• Previous Article
    Optimal individual strategies for influenza vaccines with imperfect efficacy and durability of protection
  • MBE Home
  • This Issue
  • Next Article
    Feedback control of an HBV model based on ensemble kalman filter and differential evolution
June 2018, 15(3): 653-666. doi: 10.3934/mbe.2018029

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

* 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.

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. BerettaT. HaraW. 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. CorlessC. Essex and M. A. H. Nerenberg, Numerical methods can suppress chaos, Phys. Lett. A, 157 (1991), 27-36.

[5]

Y. EnatsuY. 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. EnatsuY. NakataY. MuroyaG 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. GriffithsP. 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. JódarR. J. VillanuevaA. 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. MoghadasM. E. AlexanderB. 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. RamaniA. S. CarsteaR. 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. TokihiroD. TakahashiJ. 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. SatsumaR. WilloxA. RamaniB. 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. SirakoulisI. 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. WhiteA. 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. WilloxB. GrammaticosA. 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. BerettaT. HaraW. 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. CorlessC. Essex and M. A. H. Nerenberg, Numerical methods can suppress chaos, Phys. Lett. A, 157 (1991), 27-36.

[5]

Y. EnatsuY. 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. EnatsuY. NakataY. MuroyaG 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. GriffithsP. 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. JódarR. J. VillanuevaA. 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. MoghadasM. E. AlexanderB. 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. RamaniA. S. CarsteaR. 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. TokihiroD. TakahashiJ. 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. SatsumaR. WilloxA. RamaniB. 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. SirakoulisI. 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. WhiteA. 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. WilloxB. GrammaticosA. 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 $\omega =0$.
Figure 2.  Numerical experiments $x_{n}$ and $y_{n}$ with $\omega =10$.
Figure 3.  A solution $w^{j}_{m}$ is constructed by two solutions $u^{j}_{m}$ and $v^{j}_{m}$
[1]

Lin Zhao, Zhi-Cheng Wang, Liang Zhang. Threshold dynamics of a time periodic and two–group epidemic model with distributed delay. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1535-1563. doi: 10.3934/mbe.2017080

[2]

Songbai Guo, Wanbiao Ma. Global dynamics of a microorganism flocculation model with time delay. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1883-1891. doi: 10.3934/cpaa.2017091

[3]

Yijun Lou, Xiao-Qiang Zhao. Threshold dynamics in a time-delayed periodic SIS epidemic model. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 169-186. doi: 10.3934/dcdsb.2009.12.169

[4]

Hongying Shu, Xiang-Sheng Wang. Global dynamics of a coupled epidemic model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1575-1585. doi: 10.3934/dcdsb.2017076

[5]

F. Berezovskaya, G. Karev, Baojun Song, Carlos Castillo-Chavez. A Simple Epidemic Model with Surprising Dynamics. Mathematical Biosciences & Engineering, 2005, 2 (1) : 133-152. doi: 10.3934/mbe.2005.2.133

[6]

Zhigui Lin, Yinan Zhao, Peng Zhou. The infected frontier in an SEIR epidemic model with infinite delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2355-2376. doi: 10.3934/dcdsb.2013.18.2355

[7]

Maria do Carmo Pacheco de Toledo, Sergio Muniz Oliva. A discretization scheme for an one-dimensional reaction-diffusion equation with delay and its dynamics. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1041-1060. doi: 10.3934/dcds.2009.23.1041

[8]

Jianquan Li, Xiaoqin Wang, Xiaolin Lin. Impact of behavioral change on the epidemic characteristics of an epidemic model without vital dynamics. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1425-1434. doi: 10.3934/mbe.2018065

[9]

Meihong Qiao, Anping Liu, Qing Tang. The dynamics of an HBV epidemic model on complex heterogeneous networks. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1393-1404. doi: 10.3934/dcdsb.2015.20.1393

[10]

Aili Wang, Yanni Xiao, Huaiping Zhu. Dynamics of a Filippov epidemic model with limited hospital beds. Mathematical Biosciences & Engineering, 2018, 15 (3) : 739-764. doi: 10.3934/mbe.2018033

[11]

Jia-Feng Cao, Wan-Tong Li, Fei-Ying Yang. Dynamics of a nonlocal SIS epidemic model with free boundary. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 247-266. doi: 10.3934/dcdsb.2017013

[12]

Zhisheng Shuai, P. van den Driessche. Impact of heterogeneity on the dynamics of an SEIR epidemic model. Mathematical Biosciences & Engineering, 2012, 9 (2) : 393-411. doi: 10.3934/mbe.2012.9.393

[13]

Jianquan Li, Yicang Zhou, Jianhong Wu, Zhien Ma. Complex dynamics of a simple epidemic model with a nonlinear incidence. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 161-173. doi: 10.3934/dcdsb.2007.8.161

[14]

Fei-Ying Yang, Wan-Tong Li. Dynamics of a nonlocal dispersal SIS epidemic model. Communications on Pure & Applied Analysis, 2017, 16 (3) : 781-798. doi: 10.3934/cpaa.2017037

[15]

Zhen Jin, Zhien Ma. The stability of an SIR epidemic model with time delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 101-109. doi: 10.3934/mbe.2006.3.101

[16]

C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837

[17]

Chunhua Shan, Hongjun Gao, Huaiping Zhu. Dynamics of a delay Schistosomiasis model in snail infections. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1099-1115. doi: 10.3934/mbe.2011.8.1099

[18]

Hui Wan, Huaiping Zhu. A new model with delay for mosquito population dynamics. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1395-1410. doi: 10.3934/mbe.2014.11.1395

[19]

Monika Joanna Piotrowska, Urszula Foryś, Marek Bodnar, Jan Poleszczuk. A simple model of carcinogenic mutations with time delay and diffusion. Mathematical Biosciences & Engineering, 2013, 10 (3) : 861-872. doi: 10.3934/mbe.2013.10.861

[20]

Matthieu Hillairet, Alexei Lozinski, Marcela Szopos. On discretization in time in simulations of particulate flows. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 935-956. doi: 10.3934/dcdsb.2011.15.935

2017 Impact Factor: 1.23

Metrics

  • PDF downloads (121)
  • HTML views (571)
  • Cited by (0)

[Back to Top]