Global stability for a class of discrete SIR epidemic models

Enatsu, Yoichi; Nakata, Yukihiko; Muroya, Yoshiaki

doi:10.3934/mbe.2010.7.347

Cited by 61 publications

(26 citation statements)

References 17 publications

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“…By Perron–Frobenius theorem, there exists a positive principal eigenvector ω =( ω 1 , ω 2 ,⋯, ω m ) such that ω k >0 for k =1,2,⋯, m and ω · ρ ( M 0 )= ω · M 0 . Consider the following Lyapunov function:

L_{n} = \sum_{k = 1}^{m} \frac{ω_{k}}{h (d_{k}^{I} + ϵ_{k} + γ_{k})} ([) S_{k_{n}} - S_{k}^{0} - \int_{S_{k}^{0}}^{S_{k_{n}}} \frac{f_{k} (S_{k}^{0})}{f_{k} (η)} d η + (1 + h (d_{k}^{I} + ϵ_{k} + γ_{k})) I_{k_{n}} (]) .

Applying Lemma 4.1 in and (H1) , we obtain

\int_{x_{1}}^{x_{2}} \frac{1}{f_{k} (s)} d s ⩾ \frac{x_{2} - x_{1}}{f_{k} (x_{2})}, x_{1}, x_{2} > 0, f o r k = 1, 2, \dots, m .

Note that

g_{j} (I) ⩽ g_{j}^{'} (0) I

for all I >0. Then, the difference of L n is …”

Section: Global Stability Of the Equilibriamentioning

confidence: 99%

“… pointed out that how to choose the discrete schemes that preserve the global asymptotic stability for equilibria of the corresponding continuous‐time epidemic models was still an open problem. Recently, researchers have used nonstandard finite difference (NSFD) scheme, which was developed by Mickens , to investigate the global stability of the corresponding continuous models (for example, Ding and Ding , Enatsu et al , Hattaf and Yousfi , Liu et al . , Sekiguchi and Ishiwata , and Yang et al .…”

Section: Introductionmentioning

confidence: 99%

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Global stability of a discrete multigroup SIR model with nonlinear incidence rate

Zhou

Zhang

2017

Math Methods in App Sciences

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In this paper, by applying nonstandard finite difference scheme, we propose a discrete multigroup Susceptible‐Infective‐Removed (SIR) model with nonlinear incidence rate. Using Lyapunov functions, it is shown that the global dynamics of this model are completely determined by the basic reproduction number scriptR0. If scriptR0⩽1, then the disease‐free equilibrium is globally asymptotically stable; if scriptR0>1, then there exists a unique endemic equilibrium and it is globally asymptotically stable. Example and numerical simulations are presented to illustrate the results. Copyright © 2017 John Wiley & Sons, Ltd.

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L_{n} = \sum_{k = 1}^{m} \frac{ω_{k}}{h (d_{k}^{I} + ϵ_{k} + γ_{k})} ([) S_{k_{n}} - S_{k}^{0} - \int_{S_{k}^{0}}^{S_{k_{n}}} \frac{f_{k} (S_{k}^{0})}{f_{k} (η)} d η + (1 + h (d_{k}^{I} + ϵ_{k} + γ_{k})) I_{k_{n}} (]) .

Applying Lemma 4.1 in and (H1) , we obtain

\int_{x_{1}}^{x_{2}} \frac{1}{f_{k} (s)} d s ⩾ \frac{x_{2} - x_{1}}{f_{k} (x_{2})}, x_{1}, x_{2} > 0, f o r k = 1, 2, \dots, m .

Note that

g_{j} (I) ⩽ g_{j}^{'} (0) I

for all I >0. Then, the difference of L n is …”

Section: Global Stability Of the Equilibriamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Global stability of a discrete multigroup SIR model with nonlinear incidence rate

Zhou

Zhang

2017

Math Methods in App Sciences

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“…malaria, tuberculosis, etc. It is pointed out in [14,15] that discretetime epidemic models are much better as compared to continuous ones because discrete-time models allow arbitrary time-step units, preserving the basic features of corresponding continuous-time models. Moreover, in case of discrete-time models, we can use statistical data for numerical simulations because infection data are computed at discrete-time.…”

Section: Introductionmentioning

confidence: 99%

Qualitative behavior of a discrete SIR epidemic model

Din¹

2016

Int. J. Biomath.

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In this paper, we study the qualitative behavior of a discrete-time epidemic model. More precisely, we investigate equilibrium points, asymptotic stability of both disease-free equilibrium and the endemic equilibrium. Furthermore, by using comparison method, we obtain the global stability of these equilibrium points under certain parametric conditions. Some illustrative examples are provided to support our theoretical discussion.

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“…There is a basic reason that the statistical data of epidemic are collected or reported in discrete time, such as hourly, weekly or yearly. More importantly, we can exhibit more richer and complicated dynamical behaviors in the discrete-time models such as generating oscillations, bifurcations, chaos, see [11,12,28]. A well known method is referred to as the non-standard finite difference method, which is developed by Mickens [14][15][16] and has brought the creation of new numerical schemes that preserve the properties of the continuous model [17,18].…”

Section: Introductionmentioning

confidence: 99%

A Non-Standard Finite Difference Scheme of a Multiple Infected Compartments Model for Waterborne Disease

Zhang

Gao

Zou

2016

Differ Equ Dyn Syst

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In this paper, we study a multiple infected compartments model for waterborne diseases which is derived from the continuous case by using the well-known Mickensnonstandard discretization. The positivity of solutions with positive initial conditions and the expressions of equilibria are obtained. By applying analytic techniques and constructing discrete Lyapunov functions, we obtain the results that if R 0 ≤ 1, the disease-free equilibrium is globally asymptotically stable, and if R 0 > 1 the unique endemic equilibrium is also globally asymptotically stable when the system degenerates the fast-slow system. Furthermore, numerical simulations verify our theoretical results. Our numerical results imply that the decay rate of pathogen in the water has no influence on endemic equilibrium, but it has a significant impact on the peak value of infected individuals.

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Global stability for a class of discrete SIR epidemic models

Cited by 61 publications

References 17 publications

Global stability of a discrete multigroup SIR model with nonlinear incidence rate

Global stability of a discrete multigroup SIR model with nonlinear incidence rate

Qualitative behavior of a discrete SIR epidemic model

A Non-Standard Finite Difference Scheme of a Multiple Infected Compartments Model for Waterborne Disease

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