries and regions as of June 2003 due to the human mobility [23]. The higher interregional mobility may bring about the faster regional and global spread of infectious diseases [14]. Dynamics analysis of epidemic models in patchy environment can show how individual migration among patches affects the dynamic behaviors of epidemic disease transmission, see [5,21,33,35] and the references cited therein.
When modeling the dynamics of population, we usually assumed that coefficients of dynamical models are constant [10,12]. However, the nonautonomous phenomena are much universal in the real world and nonlinear differential equations can be used to model numerous dynamical problems [11,13,20], which could make the model be more realistic than autonomous differential equations. In the case of nonautonomous models, periodic and almost periodic coefficients are taken into consideration in the relevant researches. Moreover, as indicated in [6], almost periodic effects are more approaching to reality in a variety of real world applications than periodic effects. Some recent development on the transmission dynamics of epidemic models with almost periodic coefficients have been discussed in [19,26,31,34] and references therein.
Motivated by the above discussions, we construct the HBV transmission model with almost periodic infection rate in patchy environment based on the model of Kamyad etc [15] and study the stability for this model both in the autonomous and nonautonomous cases. The remaining parts of this paper is organized as follows. An HBV infection model with almost periodic infection rate and patch structure is formulated and some basic properties are deduced in Section 2. In Section 3, the stability analysis of corresponding autonomous system is presented. Section 4 is devoted to existence and global attractivity of almost periodic solution for this system in nonautonomous case. In Section 5, we present numerical examples to demonstrate the effectiveness of established results. Finally, in Section 6, conclusion and discussion for this paper are provided.

System description
infection for HBV, that is, HBV transmits directly from mother to offspring (vertical transmission) and people are infected by contacting with infective individuals (horizontal transmission) were considered. They also accounted for the relapse between recovered people in their paper.
Based on the model of Kamyad etc, we propose a nonautonomous model for the HBV infection which is an extended and improved version of the HBV transmission model in [15] with the inclusion of population travel between n patches and almost periodic infection rate. The total population is divided into five classes in each patch i (i = 1, 2, ..., n): susceptible class S i ; latent class E i ; acute infected class I i ; chronic infected class C i ; and recovered class R i . Thus, our model is formulated in the following form where i, j = 1, 2, ..., n. a ij , b ij , c ij , k ij and l ij represent the travel rates of susceptible individuals, latent individuals, acute infected people, chronic HBV carriers and recovered (or immune) people from patch (or group) j to patch (or group) i, respectively. The other parameters in patch i are described in Table 1. ρ i (t) is positive almost periodic with i = 1, 2, ..., n.
For the term ν i ξ i C i , it denotes the vertical transmission in patch i. And ν i η i R i denotes immune newborns from recovered class in patch i. Accordingly, the birth flow rate in the susceptible compartment in patch i is denoted by for a solution of system (2.1).

Basic properties
It is clearly that all the rates are nonnegative on the bounding planes of ∆ + . Now , and the interior of region ∆ + attracts all solution orbits of (2.1). Thus, all solutions of (2.1) always remain in ∆ + .
< 0. Moreover, we observe the ordinary differential equation with general solution where N (0) means the initial value of total population. By applying the standard comparison theorem, we have for all t ≥ 0, Hence, ∆ is positive invariant for system (2.1).

Autonomous system case
In this section, the global stability are studied for autonomous system corresponding to (2.1) by taking infection rate ρ i as a constant for i = 1, 2, ..., n. Thus, system (2.1) could be given in the following form where i, j = 1, 2, ..., n.

Global dynamics for disease-free equilibrium
Noting the fact that E i = I i = C i = 0, i = 1, 2, · · · , n at disease-free equilibrium of system (3.1), substituting it into (3.1), we drive which could be rewritten in form of matrix equation where Assume that the following hypotheses hold: (H1) ν i η i > γ i for all i = 1, 2, ..., n; (H2) a ij = a ji and l ij = l ji for all i ̸ = j; It is obvious that all off-diagonal elements of A 1 and B 2 are nonpositive, and column sums of Accordingly, system (3.1) admits a unique disease-free equilibrium Naturally, we can draw the following conclusion.

Theorem 3.1. Suppose hypotheses (H1)-(H3) hold, then system (3.1) has a unique disease-free equilibrium.
We utilize next generation matrix approach [30] as follows so as to derive the basic reproduction number of system (3.1). For simplicity, we rearrange (3.1) as following . . .

