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June  2014, 19(4): 1105-1118. doi: 10.3934/dcdsb.2014.19.1105

Global stability of a multi-group SIS epidemic model for population migration

Toshikazu Kuniya 1, and  Yoshiaki Muroya 2,

1. 

Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan
2. 

Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555

Received  December 2012 Revised  October 2013 Published  April 2014

In this paper, using an approach of Lyapunov functional, we establish the complete global stability of a multi-group SIS epidemic model in which the effect of population migration among different regions is considered. We prove the global asymptotic stability of the disease-free equilibrium of the model for $R_0\leq 1$, and that of an endemic equilibrium for $R_0>1$. Here $R_0$ denotes the well-known basic reproduction number defined by the spectral radius of an irreducible nonnegative matrix called the next generation matrix. We emphasize that the graph-theoretic approach, which is typically used for multi-group epidemic models, is not needed in our proof.
Keywords: Lyapunov functional, population migration., global stability, Multi-group SIS epidemic model.
Mathematics Subject Classification: Primary: 34D23, 37N25; Secondary: 92D3.
Citation: Toshikazu Kuniya, Yoshiaki Muroya. Global stability of a multi-group SIS epidemic model for population migration. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1105-1118. doi: 10.3934/dcdsb.2014.19.1105
References:
[1]

J. Arino, Diseases in metapopulations,, in Modeling and Dynamics of Infectious Diseases (eds. Z. Ma, (2009), 65.   Google Scholar

[2]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, (1979).   Google Scholar

[3]

N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications,, Springer, (1967).   Google Scholar

[4]

S. N. Busenberg, M. Iannelli and H. R. Thieme, Global behavior of an age-structured epidemic model,, SIAM J. Math. Anal., 22 (1991), 1065.  doi: 10.1137/0522069.  Google Scholar

[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.  doi: 10.1016/j.amc.2011.10.015.  Google Scholar

[6]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365.  doi: 10.1007/BF00178324.  Google Scholar

[7]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases,, Wiley, (2000).   Google Scholar

[8]

Z. Feng, W. Huang and C. Castillo-Chavez, Global behavior of a multi-group SIS epidemic model with age structure,, J. Diff. Equat., 218 (2005), 292.  doi: 10.1016/j.jde.2004.10.009.  Google Scholar

[9]

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.  doi: 10.1007/BF02218848.  Google Scholar

[10]

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.   Google Scholar

[11]

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.  doi: 10.1090/S0002-9939-08-09341-6.  Google Scholar

[12]

G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence,, J. Math. Biol., 63 (2011), 125.  doi: 10.1007/s00285-010-0368-2.  Google Scholar

[13]

T. Kuniya, Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model,, Nonlinear Analysis RWA., 12 (2011), 2640.  doi: 10.1016/j.nonrwa.2011.03.011.  Google Scholar

[14]

A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population,, Math. Biosci., 28 (1976), 221.  doi: 10.1016/0025-5564(76)90125-5.  Google Scholar

[15]

J. P. LaSalle, The Stability of Dynamical Systems,, SIAM, (1976).   Google Scholar

[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.  doi: 10.1016/S0025-5564(99)00030-9.  Google Scholar

[17]

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Diff. Equat., 284 (2010), 1.  doi: 10.1016/j.jde.2009.09.003.  Google Scholar

[18]

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.  doi: 10.1016/j.jmaa.2009.09.017.  Google Scholar

[19]

J. Liu and Y. Zhou, Global stability of an SIRS epidemic model with transport-related infection,, Chaos Solitons and Fractals, 40 (2009), 145.  doi: 10.1016/j.chaos.2007.07.047.  Google Scholar

[20]

X. Liu and Y. Takeuchi, Spread of disease with transport-related infection and entry screening,, J. Theoret. Biol., 242 (2006), 517.  doi: 10.1016/j.jtbi.2006.03.018.  Google Scholar

[21]

C. C. McCluskey, Complete global stability for an SIR epidemic model with delay (distributed or discrete),, Nonlinear Analysis RWA., 11 (2010), 55.  doi: 10.1016/j.nonrwa.2008.10.014.  Google Scholar

[22]

K. Mischaikow, H. L. Smith and H. R. Thieme, Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions,, Trans. Amer. Math. Soc., 347 (1995), 1669.  doi: 10.1090/S0002-9947-1995-1290727-7.  Google Scholar

