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Global dynamics for discretetime analog of viral infection model with nonlinear incidence and CTL immune response
Advances in Difference Equations volume 2016, Article number: 143 (2016)
Abstract
In this paper, a discretetime analog of a viral infection model with nonlinear incidence and CTL immune response is established by using the Micken nonstandard finite difference scheme. The two basic reproduction numbers \(R_{0}\) and \(R_{1}\) are defined. The basic properties on the positivity and boundedness of solutions and the existence of the virusfree, the noimmune, and the infected equilibria are established. By using the Lyapunov functions and linearization methods, the global stability of the equilibria for the model is established. That is, when \(R_{0}\leq1\) then the virusfree equilibrium is globally asymptotically stable, and under the additional assumption \((A_{4})\) when \(R_{0}>1\) and \(R_{1}\leq1\) then the noimmune equilibrium is globally asymptotically stable and when \(R_{0}>1\) and \(R_{1}>1\) then the infected equilibrium is globally asymptotically stable. Furthermore, the numerical simulations show that even if assumption \((A_{4})\) does not hold, the noimmune equilibrium and the infected equilibrium also may be globally asymptotically stable.
1 Introduction
As is well known, viruses have caused the abundant types of epidemics and are alive almost everywhere on Earth, infecting people, animals, plants, and so on. There are a large number of diseases, which are caused by viruses for example: influenza, hepatitis, HIV, AIDS, SARS, Ebola, MERS. Therefore, it is important to study viral infection, which can supply theoretical evidence for controlling a disease to break out. In the past years, many authors have studied continuous time viral infection models which are described by the differential equations. See, for example, [1–28] and the references cited therein.
In [1], Hattaf et al. proposed the following continuous time viral infection model:
where x, y, and v denote the densities of uninfected cells, infected cells and virus cells, respectively, λ is the rate of production of uninfected cells, d is the death rate of uninfected cells, \(f(x,y,v)\) is the rate of uninfected cells to become infected by virus, a is the rate of disappearance of infected cells, k is the rate that virus produces by infected cells, and u is the rate of virus died. The dynamic behaviors of the model are studied. Although model (1) is simple, model (1) is very important in viral epidemiology, which can show ample viral behaviors. Then, based on continuous model (1), Shi and Dong in [22] proposed a discretetime analog for the special case \(f(x,y,v)=\beta x\) of model (1) by using Micken’s nonstandard finite difference (NSFD) scheme. The authors studied the local and global stability of the equilibria and the permanence of the model. In [26], Hattaf and Yousfi proposed the following discretetime analog directly for model (1) by using the NSFD scheme:
where \(n\in N\), and N denotes the set of all nonnegative integers. The global asymptotic stability of the diseasefree equilibrium and the chronic infection equilibrium is established by constructing the suitable Lyapunov functions. In [28], the authors extended model (2) to the delayed case. By using the method of Lyapunov functions, the authors established the global asymptotic stability of the diseasefree equilibrium and the chronic infection equilibrium with no restriction on the timestep size.
In general, our target is to eliminate and control the virus and infected cells. For all this, many authors have noted that the immune response takes great effect to eliminate and control the virus and infected cells because CTL (cytotoxic T lymphocyte) cells affect the virus load. Therefore, a four dimension continuous time virus dynamical model with BeddingtonDeAngelis incidence rate and CTL immune response was studied by Wang, Tao and Song in [2]. The model proposed is as follows:
The authors established the global stability of the diseasefree equilibrium, the immunefree equilibrium, and the endemic equilibrium.
Motivated by the above works, in this paper we consider a discretetime analog of a class of continuous time virus dynamical models with nonlinear incidence and CTL immune response which is established by using NSFD scheme. The model is proposed in the following form:
where \(x_{n}\), \(y_{n}\), \(v_{n}\) and \(z_{n}\) denote the densities of uninfected cells, infected cells, virus cells, and CTL cells at time n, respectively. The parameters λ, d, a, k, and u have the same biological meanings as in model (1), p is the removed rate for the infected cells by the CTL immune response, c is the proliferated rate for the CTL cells by contact with infected cells, b is the disappearance rate for the CTL cells, and the function ϕ is a denominator function (see [29, 30]), which is defined by
It is well known that the nonstandard scheme satisfies the following important rules: the standard denominator h in standard discrete derivative is replaced by a denominator function \(0<\phi(h)<1\), where \(\phi(h)=h+o(h^{2})\) and h is the timestep size of numerical integration, and the nonlinear terms are approximated in a nonlocal way using more than one mesh point (see [31, 32]).
