# American Institute of Mathematical Sciences

2014, 11(6): 1375-1393. doi: 10.3934/mbe.2014.11.1375

## Threshold dynamics of an SIR epidemic model with hybrid of multigroup and patch structures

 1 Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501 2 Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555 3 Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914

Received  July 2013 Revised  May 2014 Published  September 2014

In this paper, we formulate an SIR epidemic model with hybrid of multigroup and patch structures, which can be regarded as a model for the geographical spread of infectious diseases or a multi-group model with perturbation. We show that if a threshold value, which corresponds to the well-known basic reproduction number $R_0$, is less than or equal to unity, then the disease-free equilibrium of the model is globally asymptotically stable. We also show that if the threshold value is greater than unity, then the model is uniformly persistent and has an endemic equilibrium. Moreover, using a Lyapunov functional technique, we obtain a sufficient condition under which the endemic equilibrium is globally asymptotically stable. The sufficient condition is satisfied if the transmission coefficients in the same groups are large or the per capita recovery rates are small.
Citation: Toshikazu Kuniya, Yoshiaki Muroya, Yoichi Enatsu. Threshold dynamics of an SIR epidemic model with hybrid of multigroup and patch structures. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1375-1393. doi: 10.3934/mbe.2014.11.1375
##### References:
 [1] R. M. Anderson and R. M. May, Infectious Diseases of Humans, Oxford University, Oxford, 1991. [2] J. Arino, Diseases in metapopulations, in Modeling and Dynamics of Infectious Diseases (eds. Z. Ma, Y. Zhou and J. Wu), Higher Education Press, 2009, 65-122. doi: 10.1142/7223. [3] M. S. Bartlet, Deterministic and stochastic models for recurrent epidemics, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, 1956, 81-109. [4] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. doi: 10.1007/978-1-4612-0873-0. [5] H. Chen and J. Sun, Global stability of delay multigroup epidemic models with group mixing nonlinear incidence rates, Appl. Math. Comput., 218 (2011), 4391-4400. doi: 10.1016/j.amc.2011.10.015. [6] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, 1st edition, John Wiley and Sons, Chichester, 2000. doi: 10.1007/978-1-4612-0873-0. [7] M. J. Faddy, A note on the behavior of deterministic spatial epidemics, Math. Biosci., 80 (1986), 19-22. doi: 10.1016/0025-5564(86)90064-7. [8] H. I. Freedman, M. X. Tang and S. G. Ruan, Uniform persistence and flows near a closed positively invariant set, J. Dynam. Diff. Equat., 6 (1994), 583-600. doi: 10.1007/BF02218848. [9] H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canadian Appl. Math. Quart., 14 (2006), 259-284. [10] H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802. doi: 10.1090/S0002-9939-08-09341-6. [11] J. M. Hyman and T. LaForce, Modeling the spread of influenza among cities, in Bioterrorism (eds. H. T. Banks and C. Castillo-Chavez), SIAM, 2003, 211-236. [12] Y. Jin and W. Wang, The effect of population dispersal on the spread of a disease, J. Math. Anal. Appl., 308 (2005), 343-364. doi: 10.1016/j.jmaa.2005.01.034. [13] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic model, Math. Med. Biol., 21 (2004), 75-83. doi: 10.1007/s11538-008-9352-z. [14] T. Kuniya and Y. Muroya, Global stability of a multi-group SIS epidemic model for population migration, Discrete Cont. Dyn. Sys. Series B, 19 (2014), 1105-1118. doi: 10.3934/dcdsb.2014.19.1105. [15] J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976. doi: 10.1007/978-1-4612-0873-0. [16] M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999), 191-213. doi: 10.1016/S0025-5564(99)00030-9. [17] M. Y. Li and Z. Shuai, Global stability of an epidemic model in a patchy environment, Canadian Appl. Math. Quart., 17 (2009), 175-187. [18] M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Diff. Equat., 284 (2010), 1-20. doi: 10.1016/j.jde.2009.09.003. [19] M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47. doi: 10.1016/j.jmaa.2009.09.017. [20] Y. Muroya, Y. Enatsu and T. Kuniya, Global stability of extended multi-group SIR epidemic models with patches through migration and cross patch infection, Acta Mathematica Scientia, 33 (2013), 341-361. doi: 10.1016/S0252-9602(13)60003-X. [21] H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. RWA, 13 (2012), 1581-1592. doi: 10.1016/j.nonrwa.2011.11.016. [22] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995. doi: 10.1007/978-1-4612-0873-0. [23] R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence, Comput. Math. Appl., 60 (2010), 2286-2291. doi: 10.1016/j.camwa.2010.08.020. [24] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [25] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. doi: 10.1007/978-1-4612-0873-0. [26] W. Wang and X. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001. [27] Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Anal. RWA, 11 (2010), 995-1004. doi: 10.1016/j.nonrwa.2009.01.040.

