
-
Previous Article
Feedback control of an HBV model based on ensemble kalman filter and differential evolution
- MBE Home
- This Issue
-
Next Article
Optimal individual strategies for influenza vaccines with imperfect efficacy and durability of protection
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.
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. |
[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] |
Hong Yang, Junjie Wei. Dynamics of spatially heterogeneous viral model with time delay. Communications on Pure and Applied Analysis, 2020, 19 (1) : 85-102. doi: 10.3934/cpaa.2020005 |
[3] |
Songbai Guo, Wanbiao Ma. Global dynamics of a microorganism flocculation model with time delay. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1883-1891. doi: 10.3934/cpaa.2017091 |
[4] |
Rong Wang, Yihong Du. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2201-2238. doi: 10.3934/dcdsb.2020360 |
[5] |
Yijun Lou, Xiao-Qiang Zhao. Threshold dynamics in a time-delayed periodic SIS epidemic model. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 169-186. doi: 10.3934/dcdsb.2009.12.169 |
[6] |
Lifeng Han, Changhan He, Yang Kuang. Dynamics of a model of tumor-immune interaction with time delay and noise. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2347-2363. doi: 10.3934/dcdss.2020140 |
[7] |
Hongying Shu, Xiang-Sheng Wang. Global dynamics of a coupled epidemic model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1575-1585. doi: 10.3934/dcdsb.2017076 |
[8] |
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 |
[9] |
Zhigui Lin, Yinan Zhao, Peng Zhou. The infected frontier in an SEIR epidemic model with infinite delay. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2355-2376. doi: 10.3934/dcdsb.2013.18.2355 |
[10] |
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 and Continuous Dynamical Systems, 2009, 23 (3) : 1041-1060. doi: 10.3934/dcds.2009.23.1041 |
[11] |
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 |
[12] |
Min Yu, Gang Huang, Yueping Dong, Yasuhiro Takeuchi. Complicated dynamics of tumor-immune system interaction model with distributed time delay. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2391-2406. doi: 10.3934/dcdsb.2020015 |
[13] |
Ning Wang, Zhi-Cheng Wang. Propagation dynamics of a nonlocal time-space periodic reaction-diffusion model with delay. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1599-1646. doi: 10.3934/dcds.2021166 |
[14] |
Meihong Qiao, Anping Liu, Qing Tang. The dynamics of an HBV epidemic model on complex heterogeneous networks. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1393-1404. doi: 10.3934/dcdsb.2015.20.1393 |
[15] |
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 |
[16] |
Jia-Feng Cao, Wan-Tong Li, Fei-Ying Yang. Dynamics of a nonlocal SIS epidemic model with free boundary. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 247-266. doi: 10.3934/dcdsb.2017013 |
[17] |
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 |
[18] |
Jianquan Li, Yicang Zhou, Jianhong Wu, Zhien Ma. Complex dynamics of a simple epidemic model with a nonlinear incidence. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 161-173. doi: 10.3934/dcdsb.2007.8.161 |
[19] |
Fei-Ying Yang, Wan-Tong Li. Dynamics of a nonlocal dispersal SIS epidemic model. Communications on Pure and Applied Analysis, 2017, 16 (3) : 781-798. doi: 10.3934/cpaa.2017037 |
[20] |
Luca Bolzoni, Rossella Della Marca, Maria Groppi, Alessandra Gragnani. Dynamics of a metapopulation epidemic model with localized culling. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : 2307-2330. doi: 10.3934/dcdsb.2020036 |
2018 Impact Factor: 1.313
Tools
Metrics
Other articles
by authors
[Back to Top]