 Research
 Open Access
Global dynamics for a class of discrete SEIRS epidemic models with general nonlinear incidence
 Xiaolin Fan^{1, 2},
 Lei Wang^{3} and
 Zhidong Teng^{1}Email author
https://doi.org/10.1186/s136620160846y
© Fan et al. 2016
 Received: 29 October 2015
 Accepted: 25 April 2016
 Published: 6 May 2016
Abstract
In this paper, a class of discrete SEIRS epidemic models with general nonlinear incidence is investigated. Particularly, a discrete SEIRS epidemic model with standard incidence is also considered. The positivity and boundedness of solutions with positive initial conditions are obtained. It is shown that if the basic reproduction number \(\mathcal{R}_{0}\leq1\), then diseasefree equilibrium is globally attractive, and if \(\mathcal{R}_{0}> 1\), then the disease is permanent. When the model degenerates into SEIR model, it is proved that if \(\mathcal{R}_{0}> 1\), then the model has a unique endemic equilibrium, which is globally attractive. Furthermore, the numerical examples verify an important open problem that when \(\mathcal{R}_{0}>1\), the endemic equilibrium of general SEIRS models is also globally attractive.
Keywords
 discrete SEIRS epidemic model
 nonlinear incidence
 basic reproduction number
 global attractivity
 permanence
MSC
 37M99
 39A11
 92D30
1 Introduction
As is well known, many infectious diseases possess a latent period, such as Hepatitis, HIV, SARS, Ebola, MERS, etc. When a susceptible individual is infected at the beginning, the disease incubates inside the susceptible for a period of time, then the susceptible becomes an exposed individual before becoming infectious. For such infectious diseases, the resulting model is SEIR (susceptible S, exposed E, infectious I, removed R) epidemic type. The study on SEIRtype epidemic dynamical models is a very important subject in the mathematical theory of epidemiology, and in the last two decades there have been a number of researches on modeling, theoretical analysis, and applications. Continuous SEIRtype epidemic models described by the differential equations have been widely studied. Many important and interesting results can be found in [1–9] and the references therein.
As we all know, it is very difficult to accurately solve a nonlinear differential equation with a given initial condition. Therefore, for many practical requirements, such as numerical calculation, it is often necessary to discretize a continuous model to obtain the corresponding discrete model. At the present time, there are various discretization methods to discretize a continuous model, including the standard methods, such as Euler method, RungeKutta method, and some other standard finite difference schemes, and the nonstandard finite difference (NSFD) scheme, which is originally developed by Mickens [10–12].
In recent years, discrete epidemic models have been widely studied. The basic and important research subjects for these models are the computing of the thresholds values and basic reproduction numbers, the local and global stability of diseasefree equilibrium and the endemic equilibrium, the persistence, permanence, and extinction of the disease, and bifurcations and chaos phenomena of the models when some parameters of the models vary, and so on. Many important and interesting results can be found in [13–36] and the references therein. Particularly, we see that in [14, 17, 18, 20–22, 27, 36] discrete SItype epidemic models are investigated, and in [19, 24, 25, 31, 32, 35] discrete SIRtype epidemic models are discussed.
However, we see that up to now there have been fewer research works on discrete SEI and SEIRtype epidemic models, where the disease has a latent period. Cao and Zhou [16] formulated and studied a discrete agestructured SEIT epidemic model, and as an application, discussed the tuberculosis transmission in China. In [26], the authors applied Micken’s discretization method to obtain a discrete SEIR epidemic model. The positivity of solutions and the existence and stability of equilibrium are discussed. The design of a state observer for the model is tackled. Some sufficient conditions to ensure the asymptotic stability of the observer are provided in terms of a matrix inequality. In [28], the authors studied a discrete plant virus disease model with roguing and replanting, which is derived from the continuous case by using the backward Euler method. The basic reproduction number \(R_{0}\) is obtained. It is showed that the diseasefree equilibrium is globally attractive if \(R_{0}\leq1\), and otherwise, the disease is permanent if \(R_{0}>1\). In [29, 30], the authors proposed a class of discrete SEIS epidemic models with bilinear incidence, which is established from the corresponding continuous SEIS epidemic model by applying the wellknown backward difference scheme. The positivity of solutions and the permanence of the model are established. Furthermore, using the Lyapunov function method, the authors proved that if the basic reproduction number \(R_{0}\leq1\), then the diseasefree equilibrium is globally asymptotically stable, and if \(R_{0}>1\), then the endemic equilibrium exists and is globally asymptotically stable.