[23]

Y. Muroya, A. Bellen, Y. Enatsu and Y. Nakata, Global stability for a discrete epidemic model for disease with immunity and latency spreading in a heterogeneous host population,, Nonlinear Analysis RWA., 13 (2012), 258.  doi: 10.1016/j.nonrwa.2011.07.031.  Google Scholar

[24]

Y. Muroya, Y. Enatsu and T. Kuniya, Global stability for a multi-group SIRS epidemic model with varying population sizes,, Nonlinear Analysis RWA., 14 (2013), 1693.  doi: 10.1016/j.nonrwa.2012.11.005.  Google Scholar

[25]

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.  doi: 10.1016/S0252-9602(13)60003-X.  Google Scholar

[26]

Y. Nakata, On the global stability of a delayed epidemic model with transport-related infection,, Nonlinear Analysis RWA., 12 (2011), 3028.  doi: 10.1016/j.nonrwa.2011.05.004.  Google Scholar

[27]

H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar

[28]

H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission,, Nonlinear Analysis RWA., 13 (2012), 1581.  doi: 10.1016/j.nonrwa.2011.11.016.  Google Scholar

[29]

R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence,, Comput. Math. Appl., 60 (2010), 2286.  doi: 10.1016/j.camwa.2010.08.020.  Google Scholar

[30]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755.  doi: 10.1007/BF00173267.  Google Scholar

[31]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188.  doi: 10.1137/080732870.  Google Scholar

[32]

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.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[33]

R. S. Varga, Matrix Iterative Analysis,, Prentice-Hall, (1962).   Google Scholar

[34]

C. Vargas-De-León, On the global stability of SIS, SIR and SIRS epidemic models with standard incidence,, Chaos Solitons and Fractals, 44 (2011), 1106.  doi: 10.1016/j.chaos.2011.09.002.  Google Scholar

[35]

W. Wang and X. Zhao, An epidemic model in a patchy environment,, Math. Biosci., 190 (2004), 97.  doi: 10.1016/j.mbs.2002.11.001.  Google Scholar

[36]

L. Wang and G. Z. Dai, Global stability of virus spreading in complex heterogeneous networks,, SIAM J. Appl. Math., 68 (2008), 1495.  doi: 10.1137/070694582.  Google Scholar

[37]

Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates,, Nonlinear Analysis RWA., 11 (2010), 995.  doi: 10.1016/j.nonrwa.2009.01.040.  Google Scholar

[38]

Z. Yuan and X. Zou, Global threshold property in an epidemic models for disease with latency spreading in a heterogeneous host population,, Nonlinear Analysis RWA., 11 (2010), 3479.  doi: 10.1016/j.nonrwa.2009.12.008.  Google Scholar

show all references

References:
[1]

J. Arino, Diseases in metapopulations,, in Modeling and Dynamics of Infectious Diseases (eds. Z. Ma, (2009), 65.   Google Scholar

[2]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, (1979).   Google Scholar

[3]

N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications,, Springer, (1967).   Google Scholar

[4]

S. N. Busenberg, M. Iannelli and H. R. Thieme, Global behavior of an age-structured epidemic model,, SIAM J. Math. Anal., 22 (1991), 1065.  doi: 10.1137/0522069.  Google Scholar

[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.  doi: 10.1016/j.amc.2011.10.015.  Google Scholar

[6]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365.  doi: 10.1007/BF00178324.  Google Scholar

[7]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases,, Wiley, (2000).   Google Scholar

[8]

Z. Feng, W. Huang and C. Castillo-Chavez, Global behavior of a multi-group SIS epidemic model with age structure,, J. Diff. Equat., 218 (2005), 292.  doi: 10.1016/j.jde.2004.10.009.  Google Scholar

[9]

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.  doi: 10.1007/BF02218848.  Google Scholar

[10]

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.   Google Scholar

[11]

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.  doi: 10.1090/S0002-9939-08-09341-6.  Google Scholar

[12]

G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence,, J. Math. Biol., 63 (2011), 125.  doi: 10.1007/s00285-010-0368-2.  Google Scholar

[13]

T. Kuniya, Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model,, Nonlinear Analysis RWA., 12 (2011), 2640.  doi: 10.1016/j.nonrwa.2011.03.011.  Google Scholar