Particularly, when \(f(x,y,v)=\frac{\beta x}{1+mx+nv}\), we can get the corresponding discretetime analog of continuous model (3) as follows:
In this paper, our main purpose is to study the threshold dynamics of model (4). The two basic reproduction numbers \(R_{0}\) and \(R_{1}\) are defined. The basic properties on the positivity and boundedness of solutions and the existence of the virusfree equilibrium, the noimmune equilibrium and the infected equilibrium are established. By using the Lyapunov functions and linearization methods, we will establish a series of criteria to ensure the stability of the equilibria for model (4). That is, we will prove that when \(R_{0}\leq1\) then model (4) only has the virusfree equilibrium and it is globally asymptotically stable, when \(R_{0}>1\) and \(R_{1}\leq1\) then model (4) has only the virusfree and the noimmune equilibria, the virusfree equilibrium is unstable and under the additional assumption \((A_{4})\) (see Section 3) the noimmune equilibrium is globally asymptotically stable, and lastly when \(R_{0}>1\) and \(R_{1}>1\) then model (4) has three equilibria: the virusfree equilibrium, the noimmune equilibrium, and the infected equilibrium; the virusfree and the noimmune equilibria are unstable and under the additional assumption \((A_{4})\) the infected equilibrium is globally asymptotically stable. Furthermore, numerical simulations are given. It is shown that even if assumption \((A_{4})\) does not hold, the noimmune equilibrium may be globally asymptotically stable only when \(R_{0}>1\) and \(R_{1}<1\), and the infected equilibrium may be globally asymptotically stable only when \(R_{0}>1\) and \(R_{1}>1\).
This paper is organized as follows. In Section 2, we will first introduce some assumptions for nonlinear incidence function \(f(x,y,v)\). Next, we will state and prove some basic results on the existence, uniqueness, positivity and ultimate boundedness of solutions with positive initial conditions for model (4). Furthermore, the existence of the virusfree, the noimmune, and the infected equilibria also is obtained. The stability of the virusfree, the noimmune, and the infected equilibria is presented in Section 3. The numerical simulations are presented in Section 4. Lastly, some concluding remarks are presented in Section 5.
2 Preliminaries
As the epidemiological background of model (4), we assume that any solution \((x_{n},y_{n},v_{n},z_{n})\) of model (4) satisfies the following initial condition:
We also require that the function \(f(x,y,z)\) satisfies the following assumptions:
 \((A_{1})\) :

\(f(0,y,v)=0\) for all \(y\geq0\) and \(v\geq0\),
 \((A_{2})\) :

\(\frac{\partial f(x,y,v)}{\partial x}>0\) for all \(x>0\), \(y\geq 0\) and \(v\geq0\),
 \((A_{3})\) :

\(\frac{\partial f(x,y,v)}{\partial y}\leq0\) and \(\frac {\partial f(x,y,v)}{\partial v}\leq0\) for all \(x\geq0\), \(y\geq0\) and \(v\geq0\).
Specially, when \(f(x,y,v)=\frac{\beta x}{1+mx+nv}\) and \(f(x,y,v)=\frac {\beta x}{1+nv^{q}}\), where \(\beta>0\), \(m\geq0\), \(q\geq0\), and \(n\geq 0\) are constants, by simple calculation we know that such \(f(x,y,v)\) satisfies the above assumptions \((A_{1})\)\((A_{3})\).
Lemma 1
Let \((A_{1})\) and \((A_{2})\) hold. Then the solution \((x_{n},y_{n},v_{n},z_{n})\) of model (4) with initial value (6) exists uniquely and is positive for all \(n\in N\). In addition, \(0< y_{n}<\frac{1+b\phi}{c\phi}\) for \(n=1,2, \ldots\) .
Proof
We know that model (4) is equivalent to the following form:
When \(n=0\), we prove that \((x_{1}, y_{1}, v_{1}, z_{1})\) exists uniquely and is positive.
