Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Dynamics of an ultra-discrete SIR epidemic model with time delay

1. Tokyo Metropolitan Ogikubo High School, 5-7-20, Ogikubo, Suginami-ku, Tokyo 167-0051, Japan
2. Department of Applied Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
3. Department of Mathematics, Shimane University, 1600 Nishikawatsu-cho, 690-8504, Matsue, Japan

We propose an ultra-discretization for an SIR epidemic model with time delay. It is proven that the ultra-discrete model has a threshold property concerning global attractivity of equilibria as shown in differential and difference equation models. We also study an interesting convergence pattern of the solution, which is illustrated in a two-dimensional lattice.

  Figure/Table
  Supplementary
  Article Metrics

References

[1] L. J. S. Allen, Some discrete-time SI, SIR and SIS epidemic models, Math. Bio., 124 (1994): 83-105.

[2] E. Beretta,Y. Takeuchi, Global stability of an SIR epidemic model with time delays, J. Math. Biol., 33 (1995): 250-260.

[3] E. Beretta,T. Hara,W. Ma,Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonl. Anal., 47 (2001): 4107-4115.

[4] R. M. Corless,C. Essex,M. A. H. Nerenberg, Numerical methods can suppress chaos, Phys. Lett. A, 157 (1991): 27-36.

[5] Y. Enatsu,Y. Nakata,Y. Muroya, Global stability for a class of discrete SIR epidemic models, Math. Bio. and Eng., 7 (2010): 347-361.

[6] Y. Enatsu,Y. Nakata,Y. Muroya,G Izzo,A Vecchio, Global dynamics of difference equations for SIR epidemic models with a class of nonlinear incidence rates, J. Diff. Equ. Appl., 18 (2012): 1163-1181.

[7] S. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 2005.

[8] D. F. Griffiths,P. K. Sweby,H. C. Yee, On spurious asymptotic numerical solutions of explicit Runge-Kutta methods, IMA J. Numer. Anal., 12 (1992): 319-338.

[9] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences Vol 99, Springer, 1993.

[10] G. Izzo,A. Vecchio, A discrete time version for models of population dynamics in the presence of an infection, J. Comput. Appl. Math., 210 (2007): 210-221.

[11] G. Izzo, Y. Muroya and A. Vecchio, A general discrete time model of population dynamics in the presence of an infection Disc. Dyn. Nat. Soc. , (2009), Art. ID 143019, 15pp.

[12] L. Jódar,R. J. Villanueva,A. J. Arenas,G. C. González, Nonstandard numerical methods for a mathematical model for influenza disease, Math. Comput. Simul., 79 (2008): 622-633.

[13] C. M. Kent, Piecewise-defined difference equations: Open Problem, 'Bridging Mathematics, Statistics, Engineering and Technology, 55-71, Springer Proc. Math. Stat., 24, Springer, New York, 2012.

[14] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay -distributed or discrete time delays, Nonl. Anal. RWA, 11 (2010): 55-59.

[15] R. E. Mickens, Discretizations of nonlinear differential equations using explicit nonstandard methods, J. Comput. Appl. Math., 110 (1999): 181-185.

[16] S. M. Moghadas,M. E. Alexander,B. D. Corbett,A. B. Gumel, A positivity-preserving Mickens-type discretization of an epidemic model, J. Diff. Equ. Appl., 9 (2003): 1037-1051.

[17] K. Matsuya,M. Murata, Spatial pattern of discrete and ultradiscrete Gray-Scott model, DCDS-B, 20 (2015): 173-187.

[18] K. Matsuya and M. Kanai, Exact solution of a delay difference equation modeling traffic flow and their ultra-discrete limit, arXiv: 1509.07861 [nlin. CG].

[19] K. Nishinari,D. Takahashi, Analytical properties of ultradiscrete Burgers equation and rule-184 cellular automaton, J. Phys. A, 31 (1998): 5439-5450.

[20] A. Ramani,A. S. Carstea,R. Willox,B. Grammaticos, Oscillating epidemics: A discrete-time model, Phys. A, 333 (2004): 278-292.

[21] T. Tokihiro,D. Takahashi,J. Matsukidaira,J. Satsuma, From soliton equations to integrable cellular automata through a limiting procedure, Phys. Rev. Lett., 76 (1996): 3247-3250.

[22] J. Satsuma,R. Willox,A. Ramani,B. Grammaticos,A. S. Carstea, Extending the SIR epidemic model, Phys. A, 336 (2004): 369-375.

[23] M. Sekiguchi, Permanence of some discrete epidemic models, Int. J. Biomath., 2 (2009): 443-461.

[24] M. Sekiguchi,E. Ishiwata, Global dynamics of a discretized SIRS epidemic model with time delay, J. Math. Anal. Appl., 371 (2010): 195-202.

[25] G. C. Sirakoulis,I. Karafyllidis,A. Thanailakis, A cellular automaton model for the effects of population movement and vaccination on epidemic propagation, Ecol. Mod., 133 (2000): 209-223.

[26] S. H. White,A. Martin del Rey,G. Rodríguez Sánchez, Modeling epidemics using cellular automata, Appl. Math. Comp., 186 (2007): 193-202.

[27] R. Willox,B. Grammaticos,A. S. Carstea,A. Ramani, Epidemic dynamics: Discrete-time and cellular automaton models, Phys. A, 328 (2003): 13-22.

[28] S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys., 55 (1983): 601-644.

[29] T. Zhang,Z. Teng, Global behavior and permanence of SIRS epidemic model with time delay, Nonl. Anal. RWA., 9 (2008): 1409-1424.

© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved