A numerical method for a delayed viral infection model with general incidence rate

Proposition 2.1

All solutions of system (2) subject to condition (3) remain nonnegative and bounded for all $n\in \mathbb{N}$ .

Proof

First, we prove the positivity of solutions by using mathematical induction. When $n=0$ , we have $$(1+\mathit{hd})x_1+\mathit{hf}(x_1\text{,}{y}_0\text{,}{v}_0)v_0=x_0+h\lambda \text{.}$$ Let $g(x)=(1+\mathit{hd})x+\mathit{hf}(x\text{,}{y}_0\text{,}{v}_0)v_0-x_0-h\lambda$ . Clearly, g(x) is a continuous function for $x\text{,}g(\mathbb{R})=\mathbb{R}$ and $$g^{\prime }(x)=1+\mathit{hd}+{\mathit{hv}}_0\frac{\partial f}{\partial x}(x\text{,}{y}_0\text{,}{v}_0)>0\text{.}$$ Then $g(x)$ is monotonically increasing on $\mathbb{R}$ . Hence, the equation $g(x)=0$ has a unique solution on $\mathbb{R}$ . Since $$g(0)=-x_0-h\lambda <0\text{and}g\left(\frac{x_0+h\lambda }{1+\mathit{hd}}\right)=\mathit{hf}\left(\frac{x_0+h\lambda }{1+\mathit{hd}}\text{,}{y}_0\text{,}{v}_0\right)v_0>0\text{,}$$ we have $\overline{x}\in \left(0\text{,}\frac{x_0+h\lambda }{1+\mathit{hd}}\right)$ . Hence, $\overline{x}=x_1>0$ . From (2), we obtain $$\begin{array}{cc}\hfill & y_1=\frac{y_0+\mathit{hf}(x_{-m_1+1}\text{,}{y}_{-m_1}\text{,}{v}_{-m_1})v_{-m_1}{e}^{-{\alpha }_1{\tau }_1}}{1+\mathit{ah}}\text{,}\hfill \\ \hfill & v_1=\frac{v_0+{\mathit{hky}}_1}{1+\mathit{uh}}\text{.}\hfill \end{array}$$ Then $y_1⩾0$ and $v_1⩾0$ . Therefore, by using the induction, we get $x_n⩾0\text{,}{y}_n⩾0$ and $v_n⩾0$ for all $n⩾0$ . This proves the positivity of solutions.

Next, we prove the boundedness of solutions. Let $$T_n=x_n+y_n+h{\displaystyle \sum _{j=n-m_1}^{n-1}}f(x_{j+1}\text{,}{y}_j\text{,}{v}_j)v_j{e}^{-h{\alpha }_1(n-j)}\text{.}$$ Then, $$\begin{array}{cc}\hfill T_{n+1}-T_n& =h(\lambda -{\mathit{dx}}_{n+1}-f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)v_n)\hfill \\ \hfill & +h(f(x_{n-m_1+1}\text{,}{y}_{n-m_1}\text{,}{v}_{n-m_1})v_{n-m_1}{e}^{-{\alpha }_1{\tau }_1}-{\mathit{ay}}_{n+1})\hfill \\ \hfill & +h{\displaystyle \sum _{j=n-m_1+1}^n}f(x_{j+1}\text{,}{y}_j\text{,}{v}_j)v_j{e}^{-h{\alpha }_1(n+1-j)}-h{\displaystyle \sum _{j=n-m_1}^{n-1}}f(x_{j+1}\text{,}{y}_j\text{,}{v}_j)v_j{e}^{-h{\alpha }_1(n-j)}\hfill \\ \hfill & =h(\lambda -{\mathit{dx}}_{n+1}-{\mathit{ay}}_{n+1})+h(1-e^{h{\alpha }_1}){\displaystyle \sum _{j=n-m_1+1}^n}f(x_{j+1}\text{,}{y}_j\text{,}{v}_j)v_j{e}^{-h{\alpha }_1(n+1-j)}\hfill \\ \hfill & ⩽h(\lambda -\delta T_{n+1})\hfill \end{array}$$ where $\delta =\min \{d\text{,}a\text{,}\frac{e^{h{\alpha }_1}-1}{h}\}$ . Hence, $$T_{n+1}⩽\frac{1}{1+h\delta }{T}_n+\frac{h\lambda }{1+h\delta }\text{.}$$ By using the induction, we get the following inequality $$T_n⩽{\left(\frac{1}{1+h\delta }\right)}^n{T}_0+\frac{\lambda }{\delta }\left[1-{\left(\frac{1}{1+h\delta }\right)}^n\right]\text{.}$$ Then, $${\displaystyle \underset{n\to +\infty }{\lim \sup }}{T}_n⩽\frac{\lambda }{\delta }\text{.}$$ This implies that $\{T_n\}$ is bounded. Therefore, $\{x_n\}$ and $\{y_n\}$ are also bounded.

