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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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