Global dynamics for discretetime analog of viral infection model with nonlinear incidence and CTL immune response
 Jianpeng Wang^{1},
 Zhidong Teng^{1}Email author and
 Hui Miao^{1}
https://doi.org/10.1186/s136620160862y
© Wang et al. 2016
Received: 25 January 2016
Accepted: 10 May 2016
Published: 23 May 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.
Keywords
MSC
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 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
 \((A_{1})\) :

\(f(0,y,v)=0\) 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
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. □
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
Lemma 3
 (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
3 Stability of equilibria
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
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
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
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
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
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
List of parameters
Parameter  Definition  Value  Source 

λ  Production rate of uninfected cells  10  
d  Death rate of uninfected cells  0.1  
β  Infection rate  0.15  
a  Death rate of infected cells  0.2  
p  CTL effectiveness  1  
n  Saturation coefficient  0.01  Reference [7] 
k  Production rate of free virus  0.1  
u  Clearance rate of free virus  0.1  
c  Proliferation rate of CTL response  0.01 
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 \(E_{1}(21.745,39.127,39.127,0)\).
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.
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.
Declarations
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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