November  2015, 20(9): 3057-3091. doi: 10.3934/dcdsb.2015.20.3057

Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection

1. 

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

2. 

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

3. 

Department of Mathematical Information Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

Received  August 2014 Revised  July 2015 Published  September 2015

In this paper, we investigate the global stability of a delayed multi-group SIRS epidemic model which includes not only nonlinear incidence rates but also rates of immunity loss and relapse of infection. The model analysis can be regarded as an extension to a multi-group epidemic analysis in [Muroya, Li and Kuniya, Complete global analysis of an SIRS epidemic model with graded cure rate and incomplete recovery rate, J. Math. Anal. Appl. 410 (2014) 719-732] is studied. Applying a Lyapunov functional approach, we prove that a disease-free equilibrium of the model, is globally asymptotically stable, if a threshold parameter $R_0 \leq 1$. For the global stability of an endemic equilibrium of the model, we establish a sufficient condition for small recovery rates $\delta_k \geq 0$, $k=1,2,\ldots,n$, if $R_0>1$. Further, by a monotone iterative approach, we obtain another sufficient condition for large $\delta_k$, $k=1,2,\ldots,n$. Both results generalize several known results obtained for, e.g., SIS, SIR and SIRS models in the recent literature. We also offer a new proof on permanence which is applicable to other multi-group epidemic models.
Citation: Yoshiaki Muroya, Toshikazu Kuniya, Yoichi Enatsu. Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3057-3091. doi: 10.3934/dcdsb.2015.20.3057
References:
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[2]

J. Arino, Disease in metapopulations, Modeling and Dynamics of Infectious Diseases,, Higher Education Press, 11 (2009), 64. doi: 10.1142/7223. Google Scholar

[3]

E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay,, Nonlinear Anal., 47 (2001), 4107. doi: 10.1016/S0362-546X(01)00528-4. Google Scholar

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

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H. Chen and J. Sun, Global stability of delay multigroup epidemic models with group mixing and nonlinear incidence rates,, Appl. Math. Comput., 218 (2011), 4391. doi: 10.1016/j.amc.2011.10.015. Google Scholar

[6]

Y. Chen, J. Yang and F. Zhang, The global stability of an SIRS model with infection age,, Math. Bios. Eng., 11 (2014), 449. doi: 10.3934/mbe.2014.11.449. Google Scholar

[7]

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

[8]

Y. Enatsu, Y. Nakata and Y. Muroya, Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model,, Nonlinear Analysis RWA, 13 (2012), 2120. doi: 10.1016/j.nonrwa.2012.01.007. Google Scholar

[9]

B. Fang, X. Li, M. Martcheva and L. Cai, Global stability for a heroin model with two distributed delays,, Discrete Cont. Dynamic. Syst. Series B, 19 (2014), 715. doi: 10.3934/dcdsb.2014.19.715. Google Scholar

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T. Faria, Global dynamics for Lotka-Volterra systems with infinite delay and patch structure,, Appl. Math. Comput., 245 (2014), 575. doi: 10.1016/j.amc.2014.08.009. Google Scholar

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T. Faria and Y. Muroya, Global attractivity and extinction for Lotka-Volterra systems with infinite delay and feedback controls,, Proceedings of the Royal Society of Edinburgh: Section A, 145 (2015), 301. doi: 10.1017/S0308210513001194. Google Scholar

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M. G. M. Gomes, A. Margheri, G. F. Medley and E. C. Rebelo, Dynamical behaviour of epidemiological models with sub-optimal immunity and nonlinear incidence,, J. Math. Biol., 51 (2005), 414. doi: 10.1007/s00285-005-0331-9. Google Scholar

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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

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J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations,, Applied Mathematical Sciences, (1993). Google Scholar

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G. Huang and A. Liu, A note on global stability for a heroin epidemic model with distributed delay,, Appl. Math. Lett., 26 (2013), 687. doi: 10.1016/j.aml.2013.01.010. Google Scholar

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W. Kermack and A. McKendrick, Contributions to the mathematical theory of epidemics I, II and III,, Bulletin of Mathematical Biology, 53 (1991), 33. Google Scholar

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T. Kuniya and Y. Muroya, Global stability of a multi-group SIS epidemic model for population migration,, Discrete and Continuous Dynamical System B, 19 (2014), 1105. doi: 10.3934/dcdsb.2014.19.1105. Google Scholar

[19]

T. Kuniya and Y. Muroya, Global stability of a multi-group SIS epidemic model with varying total population size,, Appl. Math. Comput., 265 (2015), 785. doi: 10.1016/j.amc.2015.05.124. Google Scholar

[20]

