Existence and Uniqueness of Periodic Solution for a Discrete-time SIR Epidemic Model with Time Delays and Impulses

Authors: Ling Liu, Yuan Ye

Abstract:

In this paper, a discrete-time SIR epidemic model with nonlinear incidence rate, time delays and impulses is investigated. Sufficient conditions for the existence and uniqueness of periodic solutions are obtained by using contraction theorem and inequality techniques. An example is employed to illustrate our results.

Keywords: Discrete-time SIR epidemic model, time delay, nonlinear incidence rate, impulse.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1335254

References:

[1] W.O. Kermack, A.G. McKendrick, Contributions to the mathematical theory of epidemics, Proc. Roy. Soc. A 115 (1927) 700-721.
[2] N.T.J. Bailey, The Mathematical Theory of Infectious Disease, Hafner, New York, 1975.
[3] E. Beretta, T. Hara, W. Ma, Y. Takeuchi, Convergence results in SIR epidemic models with varying population sizes, J. Nonlinear Anal. 28 (1997) 1909-1921.
[4] Y. Takeuchi, W. Ma, E. Beretta, Global asymptotic properties of a delay SIR epidemic model with ?nite incubation times, Nonlinear Anal. 42 (2000) 931-947.
[5] W. Ma, Y. Takeuchi, T. Hara, E. Beretta, Permanence of an SIR epidemic model with distributed time delays, Tohoku Math. J. 54 (2002) 581-591.
[6] W. Ma, M. Song, Y. Takeuchi, Global stability of an SIR epidemic model with time delay, Appl. Math. Lett. 17 (2004) 1141-1145.
[7] T.L. Zhang, Z.D. Teng, Global behavior and permanence of SIRS epidemic model with time delay, Nonlinear Anal. Real World Appl. 9 (2009) 1409-1424.
[8] M. Song, W.B. Ma, Y. Takeuchi, Permanence of a delayed SIR epidemic model with density dependent birth rate, J. Comput. Appl. Math. 201 (2007) 389-394.
[9] G. Z. Zeng, L.S. Chen, L.H. Sun, Complexity of an SIR epidemic dynamics model with impulsive vaccination control, Chaos, Solitons and Fractals 26 (2005) 495-505.
[10] G. Zaman, Y.H. Kang, I.H. Jung, Stability analysis and optimal vaccination of an SIR epidemic model, BioSystems 93 (2008) 240-249.
[11] X.Z. Li, W.Sh. Li, M. Ghosh, Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment, Appl. Math. Comput. 210 (2009) 141-150.
[12] R.Q Shi, X.W. Jiang, L.S. Chen, The effect of impulsive vaccination on an SIR epidemic model, Appl. Math.d Comput. 212 (2009) 305-311.
[13] X.A. Zhang, L.S. Chen, The Periodic Solution of a Class of Epidemic Models, Comput. Math. Appl. 38 (1999) 61-71.
[14] R. Xua, Z.E. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Anal. Real World Appl. 10 (2009) 3175-3189.
[15] R. Naresha, A. Tripathi, J.M. Tchuenche, D. Sharma, Stability analysis of a time delayed SIR epidemic model with nonlinear incidence rate, Comput. Math. Appl. 58 (2009) 348-359.
[16] SH.J. Gao, ZH.D. Teng, D.H. Xie, Analysis of a delayed SIR epidemic model with pulse vaccination, Chaos, Solitons and Fractals. 40 (2009) 1004-1011.