Our purpose in this paper is to investigate the dynamical behaviors of model (2). The basic reproduction number \(\mathcal{R}_{0}\) is defined. We will prove by using the linearization method and Lyapunov function that if \(\mathcal{R}_{0}\leq1\), then diseasefree equilibrium is globally attractive, and as a result, the disease is also extinct, and by using the theory of persistence for dynamical systems that if \(\mathcal {R}_{0}>1\), then the disease is permanent. Furthermore, when model (2) degenerates into the particular case \(f(S,E,I,R)=f(S,E,I)\) and \(\sigma=0\), by constructing the suitable discrete type Lyapunov function we also will prove that if \(\mathcal {R}_{0}>1\), then model (2) has a unique endemic equilibrium, which is globally attractive.
The organization of this paper is as follows. In Section 2, the model description and some basic properties are given. Section 3 deals with the global attractivity of diseasefree equilibrium of model (2). In Section 4, the criterion on the permanence of the disease for model (2) is stated and proved. In Section 5, the criterion on the global attractivity of the endemic equilibrium for model (2) in the particular case \(f(S,E,I,R)=f(S,E,I)\) and \(\sigma=0\) is stated and proved. Furthermore, in Section 6, some numerical examples are provided to illustrate the validity of main results obtained in this paper and verify the interesting open problem given in Remark 5.1. Lastly, a discussion is given in Section 7.
2 Basic properties
In model (2), \(S(n)\), \(E(n)\), \(I(n)\), and \(R(n)\) denote the numbers of susceptible, exposed, infectious, and recovered classes at nth generation, respectively, Λ is the recruitment rate of the susceptible, \(\mu_{i}\) (\(i=1,2,3,4\)) are the death rates of susceptible, exposed, infectious, and recovered individuals, respectively. Particularly, \(\mu_{3}\) includes the natural death rate and the diseaserelated death rate of the infectious class. δ is the translation rate from exposed to infectious, γ is the recovery rate of the infectious individuals, and σ is the rate of losing immunity of the recovered; \(\sigma>0\) indicates that the recovered individuals possess the provisional immunity, and \(\sigma=0\) predicates that the recovered individuals acquire permanent immunity. The incidence rate of the infectious is described by a nonlinear function \(f(S,E,I,R)\).
 (H):

\(f(S,E,I,R)\) is continuously differentiable with respect to \((S,E,I,R)\in\Omega\), \(f(S,E,I,R)\) is increasing with respect to \(S\geq0\) and decreasing with respect to \(E\geq0\) and \(R\geq0\), and \(\frac{f(S,E,I,R)}{I}\) is nonincreasing with respect to \(I>0\). Furthermore, \(f(0,E,I,R)=f(S,E,0,R)\equiv0\) and \(\frac{\partial f(S_{0},0,0,0)}{\partial I}>0\), where \(S_{0}=\frac{\Lambda }{\mu_{1}}\).
Remark 2.1
 (H^{∗}):

\(h(S)\) and \(g(I)\) are continuously differentiable with respect to \(S\geq0\) and \(I\geq0\), respectively, \(h(S)\) is increasing for \(S\geq0\), and \(\frac{g(I)}{I}\) is nonincreasing for \(I>0\). Furthermore, \(h(0)=g(0)=0\) and \(g'(0)>0\).
Theorem 2.1
Model (2) has a unique positive solution \((S(n),E(n),I(n),R(n))\) for all \(n\geq0\) with initial condition (3), and this solution is ultimately bounded.
Proof
We can prove this theorem by using an argument similar to that introduced in [33], Theorem 2.2. In fact, we only need to prove by induction that, for any integer \(n\geq0\), if \((S(n),E(n),I(n),R(n))\) exists and \(S(n)>0\), \(E(n)>0\), \(I(n)>0\), and \(R(n)\geq0\), then \((S(n+1),E(n+1),I(n+1),R(n+1))\) also exists, and \(S(n+1)>0\), \(E(n+1)>0\), \(I(n+1)>0\), and \(R(n+1)>0\).