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##### References:
 [1] R. M. Anderson and R. M. May, Infectious Diseases of Humans, Oxford University, Oxford, 1991. [2] J. Arino, Diseases in metapopulations, in Modeling and Dynamics of Infectious Diseases (eds. Z. Ma, Y. Zhou and J. Wu), Higher Education Press, 2009, 65-122. doi: 10.1142/7223. [3] M. S. Bartlet, Deterministic and stochastic models for recurrent epidemics, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, 1956, 81-109. [4] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. doi: 10.1007/978-1-4612-0873-0. [5] H. Chen and J. Sun, Global stability of delay multigroup epidemic models with group mixing nonlinear incidence rates, Appl. Math. Comput., 218 (2011), 4391-4400. doi: 10.1016/j.amc.2011.10.015. [6] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, 1st edition, John Wiley and Sons, Chichester, 2000. doi: 10.1007/978-1-4612-0873-0. [7] M. J. Faddy, A note on the behavior of deterministic spatial epidemics, Math. Biosci., 80 (1986), 19-22. doi: 10.1016/0025-5564(86)90064-7. [8] H. I. Freedman, M. X. Tang and S. G. Ruan, Uniform persistence and flows near a closed positively invariant set, J. Dynam. Diff. Equat., 6 (1994), 583-600. doi: 10.1007/BF02218848. [9] H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canadian Appl. Math. Quart., 14 (2006), 259-284. [10] H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802. doi: 10.1090/S0002-9939-08-09341-6. [11] J. M. Hyman and T. LaForce, Modeling the spread of influenza among cities, in Bioterrorism (eds. H. T. Banks and C. Castillo-Chavez), SIAM, 2003, 211-236. [12] Y. Jin and W. Wang, The effect of population dispersal on the spread of a disease, J. Math. Anal. Appl., 308 (2005), 343-364. doi: 10.1016/j.jmaa.2005.01.034. [13] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic model, Math. Med. Biol., 21 (2004), 75-83. doi: 10.1007/s11538-008-9352-z. [14] T. Kuniya and Y. Muroya, Global stability of a multi-group SIS epidemic model for population migration, Discrete Cont. Dyn. Sys. Series B, 19 (2014), 1105-1118. doi: 10.3934/dcdsb.2014.19.1105. [15] J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976. doi: 10.1007/978-1-4612-0873-0. [16] M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999), 191-213. doi: 10.1016/S0025-5564(99)00030-9. [17] M. Y. Li and Z. Shuai, Global stability of an epidemic model in a patchy environment, Canadian Appl. Math. Quart., 17 (2009), 175-187. [18] M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Diff. Equat., 284 (2010), 1-20. doi: 10.1016/j.jde.2009.09.003. [19] M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47. doi: 10.1016/j.jmaa.2009.09.017. [20] Y. Muroya, Y. Enatsu and T. Kuniya, Global stability of extended multi-group SIR epidemic models with patches through migration and cross patch infection, Acta Mathematica Scientia, 33 (2013), 341-361. doi: 10.1016/S0252-9602(13)60003-X. [21] H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. RWA, 13 (2012), 1581-1592. doi: 10.1016/j.nonrwa.2011.11.016. [22] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995. doi: 10.1007/978-1-4612-0873-0. [23] R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence, Comput. Math. Appl., 60 (2010), 2286-2291. doi: 10.1016/j.camwa.2010.08.020. [24] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [25] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. doi: 10.1007/978-1-4612-0873-0. [26] W. Wang and X. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001. [27] Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Anal. RWA, 11 (2010), 995-1004. doi: 10.1016/j.nonrwa.2009.01.040.
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