We first consider \(x_{1}\). According to the first equation of model (7), we have
Owing to \(\varphi(0)=x_{0}\phi\lambda<0\), \(\lim_{x_{1}\to\infty}\varphi (x_{1})=\infty\) and from \((A_{2})\),
Hence, there is a unique \(x_{1}>0\) such that \(x_{1}=x_{0}+\phi[\lambdad x_{1}f(x_{1},y_{0},v_{0})v_{0}]\).
Next, we consider \(z_{1}\). According to the second and fourth equations of model (7), we have
Let
This is a quadratic function. Since \(\varphi(0)=z_{0}(1+\phi a)<0\) and \(\lim_{z_{1}\to\infty}\varphi (z_{1})=\infty\), there is a unique \(z_{1}>0\) such that \(\varphi(z_{1})=0\). That is, (8) holds.
In the following, we consider \(y_{1}\). According to the second and last equations of model (7), we have
Let
Owing to \(z_{1}>0\), from the last equation of model (7) we have \(y_{1}<\frac{1+\phi b}{\phi c}\). Then we have
Since \(\varphi(y_{1})\) is a quadratic function, there is a unique \(y_{1} \in(0,\frac{1+\phi b}{\phi c})\) such that \(\varphi(y_{1})=0\). That is, (9) holds.
Finally, we consider \(v_{1}\). According to the third equation of model (7), we have \(v_{1}=\frac{v_{0}+\phi ky_{1}}{1+\phi u}\). Hence, we know that \(v_{1}\) uniquely exists and is positive. Therefore, \((x_{1},y_{1},v_{1},z_{1})\) exists uniquely and is positive.
When \(n=1\), by a similar argument to the above, we can prove that \((x_{2}, y_{2}, v_{2}, z_{2})\) exists uniquely and is positive. Owing to \(z_{2}>0\), we also have \(y_{2}<\frac{1+\phi b}{\phi c}\). Using the mathematical induction, for any \(n\geq0\), we know that \((x_{n},y_{n},v_{n},z_{n})\) exists uniquely and is positive. Furthermore, we also have \(y_{n}<\frac{1+\phi b}{\phi c}\). This completes the proof. □
Let us consider the region
where \(\xi=\min\{d,\frac{a}{2},u,b\}\). We have the following result.
Lemma 2
Any solution \((x_{n},y_{n},v_{n},z_{n})\) of model (4) with initial condition (6) converges on Γ as \(n\to\infty\), and Γ is positive invariable for model (4).
Proof
Define a sequence \(M_{n}\) as follows:
We have
Hence,
By using the induction, we have
Consequently, \(\limsup_{n \rightarrow\infty} M_{n}\leq\frac{\lambda}{\xi}\). Owing to the positivity of solution \(({x_{n}},{y_{n}},{v_{n}},{z_{n}})\), we see that \(({x_{n}},{y_{n}},{v_{n}},{z_{n}})\) converges on Γ as \(n\to \infty\). Furthermore, from Lemma 1 and (10), we easily see that Γ is positive invariable for model (4). This completes the proof. □
The basic reproductive numbers for model (4) are given by
where \(y^{*}_{1}\) is given in the following conclusion (ii) of Lemma 3. \(R_{0}\) is defined as the average number of secondary infected cells generated by a single infected cell put in an uninfected cell (or free virus) population, \(R_{1}\) is defined as the average number of killed infected cells by a single CTL cell contacting the infected cells. Based on these basic reproductive numbers, we give the following lemma.
Lemma 3
Let \((A_{1})\)\((A_{3})\) hold.

(i)
Model (4) always has a virusfree equilibrium \(E_{0}(\frac {\lambda}{d},0,0,0)\).

(ii)
If \(R_{0}\leq1\), then model (4) has only a virusfree equilibrium \(E_{0}\), and if \(R_{0}>1\), then model (4) has a noimmune equilibrium \(E_{1}(x^{*}_{1},y^{*}_{1},v^{*}_{1},0)\), except for equilibrium \(E_{0}\).

(iii)
If \(R_{0}>1\) and \(R_{1}\leq1\), then model (4) has only the virusfree equilibrium \(E_{0}\) and the noimmune equilibrium \(E_{1}\), and if \(R_{0}>1\) and \(R_{1}>1\), then model (4) has an infected equilibrium \(E_{2}(x^{*}_{2},y^{*}_{2},v^{*}_{2},z^{*}_{2})\), except for equilibria \(E_{0}\) and \(E_{1}\).