By (4), when \(E(n+1)>0\) exists, then \(I(n+1)\) also exists, and \(I(n+1)>0\). By the fourth equation of model (2) we further have that \(R(n+1)\) exists and \(R(n+1)>0\).
From the previous discussions we finally obtain that \((S(n+1),E(n+1),I(n+1), R(n+1))\) exists and is positive. Therefore, solution \((S(n),E(n),I(n),R(n))\) uniquely exists and is positive for all \(n>0\).
Remark 2.2
Theorem 2.2
Proof
If \(\mathcal{R}_{0}\leq1\), then \(\lim_{I\to0^{+}}\Phi(I)\leq0\). Hence, \(\Phi(I)=0\) has no positive roots. This shows that model (2) has only a diseasefree equilibrium \(P_{0}\).
We have the following result on the local stability of the diseasefree equilibrium and endemic equilibrium.
Theorem 2.3
When \(\mathcal{R}_{0}<1\), the diseasefree equilibrium \(P_{0}\) of model (2) is locally asymptotically stable, and when \(\mathcal {R}_{0}>1\), \(P_{0}\) is unstable.
Proof
When \(\mathcal{R}_{0}>1\), we easily prove that two eigenvalues \(\rho_{i} \) (\(i=1,2\)) of the matrix \(A^{1}\) are real numbers and \(\rho_{1}<1\) and \(\rho_{2}>1\). Hence, the equilibrium \((0,0)\) of system (8) is unstable. This shows that the equilibrium \(P_{0}\) is unstable when \(\mathcal {R}_{0}>1\). □
Remark 2.3
Therefore, when \(\mathcal{R}_{0}>1\), whether the endemic equilibrium \(P_{*}\) of model (2) also is locally asymptotically stable still is an interesting open problem.
3 Global attractivity of diseasefree equilibrium
In this section, we discuss the global attractivity of diseasefree equilibrium of model (2). We have the following result.
Theorem 3.1
The diseasefree equilibrium \(P_{0}\) of model (2) is globally attractive if and only if \(\mathcal{R}_{0}\leq1\).
Proof
Therefore, using the theorems of stability of difference equations (see Theorem 6.3 in [37]), we finally obtain that the diseasefree equilibrium \(P_{0}\) of model (2) is globally attractive. This completes the proof. □
4 Permanence of disease
For model (2), disease \(I(n)\) is said to be permanent if there exists constants \(M>m>0\) such that for any solution \((S(n),E(n),I(n),R(n))\) of model (2) with initial condition (3), \(m\leq\liminf_{n\rightarrow\infty}I(n) \leq\limsup_{n\rightarrow\infty}I(n)\leq M\). We have the following result.
Theorem 4.1
Disease \(I(n)\) in model (2) is permanent if and only if \(\mathcal {R}_{0}>1\).
Proof
The necessity is obvious. In fact, if \(\mathcal{R}_{0}\leq1\), then by Theorem 3.1 the diseasefree equilibrium \(P_{0}\) is globally attractive.
Remark 4.1
When \(f(S,E,I,R)=\frac{SI}{N}\), by Theorem 4.1, if \(\mathcal{R}_{0}=\frac{\beta\delta}{(\mu_{3}+\gamma)(\mu_{2}+\delta)}>1\), then the disease in model (2) is permanent.
Remark 4.2
Theorem 4.1 only obtains the permanence of the disease for model (2). However, whether we can also prove that an endemic equilibrium \(P_{*}\) is globally attractive for model (2) when \(\mathcal{R}_{0}>1\)? In the following section, we will give a partial positive answer. We will prove that, for special case \(\sigma=0\) of model (2), an endemic equilibrium \(P_{*}\) is globally attractive only when \(\mathcal{R}_{0}>1\).
5 Global attractivity of endemic equilibrium in a particular case
Theorem 5.1
If \(\mathcal{R}_{0}>1\), then the endemic equilibrium \(P_{*}\) of model (12) is globally attractive.