By the third equation of (2), we obtain $$v_{n+1}=\frac{1}{1+\mathit{hu}}{v}_n+\frac{{\mathit{hke}}^{-{\alpha }_2{\tau }_2}}{1+\mathit{hu}}{y}_{n-m_2+1}\text{.}$$ As $\{y_n\}$ is bounded, then there exists a M such that $y_n⩽M$ for all $n\in \{-m_2\text{,}-m_2+1\text{,}\dots \text{,}0\text{,}1\text{,}\dots \}$ . Thus, $$v_{n+1}⩽\frac{1}{1+\mathit{hu}}{v}_n+\frac{{\mathit{hke}}^{-{\alpha }_2{\tau }_2}}{1+\mathit{hu}}M\text{.}$$ By induction, we get $$v_n⩽{\left(\frac{1}{1+\mathit{hu}}\right)}^n{v}_0+\frac{\mathit{kM}}{u}{e}^{-{\alpha }_2{\tau }_2}\left[1-{\left(\frac{1}{1+\mathit{hu}}\right)}^n\right]⩽v_0+\frac{\mathit{kM}}{u}{e}^{-{\alpha }_2{\tau }_2}\text{,}$$ Therefore, $\{v_n\}$ is bounded. This completes the proof. □

Remark 2.2

If $x_0>0\text{,}{y}_0>0$ and $v_0>0$ , then all solutions of system (2) subject to condition (3) are positive for any $n>0$ . In this case, the mixed Euler numerical scheme for the system (1) is called unconditionally positive.

Theorem 2.3

Let us define $R_0=\frac{k}{\mathit{au}}f\left(\frac{\lambda }{d}\text{,}0\text{,}0\right)e^{-{\alpha }_1{\tau }_1-{\alpha }_2{\tau }_2}$ .

1. If $R_0⩽1$ , then the system (2) has a unique disease-free equilibrium of the form $E_f\left(\frac{\lambda }{d}\text{,}0\text{,}0\right)$ .

2. If $R_0>1$ , then the system (2) has a unique chronic infection equilibrium of the form $E^{\ast }(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })$ besides $E_f$ , where $x^{\ast }\in (\frac{\lambda }{d}\text{,}0)\text{,}{y}^{\ast }=\frac{\lambda -{\mathit{dx}}^{\ast }}{{\mathit{ae}}^{{\alpha }_1{\tau }_1}}$ and $v^{\ast }=\frac{{\mathit{ky}}^{\ast }}{{\mathit{ue}}^{{\alpha }_2{\tau }_2}}$ .

Theorem 3.1

The disease-free equilibrium $E_f$ of system (2) is globally asymptotically stable whenever $R_0⩽1$ , and unstable otherwise.

Proof

We construct a discrete Lyapunov functional as follows $$\begin{array}{cc}\hfill L_n& =x_n-x^0-{\int }_{x^0}^{x_n}\frac{f(x^0\text{,}0\text{,}0)}{f(s\text{,}0\text{,}0)}\mathit{ds}+e^{{\alpha }_1{\tau }_1}{y}_n+\frac{a}{k}{e}^{{\alpha }_1{\tau }_1+{\alpha }_2{\tau }_2}(1+\mathit{uh})v_n\hfill \\ \hfill & +h{\displaystyle \sum _{j=n-m_1}^{n-1}}f(x_{j+1}\text{,}{y}_j\text{,}{v}_j)v_j+{\mathit{ae}}^{{\alpha }_1{\tau }_1}h{\displaystyle \sum _{j=n-m_2}^{n-1}}{y}_{j+1}\text{,}\hfill \end{array}$$ Calculating the first difference of $L_n$ along the positive solution of system (2), we have $$\begin{array}{cc}\hfill \Delta L_n& =L_{n+1}-L_n\hfill \\ \hfill & =x_{n+1}-x_n-{\int }_{x_n}^{x_{n+1}}\frac{f(x^0\text{,}0\text{,}0)}{f(s\text{,}0\text{,}0)}\mathit{ds}+e^{{\alpha }_1{\tau }_1}(y_{n+1}-y_n)\hfill \\ \hfill & +\frac{a}{k}(1+\mathit{uh})e^{{\alpha }_1{\tau }_1+{\alpha }_2{\tau }_2}(v_{n+1}-v_n)\hfill \\ \hfill & +h\left(f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)v_n-f(x_{n-m_1+1}\text{,}{y}_{n-m_1}\text{,}{v}_{n-m_1})v_{n-m_1}\right)\hfill \\ \hfill & +{\mathit{ae}}^{{\alpha }_1{\tau }_1}h\left(y_{n+1}-y_{n-m_2+1}\right)\hfill \\ \hfill & ⩽\left(1-\frac{f(x^0\text{,}0\text{,}0)}{f(x_{n+1}\text{,}0\text{,}0)}\right)(x_{n+1}-x_n)+e^{{\alpha }_1{\tau }_1}(y_{n+1}-y_n)\hfill \\ \hfill & +\frac{a}{k}(1+\mathit{uh})e^{{\alpha }_1{\tau }_1+{\alpha }_2{\tau }_2}(v_{n+1}-v_n)\hfill \\ \hfill & +h\left(f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)v_n-f(x_{n-m_1+1}\text{,}{y}_{n-m_1}\text{,}{v}_{n-m_1})v_{n-m_1}\right)\hfill \\ \hfill & +{\mathit{ae}}^{{\alpha }_1{\tau }_1}h\left(y_{n+1}-y_{n-m_2+1}\right)\text{.}\hfill \end{array}$$ Using the equality $\lambda ={\mathit{dx}}^0$ , we get $$\begin{array}{cc}\hfill \Delta L_n& ⩽{\mathit{hdx}}^0\left(1-\frac{x_{n+1}}{x^0}\right)\left(1-\frac{f(x^0\text{,}0\text{,}0)}{f(x_{n+1}\text{,}0\text{,}0)}\right)+\frac{\mathit{auh}}{k}{e}^{{\alpha }_1{\tau }_1+{\alpha }_2{\tau }_2}\left(\frac{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)}{f(x_{n+1}\text{,}0\text{,}0)}{R}_0-1\right)v_n\hfill \\ \hfill & ⩽{\mathit{hdx}}^0\left(1-\frac{x_{n+1}}{x^0}\right)\left(1-\frac{f(x^0\text{,}0\text{,}0)}{f(x_{n+1}\text{,}0\text{,}0)}\right)+\frac{\mathit{auh}}{k}{e}^{{\alpha }_1{\tau }_1+{\alpha }_2{\tau }_2}\left(R_0-1\right)v_n\text{.}\hfill \end{array}$$ Since $f(x\text{,}y\text{,}v)$ is strictly monotonically increasing with respect to x, we have $$\left(1-\frac{x_{n+1}}{x^0}\right)\left(1-\frac{f(x^0\text{,}0\text{,}0)}{f(x_{n+1}\text{,}0\text{,}0)}\right)⩽0\text{.}$$ Since $R_0\le 1$ , we have $\mathrm{\Delta }{L}_n=L_{n+1}-L_n⩽0$ for all $n⩾0$ . Then, $\{L_n\}$ is a monotonically decreasing sequence. Since $L_n⩾0$ , we have ${\lim }_{n\to +\infty }{L}_n⩾0$ . Hence ${\lim }_{n\to +\infty }(L_{n+1}-L_n)=0$ , from which we obtain ${\lim }_{n\to +\infty }{x}_{n+1}=x^0$ and ${\lim }_{n\to +\infty }\left((R_0-1)v_n\right)=0$ . We discuss two cases:

• If $R_0<1$ , then ${\lim }_{n\to +\infty }{v}_n=0$ . By the third equation of (2), we get ${\lim }_{n\to +\infty }{y}_n=0$ .

• If $R_0=1$ . Using ${\lim }_{n\to +\infty }{x}_n=x^0$ and the first equation of (2), it is not hard to have ${\lim }_{n\to +\infty }{v}_n=0$ .

By the above discussion, we deduce that $E_f$ is globally asymptotically stable if $R_0\le 1$ .

Now, we prove that the disease-free equilibrium $E_f$ is unstable when $R_0>1$ . Calculating the linearization system of model (2) at equilibrium $E_f$ , we get a new system of the form

$$\left\{\begin{array}{c}{X}_{n+1}=X_n+h\left(-{\mathit{dX}}_{n+1}-f(\frac{\lambda }{d}\text{,}0\text{,}0)V_n\right)\text{,}\hfill \\ Y_{n+1}=Y_n+h\left(e^{-{\alpha }_1{\tau }_1}f(\frac{\lambda }{d}\text{,}0\text{,}0)V_{n-m_1}-{\mathit{aY}}_{n+1}\right)\text{,}\hfill \\ V_{n+1}=V_n+h\left({\mathit{ke}}^{-{\alpha }_2{\tau }_2}{Y}_{n-m_2+1}-{\mathit{uV}}_{n+1}\right)\text{,}\hfill \end{array}\right.$$ where $X_n=x_n-\frac{\lambda }{d}\text{,}{Y}_n=y_n$ and $V_n=v_n$ . Let $Z_n={(X_n\text{,}{Y}_n\text{,}{V}_n)}^T$ . Then, system (4) is equivalent to

(5)

$$Z_{n+1}={\mathit{AZ}}_n+{\mathit{BZ}}_{n-m_1}+{\mathit{CZ}}_{n-m_2}+{\mathit{DZ}}_{n-m_1-m_2}\text{,}$$ where $$\begin{array}{cc}\hfill & A=\left(\begin{array}{ccc}\hfill \frac{1}{1+\mathit{hd}}\hfill & \hfill 0\hfill & \hfill -\frac{\mathit{hf}\left(\frac{\lambda }{d}\text{,}0\text{,}0\right)}{1+\mathit{hd}}\hfill \\ \hfill 0\hfill & \hfill \frac{1}{1+\mathit{ha}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{1+\mathit{hu}}\hfill \\ \hfill \hfill \end{array}\right)\text{,}B=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{\mathit{hf}\left(\frac{\lambda }{d}\text{,}0\text{,}0\right)}{1+\mathit{ha}}{e}^{-{\alpha }_1{\tau }_1}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill \end{array}\right)\text{,}\hfill \\ \hfill & C=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{\mathit{hk}}{(1+\mathit{ha})(1+\mathit{hu})}{e}^{-{\alpha }_2{\tau }_2}\hfill & \hfill 0\hfill \\ \hfill \hfill \end{array}\right)\text{,}D=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{h^2\mathit{kf}\left(\frac{\lambda }{d}\text{,}0\text{,}0\right)}{(1+\mathit{ha})(1+\mathit{hu})}{e}^{-{\alpha }_1{\tau }_1-{\alpha }_2{\tau }_2}\hfill \\ \hfill \hfill \end{array}\right)\text{.}\hfill \end{array}$$ Hence, the characteristic equation corresponding to linearized system (5) is given by $$\det (A-\xi I+{\xi }^{-m_1}B+{\xi }^{-m_2}C+{\xi }^{-m_1-m_2}D)=0\text{.}$$ Therefore, this characteristic equation can be rewritten as $$\left(\frac{1}{1+\mathit{hd}}-\xi \right)P(\xi )=0\text{,}$$ where $$P(\xi )=(1+\mathit{ha})(1+\mathit{hu}){\xi }^{m_1+m_2+1}-(2+\mathit{ha}+\mathit{hu}){\xi }^{m_1+m_2}+{\xi }^{m_1+m_2-1}-h^2\mathit{kf}(\frac{\lambda }{d}\text{,}0\text{,}0)e^{-{\alpha }_1{\tau }_1-{\alpha }_2{\tau }_2}\text{.}$$ Clearly, $\xi =\frac{1}{1+\mathit{hd}}$ is the root of this equation. The remaining roots are given by the solutions of the equation $P(\xi )=0$ .