Proof
It is clear that the equilibrium of model (4) satisfies the following equation:
Obviously, (11) has a solution \((\frac{\lambda}{d},0,0,0)\). Hence, model (4) always has a virusfree equilibrium \(E_{0}(\frac {\lambda}{d},0,0,0)\). This shows conclusion (i).
Let \(z=0\), from (11) we have \(y=\frac{\lambdad x}{a}\), \(v=\frac{k(\lambdad x)}{au}\), and \(f(x,y,v)v=ay\). Hence,
We have \(g_{1}(0)=\frac{au}{k}<0\) and \(g_{1}(\frac{\lambda}{d})=\frac {au}{k}(R_{0}1)\). Based on \((A_{2})\) and \((A_{3})\), we know that \(g_{1}(x)\) is monotonously increasing for all \(x\in(0,\frac{\lambda}{d})\). When \(R_{0}>1\), then \(g_{1}(\frac{\lambda}{d})>0\). Hence, \(g_{1}(x)=0\) has a unique solution \(x_{1}^{*}\in(0,\frac{\lambda}{d})\). This shows that model (4) has a unique noimmune equilibrium \(E_{1}(x^{*}_{1},y^{*}_{1},v^{*}_{1},0)\) with \(y^{*}_{1}=\frac{\lambda d x^{*}_{1}}{a}\) and \(v^{*}_{1}=\frac{k(\lambdad x^{*}_{1})}{au}\). When \(R_{0}\leq1\), then \(g_{1}(\frac{\lambda}{d})\leq0\). Hence, \(g_{1}(x)=0\) has no solution in \((0,\frac{\lambda}{d})\). This shows that model (4) has only equilibrium \(E_{0}\). Therefore, conclusion (ii) is true.
Let \(z\neq0\), from (11) we have \(y=\frac{b}{c}\), \(v=\frac {kb}{uc}\), and the following equation:
We have \(g_{2}(0)=\frac{uc\lambda}{kb}<0\). When \(R_{0}>1\) and \(R_{1}>1\), we know \(y^{*}_{1}>\frac{b}{c}\). Owing to \(y^{*}_{1}=\frac{\lambdad x^{*}_{1}}{a}\), by simply calculating we can obtain \(x^{*}_{1}<\frac{\lambda}{d }\frac{ab}{d c}\). Hence,
From \((A_{2})\), we know that \(g_{2}(x)\) is monotonously increasing for \(x>0\). Hence, there is a unique \(x^{*}_{2}\in(0,\frac{\lambda }{d}\frac{ab}{d c})\) such that \(g_{2}(x^{*}_{2})=0\). This shows that model (4) has a unique infected equilibrium \(E_{2}(x^{*}_{2},y^{*}_{2},v^{*}_{2},z^{*}_{2})\) with \(y^{*}_{2}=\frac {b}{c}\), \(v^{*}_{2}=\frac{kb}{uc}\) and
When \(R_{0}>1\) and \(R_{1}\leq1\), similarly to above discussion we can see that if \(g_{2}(x)=0\) has a positive solution \(x_{2}^{*}\), then \(x_{2}^{*}>\frac{\lambda}{d}\frac{ab}{d c}\). But, if model (4) has a positive equilibrium \(E_{2}(x^{*}_{2},y^{*}_{2},v^{*}_{2},z^{*}_{2})\), then \(z^{*}_{2} =\frac{1}{py^{*}_{2}}(\lambdad x^{*}_{2}ay^{*}_{2})>0\). Hence, we must have \(x_{2}^{*}<\frac{\lambda}{d}\frac{ab}{d c}\), which leads to a contradiction. Therefore, conclusion (iii) is true. This completes the proof. □
3 Stability of equilibria
First of all, we introduce the following assumption:
for all \((x,y,v,z)\in\Gamma\), and \((x^{*}_{i},y^{*}_{i},v^{*}_{i})\) is the coordinate of equilibrium \(E_{i}\) for \(i=1,2 \), respectively.
Specially, when \(f(x,y,v)=\frac{\beta x}{1+mx+nv}\), by simple calculation we know that \(f(x,y,v)\) satisfies assumption \((A_{4})\).
However, when \(f(x,y,v)=\frac{\beta x}{1+nv^{2}}\), in Section 4, we will give the numerical examples to indicate that assumption \((A_{4})\) may not be satisfied.