Proof
Therefore, using the theorems of stability of difference equations, we finally obtain that the endemic equilibrium \(P_{*}\) of model (12) is globally attractive. This completes the proof. □
Remark 5.1
In Remark 4.2, we indicated that for SEIRStype model (2), an important problem is to prove that the endemic equilibrium is globally attractive only when \(\mathcal{R}_{0}>1\). From Theorem 5.1 we see that only for the particular case \(\sigma=0\) of model (2), that is, SEIRtype model, we get a positive answer. Therefore, an interesting open problem for general SEIRS model (2) is whether the endemic equilibrium is also globally attractive only when \(\mathcal{R}_{0}>1\).
Remark 5.2
From the proofs of Theorem 3.1, Theorem 4.1, and Theorem 5.1 we easily see that the condition \(\mu_{1}\leq\min\{\mu_{2},\mu_{3},\mu_{4}\}\) is not used. In fact, this condition is only used in Theorem 2.1 to obtain the positivity of solutions of model (2). Therefore, an interesting question is whether the condition \(\mu_{1}\leq\min\{\mu_{2},\mu_{3},\mu_{4}\}\) can be taken out in the proof of the positivity of solutions of model (2).
6 Numerical examples
Now, we give numerical examples to show that for SEIRStype model (2), the endemic equilibrium may be globally attractive for different incidence function \(f(S,E,I,R)\), which satisfies (H) only when the basic reproduction number \(\mathcal{R}_{0}>1\).
Example 6.1
In model (2), we take \(f(S,E,I,R)=\frac{\beta SI}{1+\alpha I+\omega S}\), \(\Lambda=1.5\), \(\mu_{1}=0.2\), \(\mu_{2}=0.35\), \(\mu_{3}=0.5\), \(\beta=0.36\), \(\delta=0.3\), \(\omega=0.1\), and \(\gamma=0.1\). The parameters \(\mu_{4}\), α, and σ will be chosen later.
Example 6.2
In model (2), we take \(f(S,E,I,R)=\frac{\beta SI}{1+\alpha I^{2}}\), \(\Lambda=2.5\), \(\mu_{1}=0.2\), \(\mu_{2}=0.35\), \(\mu_{3}=0.5\), \(\beta=0.3\), \(\delta=0.4\), and \(\gamma=0.6\). The parameters \(\mu_{4}\), α, and σ will be chosen later.
Example 6.3
In model (2), we take \(f(S,E,I,R)=\frac{\beta S^{2}I}{(1+\omega S)(1+\alpha I)}\), \(\Lambda=5\), \(\mu_{1}=0.9\), \(\mu_{2}=0.6\), \(\mu_{3}=0.5\), \(\beta=0.3\), \(\delta=0.2\), \(\omega=0.3\), and \(\gamma=0.3\). The parameters \(\mu_{4}\), α, and σ will be chosen later.
Example 6.4
In model (2), we take \(f(S,E,I,R)=\frac{\beta S^{2}I}{(1+\omega S)(1+\alpha I^{2})}\), \(\Lambda=1.5\), \(\mu_{1}=0.2\), \(\mu_{2}=0.35\), \(\mu_{3}=0.5\), \(\beta=0.32\), \(\delta=0.4\), \(\omega=0.3\), and \(\gamma=0.6\). The parameters \(\mu_{4}\), α, and σ will be chosen later.
All these examples of numerical simulations show that when \(\mathcal{R}_{0}>1\), no matter sufficiently greater than one or closer to one but still greater than one, we always obtain that the endemic equilibrium \(P_{*}\) is globally attractive, which may offer an affirmative conjecture to the open problem given in Remark 5.1, that is, for the general SEIRS model (2) the endemic equilibrium \(P_{*}\) is globally attractive only when \(\mathcal{R}_{0}>1\). Therefore, in our future work, we expect to obtain the corresponding theoretical results for this open problem.
7 Discussion
In this paper, we proposed a discrete SEIRS epidemic model (2) with general nonlinear incidence, which is described by the backward difference scheme. By our discussions presented in this paper, necessary and sufficient conditions for the global attractivity of the diseasefree equilibrium and the permanence of the disease are established, that is, if the basic reproduction number \(\mathcal {R}_{0}\leq1\), then the diseasefree equilibrium is globally attractive, and if \(\mathcal{R}_{0}> 1\), then the disease is permanent. Furthermore, when the model degenerates into SEIR model, it is proved that when \(\mathcal{R}_{0}> 1\), the model has a unique globally attractive endemic equilibrium.