T. Kuniya, Y. Muroya and Y. Enatsu, Threshold dynamics of an SIR epidemic model with hybrid of multi-group and patch structures,, Math. Bios. Eng., 11 (2014), 1375. doi: 10.3934/mbe.2014.11.1375. Google Scholar

[21]

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

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[23]

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Diff. Equat., 248 (2010), 1. doi: 10.1016/j.jde.2009.09.003. Google Scholar

[24]

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

[25]

J. Liu and T. Zhang, Global behaviour of a heroin epidemic model with distributed delays,, Appl. Math. Lett., 24 (2011), 1685. doi: 10.1016/j.aml.2011.04.019. Google Scholar

[26]

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

[27]

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

[28]

J. Mena-Lorca and H. W. Hethcote, Dynamic models of infectious diseases as regulators of population sizes,, J. Math. Biol., 30 (1992), 693. doi: 10.1007/BF00173264. Google Scholar

[29]

G. Mulone and B. Straughan, A note on heroin epidemics,, Math. Biosci., 218 (2009), 138. doi: 10.1016/j.mbs.2009.01.006. Google Scholar

[30]

Y. Muroya, Practical monotonous iterations for nonlinear equations,, Memoirs of the Faculty of Science, 22 (1968), 56. doi: 10.2206/kyushumfs.22.56. Google Scholar

[31]

Y. Muroya, A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model),, Disc. Cont. Dyn. Sys. Supplement, 8 (2015), 999. doi: 10.3934/dcdss.2015.8.999. Google Scholar

[32]

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

[33]

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

[34]

Y. Muroya, Y. Enatsu and Y. Nakata, Monotone iterative techniques to SIRS epidemic models with nonlinear incidence rates and distributed delays,, Nonlinear Analysis RWA, 12 (2011), 1897. doi: 10.1016/j.nonrwa.2010.12.002. Google Scholar

[35]

Y. Muroya and T. Kuniya, Global stability of nonresident computer virus models,, Math. Methods Appl. Sciences, 38 (2015), 281. doi: 10.1002/mma.3068. Google Scholar

[36]

Y. Muroya and T. Kuniya, Further stability analysis for a multi-group SIRS epidemic model with varying total population sizes,, Appl. Math. Lett., 38 (2014), 73. doi: 10.1016/j.aml.2014.07.005. Google Scholar

[37]

Y. Muroya and T. Kuniya, Global stability for a delayed multi-group SIRS epidemic model with cure rate and incomplete recovery rate,, Intern. J. Biomath., 8 (2015). doi: 10.1142/S1793524515500485. Google Scholar

[38]

Y. Muroya, T. Kuniya and J. Wang, Stability analysis of a delayed multi-group SIS epidemic model with nonlinear incidence rates and patch structure,, J. Math. Anal. Appl., 425 (2015), 415. doi: 10.1016/j.jmaa.2014.12.019. Google Scholar

[39]

Y. Muroya, H. Li and T. Kuniya, Complete global analysis of an SIRS epidemic model with graded cure and incomplete recovery rates,, J. Math. Anal. Appl., 410 (2014), 719. doi: 10.1016/j.jmaa.2013.08.024. Google Scholar

[40]

Y. Muroya, H. Li and T. Kuniya, On global stability of a nonresident computer virus model,, Acta. Math. Scientia., 34 (2014), 1427. doi: 10.1016/S0252-9602(14)60094-1. Google Scholar

[41]

Y. Nakata, Y. Enatsu, H. Inaba, T. Kuniya, Y. Muroya and Y. Takeuchi, Stability of epidemic models with waning immunity,, SUT Journal of Mathematics, 50 (2015), 205. Google Scholar

[42]

Y. Nakata, Y. Enatsu and Y. Muroya, On the global stability of an SIRS epidemic model with distributed delays,, Disc. Cont. Dyn. Sys. Supplement, 2 (2011), 1119. Google Scholar

[43]

Y. Nakata and G. Röst, Global analysis for spread of infectious diseases via transportation networks,, J. Math. Biol., 70 (2015), 1411. doi: 10.1007/s00285-014-0801-z. Google Scholar

[44]

J. Ortega and W. Rheinboldt, Monotone iterations for nonlinear equations with application to Gauss-Seidel methods,, SIAM J. Numer. Anal., 4 (1967), 171. doi: 10.1137/0704017. Google Scholar

[45]

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

[46]

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

[47]

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

[48]

J. Wang, Y. Muroya and T. Kuniya, Global stability of a time-delayed multi-group SIS epidemic model with nonlinear incidence rates and patch structure,, Journal of Nonlinear Science and Applications, 8 (2015), 578. Google Scholar

[49]

