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

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June 2018, 15(3): 653-666. doi: 10.3934/mbe.2018029

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

Masaki Sekiguchi ^1,, , Emiko Ishiwata ^2, and Yukihiko Nakata ^3,

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

* Corresponding author: Masaki Sekiguchi

Received March 12, 2017 Published December 2017

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.

Keywords: Epidemic model, ultra-discretization, time delay, dynamics.

Mathematics Subject Classification: Primary: 37N25; Secondary: 39B82.

Citation: Masaki Sekiguchi, Emiko Ishiwata, Yukihiko Nakata. Dynamics of an ultra-discrete SIR epidemic model with time delay. Mathematical Biosciences & Engineering, 2018, 15 (3) : 653-666. doi: 10.3934/mbe.2018029

References:

[1]	L. J. S. Allen, Some discrete-time SI, SIR and SIS epidemic models, Math. Bio., 124 (1994), 83-105. doi: 10.1016/0025-5564(94)90025-6.
[2]	E. Beretta and Y. Takeuchi, Global stability of an SIR epidemic model with time delays, J. Math. Biol., 33 (1995), 250-260. doi: 10.1007/BF00169563.
[3]	E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonl. Anal., 47 (2001), 4107-4115. doi: 10.1016/S0362-546X(01)00528-4.
[4]	R. M. Corless, C. Essex and M. A. H. Nerenberg, Numerical methods can suppress chaos, Phys. Lett. A, 157 (1991), 27-36.
[5]	Y. Enatsu, Y. Nakata and Y. Muroya, Global stability for a class of discrete SIR epidemic models, Math. Bio. and Eng., 7 (2010), 347-361. doi: 10.3934/mbe.2010.7.347.
[6]	Y. Enatsu, Y. Nakata, Y. Muroya, G Izzo and 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. doi: 10.1080/10236198.2011.555405.
[7]	S. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 2005.
[8]	D. F. Griffiths, P. K. Sweby and H. C. Yee, On spurious asymptotic numerical solutions of explicit Runge-Kutta methods, IMA J. Numer. Anal., 12 (1992), 319-338. doi: 10.1093/imanum/12.3.319.
[9]	J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences Vol 99, Springer, 1993. doi: 10.1007/978-1-4612-4342-7.
[10]	G. Izzo and 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. doi: 10.1016/j.cam.2006.10.065.
[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. doi: 10.1155/2009/143019.
[12]	L. Jódar, R. J. Villanueva, A. J. Arenas and G. C. González, Nonstandard numerical methods for a mathematical model for influenza disease, Math. Comput. Simul., 79 (2008), 622-633. doi: 10.1016/j.matcom.2008.04.008.
[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. doi: 10.1007/978-1-4614-4559-3_7.
[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. doi: 10.1016/j.nonrwa.2008.10.014.
[15]	R. E. Mickens, Discretizations of nonlinear differential equations using explicit nonstandard methods, J. Comput. Appl. Math., 110 (1999), 181-185. doi: 10.1016/S0377-0427(99)00233-2.
[16]	S. M. Moghadas, M. E. Alexander, B. D. Corbett and A. B. Gumel, A positivity-preserving Mickens-type discretization of an epidemic model, J. Diff. Equ. Appl., 9 (2003), 1037-1051. doi: 10.1080/1023619031000146913.
[17]	K. Matsuya and M. Murata, Spatial pattern of discrete and ultradiscrete Gray-Scott model, DCDS-B, 20 (2015), 173-187. doi: 10.3934/dcdsb.2015.20.173.
[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 and D. Takahashi, Analytical properties of ultradiscrete Burgers equation and rule-184 cellular automaton, J. Phys. A, 31 (1998), 5439-5450. doi: 10.1088/0305-4470/31/24/006.
[20]	A. Ramani, A. S. Carstea, R. Willox and B. Grammaticos, Oscillating epidemics: A discrete-time model, Phys. A, 333 (2004), 278-292. doi: 10.1016/j.physa.2003.10.051.
[21]	T. Tokihiro, D. Takahashi, J. Matsukidaira and J. Satsuma, From soliton equations to integrable cellular automata through a limiting procedure, Phys. Rev. Lett., 76 (1996), 3247-3250. doi: 10.1103/PhysRevLett.76.3247.
[22]	J. Satsuma, R. Willox, A. Ramani, B. Grammaticos and 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. doi: 10.1142/S1793524509000807.
[24]	M. Sekiguchi and E. Ishiwata, Global dynamics of a discretized SIRS epidemic model with time delay, J. Math. Anal. Appl., 371 (2010), 195-202. doi: 10.1016/j.jmaa.2010.05.007.
[25]	G. C. Sirakoulis, I. Karafyllidis and A. Thanailakis, A cellular automaton model for the effects of population movement and vaccination on epidemic propagation, Ecol. Mod., 133 (2000), 209-223. doi: 10.1016/S0304-3800(00)00294-5.
[26]	S. H. White, A. Martin del Rey and G. Rodríguez Sánchez, Modeling epidemics using cellular automata, Appl. Math. Comp., 186 (2007), 193-202. doi: 10.1016/j.amc.2006.06.126.
[27]	R. Willox, B. Grammaticos, A. S. Carstea and A. Ramani, Epidemic dynamics: Discrete-time and cellular automaton models, Phys. A, 328 (2003), 13-22. doi: 10.1016/S0378-4371(03)00552-1.
[28]	S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys., 55 (1983), 601-644. doi: 10.1103/RevModPhys.55.601.
[29]	T. Zhang and Z. Teng, Global behavior and permanence of SIRS epidemic model with time delay, Nonl. Anal. RWA., 9 (2008), 1409-1424. doi: 10.1016/j.nonrwa.2007.03.010.

