Global analysis of discrete-time SI and SIS epidemic models

Jianquan Li; Zhien Ma; Fred Brauer; Jianquan Li; Zhien Ma; Fred Brauer

doi:10.3934/mbe.2007.4.699

Mathematical Biosciences and Engineering

2007, Volume 4, Issue 4: 699-710. doi: 10.3934/mbe.2007.4.699

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Global analysis of discrete-time SI and SIS epidemic models

1.
Department of Applied Mathematics and Physics, Air Force Engineering University, Xi'an 710051
2.
Department of Mathematics, Xi’an Jiaotong University, Xi’an, 710049
3.
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2

Received: 01 February 2007 Accepted: 29 June 2018 Published: 01 August 2007
MSC : 92D30, 39A11.

Discrete-time SI and SIS models formulated as the discretization of a continuous-time model may exhibit behavior different from that of the continuous-time model such as period-doubling and chaotic behavior unless the step size in the model is sufficiently small. Some new discrete-time SI and SIS epidemic models with vital dynamics are formulated and analyzed. These new models do not exhibit period doubling and chaotic behavior and are thus better approximations to continuous models. However, their reproduction numbers and therefore their asymptotic behavior can differ somewhat from that of the corresponding continuous-time model.
- discrete-time epidemic model,
- dynamic behavior,
- equilibrium,
- stability.
Citation: Jianquan Li, Zhien Ma, Fred Brauer. Global analysis of discrete-time SI and SIS epidemic models[J]. Mathematical Biosciences and Engineering, 2007, 4(4): 699-710. doi: 10.3934/mbe.2007.4.699

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Abstract

Discrete-time SI and SIS models formulated as the discretization of a continuous-time model may exhibit behavior different from that of the continuous-time model such as period-doubling and chaotic behavior unless the step size in the model is sufficiently small. Some new discrete-time SI and SIS epidemic models with vital dynamics are formulated and analyzed. These new models do not exhibit period doubling and chaotic behavior and are thus better approximations to continuous models. However, their reproduction numbers and therefore their asymptotic behavior can differ somewhat from that of the corresponding continuous-time model.
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