Theorem 1
Suppose that \((A_{1})\)\((A_{3})\) hold. If \(R_{0}\leq1\), then the virusfree equilibrium \(E_{0}(\frac{\lambda}{d},0,0,0)\) of model (4) is globally asymptotically stable.
Proof
Let \(x^{*}=\frac{\lambda}{d}\) and \((x_{n},y_{n},v_{n},z_{n})\) be any solution of model (3) with initial condition (6). Choosing a Lyapunov function as follows:
we let
According to \((A_{2})\), we easily obtain \(m(x)\geq m(x^{*})=0\) for all \(x\geq0\). Therefore, \(W_{n}\geq0\) for all \(x_{n}\geq0\), \(y_{n}\geq0\), \(v_{n}\geq0\), and \(z_{n}\geq0\). In addition, \(W_{n}=0\) if and only if \(x_{n}=x^{*}\), \(y_{n}=0\), \(v_{n}=0\) and \(z_{n}=0\). Computing \(\Delta W_{n}\), we have
Substituting model (4), we have
Substituting \(\lambda=d x^{*}\) and \(v_{n+1}=\frac{v_{n}+\phi ky_{n+1}}{1+\phi u}\) from model (7), we further obtain
Based on \((A_{3})\), we have
According to \((A_{2})\), we know
Therefore, when \(R_{0}\leq1\) and \(z_{n}>0\), \(v_{n}>0\), we get \(\Delta W_{n}\leq0\). It is obvious that \(\Delta W_{n}=0\) if and only if \(x_{n}=x^{*}\), \(y_{n}=0\), \(v_{n}=0\) and \(z_{n}=0\). Based on LaSalle’s invariance principle (see [33]), we finally see that the virusfree equilibrium \(E_{0}(x^{*},0,0,0)\) is globally asymptotically stable. This completes the proof. □
Theorem 2
Suppose that \((A_{1})\)\((A_{3})\) hold. If \(R_{0}>1\), then the virusfree equilibrium \(E_{0}(\frac{\lambda}{d},0,0,0)\) of model (4) is unstable.
Proof
By calculating, we can see that the linearization system of model (4) at equilibrium \(E_{0}\) is
By calculating we can obtain the characteristic equation of system (12),
Solving this equation, we get \(\lambda_{1}=\frac{1}{1+\phi d}\), \(\lambda_{2}=\frac{1}{1+\phi b}\), \(\lambda_{3}\) and \(\lambda_{4}\) are determined by the following equation:
Since when \(R_{0}>1\), we have \(g(1)=\phi^{2}au(1R_{0})<0\) and \(\lim_{\lambda\to\infty}g(\lambda)=\infty\), there exists an \(\eta\in(1,+\infty)\) such that \(g(\eta)=0\). This shows that \(\lambda_{3}\) or \(\lambda_{4}\) is greater than 1. Therefore, the virusfree equilibrium \(E_{0}(\frac{\lambda }{d},0,0,0)\) is unstable. This completes the proof. □
For model (5), by calculating, we see that the basic reproductive numbers \(R_{0}\) and \(R_{1}\) are given by
As a consequence of Theorem 1 and Theorem 2 we have the following result for model (5).
Corollary 1
If \(R_{0}\leq1\), then the virusfree equilibrium \(E_{0}(\frac{\lambda }{d},0,0,0)\) of model (5) is globally asymptotically stable. Otherwise, if \(R_{0}>1\), then equilibrium \(E_{0}\) is unstable.
Theorem 3
Suppose that \((A_{1})\)\((A_{3})\) and \((A_{4})\) for \(i=1\) hold. If \(R_{0}>1\) and \(R_{1}\leq1\), then the noimmune equilibrium \(E_{1}(x^{*}_{1},y^{*}_{1},v^{*}_{1},0)\) of model (4) is globally asymptotically stable.