−νn+ρn(In+θnCn)Sn+(νn+αn)Sn+νnξnCn−(γn−νnηn)Rn
The Jacobian matrices of F ( x) and V ( x) at the disease-free equilibrium P 0 are, respectively, Since column sums of V are positive and all off-diagonal elements of V are nonpositive, then V is a nonsingular M-matrix (M 35 in [1, p137]). Furthermore, we get Applying the approach in [30], the basic reproductive number is shown by represents the spectral radius of matrix.
The following result follows by applying Theorem 2 of [30].
Theorem 3.2. The disease-free equilibrium, P 0 , is local asymptotical stable as R 0 < 1 and unstable as R 0 > 1. Proof. Firstly, we show that Let E i = 0, invoking the second equation of (3.1), we have according to the third and forth equations of (3.1), it holds that Thus, for any i, j = 1, 2, ..., n, Since B = (b ij ), C = (c ij ) and K = (k ij ) are irreducible and directed graph of irreducible matrix is strongly connected (Theorem 2.7 in [1, p30]), there exist sequences of ordered pairs We next show that there exist no equilibria on the boundary ∂∆ of non zero elements. It is obvious that (2.2) takes the equal sign if and only if ν 1 To find the equilibrium of system (3.1), we set the right side of (3.1) equal to 0.
Therefore, no non zero equilibrium lies on the boundary ∂∆ when there exist i ̸ = j such that ν i ̸ = ν j .
Using the property of Mittage-Leffler function given in [8] Thus, we define the following auxiliary linear system (3.7) According to [30], we have It is evident the right side of (3.7) has efficient matrix F − V , then all non-negative solutions of (3.7) satisfies that lim performs the limiting system of dS dt , dR dt terms of (3.1). From [25] and Theorem 2.3 in [3], we conclude that all the solutions to (3.1) satisfy that lim , P 0 is the unique equilibrium lies on ∂∆. Hence, equilibrium P 0 is global asymptotic stable whenever R 0 < 1. Proof. If R 0 > 1, according to Theorem 3.2, P 0 is unstable. Choose X = R 5n and E = ∆ for Theorem 4.3 in [7]. When B, C and K are irreducible, by Theorem 3.3, singleton {P 0 } is isolated as the maximal invariant set on ∂∆. Accordingly, hypothesis (H) in [7] is valid for system (3.1). Note that the instability of P 0 is equivalent to the necessary and sufficient condition of Theorem 4.3 in [7], which indicates the uniformly persistence of system (3.1).
From the positive invariance of ∆, we get that solutions in∆ are uniform bounded. Then, by Theorem 2.8.6 in [2], and according to the uniform persistence of system (3.1), we draw the conclusion that there exists an equilibrium in∆.

Local dynamics for endemic equilibrium
Let R where S 0 i = νi(1−ηi)+γi νi(1−ηi)+γi+αi . To find the endemic equilibrium , let the right side of system (3.1) with a ij = b ij = c ij = k ij = l ij = 0 be equal to zero. Then, we get where J i for i = 1, 2, ..., n is 5 × 5 matrix and takes the following form The characteristic equation of J i is given by It is obvious one of the eigenvalues for J i , −ν i , is negative. To proceed, we consider the following equation In view of the Routh-Hurwitz criteria [18], all roots of (3.10) possess negative real parts iff f i > 0 for i = 1, 2, 3, 4 and f 1 f 2 f 3 > f 3 2 + f 1 2 f 4 . From (3.9), we obtain (3.11) Thus, we have Furthermore, by (3.8) and (3.9), we get .
and d 4 > δ i , d 5 > σ i , (3.11) and (3.12) imply that where h 1 = d 1 +d 2 and h 2 = d 4 +d 5 +d 0 . Thus, J i only has eigenvalues with negative real part as R (i) 0 > 1 for all i = 1, 2, ..., n, and it is an immediate consequence the endemic equilibrium P * is locally asymptotically stable.
We consider the Lyapunov function
Theorem 4.1 implies any solution of (2.1) converges to X, which reveals the global attractivity of this system.

Numerical simulations
To proceed, several numerical examples are provided to validate the theoretical findings derived in previous sections. We focus on the following HBV transmission model with two patches which is a special case of system (2.1) 0 ≈ 1.3293 > 1. Hence, system (5.1) admits unique endemic equilibrium which is local asymptotic stable from Theorem 3.6, see Fig. 3.

Conclusion
A nonautonomous model for HBV infection in a patchy environment has been constructed to reveal the influences of population migration and almost periodicity for infection rate on the spread of HBV in this paper. Compared with the HBV transmission model presented by Kamyad etc [15], we have taken into account the population travel between n patches and almost periodic infection rate. Firstly, the qualitative behaviour of autonomous model (3.1) associated with model system (2.1) has been carried out. The basic reproduction number has been determined and sufficient conditions guaranteeing the global stability for diseasefree equilibrium have been derived by combining the stability theory of asymptotically autonomous systems with basic comparison theorem of differential equations. Furthermore, conditions under which system admits unique and locally asymptotically stable endemic equilibrium have been obtained, respectively. Secondly, we have studied the existence and global attractivity for almost periodic solution of system in nonautonomous case. Moreover, we have deduced that the almost periodicity of time evolution for all the populations is ensured when model parameters satisfy the conditions of Theorem 4.2. Finally, to illustrate the analytical findings, numerical simulations of the model with two patches has been done in cases of autonomous and nonautonomous system.
There yet have many challenging and interesting issues remain to be investigated in future work. From Example 5.5, we find the existence of almost periodic solution are ensured though condition (4.1) of Theorem 4.1 is dissatisfied. Nevertheless, we are unable to prove it at present. In addition, it is known that there may exist timelag when susceptible individual to be immune after vaccination and to be infected after contacting with HBV carriers. We leave these issues for future research.