J. Wang, Y. Takeuchi and S. Liu, A multi-group SVEIR epidemic model with distributed delay and vaccination,, Inter. J. Biomath., 5 (2012). doi: 10.1142/S1793524512600017. Google Scholar

[50]

E. White and C. Comiskey, Heroin epidemics, treatment and ODE modelling,, Mathematical Biosciences, 208 (2007), 312. doi: 10.1016/j.mbs.2006.10.008. Google Scholar

[51]

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

[52]

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

[53]

X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations,, Cann. Appl. Math. Quart., 4 (1996), 421. Google Scholar

show all references

References:
[1]

R. M. Anderson and R. M. May, Population biology of infectious diseases: Part I,, Nature, 280 (1979), 361. doi: 10.1038/280361a0. Google Scholar

[2]

J. Arino, Disease in metapopulations, Modeling and Dynamics of Infectious Diseases,, Higher Education Press, 11 (2009), 64. doi: 10.1142/7223. Google Scholar

[3]

E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay,, Nonlinear Anal., 47 (2001), 4107. doi: 10.1016/S0362-546X(01)00528-4. Google Scholar

[4]

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

[5]

H. Chen and J. Sun, Global stability of delay multigroup epidemic models with group mixing and nonlinear incidence rates,, Appl. Math. Comput., 218 (2011), 4391. doi: 10.1016/j.amc.2011.10.015. Google Scholar

[6]

Y. Chen, J. Yang and F. Zhang, The global stability of an SIRS model with infection age,, Math. Bios. Eng., 11 (2014), 449. doi: 10.3934/mbe.2014.11.449. Google Scholar

[7]

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

[8]

Y. Enatsu, Y. Nakata and Y. Muroya, Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model,, Nonlinear Analysis RWA, 13 (2012), 2120. doi: 10.1016/j.nonrwa.2012.01.007. Google Scholar

[9]

B. Fang, X. Li, M. Martcheva and L. Cai, Global stability for a heroin model with two distributed delays,, Discrete Cont. Dynamic. Syst. Series B, 19 (2014), 715. doi: 10.3934/dcdsb.2014.19.715. Google Scholar

[10]

T. Faria, Global dynamics for Lotka-Volterra systems with infinite delay and patch structure,, Appl. Math. Comput., 245 (2014), 575. doi: 10.1016/j.amc.2014.08.009. Google Scholar

[11]

T. Faria and Y. Muroya, Global attractivity and extinction for Lotka-Volterra systems with infinite delay and feedback controls,, Proceedings of the Royal Society of Edinburgh: Section A, 145 (2015), 301. doi: 10.1017/S0308210513001194. Google Scholar

[12]

M. G. M. Gomes, A. Margheri, G. F. Medley and E. C. Rebelo, Dynamical behaviour of epidemiological models with sub-optimal immunity and nonlinear incidence,, J. Math. Biol., 51 (2005), 414. doi: 10.1007/s00285-005-0331-9. Google Scholar

[13]

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

[14]

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

[15]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations,, Applied Mathematical Sciences, (1993). Google Scholar

[16]

G. Huang and A. Liu, A note on global stability for a heroin epidemic model with distributed delay,, Appl. Math. Lett., 26 (2013), 687. doi: 10.1016/j.aml.2013.01.010. Google Scholar

[17]

W. Kermack and A. McKendrick, Contributions to the mathematical theory of epidemics I, II and III,, Bulletin of Mathematical Biology, 53 (1991), 33. Google Scholar

[18]

T. Kuniya and Y. Muroya, Global stability of a multi-group SIS epidemic model for population migration,, Discrete and Continuous Dynamical System B, 19 (2014), 1105. doi: 10.3934/dcdsb.2014.19.1105. Google Scholar

[19]

T. Kuniya and Y. Muroya, Global stability of a multi-group SIS epidemic model with varying total population size,, Appl. Math. Comput., 265 (2015), 785. doi: 10.1016/j.amc.2015.05.124. Google Scholar

[20]

T. Kuniya, Y. Muroya and Y. Enatsu, Threshold dynamics of an SIR epidemic model with hybrid of multi-group and patch structures,, Math. Bios. Eng., 11 (2014), 1375. doi: 10.3934/mbe.2014.11.1375. Google Scholar

[21]

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

[22]

J. P. LaSalle, The Stability of Dynamical Systems,, Regional Conference Series in Applied Mathematics, (1976). Google Scholar

[23]

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Diff. Equat., 248 (2010), 1. doi: 10.1016/j.jde.2009.09.003. Google Scholar

[24]

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

[25]

J. Liu and T. Zhang, Global behaviour of a heroin epidemic model with distributed delays,, Appl. Math. Lett., 24 (2011), 1685. doi: 10.1016/j.aml.2011.04.019. Google Scholar

