Stability and bifurcation analysis in a viral infection model with delays
- Xinguo Sun^{1, 2} and
- Junjie Wei^{1}Email author
https://doi.org/10.1186/s13662-015-0664-7
© Sun and Wei 2015
Received: 7 May 2015
Accepted: 6 October 2015
Published: 23 October 2015
Abstract
In this paper, a class of virus infection models with CTLs response is considered. We incorporate an immune delay and two intracellular delays into the virus infection model. It is found that only incorporating two intracellular delays almost does not change the dynamics of the system, but incorporating an immune delay changes the dynamics of the system very greatly, namely, a Hopf bifurcation and oscillations can appear. Those results show immune delay dominates intracellular delays in some viral infection models, which indicates the human immune system has a special effect in virus infection models with CTLs response, and the human immune system itself is very complicated. In fact, people are aware of the complexity of the human immune system in medical science, which coincides with our investigating. We also investigate the global Hopf bifurcation of the system with the immune delay as a bifurcation parameter.
Keywords
1 Introduction
A question is how the intracellular delays \(\tau_{1}\), \(\tau_{2}\) and the immune delay \(\tau_{3}\) affect the dynamics of the system (1.4). That is the main goal of this paper. We find that only incorporating two intracellular delays \(\tau_{1}\) and \(\tau_{2}\) almost does not change the dynamics of the system, but incorporating the immune delay \(\tau_{3}\) changes the dynamics of the system very greatly, namely, a Hopf bifurcation and oscillations can occur. Those results show that the immune delay dominates the intracellular delays in this class of viral infection models, which indicates the human immune system has a special effect in virus infection models with CTLs response, and the human immune system itself is very complicated. People are aware of the complexity of the human immune system in medical science, which coincides with our investigation.
The paper is organized as follows. In the next section, the attractive region and equilibria for system (1.4) are discussed and the two threshold parameters \(R_{0}\) and \(R_{1}\) are introduced. In Section 3, by combining the linear stability theory and the LaSalle-Lyapunov theorem, the global stability of \(P_{0}\) and \(P_{1}\) when \(R_{0}<1\) and \(R_{1}<1<R_{0}\) is discussed, respectively. By analyzing the distribution of the eigenvalues, the dynamics of the system when \(R_{1}>1\) is investigated. In Section 4, we study the global Hopf branch of the system. Numerical simulations are presented in Section 5 to illustrate the analysis results. The paper ends with a brief conclusion.
2 Attractive region and equilibria
Proposition 2.1
Proof
First, we prove that \(x(t)\) is positive for \(t\geq0\). Assuming the contrary and letting \(t_{1}>0\) be the first time such that \(x(t_{1})=0\), by the first equation of system (1.4), we have \(x'(t_{1})=\lambda>0\), and hence \(x(t)<0\) for \(t\in(t_{1}-\eta ,t_{1})\) and sufficiently small \(\eta>0\). This contradicts \(x(t)>0\) for \(t\in[0,t_{1})\). It follows that \(x(t)>0\) for \(t>0\). From the fourth equation of (1.4), we use the method of the steps to prove \(z(t)>0\) for \(t>0\). From the third equation of (1.4), we can prove \(v(t)>0\) for \(t>0\). From the second equation of (1.4), we can obtain \(y(t)>0\) for \(t>0\).
3 Global stability and Hopf bifurcation
We investigate stability of the equilibria and the Hopf bifurcation in this section. First, \(P_{0}\) is considered in the following.
3.1 Global stability of \(P_{0}\)
In this subsection, we rigorously show that when \(R_{0}<1\), the infection-free equilibrium \(P_{0}\) is globally asymptotically stable in Γ.
Theorem 3.1
If \(R_{0}<1\), the infection-free equilibrium \(P_{0}\) of system (1.4) is globally asymptotically stable in Γ. If \(R_{0}>1\), \(P_{0}\) is unstable.
Proof
Therefore, \(P_{0}\) is globally asymptotically stable in Γ.
We can easily see that (3.1) has a root with a positive real part when \(R_{0}>1\). \(P_{0}\) is unstable when \(R_{0}>1\). □
Remark 3.2
Obviously \(P_{0}\) is globally asymptotically stable without any delays when \(R_{0}<1\), but after incorporating three delays (a immune delay and two intracellular delays), \(P_{0}\) is still globally asymptotically stable. Delays do not destroy the globally asymptotical stability of \(P_{0}\).
3.2 Global stability of \(P_{1}\)
Theorem 3.3
If \(R_{1}<1<R_{0}\), then the equilibrium \(P_{1}\) is globally asymptotically stable. If \(R_{1}>1\), \(P_{1}\) is unstable.
Proof
Further, \(P_{1}\) is globally asymptotically stable.
For \(R_{1}>1\), we can find the characteristic equation (3.9) has positive root. Thus \(P_{1}\) is unstable when \(R_{1}>1\). □
Remark 3.4
\(P_{1}\) is globally asymptotically stable without any delays when \(R_{1}<1<R_{0}\). Although incorporating three delays (a immune delay and two intracellular delays), \(P_{1}\) is still globally asymptotically stable. Delays do not destroy the globally asymptotical stability of \(P_{1}\).