Proof
Let \((x_{n},y_{n},v_{n},z_{n})\) be any solution of model (4) with initial condition (6). From Lemma 2, we can assume \((x_{n},y_{n},v_{n},z_{n})\in\Gamma\) for all \(n\geq0\). Define a Lyapunov function as follows:
According to \((A_{2})\), we easily obtain
for all \(x\geq0\) and \(x\neq x^{*}\). Hence, \(L_{n}\geq0\) for all \(x_{n}\geq0\), \(y_{n}\geq0\), \(v_{n}\geq0\) and \(z_{n}\geq0\). Obviously, \(L_{n}=0\) if and only if \(x_{n}=x^{*}_{1}\), \(y_{n}=y^{*}_{1}\), \(v_{n}=v^{*}_{1}\), and \(z_{n}=0\). Computing \(\Delta L_{n}\), we have
From \(\ln x\leq x1\) for \(x>0\), we further have
Substituting model (4), owing to \(x^{*}_{1}\), \(y^{*}_{1}\), and \(v^{*}_{1}\) satisfying the equations
we obtain
Since \(\lambda=d x^{*}_{1}+ay^{*}_{1}\), the first equation model (4) becomes
We have
Let \(g(x)=x1\ln x\), then \(g(x)\geq0\) for all \(x>0\). Hence, we can get
According to \((A_{2})\), we know
Since \((A_{4})\) holds for \(i=1\), we further have
Therefore, when \(R_{1}\leq1\), from (13), (14), and (15) we finally obtain \(\Delta L_{n}\leq0\) and \(\Delta L_{n}=0\) if and only if \(x_{n}=x^{*}_{1}\), \(y_{n}=y^{*}_{1}\), \(v_{n}=v^{*}_{1}\) and \(z_{n}=0\). Based on LaSalle’s invariance principle, we see that the noimmune equilibrium \(E_{1}(x^{*}_{1}, y^{*}_{1}, v^{*}_{1},0)\) is globally asymptotically stable. This completes the proof. □
Theorem 4
Suppose that \((A_{1})\)\((A_{3})\) hold. If \(R_{0}>1\) and \(R_{1}>1\), then the noimmune equilibrium \(E_{1}(x^{*}_{1},y^{*}_{1},v^{*}_{1},0)\) of model (4) is unstable.
Proof
By calculating, we easily see that the linearization system of model (4) at equilibrium \(E_{1}(x^{*}_{1},y^{*}_{1},v^{*}_{1},0)\) is
where
By calculating we obtain the characteristic equation of equation (16),
where
Let \(\lambda_{i}\) (\(i=1,2,3,4\)) be the roots of \(f(\lambda)=0\), then \(\lambda_{1}=\frac{1}{1+\phi(bcy^{*}_{1})}\) and \(\lambda_{2}\), \(\lambda_{3}\) and \(\lambda_{4}\) satisfy the equation \(\lambda^{3}+m\lambda^{2}+n\lambda+l=0\). From \(Z_{n}>0\), we know \(\frac{1}{1+\phi(bcy^{*}_{1})}>0\). By \(R_{1}>1\), we have \(\frac{b}{c}< y^{*}_{1}\). Hence, we get \(\frac {1}{1+\phi(bcy^{*}_{1})}>1\). This shows that when \(R_{1}>1\), the noimmune equilibrium \(E_{1}(x^{*}_{1},y^{*}_{1},v^{*}_{1},0)\) is unstable. This completes the proof. □
As a consequence of Theorems 3 and 4 we have the following result for model (5).
Corollary 2
If \(R_{0}>1\) and \(R_{1}\leq1\), then the noimmune equilibrium \(E_{1}(x^{*}_{1},y^{*}_{1},v^{*}_{1},0)\) of model (5) is globally asymptotically stable. Otherwise, if \(R_{0}>1\) and \(R_{1}>1\), then equilibrium \(E_{1}\) is unstable.
Theorem 5
Suppose that \((A_{1})\)\((A_{3})\) and \((A_{4})\) for \(i=2\) hold. If \(R_{0}>1\) and \(R_{1}>1\), then the infected equilibrium \(E_{2}(x^{*}_{2},y^{*}_{2},v^{*}_{2},z^{*}_{2})\) of model (4) is globally asymptotically stable.