We have $P(1)=h^2\mathit{au}(1-R_0)<0\text{,}{\lim }_{\xi \to +\infty }P(\xi )=+\infty$ and P is a continuous function on interval $\in [1\text{,}+\infty )$ . Then there exists a $\overline{\xi }\in (1\text{,}+\infty )$ such that $P(\overline{\xi })=0$ . Therefore, $E_f$ is unstable when $R_0>1$ . This completes the proof.  □

Remark 4.1

The assumption (H4) is satisfied by various types of the incidence rate including the mass action when $f(x\text{,}y\text{,}v)=\beta x$ , the saturation incidence when $f(x\text{,}y\text{,}v)=\frac{\beta x}{1+\alpha v}$ , the incidence function was used in Zhuo (2012) and Sun and Min (2014) when $f(x\text{,}y\text{,}v)=\frac{\beta x}{x+v}$ , Beddington–DeAngelis response when $f(x\text{,}y\text{,}v)=\frac{\beta x}{1+{\alpha }_{1}x+{\alpha }_{2}v}$ , Crowley-Martin response when $f(x\text{,}y\text{,}v)=\frac{\beta x}{1+{\alpha }_{1}x+{\alpha }_{2}v+{\alpha }_1{\alpha }_2\mathit{xv}}$ and the more generalized response introduced by Hattaf et al. (see Section 5 in Hattaf et al. (2013)) when $f(x\text{,}y\text{,}v)=\frac{\beta x}{1+{\alpha }_{1}x+{\alpha }_{2}v+{\alpha }_3\mathit{xv}}$ , where $\beta$ is a positive constant rate describing the infection process, $\alpha \text{,}{\alpha }_1\text{,}{\alpha }_2$ and ${\alpha }_3$ are nonnegative constants. Further, the fourth hypothesis given in Wang et al. (2013) on the incidence function depending only of x and v is a particular case of the assumption (H4).

Theorem 4.2

Assume $R_0>1$ and (H4) hold. Then the chronic infection equilibrium $E^{\ast }$ of system (2) is globally asymptotically stable.

Proof

We define a discrete Lyapunov functional as follows $$\begin{array}{cc}\hfill W_n& =x_n-x^{\ast }-{\int }_{x^{\ast }}^{x_n}\frac{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(s\text{,}{y}^{\ast }\text{,}{v}^{\ast })}\mathit{ds}+e^{{\alpha }_1{\tau }_1}{y}^{\ast }\varphi \left(\frac{y_n}{y^{\ast }}\right)+\frac{a}{k}(1+\mathit{uh})e^{{\alpha }_1{\tau }_1+{\alpha }_2{\tau }_2}{v}^{\ast }\varphi \left(\frac{v_n}{v^{\ast }}\right)\hfill \\ \hfill & +f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })v^{\ast }h{\displaystyle \sum _{j=n-m_1}^{n-1}}\varphi \left(\frac{f(x_{j+1}\text{,}{y}_j\text{,}{v}_j)v_j}{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })v^{\ast }}\right)+{\mathit{ae}}^{{\alpha }_1{\tau }_1}{y}^{\ast }h{\displaystyle \sum _{j=n-m_2}^{n-1}}\varphi \left(\frac{y_{j+1}}{y^{\ast }}\right)\text{,}\hfill \end{array}$$ where $\varphi (x)=x-1-\ln x\text{,}x\in {\mathbb{R}}^{+}$ . Clearly, $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ attains its strict global minimum at $x=1$ and $\varphi (1)=0$ .

