Stability analysis of a disease resistance SEIRS model with nonlinear incidence rate
 Jianwen Jia^{1}Email authorView ORCID ID profile and
 Jing Xiao^{1}
https://doi.org/10.1186/s1366201814941
© The Author(s) 2018
Received: 4 September 2017
Accepted: 17 January 2018
Published: 27 February 2018
Abstract
In this paper, we study a new SEIRS epidemic model describing nonlinear incidence with a more general form and the transmission of influenza virus with disease resistance. The basic reproductive number \(\Re_{0}\) is obtained by using the method of next generating matrix. If \(\Re_{0}<1\), the diseasefree equilibrium is globally asymptotically stable, and if \(\Re_{0}>1\), by using the geometric method, we obtain some sufficient conditions for global stability of the unique endemic equilibrium. Finally, numerical simulations are provided to support our theoretical results.
Keywords
1 Introduction
There are lots of people dying because of infectious diseases every day. From an epidemiological viewpoint, it is important to study the global stability of disease transmission. Mathematical models describing the infectious disease dynamics have played an important role and provided the preventive strategies in a period. The SEIRS epidemic model is an important model. It shows that the total population is divided into four classes: the susceptible S, the exposed E, the infectious I and the removed R. This model has been studied by many authors [1–11]; however, many literature works did not consider disease resistance in humans. With the development and progress of society, people begin to understand the importance of health and exercise. In other words, people’s resistance has improved greatly. So, disease resistance has become an indispensable factor in the study of infectious disease models. Nguyen Huu Khanh considered the disease resistance and formulated a mathematical model [3]. In the model, a person in the exposed group or infected group can return to the susceptible group without treatment.
 (\(A_{1}\)):

for \(x\geq0\), \(h(x)\geq0\), with equality if and only if \(x=0\), \(h'(x)>0\) and \(h''(x)\leq0\) (where ′ represents differentiation with respect to x).
Our paper is organized as follows. In Section 2, we formulate an SEIRS mathematical model and obtain the basic reproductive number \(\Re_{0}\). Furthermore, the existence of equilibria is given. In Section 3, we prove the global stability of the diseasefree equilibrium. Section 4 is devoted to the stability analysis of the endemic equilibria of the model. In Section 5, some numerical simulations are given to justify the theoretical analysis. Finally, we summarize this work.
2 The model and its basic properties
2.1 The structure of the model
2.2 Basic reproduction number
2.3 Existence of equilibria
Theorem 2.1
Proof
It is easy to see that the diseasefree equilibrium \(E_{0}\) always exists.
The proof of Theorem 2.1 is completed. □
3 The stability of the diseasefree equilibrium
In this section, we analyze the stability of the diseasefree equilibrium.
Theorem 3.1
\(E_{0}\) is locally asymptotically stable if \(\Re_{0}<1\), whereas \(E_{0}\) is unstable if \(\Re_{0}>1\).
Proof
When \(\Re_{0}<1\), we have \(\beta h(S_{0})(\gamma+b+\mu+\varepsilon )<(c+\varepsilon+\mu)(\gamma+b+\mu)\) and \(\beta h(S_{0})< c+\varepsilon+\mu\). It is clear that \(a_{i}>0\); \(i=1,2\). By Vieta’s theorem, all roots of (3.2) are negative. Hence, \(E_{0}\) is locally asymptotically stable.
When \(\Re_{0}>1\), we have \(a_{2}<0\), (3.2) has a positive root, so \(E_{0}\) is unstable. □
In the following, applying LaSalle’s invariance principle and the Lyapunov direct method, we prove the global stability of the diseasefree equilibrium.
Theorem 3.2
The diseasefree equilibrium \(E_{0}\) is globally asymptotically stable if \(\Re_{0}<1\).
Proof
4 The stability of the endemic equilibrium
In this section, we analyze the stability of the endemic equilibrium.
Theorem 4.1
If \(\Re_{0}>1\), \(\beta h(S^{*})\leq c+\varepsilon\) and \(\beta h'(S^{*})I^{*}\geq\frac {b+\gamma+\mu\varepsilon}{b+\gamma+\mu+\varepsilon}(\mu+b)\), then the endemic equilibrium \(E_{c}\) is locally asymptotically stable.
Proof
From Eq. (2.2), we have \((\gamma+b+\mu)(c+\varepsilon+\mu)\beta h(S^{*})(b+\gamma+\mu+\varepsilon)=0\) and \(\beta h(S^{*})< c+\varepsilon+\mu\), so we get \(M=0\), \(F>0\).
When \(\beta h(S^{*})\geq c+\varepsilon\) and \(b< c\), we have \(\beta h(S^{*})>b+\varepsilon\). Take notice of condition \(\beta h'(S^{*})I^{*}\geq\frac{b+\gamma+\mu\varepsilon}{b+\gamma+\mu +\varepsilon}(\mu+b)\). It is easy to see that \(a_{i}>0\); \(i=1,2,3,4\), \(a_{1}a_{2}a_{3}>0\) and \(a_{4}(a_{3}(a_{1}a_{2}a_{3})a_{1}^{2}a_{4})>0\). By the RouthHurwitz criterion, all roots of (4.2) have negative real parts. Hence, the endemic equilibrium \(E_{c}\) of system (2.1) is locally asymptotically stable.
The proof of Theorem 4.1 is completed. □
In the following, we use the geometric approach to discuss the global stability of the endemic equilibrium. We will expand its application to fourdimensional systems, which can also be seen in [21].
Firstly, we present some preliminaries on the geometric approach to prove global stability [22].
Definition 4.1
Similar to [7], we can get the following.
Theorem 4.2
System (2.1) is uniformly persistent in Ω if and only if \(\Re_{0}>1\).
Remark 4.1
 (H1):

There is a compact absorbing set \(K\subset\Omega_{1}\);
 (H2):

Differential equation (4.3) has a unique equilibrium \(x^{*}\) in \(\Omega_{1}\).
Lemma 4.1
([22])
Suppose that \(\Omega_{1}\) is simply connected and that assumptions (H1) and (H2) hold, then the unique equilibrium \(x^{*}\) of system (4.3) is globally stable in \(\Omega _{1}\) if \(q < 0\).
Now we apply Lemma 4.1 to prove the global stability of \(E_{c}\).
Theorem 4.3
Proof
The proof of Theorem 4.1 is completed. □
5 Numerical simulation
To support our main results, we perform some numerical simulations. We choose \(h(S)=\frac{S}{1+gS}\) and consider the set of parameters:
6 Conclusions
In this paper, we have proposed a nonlinear mathematical model for influenza virus transmission with disease resistance; nonlinear incidence has a more general form. Through mathematical analysis we obtain the dynamic behaviors of the model. The basic reproduction number \(\Re_{0}\) is obtained. If \(\Re_{0}<1\), the diseasefree equilibrium is globally asymptotically stable. It implies that the disease dies out eventually. When \(\Re_{0}>1\), the endemic equilibrium is globally asymptotically stable under some conditions. It implies that the disease persists in the population. All of these results imply that the disease resistance and nonlinear incidence can influence the dynamic behaviors of the SEIRS model. From the expression of \(\Re_{0}\), it is easy to see that when β is increased, b, c decrease and then \(\Re_{0}\) increases. \(\Re_{0}>1\) leads to the stability of the endemic equilibrium and then the prevalence of the disease. So we can get some effective strategies for controlling the disease such as reducing the contact rate β and increasing the b, c. That is to say, by taking proper isolation of the population and increasing the resistance of people, we can avoid the development of infectious diseases into endemic diseases. In reality, the exposed and infected individuals have different infection rates, which will be the focus of our future research.
Declarations
Acknowledgements
We greatly appreciate the editor and the anonymous referees’ careful reading and valuable comments, their critical comments and helpful suggestions have greatly improved the presentation of this paper.
Authors’ contributions
All authors contributed equally to the writing of this paper. The authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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