Unfortunately, for SEIRS model (2), when the basic reproduction number is greater than one, we do not obtain the local asymptotic stability and global attractivity of the endemic equilibrium. But the numerical examples given in Section 5 show that the endemic equilibrium for general SEIRS model (2) may be globally attractive. Therefore, it is still an important and interesting open problem how to apply the linearization method to establish the local asymptotic stability of the endemic equilibrium and how to construct the discrete analogue Lyapunov functions to study the global attractivity of the endemic equilibrium for general SEIRS model (2).
In addition, the dynamical behaviors for the nonautonomous discrete SEIRS epidemic models, discrete SEIRS epidemic models with vaccination, stagestructured discrete SEIRS epidemic models, and delayed discrete SEIRS epidemic models with nonlinear incidence described by the backward difference scheme are rarely considered. Whether similar results on the permanence and extinction of the disease and the global attractivity of the diseasefree equilibrium for these models can be obtained is also an interesting open question.
Declarations
Acknowledgements
This work is supported by the Doctoral Subject Science Foundation (Grant No. 20136501110001), and the National Natural Science Foundation of China (Grant Nos. 11271312, 11401512, 11261056, 11301451, 61473244, 11402223).
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
References
 Bai, Z, Zhou, Y: Global dynamics of an SEIRS epidemic model with periodic vaccination and seasonal contact rate. Nonlinear Anal., Real World Appl. 13, 10601068 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Gao, S, Chen, L, Teng, Z: Impulsive vaccination of an SEIRS model with time delay and varying total population size. Bull. Math. Biol. 68, 731745 (2007) View ArticleMATHGoogle Scholar
 Korobeinikov, A: Lyapunov functions and global properties for SEIR and SEIS epidemic models. Math. Med. Biol. 21, 7583 (2004) View ArticleMATHGoogle Scholar
 Korobeinikov, A: Global properties of infectious disease models with nonlinear incidence. Bull. Math. Biol. 69, 18711886 (2007) MathSciNetView ArticleMATHGoogle Scholar
 Korobeinikov, A, Maini, P: Nonlinear incidence and stability of infectious disease models. Math. Med. Biol. 22, 113128 (2005) View ArticleMATHGoogle Scholar
 Kuniya, T, Nakata, Y: Permanence and extinction for a nonautonomous SEIRS epidemic model. Appl. Math. Comput. 218, 93219331 (2012) MathSciNetMATHGoogle Scholar
 Melesse, DY, Gumel, AB: Global asymptotic properties of an SEIRS model with multiple infectious stages. J. Math. Anal. Appl. 366, 202217 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Nakata, Y, Kuniya, T: Global dynamics of a class of SEIRS epidemic models in a periodic environment. J. Math. Anal. Appl. 363, 230237 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, T, Teng, Z: On a nonautonomous SEIRS model in epidemiology. Bull. Math. Biol. 69, 25372559 (2007) MathSciNetView ArticleMATHGoogle Scholar
 Mickens, RE: Application of Nonstandard Finite Difference Scheme. World Scientific, Singapore (2000) View ArticleMATHGoogle Scholar
 Mickens, RE: Dynamics consistency: a fundamental principle for constructing nonstandard finite difference scheme for differential equation. J. Differ. Equ. Appl. 9, 10371051 (2003) View ArticleGoogle Scholar
 Mickens, RE: Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition. Numer. Methods Partial Differ. Equ. 23, 672691 (2007) MathSciNetView ArticleMATHGoogle Scholar
 Allen, LJS, Driessche, P: The basic reproduction number in some discretetime epidemic models. J. Differ. Equ. Appl. 14, 11271147 (2008) MathSciNetView ArticleMATHGoogle Scholar
 Allen, LJS, Lou, Y, Nevai, AL: Spatial patterns in a discretetime SIS patch model. J. Math. Biol. 58, 339375 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Brauer, F, Feng, Z, CastilloChavez, C: Discrete epidemic models. Math. Biosci. Eng. 7, 115 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Cao, H, Zhou, Y: The discrete agestructured SEIT model with application to tuberculosis transmission in China. Math. Comput. Model. 55, 385395 (2012) MathSciNetView ArticleMATHGoogle Scholar