[14]

A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population,, Math. Biosci., 28 (1976), 221.  doi: 10.1016/0025-5564(76)90125-5.  Google Scholar

[15]

J. P. LaSalle, The Stability of Dynamical Systems,, SIAM, (1976).   Google Scholar

[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.  doi: 10.1016/S0025-5564(99)00030-9.  Google Scholar

[17]

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Diff. Equat., 284 (2010), 1.  doi: 10.1016/j.jde.2009.09.003.  Google Scholar

[18]

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.  doi: 10.1016/j.jmaa.2009.09.017.  Google Scholar

[19]

J. Liu and Y. Zhou, Global stability of an SIRS epidemic model with transport-related infection,, Chaos Solitons and Fractals, 40 (2009), 145.  doi: 10.1016/j.chaos.2007.07.047.  Google Scholar

[20]

X. Liu and Y. Takeuchi, Spread of disease with transport-related infection and entry screening,, J. Theoret. Biol., 242 (2006), 517.  doi: 10.1016/j.jtbi.2006.03.018.  Google Scholar

[21]

C. C. McCluskey, Complete global stability for an SIR epidemic model with delay (distributed or discrete),, Nonlinear Analysis RWA., 11 (2010), 55.  doi: 10.1016/j.nonrwa.2008.10.014.  Google Scholar

[22]

K. Mischaikow, H. L. Smith and H. R. Thieme, Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions,, Trans. Amer. Math. Soc., 347 (1995), 1669.  doi: 10.1090/S0002-9947-1995-1290727-7.  Google Scholar

[23]

Y. Muroya, A. Bellen, Y. Enatsu and Y. Nakata, Global stability for a discrete epidemic model for disease with immunity and latency spreading in a heterogeneous host population,, Nonlinear Analysis RWA., 13 (2012), 258.  doi: 10.1016/j.nonrwa.2011.07.031.  Google Scholar

[24]

Y. Muroya, Y. Enatsu and T. Kuniya, Global stability for a multi-group SIRS epidemic model with varying population sizes,, Nonlinear Analysis RWA., 14 (2013), 1693.  doi: 10.1016/j.nonrwa.2012.11.005.  Google Scholar

[25]

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.  doi: 10.1016/S0252-9602(13)60003-X.  Google Scholar

[26]

Y. Nakata, On the global stability of a delayed epidemic model with transport-related infection,, Nonlinear Analysis RWA., 12 (2011), 3028.  doi: 10.1016/j.nonrwa.2011.05.004.  Google Scholar

[27]

H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar

[28]

H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission,, Nonlinear Analysis RWA., 13 (2012), 1581.  doi: 10.1016/j.nonrwa.2011.11.016.  Google Scholar

[29]

R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence,, Comput. Math. Appl., 60 (2010), 2286.  doi: 10.1016/j.camwa.2010.08.020.  Google Scholar

[30]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755.  doi: 10.1007/BF00173267.  Google Scholar

[31]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188.  doi: 10.1137/080732870.  Google Scholar

[32]

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.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[33]

R. S. Varga, Matrix Iterative Analysis,, Prentice-Hall, (1962).   Google Scholar

[34]

C. Vargas-De-León, On the global stability of SIS, SIR and SIRS epidemic models with standard incidence,, Chaos Solitons and Fractals, 44 (2011), 1106.  doi: 10.1016/j.chaos.2011.09.002.  Google Scholar

[35]

W. Wang and X. Zhao, An epidemic model in a patchy environment,, Math. Biosci., 190 (2004), 97.  doi: 10.1016/j.mbs.2002.11.001.  Google Scholar

[36]

L. Wang and G. Z. Dai, Global stability of virus spreading in complex heterogeneous networks,, SIAM J. Appl. Math., 68 (2008), 1495.  doi: 10.1137/070694582.  Google Scholar

[37]

Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates,, Nonlinear Analysis RWA., 11 (2010), 995.  doi: 10.1016/j.nonrwa.2009.01.040.  Google Scholar

[38]

Z. Yuan and X. Zou, Global threshold property in an epidemic models for disease with latency spreading in a heterogeneous host population,, Nonlinear Analysis RWA., 11 (2010), 3479.  doi: 10.1016/j.nonrwa.2009.12.008.  Google Scholar

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