[26]

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

[27]

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

[28]

J. Mena-Lorca and H. W. Hethcote, Dynamic models of infectious diseases as regulators of population sizes,, J. Math. Biol., 30 (1992), 693. doi: 10.1007/BF00173264. Google Scholar

[29]

G. Mulone and B. Straughan, A note on heroin epidemics,, Math. Biosci., 218 (2009), 138. doi: 10.1016/j.mbs.2009.01.006. Google Scholar

[30]

Y. Muroya, Practical monotonous iterations for nonlinear equations,, Memoirs of the Faculty of Science, 22 (1968), 56. doi: 10.2206/kyushumfs.22.56. Google Scholar

[31]

Y. Muroya, A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model),, Disc. Cont. Dyn. Sys. Supplement, 8 (2015), 999. doi: 10.3934/dcdss.2015.8.999. Google Scholar

[32]

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

[33]

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

[34]

Y. Muroya, Y. Enatsu and Y. Nakata, Monotone iterative techniques to SIRS epidemic models with nonlinear incidence rates and distributed delays,, Nonlinear Analysis RWA, 12 (2011), 1897. doi: 10.1016/j.nonrwa.2010.12.002. Google Scholar

[35]

Y. Muroya and T. Kuniya, Global stability of nonresident computer virus models,, Math. Methods Appl. Sciences, 38 (2015), 281. doi: 10.1002/mma.3068. Google Scholar

[36]

Y. Muroya and T. Kuniya, Further stability analysis for a multi-group SIRS epidemic model with varying total population sizes,, Appl. Math. Lett., 38 (2014), 73. doi: 10.1016/j.aml.2014.07.005. Google Scholar

[37]

Y. Muroya and T. Kuniya, Global stability for a delayed multi-group SIRS epidemic model with cure rate and incomplete recovery rate,, Intern. J. Biomath., 8 (2015). doi: 10.1142/S1793524515500485. Google Scholar

[38]

Y. Muroya, T. Kuniya and J. Wang, Stability analysis of a delayed multi-group SIS epidemic model with nonlinear incidence rates and patch structure,, J. Math. Anal. Appl., 425 (2015), 415. doi: 10.1016/j.jmaa.2014.12.019. Google Scholar

[39]

Y. Muroya, H. Li and T. Kuniya, Complete global analysis of an SIRS epidemic model with graded cure and incomplete recovery rates,, J. Math. Anal. Appl., 410 (2014), 719. doi: 10.1016/j.jmaa.2013.08.024. Google Scholar

[40]

Y. Muroya, H. Li and T. Kuniya, On global stability of a nonresident computer virus model,, Acta. Math. Scientia., 34 (2014), 1427. doi: 10.1016/S0252-9602(14)60094-1. Google Scholar

[41]

Y. Nakata, Y. Enatsu, H. Inaba, T. Kuniya, Y. Muroya and Y. Takeuchi, Stability of epidemic models with waning immunity,, SUT Journal of Mathematics, 50 (2015), 205. Google Scholar

[42]

Y. Nakata, Y. Enatsu and Y. Muroya, On the global stability of an SIRS epidemic model with distributed delays,, Disc. Cont. Dyn. Sys. Supplement, 2 (2011), 1119. Google Scholar

[43]

Y. Nakata and G. Röst, Global analysis for spread of infectious diseases via transportation networks,, J. Math. Biol., 70 (2015), 1411. doi: 10.1007/s00285-014-0801-z. Google Scholar

[44]

J. Ortega and W. Rheinboldt, Monotone iterations for nonlinear equations with application to Gauss-Seidel methods,, SIAM J. Numer. Anal., 4 (1967), 171. doi: 10.1137/0704017. Google Scholar

[45]

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

[46]

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

[47]

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

[48]

J. Wang, Y. Muroya and T. Kuniya, Global stability of a time-delayed multi-group SIS epidemic model with nonlinear incidence rates and patch structure,, Journal of Nonlinear Science and Applications, 8 (2015), 578. Google Scholar

[49]

J. Wang, Y. Takeuchi and S. Liu, A multi-group SVEIR epidemic model with distributed delay and vaccination,, Inter. J. Biomath., 5 (2012). doi: 10.1142/S1793524512600017. Google Scholar

[50]

E. White and C. Comiskey, Heroin epidemics, treatment and ODE modelling,, Mathematical Biosciences, 208 (2007), 312. doi: 10.1016/j.mbs.2006.10.008. Google Scholar

[51]

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

[52]

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

[53]

X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations,, Cann. Appl. Math. Quart., 4 (1996), 421. Google Scholar

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