 CastilloChavez, C: Discretetime SIS models with complex dynamics. Nonlinear Anal. TMA 47, 47534762 (2001) MathSciNetView ArticleMATHGoogle Scholar
 Chen, Q, Teng, Z, Wang, L: The existence of codimensiontwo bifurcation in a discrete SIS epidemic model with standard incidence. Nonlinear Dyn. 71, 5573 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Enatsu, Y, Nakata, Y, Muroya, Y: Global stability for a class of discrete SIR epidemic models. Math. Biosci. Eng. 7, 347361 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Franke, J, Yakubu, A: Discretetime SIS epidemic model in a seasonal environment. SIAM J. Appl. Math. 66, 15631587 (2006) MathSciNetView ArticleMATHGoogle Scholar
 Franke, JE, Yakubu, AA: Diseaseinduced mortality in densitydependent discretetime SIS epidemic models. J. Math. Biol. 57, 755790 (2008) MathSciNetView ArticleMATHGoogle Scholar
 Franke, JE, Yakubu, AA: Periodically forced discretetime SIS epidemic model with disease induced mortality. Math. Biosci. Eng. 8, 385408 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Garba, SM, Gumel, AB, Lubuma, JMS: Dynamicallyconsistent nonstandard finite difference method for an epidemic model. Math. Comput. Model. 53, 131150 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Hu, Z, Teng, Z, Jia, C, Chen, X: Complex dynamical behaviors in a discrete ecoepidemiological model with disease in prey. Adv. Differ. Equ. 2014, 265 (2014) MathSciNetView ArticleGoogle Scholar
 Hu, Z, Teng, Z, Jiang, H: Stability analysis in a class of discrete SIRS epidemic models. Nonlinear Anal., Real World Appl. 13, 20172033 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Ibeas, A, de la Sen, M, AlonsoQuesada, S, Zamani, I: Stability analysis and observer design for discretetime SEIR epidemic models. Adv. Differ. Equ. 2015, 22 (2015) MathSciNetView ArticleGoogle Scholar
 Li, J, Ma, Z, Brauer, F: Global analysis of discretetime SI and SIS epidemic models. Math. Biosci. Eng. 4, 699710 (2007) MathSciNetView ArticleMATHGoogle Scholar
 Luo, Y, Gao, S, Xie, D, Dai, Y: A discrete plant disease model with roguing and repanting. Adv. Differ. Equ. 2015, 12 (2015) MathSciNetView ArticleGoogle Scholar
 Muroya, Y, Bellen, A, Enatsu, Y, Nakata, Y: Global stability for a discrete epidemic model for disease with immunity and latency spreading in a heterogeneous host population. Nonlinear Anal., Real World Appl. 13, 258274 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Muroya, Y, Enatsu, Y: A discretetime analogue preserving the global stability of a continuous SEIS epidemic model. J. Differ. Equ. Appl. 19, 14631482 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Muroya, Y, Nakata, Y, Izzo, G, Vecchio, A: Permanence and global stability of a class of discrete epidemic models. Nonlinear Anal., Real World Appl. 12, 21052117 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Salceanu, PL, Smith, HL: Persistence in a discretetime stagestructured epidemic model. J. Differ. Equ. Appl. 16, 73103 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Sekiguchi, M: Permanence of a discrete SIRS epidemic model with delays. Appl. Math. Lett. 23, 12801285 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Wang, L, Cui, Q, Teng, Z: Global dynamics in a class of discretetime epidemic models with disease courses. Adv. Differ. Equ. 2013, 57 (2013) MathSciNetView ArticleGoogle Scholar
 Wang, L, Teng, Z, Jiang, H: Global attractivity of a discrete SIRS epidemic model with standard incidence rate. Math. Methods Appl. Sci. 36, 601619 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Yakubu, AA: Alee effects in a discretetime SIS epidemic model with infected newborns. J. Differ. Equ. Appl. 13, 341356 (2007) MathSciNetView ArticleMATHGoogle Scholar
 LaSalle, JP: The Stability of Dynamical Systems. SIAM, Philadelphia (1976) View ArticleMATHGoogle Scholar
 Zhao, XQ: Dynamical Systems in Population Biology. Springer, New York (2003) View ArticleMATHGoogle Scholar