Lyapunov functions and global stability for a discretized  multigroup SIR epidemic model

Ding, Deqiong; Qin, Wendi; Ding, Xiaohua

doi:10.3934/dcdsb.2015.20.1971

Cited by 18 publications

(16 citation statements)

References 23 publications

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“…Our model contains the model found in the literature of Ding et al. . Applying this method to other types of discrete multigroup epidemic models with time delay is our future work.…”

Section: Discussionmentioning

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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“…Our model contains the model found in the literature of Ding et al. . Applying this method to other types of discrete multigroup epidemic models with time delay is our future work.…”

Section: Discussionmentioning

confidence: 99%

“…By using NSFD scheme, recently, Ding et al . studied a discrete multigroup SIR epidemic model. They concluded that the discrete model has the same dynamics as the original continuous model independent of time step.…”

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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“…To our knowledge, there is no investigation for discrete multi‐group model with vaccination and nonlinear incidence. Hence, motivated by , we construct a discrete multi‐group SVIR epidemic model by applying Mickens' nonstandard finite difference methods to the continuous model :

({, \begin{matrix} array \frac{S_{k_{n + 1}} - S_{k_{n}}}{h} = (1 - p_{k}) b_{k} - \sum_{j = 1}^{m} β_{k j} S_{k_{n + 1}} f_{j} (I_{j_{n}}) - (d_{k}^{S} + ξ_{k}) S_{k_{n + 1}}, \\ array \frac{V_{k_{n + 1}} - V_{k_{n}}}{h} = ξ_{k} S_{k_{n + 1}} - \sum_{j = 1}^{m} {\hat{β}}_{k j} V_{k_{n + 1}} f_{j} (I_{j_{n}}) - d_{k}^{V} V_{k_{n + 1}}, \\ array \frac{I_{k_{n + 1}} - I_{k_{n}}}{h} = \sum_{j = 1}^{m} (β_{k j} S_{k_{n + 1}} + {\hat{β}}_{k j} V_{} \end{matrix})

…”

Section: Introductionmentioning

confidence: 99%

“…Meanwhile, the obtained discrete model should preserve the dynamical properties of the original continuous model as much as possible. Recently, the nonstandard finite difference scheme has been proposed by Mickens [19] and received much attention ( [20][21][22][23][24][25][26] and references therein). One important advantage of Mickens's method is that it can be more efficient in preserving the global asymptotic stability for equilibria to the corresponding continuous time-epidemic models [24][25][26].…”

Section: Introductionmentioning

confidence: 99%

Stability preserving NSFD scheme for a multi-group SVIR epidemic model

Geng

2017

Math. Meth. Appl. Sci.

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A discrete multi‐group SVIR epidemic model with general nonlinear incidence rate and vaccination is investigated by utilizing Mickens' nonstandard finite difference scheme to a corresponding continuous model. Mathematical analysis shows that the global asymptotic stability of the equilibria is fully determined by the basic reproduction number scriptR0vac by constructing Lyapunov functions. The results imply that the discretization scheme can efficiently preserves the global asymptotic stability of the equilibria for corresponding continuous model, and numerical simulations are carried out to illustrate the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

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“…However, how to select a proper discrete method so that the global properties of solutions of the corresponding continuous models can be efficiently preserved is still an open problem [14]. Recently, Mickens has made an attempt in this regard, by proposing a robust nonstandard finite difference (NSFD) scheme [15,16], which has been widely employed in the study of different kinds of epidemic models and one important advantage of Mickens' method is that it can be more efficient in preserving the global dynamics to the corresponding continuous epidemic models [10,[17][18][19][20][21][22][23]. However, there is no result about discrete viral infection model with time delays and immune response.…”

Section: Discrete Dynamics In Nature and Societymentioning

confidence: 99%

Dynamic Consistent NSFD Scheme for a Delayed Viral Infection Model with Immune Response and Nonlinear Incidence

Geng

2017

Discrete Dynamics in Nature and Society

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In this paper, a discrete-time model has been proposed by applying nonstandard finite difference (NSFD) scheme to solve a delayed viral infection model with immune response and general nonlinear incidence. It is shown that the discrete model has equilibria which are exactly the same as those of the original continuous model. Using discrete-time analogue of Lyapunov functionals, the global asymptotic stability of the equilibria of the discrete model is fully determined by the basic reproduction number of the virus and immune response, R 0 and R 1 , with no restriction on the time step size, which implies that the NSFD scheme preserves the qualitative dynamics of the corresponding continuous model.

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Lyapunov functions and global stability for a discretized multigroup SIR epidemic model

Cited by 18 publications

References 23 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

Stability preserving NSFD scheme for a multi-group SVIR epidemic model

Dynamic Consistent NSFD Scheme for a Delayed Viral Infection Model with Immune Response and Nonlinear Incidence

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