The function $\psi :x\mapsto x-x^{\ast }-{\int }_{x^{\ast }}^x\frac{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(s\text{,}{y}^{\ast }\text{,}{v}^{\ast })}\mathit{ds}$ has the global minimum at $x=x^{\ast }$ and $\psi (x^{\ast })=0$ . So, $\psi (x)⩾0$ for all $x>0$ . Thus, $W_n⩾0$ with equality holding if and only if $\frac{x_n}{x^{\ast }}=\frac{y_n}{y^{\ast }}=\frac{v_n}{v^{\ast }}=1$ for all $n⩾0$ .

The first difference of $W_n$ satisfies $$\begin{array}{cc}\hfill \Delta W_n& =W_{n+1}-W_n\hfill \\ \hfill & =x_{n+1}-x_n-{\int }_{x_n}^{x_{n+1}}\frac{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(s\text{,}{y}^{\ast }\text{,}{v}^{\ast })}\mathit{ds}+e^{{\alpha }_1{\tau }_1}\left(y_{n+1}-y_n+y^{\ast }\ln \left(\frac{y_n}{y_{n+1}}\right)\right)\hfill \\ \hfill & +\frac{a}{k}(1+\mathit{uh})e^{{\alpha }_1{\tau }_1+{\alpha }_2{\tau }_2}\left(v_{n+1}-v_n+v^{\ast }\ln \left(\frac{v_n}{v_{n+1}}\right)\right)\hfill \\ \hfill & +f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })v^{\ast }h\left(\varphi \left(\frac{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)v_n}{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })v^{\ast }}\right)-\varphi \left(\frac{f(x_{n-m_1+1}\text{,}{y}_{n-m_1}\text{,}{v}_{n-m_1})v_{n-m_1}}{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })v^{\ast }}\right)\right)\hfill \\ \hfill & +{\mathit{ae}}^{{\alpha }_1{\tau }_1}{y}^{\ast }h\left(\varphi \left(\frac{y_{n+1}}{y^{\ast }}\right)-\varphi \left(\frac{y_{n-m_2+1}}{y^{\ast }}\right)\right)\hfill \\ \hfill & ⩽\left(1-\frac{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}\right)\left(x_{n+1}-x_n\right)+e^{{\alpha }_1{\tau }_1}\left(y_{n+1}-y_n+y^{\ast }\ln \left(\frac{y_n}{y_{n+1}}\right)\right)\hfill \\ \hfill & +\frac{a}{k}(1+\mathit{uh})e^{{\alpha }_1{\tau }_1+{\alpha }_2{\tau }_2}\left(v_{n+1}-v_n+v^{\ast }\ln \left(\frac{v_n}{v_{n+1}}\right)\right)\hfill \\ \hfill & +\mathit{hf}(x_{n+1}\text{,}{y}_n\text{,}{v}_n)v_n-\mathit{hf}(x_{n-m_1+1}\text{,}{y}_{n-m_1}\text{,}{v}_{n-m_1})v_{n-m_1}\hfill \\ \hfill & +f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })v^{\ast }h\ln \left(\frac{f(x_{n-m_1+1}\text{,}{y}_{n-m_1}\text{,}{v}_{n-m_1})v_{n-m_1}}{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)v_n}\right)\hfill \\ \hfill & +{\mathit{ae}}^{{\alpha }_1{\tau }_1}h\left(y_{n+1}-y_{n-m_2+1}+y^{\ast }\ln \left(\frac{y_{n-m_2+1}}{y_{n+1}}\right)\right)\text{.}\hfill \end{array}$$ By the inequality $\ln x⩽x-1$ , we get $$\begin{array}{cc}\hfill \Delta W_n& ⩽(1-\frac{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })})(x_{n+1}-x_n)\hfill \\ \hfill & +e^{{\alpha }_1{\tau }_1}(1-\frac{y^{\ast }}{y_{n+1}})(y_{n+1}-y_n)\hfill \\ \hfill & +\frac{a}{k}{e}^{{\alpha }_1{\tau }_1+{\alpha }_2{\tau }_2}(1-\frac{v^{\ast }}{v_{n+1}})(v_{n+1}-v_n)\hfill \\ \hfill & +\frac{\mathit{auh}}{k}{e}^{{\alpha }_1{\tau }_1+{\alpha }_2{\tau }_2}\left(v_{n+1}-v_n+v^{\ast }\ln \left(\frac{v_n}{v_{n+1}}\right)\right)\hfill \\ \hfill & +\mathit{hf}(x_{n+1}\text{,}{y}_n\text{,}{v}_n)v_n-\mathit{hf}(x_{n-m_1+1}\text{,}{y}_{n-m_1}\text{,}{v}_{n-m_1})v_{n-m_1}\hfill \\ \hfill & +f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })v^{\ast }h\ln \left(\frac{f(x_{n-m_1+1}\text{,}{y}_{n-m_1}\text{,}{v}_{n-m_1})v_{n-m_1}}{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)v_n}\right)\hfill \\ \hfill & +{\mathit{ae}}^{{\alpha }_1{\tau }_1}h\left(y_{n+1}-y_{n-m_2+1}+y^{\ast }\ln \left(\frac{y_{n-m_2+1}}{y_{n+1}}\right)\right)\text{.}\hfill \end{array}$$ Using $\lambda ={\mathit{dx}}^{\ast }+{\mathit{ay}}^{\ast }{e}^{{\alpha }_1{\tau }_1}\text{,}f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })v^{\ast }={\mathit{ay}}^{\ast }{e}^{{\alpha }_1{\tau }_1}$ and $\frac{u}{k}{e}^{{\alpha }_2{\tau }_2}=\frac{y^{\ast }}{v^{\ast }}$ , we obtain $$\begin{array}{cc}\hfill \Delta W_n& ⩽{\mathit{hdx}}^{\ast }\left(1-\frac{x_{n+1}}{x^{\ast }}\right)\left(1-\frac{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}\right)\hfill \\ \hfill & +{\mathit{hay}}^{\ast }{e}^{{\alpha }_1{\tau }_1}\left(1-\frac{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}+\frac{v_n}{v^{\ast }}\frac{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)}{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}\right)\hfill \\ \hfill & +{\mathit{hay}}^{\ast }{e}^{{\alpha }_1{\tau }_1}\left(1-\frac{y^{\ast }}{y_{n+1}}\frac{v_{n-m_1}}{v^{\ast }}\frac{f(x_{n-m_1+1}\text{,}{y}_{n-m_1}\text{,}{v}_{n-m_1})}{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })}\right)\hfill \\ \hfill & +{\mathit{hay}}^{\ast }{e}^{{\alpha }_1{\tau }_1}\left(1-\frac{v_n}{v^{\ast }}-\frac{y_{n-m_2+1}{v}^{\ast }}{y^{\ast }{v}_{n+1}}\right)\hfill \\ \hfill & +{\mathit{hay}}^{\ast }{e}^{{\alpha }_1{\tau }_1}\ln \left(\frac{f(x_{n-m_1+1}\text{,}{y}_{n-m_1}\text{,}{v}_{n-m_1})v_{n-m_1}{y}_{n-m_2+1}}{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)v_{n+1}{y}_{n+1}}\right)\hfill \\ \hfill & ={\mathit{hdx}}^{\ast }{e}^{{\alpha }_1{\tau }_1}\left(1-\frac{x_{n+1}}{x^{\ast }}\right)\left(1-\frac{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}\right)\hfill \\ \hfill & +{\mathit{hay}}^{\ast }{e}^{{\alpha }_1{\tau }_1}\left[4-\frac{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}-\frac{y^{\ast }}{y_{n-m_2+1}}\frac{v_n}{v^{\ast }}\frac{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)}{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })}-\frac{y_{n-m_2+1}{v}^{\ast }}{y^{\ast }{v}_{n+1}}\right.\hfill \\ \hfill & \left.-\frac{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)}\right]+{\mathit{hay}}^{\ast }{e}^{{\alpha }_1{\tau }_1}\left(-1-\frac{v_n}{v^{\ast }}+\frac{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)}+\frac{v_n}{v^{\ast }}\frac{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)}{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}\right)\hfill \\ \hfill & +{\mathit{hay}}^{\ast }{e}^{{\alpha }_1{\tau }_1}\left(\frac{y^{\ast }}{y_{n-m_2+1}}\frac{v_n}{v^{\ast }}\frac{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)}{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })}-\frac{y^{\ast }}{y_{n+1}}\frac{v_{n-m_1}}{v^{\ast }}\frac{f(x_{n-m_1+1}\text{,}{y}_{n-m_1}\text{,}{v}_{n-m_1})}{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })}\right)\hfill \\ \hfill & +{\mathit{hay}}^{\ast }{e}^{{\alpha }_1{\tau }_1}\ln \left(\frac{f(x_{n-m_1+1}\text{,}{y}_{n-m_1}\text{,}{v}_{n-m_1})v_{n-m_1}{y}_{n-m_2+1}}{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)v_{n+1}{y}_{n+1}}\right)\hfill \\ \hfill & ={\mathit{hdx}}^{\ast }{e}^{{\alpha }_1{\tau }_1}\left(1-\frac{x_{n+1}}{x^{\ast }}\right)\left(1-\frac{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}\right)\hfill \\ \hfill & +{\mathit{hay}}^{\ast }{e}^{{\alpha }_1{\tau }_1}\left(-1-\frac{v_n}{v^{\ast }}+\frac{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)}+\frac{v_n}{v^{\ast }}\frac{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)}{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}\right)\hfill \\ \hfill & -{\mathit{hay}}^{\ast }{e}^{{\alpha }_1{\tau }_1}\left[\varphi \left(\frac{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}\right)+\varphi \left(\frac{y_{n-m_2+1}{v}^{\ast }}{y^{\ast }{v}_{n+1}}\right)+\varphi \left(\frac{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)}\right)\right.\hfill \\ \hfill & \left.+\varphi \left(\frac{y^{\ast }}{y_{n+1}}\frac{v_{n-m_1}}{v^{\ast }}\frac{f(x_{n-m_1+1}\text{,}{y}_{n-m_1}\text{,}{v}_{n-m_1})}{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })}\right)\right]\text{.}\hfill \end{array}$$ Since $f(x\text{,}y\text{,}v)$ is strictly monotonically increasing with respect to x, we obtain that $$\left(1-\frac{x_{n+1}}{x^{\ast }}\right)\left(1-\frac{f(x^{\ast }\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}\right)⩽0\text{.}$$ According to (H4), we have $$-1-\frac{v_n}{v^{\ast }}+\frac{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)}+\frac{v_n}{v^{\ast }}\frac{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)}{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}=\left(1-\frac{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)}{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}\right)\left(\frac{f(x_{n+1}\text{,}{y}^{\ast }\text{,}{v}^{\ast })}{f(x_{n+1}\text{,}{y}_n\text{,}{v}_n)}-\frac{v_n}{v^{\ast }}\right)⩽0\text{.}$$ Since $\varphi (x)⩾0$ for $x>0$ , we deduce that $\mathrm{\Delta }{W}_n⩽0$ for all $n⩾0$ . Then, $\{W_n\}$ is a monotonically decreasing sequence. Since $W(n)⩾0$ , it follows that ${\lim }_{n\to +\infty }W(n)⩾0$ . Hence, we obtain that ${\lim }_{n\to +\infty }\left(W_{n+1}-W_n\right)=0$ , which implies that ${\lim }_{n\to +\infty }{x}_n=x^{\ast }$ and ${\lim }_{n\to +\infty }\frac{y_{n-m_2+1}}{v_{n+1}}=\frac{y^{\ast }}{v^{\ast }}$ . From system (2), we obtain ${\lim }_{n\to +\infty }{v}_n=v^{\ast }$ and ${\lim }_{n\to +\infty }{y}_n=y^{\ast }$ . Therefore, we conclude that $E^{\ast }$ is globally asymptotically stable.  □