Proof
Let \((x_{n},y_{n},v_{n},z_{n})\) be any solution of model (4) with initial condition (6). We can assume by Lemma 2 \((x_{n},y_{n},v_{n},z_{n})\in\Gamma\) for all \(n\geq0\). Define a Lyapunov function as follows:
Obviously, we know \(L_{n}\geq0\) for all \(x_{n}\geq0\), \(y_{n}\geq0\), \(v_{n}\geq0\) and \(z_{n}\geq0\), and \(L_{n}=0\) if and only if \(x_{n}=x^{*}_{2}\), \(y_{n}=y^{*}_{2}\), \(v_{n}=v^{*}_{2}\), and \(z_{n}=z^{*}_{2}\). Computing \(\Delta L_{n}\), we have
Using \(\ln x\leq x1\) for \(x>0\), we further have
Since equilibrium \(E_{2}(x_{2}^{*},y_{2}^{*},v_{2}^{*},z_{2}^{*})\) satisfies the equations
we have
Substituting model (4) and \(\lambda =d x^{*}_{2}+ay^{*}_{2}+py^{*}_{2}z^{*}_{2}\), we have
Let \(g(x)=x1\ln x\), we have
According to \((A_{2})\), we know
Since \((A_{4})\) holds for \(i=2\), we further obtain
Therefore, when \(R_{1}>1\), from (17), (18), and (19) we finally have \(\Delta L_{n}\leq0\). Obviously, \(\Delta L_{n}=0\) if and only if \(x_{n}=x^{*}_{2}\), \(y_{n}=y^{*}_{2}\), \(v_{n}=v^{*}_{2}\), and \(z_{n}=z^{*}_{2}\). Based on LaSalle’s invariance principle, we finally see that the infected equilibrium \(E_{2}(x^{*}_{2},y^{*}_{2},v^{*}_{2},z^{*}_{2})\) is globally asymptotically stable. This completes the proof. □
As a consequence of Theorem 5 we have the following result for model (5).
Corollary 3
Let \(R_{1}>1\). Then the infected equilibrium \(E_{2}(x^{*}_{2},y^{*}_{2},v^{*}_{2},z^{*}_{2})\) of model (5) is globally asymptotically stable.
4 Numerical examples
In this section, we give the numerical examples to discuss assumption \((A_{4})\). In model (4), we choose a nonlinear incidence \(f(x,y,v)=\frac{\beta x}{1+nv^{2}}\). Furthermore, \(h=1\) in the denominator function ϕ. The mortality rate of the CTL response b in model (4) is chosen as a free parameter. All remaining parameters in model (4) are chosen as in Table 1.
We first take the mortality rate of CTL response \(b=0.75\). By calculating, we see that the basic reproduction numbers \(R_{0}\doteq 75>1\) and \(R_{1}\doteq0.5216<1\). Furthermore, we also have \(\frac {\lambda}{\xi}=100\). Hence, model (4) has only the virusfree equilibrium \(E_{0}(100,0,0,0,)\) and the noimmune equilibrium \(E_{1}(21.745,39.127,39.127,0)\).
Consider assumption \((A_{4})\). By calculating we obtain
For \(i=1\), since \(n\frac{\lambda}{\xi}v^{*}_{1}1\doteq38.127>0\), where \(v^{*}_{1}\doteq39.127\), from (20) we see that assumption \((A_{4})\) for \(i=1\) is not satisfied.
However, the numerical simulations given in Figure 1 show that equilibrium \(E_{1}\) is globally asymptotically stable.
We next take the mortality rate of CTL response \(b=0.15\). By calculating, we see that the basic reproduction numbers \(R_{0}\doteq 75>1\) and \(R_{1}\doteq2.608>1\). Furthermore, we also have \(\frac {\lambda}{\xi}=100\). Hence, model (4) has the virusfree equilibrium \(E_{0}(100,0,0,0)\), the noimmune equilibrium \(E_{1}(21.746,39.127,39.127,0)\), and the infected equilibrium \(E_{2}(12.621,15,15,0.383)\).
Consider assumption \((A_{4})\). Since \(n\frac{\lambda}{\xi}v^{*}_{2}1\doteq 14>0\), where \(v^{*}_{2}\doteq15\), from (20) we see that assumption \((A_{4})\) for \(i=2\) is not satisfied.
However, the numerical simulations given in Figure 2 show that equilibrium \(E_{2}\) is globally asymptotically stable.
The above numerical examples show that even if assumption \((A_{4})\) does not hold, the noimmune equilibrium may be globally asymptotically stable only when \(R_{0}>1\) and \(R_{1}<1\), and the infected equilibrium may be globally asymptotically stable only when \(R_{0}>1\) and \(R_{1}>1\).
