On the global asymptotic stability of a hepatitis B epidemic model and its solutions by nonstandard numerical schemes

Abstract

Very recently, a hepatitis B epidemic model with saturated incidence rate has been proposed and analyzed in Khan et al. (Chaos Solitons Fractals 124:1–9, 2019). The local asymptotic stability of the disease endemic equilibrium (DEE) point of model has been established theoretically but its global asymptotic stability has not been studied. In this paper, we present a mathematically rigorous analysis for the global asymptotic stability of the DEE point of the model. More precisely, we prove that if the DEE point exists, then it is not only locally asymptotically stable but also globally asymptotically stable. Furthermore, we present an alternative proof for the global stability of the disease-free equilibrium point. The main result is that we obtain the complete global stability of the hepatitis B virus model. Besides, we construct nonstandard finite difference (NSFD) schemes preserving the essential qualitative properties of the continuous model. These properties include the positivity of solutions and the stability of the model. Dynamical properties of the proposed schemes are rigorously investigated by mathematical analyses and numerical simulations. Finally, numerical simulations are performed to confirm the validity of the theoretical results as well as the advantages of the NSFD schemes. Numerical simulations also indicate that the NSFD schemes are appropriate and effective to solve the continuous model. Meanwhile, the standard finite difference schemes such as the Euler scheme, the classical fourth-order Runge–Kutta scheme cannot preserve the essential properties of the continuous model. Consequently, they can generate numerical approximations which are completely different from the solutions of the model.

Appendices

Appendix 1: The basic reproduction number of NFSD scheme (10)

We compute the basic reproduction number for NSFD scheme (10) by the next-generation matrix approach proposed in [3].

Note that the first and second equations of system (10) do not include the component R. Hence, it is sufficient to consider the following sub-system:

$$\begin{aligned} \dfrac{S_{k + 1} - S_k}{\phi (h)}= & {} \Lambda - \dfrac{\alpha S_k I_k}{1 + \gamma I_k} - (\mu _0 + \nu )S_k,\nonumber \\ \dfrac{I_{k + 1} - I_k}{\phi (h)}= & {} \dfrac{\alpha S_k I_k}{1 + \gamma I_k} - (\mu _0 + \mu _1 + \beta )I_k. \end{aligned}$$

(20)

We reorder the states in system (20) as $(I_k, S_k)$. Then the unique disease-free equilibrium point of system (20) becomes $(0, S^0)$. The Jacobian matrix of system (20) at this equilibrium point is

$$\begin{aligned} J = \begin{pmatrix} 1 - \phi (\mu _0 + \mu _1 + \beta ) + \phi \alpha S^0&{}0\\ &{}&{}\\ -\phi \alpha S^0&{}1 - \phi (\mu _0 + \nu ) \end{pmatrix}. \end{aligned}$$

Next, we represent the matrix J in the form

$$\begin{aligned} J = \begin{pmatrix} F + T&{} 0\\ &{}\\ A&{} C \end{pmatrix}, \end{aligned}$$

where

$$\begin{aligned} T := 1 - \phi (\mu _0 + \mu _1 + \beta ), \quad F := \phi \alpha S^0, \quad A := -\phi \alpha S^0, \quad C:= 1 - \phi (\mu _0 + \nu ). \end{aligned}$$

It is easy to see that if condition (11) holds, then F and T are non-negative, $F + T$ is irreducible, and matrices C and T satisfy $\rho (T) < 1$ and $\rho (C) < 1$. Consequently, all assumptions of Theorem 2.1 in [3] are met and hence, the basic reproduction number of system (10) can be computed by

$$\begin{aligned} {\mathcal {R}}_0 = \rho \Big (F(1 - T)^{-1}\Big ) = \dfrac{\alpha \Lambda }{(\mu _0 + \nu )(\mu _0 + \mu _1 + \beta )}. \end{aligned}$$

(21)

Appendix 2: An extension of the Lyapunov stability theorem for discrete-time systems

We recall briefly from [25] an extension of the Lyapunov stability theorem for discrete-time systems.

Let us consider a system of difference equations

$$\begin{aligned} {\left\{ \begin{array}{ll} x(k + 1) = f\big (x(k)\big ), &{}\quad x \in {\mathcal {U}},\\ f(0) = 0, \end{array}\right. } \end{aligned}$$

(22)

5 where ${\mathcal {U}}$ is a neighborhood of the origin in ${\mathbb {R}}^n$ and $f: {\mathbb {R}}^n \rightarrow {\mathbb {R}}^n$ is a continuous function.

For each $p \in {\mathcal {U}}$, let $f^k(p)$ denote the value at time k of the solution of (22) starting at p. We denote

$$\begin{aligned} f^+(p) := \Big \{f^k(p)\big |k \in {\mathbb {N}}\Big \}, \quad B_{\epsilon } := \Big \{x \in {\mathbb {R}}^n\big |\Vert x \Vert < \epsilon \Big \}. \end{aligned}$$

(23)

Definition

[25, Definition 2.3] A set Y is invariant if $f(Y) = Y$, positively invariant if $f(Y) \subset Y$, and negatively invariant if $Y \subset f(Y)$.

Definition

[25, Definition 2.4] Let Y be a closed positively invariant set such that $0 \in Y$. The origin is said to be:

• (a) Y-stable if $\forall \epsilon > 0$, $\exists \delta > 0$: $f^+\big (B_{\delta } \cap Y) \subset B_{\epsilon }$.

• (b) Y-asymptotically stable if it is Y-stable and there exists $\delta > 0$ such that

  $$\begin{aligned} \lim _{k \rightarrow \infty }f^{k}(x) = 0, \quad \forall x \in B_{\delta } \cap Y. \end{aligned}$$

Definition

[25, Definition 2.6] A real-valued function V is a Lyapunov function for system (22) in a neighborhood ${\mathcal {U}}$ of the origin if $V(0) = 0$ and $V\big (f(x)\big ) - V(x) \le 0$ for all $x \in {\mathcal {U}}$.

Theorem 8

[25, Theorem 3.3]) If there exists a function $V \in C^0\big ({\mathbb {R}}^n, {\mathbb {R}}^+\big )$ satisfying

• 1. $V(x) \ge 0$ for all $x \in {\mathbb {R}}^n$ and $V(0) = 0$,

• 2. $\Delta V(x) = V\big (f(x)\big ) - V(x) \le 0$ for all $x \in {\mathbb {R}}^n$,

• 3. 0 is $G^*$-globally asymptotically stable where $G^*$ is the largest positively invariant set contained in $G : = \bigg \{x \in {\mathbb {R}}^n\big |\Delta V(x) = V\big (f(x)\big ) - V(x) = 0\bigg \}$,

• 4. All the solutions of system (22) are bounded, then the origin is globally asymptotically stable.

The stability version and the local stability version of Theorem 8 can be found in [25].