show all references

References:

[1]	L. J. S. Allen, Some discrete-time SI, SIR and SIS epidemic models, Math. Bio., 124 (1994), 83-105. doi: 10.1016/0025-5564(94)90025-6.
[2]	E. Beretta and Y. Takeuchi, Global stability of an SIR epidemic model with time delays, J. Math. Biol., 33 (1995), 250-260. doi: 10.1007/BF00169563.
[3]	E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonl. Anal., 47 (2001), 4107-4115. doi: 10.1016/S0362-546X(01)00528-4.
[4]	R. M. Corless, C. Essex and M. A. H. Nerenberg, Numerical methods can suppress chaos, Phys. Lett. A, 157 (1991), 27-36.
[5]	Y. Enatsu, Y. Nakata and Y. Muroya, Global stability for a class of discrete SIR epidemic models, Math. Bio. and Eng., 7 (2010), 347-361. doi: 10.3934/mbe.2010.7.347.
[6]	Y. Enatsu, Y. Nakata, Y. Muroya, G Izzo and 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. doi: 10.1080/10236198.2011.555405.
[7]	S. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 2005.
[8]	D. F. Griffiths, P. K. Sweby and H. C. Yee, On spurious asymptotic numerical solutions of explicit Runge-Kutta methods, IMA J. Numer. Anal., 12 (1992), 319-338. doi: 10.1093/imanum/12.3.319.
[9]	J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences Vol 99, Springer, 1993. doi: 10.1007/978-1-4612-4342-7.
[10]	G. Izzo and 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. doi: 10.1016/j.cam.2006.10.065.
[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. doi: 10.1155/2009/143019.
[12]	L. Jódar, R. J. Villanueva, A. J. Arenas and G. C. González, Nonstandard numerical methods for a mathematical model for influenza disease, Math. Comput. Simul., 79 (2008), 622-633. doi: 10.1016/j.matcom.2008.04.008.
[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. doi: 10.1007/978-1-4614-4559-3_7.
[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. doi: 10.1016/j.nonrwa.2008.10.014.
[15]	R. E. Mickens, Discretizations of nonlinear differential equations using explicit nonstandard methods, J. Comput. Appl. Math., 110 (1999), 181-185. doi: 10.1016/S0377-0427(99)00233-2.
[16]	S. M. Moghadas, M. E. Alexander, B. D. Corbett and A. B. Gumel, A positivity-preserving Mickens-type discretization of an epidemic model, J. Diff. Equ. Appl., 9 (2003), 1037-1051. doi: 10.1080/1023619031000146913.
[17]	K. Matsuya and M. Murata, Spatial pattern of discrete and ultradiscrete Gray-Scott model, DCDS-B, 20 (2015), 173-187. doi: 10.3934/dcdsb.2015.20.173.
[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 and D. Takahashi, Analytical properties of ultradiscrete Burgers equation and rule-184 cellular automaton, J. Phys. A, 31 (1998), 5439-5450. doi: 10.1088/0305-4470/31/24/006.
[20]	A. Ramani, A. S. Carstea, R. Willox and B. Grammaticos, Oscillating epidemics: A discrete-time model, Phys. A, 333 (2004), 278-292. doi: 10.1016/j.physa.2003.10.051.
[21]	T. Tokihiro, D. Takahashi, J. Matsukidaira and J. Satsuma, From soliton equations to integrable cellular automata through a limiting procedure, Phys. Rev. Lett., 76 (1996), 3247-3250. doi: 10.1103/PhysRevLett.76.3247.
[22]	J. Satsuma, R. Willox, A. Ramani, B. Grammaticos and 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. doi: 10.1142/S1793524509000807.
[24]	M. Sekiguchi and E. Ishiwata, Global dynamics of a discretized SIRS epidemic model with time delay, J. Math. Anal. Appl., 371 (2010), 195-202. doi: 10.1016/j.jmaa.2010.05.007.
[25]	G. C. Sirakoulis, I. Karafyllidis and A. Thanailakis, A cellular automaton model for the effects of population movement and vaccination on epidemic propagation, Ecol. Mod., 133 (2000), 209-223. doi: 10.1016/S0304-3800(00)00294-5.
[26]	S. H. White, A. Martin del Rey and G. Rodríguez Sánchez, Modeling epidemics using cellular automata, Appl. Math. Comp., 186 (2007), 193-202. doi: 10.1016/j.amc.2006.06.126.
[27]	R. Willox, B. Grammaticos, A. S. Carstea and A. Ramani, Epidemic dynamics: Discrete-time and cellular automaton models, Phys. A, 328 (2003), 13-22. doi: 10.1016/S0378-4371(03)00552-1.
[28]	S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys., 55 (1983), 601-644. doi: 10.1103/RevModPhys.55.601.
[29]	T. Zhang and Z. Teng, Global behavior and permanence of SIRS epidemic model with time delay, Nonl. Anal. RWA., 9 (2008), 1409-1424. doi: 10.1016/j.nonrwa.2007.03.010.

Figure 1. Numerical experiments $x_{n}$ and $y_{n}$ with $\omega =0$.

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Figure 2. Numerical experiments $x_{n}$ and $y_{n}$ with $\omega =10$.

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Figure 3. A solution $w^{j}_{m}$ is constructed by two solutions $u^{j}_{m}$ and $v^{j}_{m}$

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