 Research
 Open Access
Global dynamics of SEIRS epidemic model with nonlinear generalized incidences and preventive vaccination
 Muhammad Altaf Khan^{1},
 Qaisar Badshah^{1},
 Saeed Islam^{1},
 Ilyas Khan^{2},
 Sharidan Shafie^{3}Email author and
 Sher Afzal Khan^{4}
https://doi.org/10.1186/s1366201504293
© Khan et al.; licensee Springer. 2015
 Received: 3 November 2014
 Accepted: 27 February 2015
 Published: 17 March 2015
Abstract
In this paper, we present the global dynamics of an SEIRS epidemic model for an infectious disease not containing the permanent acquired immunity with nonlinear generalized incidence rate and preventive vaccination. The model exhibits two equilibria: the diseasefree and endemic equilibrium. The diseasefree equilibrium is stable locally as well as globally when the basic reproduction number \({\mathcal{R}}_{0}<1\) and an unstable equilibrium occurs for \({\mathcal{R}}_{0}>1\). Moreover, the endemic equilibrium is stable both locally and globally when \({\mathcal{R}}_{0}>1\). We show the global stability of an endemic equilibrium by a geometric approach. Further, numerical results are presented to validate the theoretical results. Finally, we conclude our work with a brief discussion.
Keywords
 SEIRS epidemic model
 generalized nonlinear incidence rate
 basic reproduction number
 global stability
 numerical simulations
1 Introduction
To reduce the spread and increase control of an infectious disease the quarantine and vaccination methods are used commonly. For a cost effective strategy and successful intervention policy, vaccination is often considered the best tool for eradication of the morbidity and mortality of people. For the diseases measles, rubella, diphtheria, mumps, influenza, tetanus, and hepatitis B, it has been used to tackle them. In some cases for a vaccinated person it is not necessary to have lifelong immunity; see [1, 2]. In a certain community it is sometimes very difficult or impossible to vaccinate the susceptible individuals. The main reason behind this is the unavailability (or not easy availability) of such vaccine in those countries. So, it is reasonable to obtain a fraction of immune individuals in the community for which the disease does not become epidemic; that fraction is known as the herd immunity threshold [1].
A variety of incidence rates have been used in the literature, for instance, [2–6]. In all these models the incidence rate has been considered as a law of mass action. For the communicable diseases, the incidence rate in the form of \(\beta S I\) is used, where β shows the per capita contact rate. For the first time [7] introduced a saturated incidence after the cholera epidemic in Bari in 1937. This reference used the incidence rate in the form of \(Sg(I)\). Muroya et al. [8] presented a SIRS epidemic with graded curve and incomplete recovery rates. The global dynamics is completely described by the basic reproduction number \({\mathcal{R}}_{0}\). Denphedtnong et al. [9] proposed a SEIRS mathematical model with transport related infection. According to this reference, the transportation among cities is one of the main factors to affect the outbreak of diseases. A SEIRS mathematical model for disease transmission incorporating the immigration of infected, susceptible, and exposed persons is analyzed by [10]. A delay SEIRS epidemic model for computer virus network is studied by [11]. The effect of behavioral changes for the susceptible individuals have been incorporated by Liu et al. [12] in this model. They used the incidence rate in the form \(\frac{\beta SI^{p}}{1+kI^{q}}\), where k is negative constant, while p and q are positive. Different cases have been studied for p, q, and k; see, for example [7, 13–15].
In mathematical models the global stability is very important. Many methods have been used in the literature to obtain the global stability for the epidemic models. For example, [16] used the second Lyapunov function in his model. In population biology models, the Lyapunov function candidate is used as a Volterra function \((yy^{*}y^{*}\ln\frac{y}{y^{*}})\). Beretta and Capasso used this function in [17] and also the authors used [4, 7, 18–26] for the global stability of epidemic models. In this work, we modified the model of [22], to incorporate the exposed class. This new class is very important, because many infectious diseases like, dengue, yellow fever, hepatitis B, etc., have a specific incubation period. For this reason, the analysis of the exposed class is very important.
Based on the above motivation, the present paper studies the global dynamics of a SEIRS epidemic model with nonlinear generalized incidences and preventive vaccination. The structure of the paper is as follows: We formulate the basic problem, with their properties in Section 2. In Section 3, we find the local stability of disease free and endemic equilibrium. The global stability of both the diseasefree and the endemic equilibrium discussed in Section 4. Numerical results with a brief discussion are presented in Section 5.
2 Model formulation
Parameter descriptions
Parameter  Description 

Λ  The growth rate of the individuals 
q  The fraction of individuals to be vaccinated 
μ  Natural death rate 
\(\mu_{0}\)  Disease related mortality rate 
\(\mu_{1}\)  The rate at which the individuals infected 
\(\mu_{2}\)  Rate of recovery 
\(\mu_{3}\)  The rate by which the individuals susceptible again 
β  The disease contact rate 
We assumed the same transmission rate in the form of \(\frac{\beta S I}{\psi(I)}\), where ψ is a positive function with \(\psi(0)=1\) and \(\psi'\geq0\), as used by [22]. This is the generalization of mass action incidences, that is, \(\psi(I)=1\), and the incidence rate \(\frac{\beta S I}{1+kI^{q}}\). For small I, the function \(\frac{I}{\psi(I)}\) is increasing, while it is decreasing for large I, that is \(\psi(I)=1+I^{2}\). This shows the ‘psychological’ effect: when the number of infective individuals is high, the increase in the number of infectives varies inversely to the force of infection, due to the presence of the large number of infectives in the population, which then tends to decrease the individuals’ contacts per unit time [7, 15].
2.1 Basic properties of the model
2.2 Endemic equilibria
Proposition
Suppose the conditions imposed on the function \(\psi(I)\) are satisfied. Then there exists a diseasefree state for system (1), which is \(E^{0}=(\frac{\Lambda((1q)\mu+\mu_{3})}{\mu(\mu+\mu_{3})},0,0, \frac{q\Lambda}{\mu+\mu_{3}})\), which exists for all parameter values. For \({\mathcal{R}}_{0}>1\), the endemic equilibrium \(E^{1}\) admits the unique positive equilibrium for the system (1).
3 Local stability
In this section, we investigate the local stability analysis of the model (1). First, we find the local stability of the diseasefree and then the endemic equilibrium as will be discussed.
Theorem 3.1
 (i)
It is stable locally asymptotically if \({\mathcal{R}}_{0}\leq1\).
 (ii)
An unstable equilibrium exists if \({\mathcal{R}}_{0}>1\).
Proof
(ii) When \({\mathcal{R}}_{0}>1\), then \(Q_{2}<0\) and \(Q_{3}<0\), which is a failure of the RouthHurwitz criterion. So, in this case the equilibrium is unstable. □
In the next theorem, we will prove that the system (1) around the endemic equilibrium point \(E^{1}\) is stable locally asymptotically, when the basic reproduction number \({\mathcal{R}}_{0}> 1\).
Theorem 3.2
If \({\mathcal{R}}_{0}> 1\), then the endemic equilibrium point \(E^{1}\) of the system (1) is locally asymptotically stable, otherwise it is unstable.
Proof
4 Global stability
 (\({\mathcal{H}}_{1}\)):

for \(\frac{d\overline{X}}{dt}=F(\overline{X},0)\), \({\overline{X}^{0}}\) is globally asymptotically stable (g.a.s.),
 (\({\mathcal{H}}_{2}\)):

\(G(X,Z)=BZ\overline{G}(\overline {X},Z)\), where \(\overline{G}(\overline{X},Z)\geq0\), for \((\overline{X}, Z)\in \Omega\),
Lemma 4.1
If \({\mathcal{R}}_{0} < 1\), then the fixed point \(\overline {Q}^{0}=(\overline{X}^{0},0)\) of reduced system (4) is said to be globally asymptotically stable if the conditions (\({\mathcal{H}}_{1}\)) and (\({\mathcal{H}}_{2}\)) are satisfied.
Now we prove the following theorem.
Theorem 4.1
Suppose \({\mathcal{R}}_{0} < 1\), then the equilibrium point \(E^{0}\) is globally asymptotically stable.
Proof
4.1 Global stability of endemic equilibrium
 (\({\mathcal{H}}_{3}\)):

A compact absorbing set exists, i.e., \(K \subset D\).
 (\({\mathcal{H}}_{4}\)):

A unique equilibrium for (5) is \(X^{*}\) in D.
A point \(x_{0} \in D\) is wandering for (5) if there exists a neighborhood U of \(x_{0}\) and \(T>0\) such that \(U\cap x(t, U)\) is null \(\forall t>T\). We present the globalstability principle of [29] for an autonomous systems as follows.
Lemma 4.2
Let the two conditions (\({\mathcal{H}}_{3}\)) and (\({\mathcal{H}}_{4}\)) hold, assuming that (5) satisfies the Bendixson criterion, i.e., robustness under \(C^{1}\), for the local perturbations of \(f(x)\) at all nonequilibrium nonwandering points for (5). Then \(x^{*}\) is globally stable in D, provided it is stable.
Lemma 4.3
Let the simple connectivity of D together with the conditions (\({\mathcal{H}}_{3}\)) and (\({\mathcal{H}}_{4}\)) hold. Then \(x^{*}\), the equilibrium point of (5), is stable globally in D if \(\bar{q} < 0\).
Theorem 4.2
If \({\mathcal{R}}_{0} > 1\), then the endemic equilibrium \(E_{1}\) of the system (4) is globally stable in Ω.
Proof
5 Discussion
RouthHurwitz criteria for the system of \(3\times3\) matrix: \(Q_{1}>0\), \(Q_{2}>0\), \(Q_{3}>0\), and \(Q_{1} Q_{2}Q_{3}>0\).
Declarations
Acknowledgements
The authors would like to acknowledge the Ministry of Education Malaysia (MOEM) and Research Management Centre UTM for the financial support through vote numbers 06H67 and 4F255 for this research. The authors would also like to thank to the anonymous referees for their valuable suggestions.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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