5 Discussions
In this paper, we studied a four dimensional discretetime virus infected model (4) with general nonlinear incidence function \(f(x,y,v)v\) and CTL immune response obeying Micken’s nonstandard finite difference (NSFD) scheme. Assumptions \((A_{1})\)\((A_{4})\) for nonlinear function \(f(x,y,v)\) are introduced and two basic reproduction numbers \(R_{0}\) and \(R_{1}\) also are defined. The basic properties of model (4) on the existence of the virusfree equilibrium \(E_{0}\), the noimmune equilibrium \(E_{1}\), and the infected equilibrium \(E_{2}\), and the positivity and ultimate boundedness of the solutions are established. Under \((A_{1})\)\((A_{4})\), the global stability and instability of the equilibria are completely determined by the basic reproduction numbers \(R_{0}\) and \(R_{1}\). That is, if \(R_{0}\leq1\) then \(E_{0}\) is globally asymptotically stable, if \(R_{0}>1\) and \(R_{1}\leq1\) then \(E_{0}\) is unstable and \(E_{1}\) is globally asymptotically stable and if \(R_{0}>1\) and \(R_{1}>1\) then \(E_{0}\) and \(E_{1}\) are unstable and \(E_{2}\) is globally asymptotically stable.
We see that \((A_{1})\)\((A_{3})\) are basic for model (4). Particularly, when \(f(x,y,v)=\frac{\beta x}{1+mx+nv}\) and \(f(x,y,v)=\frac{\beta x}{1+nv^{q}}\) then \((A_{1})\)\((A_{3})\) naturally hold. But \((A_{4})\) is a mathematical assumption. It is only used in the proofs of theorems on the global stability of the noimmune equilibrium \(E_{1}\) and the infected equilibrium \(E_{2}\) to obtain \(\Delta L_{n}\leq0\) for the Lyapunov function \(L_{n}\) (see the proofs of Theorem 4 and Theorem 5). However, we also see that when \(f(x,y,v)=\frac{\beta x}{1+mx+nv}\), \((A_{4})\) naturally hold. Furthermore, the numerical simulations given in Section 4 show that even if \((A_{4})\) does not hold, the noimmune equilibrium \(E_{1}\) may be globally asymptotically stable only when \(R_{0}>1\) and \(R_{1}<1\), and the infected equilibrium \(E_{2}\) may be globally asymptotically stable only when \(R_{0}>1\) and \(R_{1}>1\).
Generally, we expect that the global stability of the equilibria for model (4) can be completely determined only by the basic reproduction numbers \(R_{0}\) and \(R_{1}\). Therefore, an open problem is whether \((A_{4})\) can be thrown off in Theorem 4 and Theorem 5. Furthermore, we also do not obtain the local asymptotic stability of the infected equilibrium \(E_{2}\) only under \((A_{1})\)\((A_{3})\). The cause is that the characteristic equation of linearized system of model (4) at equilibrium \(E_{2}\) is very complicated.
When the incidence function \(f(x,y,v)=\frac{\beta x}{1+mx+nv}\), we know that \((A_{1})\)\((A_{4})\) are satisfied. The global stability of the equilibria of the discrete model (5) only depends on the basic reproduction numbers \(R_{0}\) and \(R_{1}\). This shows that the global stability of the equilibria for the discrete model (5) is equal to the corresponding continuous model (3). This implies that the NSFD scheme preserves the stability of the continuous model.
As is well known, in our body the immune response is made up of both a cellular response and a humoral response. The cellular response is that T cells kill the infected cells, the humoral response is that B cells produce an antibody to neutralize the virus. In this paper, we only consider the cellular response. In the future, our work will focus on the idea that the two kinds of immune response simultaneously play a role.
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Acknowledgements
The authors would like to thank the anonymous referees for their very helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant No. 11271312), the Doctoral Subjects Foundation of the Ministry of Education of China (Grant No. 20136501110001).
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Wang, J., Teng, Z. & Miao, H. Global dynamics for discretetime analog of viral infection model with nonlinear incidence and CTL immune response. Adv Differ Equ 2016, 143 (2016). https://doi.org/10.1186/s136620160862y
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DOI: https://doi.org/10.1186/s136620160862y
MSC
 37M99
 39A11
 92D30
Keywords
 viral infection model
 CTL immune response
 NSFD scheme
 basic reproduction number
 local and global stability