Abstract
A mathematical epidemic discrete equation, which appears as a model for the spread of disease-causing, is treated. In this paper, we consider the asymptotic stability of a discrete SIR epidemic model by using the classical linearization method and some Liapunov functions.
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References
Allen, L.J.: Some discrete-time SI, SIR, and SIS epidemic models. Math. Biosci. 124, 83–105 (1994)
Anderson, R.M., May, R.M.: Population biology of infectious diseases. Part1. Nature 280, 361–367 (1979)
Elaydi, S.: An Introduction to Difference Equations, Third edn. Springer, Berlin (2005)
Enatsu, Y., Nakata, Y., Muroya, Y.: Global stability for a class of discrete SIR epidemic models. Math. Biosci. Eng. 7, 347–361 (2010)
Hamaya, Y., Saito, K.: Global asymptotic stability of a delayed SIR epidemic model with diffusion. Libertas Math. 36(1), 53–72 (2016)
Inaba, H.: Mathematical Models for Demography and Epidemics, University of Tokyo Press (2002)
Jang, S., Elaydi, S.: Difference equations from discretization of a continuous epidemic model with immigration of infectives. Canad. Appl. Math. Quart. 11(1), 93–105 (2003)
Mickens, R.: Nonstandard Finite Difference Methods of Differential Equations. World Scientific, Singapore (1994)
Murray, J.D.: Mathematical Biology, Third edn. Springer (2002)
Roeger, L.-I.W.: Dynamically consistent discrete-time SI and SIS epidemic models. Discret. Contin. Dyn. Syst. Suppl. 653–662 (2013)
Saito, K.: On the stability of SIR epidemic discrete models. To be submitted
Acknowledgements
I would like to thank Professor Saber Elaydi for useful comments in this paper. Moreover, I appreciate Professor Jim M. Cushing for useful advices. I am also grateful to Professor Toshiyuki Kohno for assistance with the numerical simulations. Finally, I am grateful to the referees for useful suggestions. The research of this paper is partially supported by the JSPS KAKENHI Grant Number 26400181.
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Saito, K. (2017). On the Stability of an SIR Epidemic Discrete Model. In: Elaydi, S., Hamaya, Y., Matsunaga, H., Pötzsche, C. (eds) Advances in Difference Equations and Discrete Dynamical Systems. ICDEA 2016. Springer Proceedings in Mathematics & Statistics, vol 212. Springer, Singapore. https://doi.org/10.1007/978-981-10-6409-8_15
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DOI: https://doi.org/10.1007/978-981-10-6409-8_15
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