3.3 Dynamics when \(R_{1}>1\)
3.3.1 When \(\tau_{1}\geq0\), \(\tau_{2}\geq0\), \(\tau_{3}=0\)
Theorem 3.5
Proof
Remark 3.6
It is very difficult to analyze the characteristic roots of the characteristic equation (3.12). But we conjecture that all characteristic roots of the characteristic equation (3.12) have negative real parts when \(\tau_{1}>0\), \(\tau_{2}>0\), \(\tau_{3}=0\). Namely, \(P_{2}\) is locally asymptotically stable, and \(P_{2}\) is also globally asymptotically stable when \(\tau_{1}\geq0\), \(\tau_{2}\geq0\), \(\tau_{3}=0\). We find the intracellular delays \(\tau_{1}\) and \(\tau_{2}\) do not destroy global attractability of \(P_{2}\).
3.3.2 When \(\tau_{1}= 0\), \(\tau_{2}= 0\), \(\tau_{3}>0\)
Lemma 3.7
- (i)
If \(v<0\), then (3.18) has at least one positive root.
- (ii)
If \(v\geq0\) and \(\Delta\geq0\), then (3.18) has positive roots if and only if \(s_{1} > 0\) and \(F(s_{1})<0\).
- (iii)
If \(v\geq0\) and \(\Delta< 0\), then (3.18) has positive roots if and only if there exists at least one \(s^{*}\in\{s_{1},s_{2},s_{3}\}\) such that \(s^{*} > 0\) and \(F(s^{*})<0\).
After a long and tedious computation, we get the following lemma.
Lemma 3.8
Remark 3.9
From Lemma 3.8, we can get the following result.
Theorem 3.10
Remark 3.11
We find that incorporating an immune delay can destroy the global attractability of \(P_{2}\) on proper conditions when \(R_{1}>1\), and a Hopf bifurcation occurs. That is, a periodic oscillation appears. Stability switches can appear when \(k\geq2\). Those results show immune delay dominates intracellular delays in this class of viral infection models. Those indicate the human immune system has a special effect in virus infection models with a CTLs response, and the human immune system itself is very complicated.
4 Global Hopf bifurcation analysis
From Lemma 3.8, we obtain the following lemma.
Lemma 4.1
Lemma 4.2
System (4.1) has no nonconstant periodic solution when \(\tau=0\).
Proof
Theorem 3.5 shows \(P_{2}\) is globally attractive when \(\tau=0\). This lemma follows from the fact that \(P_{2}\) is globally attractive when \(\tau=0\). □
Lemma 4.3
All the nontrivial periodic solutions of (4.1) are positive and uniformly bounded.
Proof
The proof of this lemma can be obtained from Proposition 2.1. □
Lemma 4.4
When \(R_{1}>1\), system (4.1) has no nonconstant periodic solution of period τ. Furthermore, system (4.1) has no nonconstant periodic solution of period \(\frac{\tau}{j}\), \(j=2,3,4,\ldots \) .
Proof
Note that the periodic solutions are all bounded away from zero, which follows from Lemma 4.2, thus we need not to consider the boundary equilibria \(P_{0}\) and \(P_{1}\).
Let \(C(P_{2}, \tau^{j}, \frac{2\pi}{\omega^{*}})\) denote the connected component of \((P_{2}, \tau^{j},\frac{2\pi}{\omega^{*}})\) in Σ, where \(\tau^{j}\), \(\omega^{*}\) are defined in (4.2).
Now, we are in a position to state the following global Hopf bifurcation results.
Theorem 4.5
When \(R_{1}>1\), for each \(\tau>\tau^{j}\), \(j=1,2,3,\ldots \) , system (4.1) has at least \(j+1\) positive periodic solutions, where \(\tau^{j}\) is defined in (4.2).
Proof
Lemma 4.3 shows the projection of \(C(P_{2}, \tau^{j}, \frac{2\pi}{\omega ^{*}})\) onto W-space is bounded. Lemma 4.2 implies the projection of \(C(P_{2}, \tau^{j}, \frac{2\pi}{\omega^{*}})\) onto τ-space is bounded below.
5 Numerical simulations
In this section, we shall carry out some numerical simulations for illustrating our theoretical analysis. As regards the selected parameters in this section, we refer to [15, 22].
6 Conclusion
In this paper, we considered a class of virus infection models with three time lags, two intracellular delays and one immune delay. We have carried out a mathematical analysis of the dynamics of the model. We proved that \(P_{0}\) is globally asymptotically stable when \(R_{0} < 1\), and the three delays do not destroy the globally asymptotical stability of \(P_{0}\). \(P_{1}\) is globally asymptotically stable when \(R_{1} < 1 < R_{0}\), and the three delays also do not destroy the globally asymptotical stability of \(P_{1}\). When \(R_{1} > 1\), we found \(P_{2}\) has still global attractability under only incorporating two intracellular delays \(\tau _{1}\) and \(\tau_{2}\). But on only incorporating the immune delay \(\tau _{3}\), \(P_{2}\) can undergo a Hopf bifurcation on proper conditions, furthermore, oscillations and stability switches can appear. The immune delay can destroy the global attractability of \(P_{2}\). Those results show immune delay dominates intracellular delays in some viral infection models, which indicates the human immune system has a special effect in virus infection models with CTLs response, and the human immune system itself is very complicated. People are aware of the complexity of human immune system in medical science, which coincides with our investigation. Finally, we studied the global Hopf bifurcation of the system, and we obtained the global existence of periodic solutions.
Declarations
Acknowledgements
The authors would like to thank the anonymous referees and the editor for pertinent suggestions and comments, which led to improvements of our paper. This research is supported by National Natural Science Foundation of China (No. 11371111), Research Fund for the Doctoral Program of Higher Education of China (No. 20122302110044) and Shandong Provincial Natural Science Foundation, China (No. ZR2013AQ023).
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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