# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems - B, 2015, 20 (9) : 3057-3091. doi: 10.3934/dcdsb.2015.20.3057
##### References:
 [1] R. M. Anderson and R. M. May, Population biology of infectious diseases: Part I, Nature, 280 (1979), 361-367. doi: 10.1038/280361a0. [2] J. Arino, Disease in metapopulations, Modeling and Dynamics of Infectious Diseases, Higher Education Press, Beijing, 11 (2009), 64-122. doi: 10.1142/7223. [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), 4107-4115. doi: 10.1016/S0362-546X(01)00528-4. [4] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. [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), 4391-4400. doi: 10.1016/j.amc.2011.10.015. [6] Y. Chen, J. Yang and F. Zhang, The global stability of an SIRS model with infection age, Math. Bios. Eng., 11 (2014), 449-469. doi: 10.3934/mbe.2014.11.449. [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), 365-382. doi: 10.1007/BF00178324. [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), 2120-2133. doi: 10.1016/j.nonrwa.2012.01.007. [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), 715-733. doi: 10.3934/dcdsb.2014.19.715. [10] T. Faria, Global dynamics for Lotka-Volterra systems with infinite delay and patch structure, Appl. Math. Comput., 245 (2014), 575-590. doi: 10.1016/j.amc.2014.08.009. [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), 301-330. doi: 10.1017/S0308210513001194. [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), 414-430. doi: 10.1007/s00285-005-0331-9. [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), 259-284. [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), 2793-2802. doi: 10.1090/S0002-9939-08-09341-6. [15] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Vol. 99, Springer, New York, 1993. [16] G. Huang and A. Liu, A note on global stability for a heroin epidemic model with distributed delay, Appl. Math. Lett., 26 (2013), 687-691. doi: 10.1016/j.aml.2013.01.010. [17] W. Kermack and A. McKendrick, Contributions to the mathematical theory of epidemics I, II and III, Bulletin of Mathematical Biology, 53 (1991), 33-55, 57-87 and 89-118. [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), 1105-1118. doi: 10.3934/dcdsb.2014.19.1105. [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), 785-798. doi: 10.1016/j.amc.2015.05.124. [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), 1375-1393. doi: 10.3934/mbe.2014.11.1375. [21] A. Lajmanovich and J. A. Yorke, A deterministic model for Gonorrhea in a nonhomogeneous population, Math. Biosci, 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5. [22] J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976. [23] M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Diff. Equat., 248 (2010), 1-20. doi: 10.1016/j.jde.2009.09.003. [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), 38-47. doi: 10.1016/j.jmaa.2009.09.017. [25] J. Liu and T. Zhang, Global behaviour of a heroin epidemic model with distributed delays, Appl. Math. Lett., 24 (2011), 1685-1692. doi: 10.1016/j.aml.2011.04.019. [26] J. Liu and Y. Zhou, Global stability of an SIRS epidemic model with transport-related infection, Chaos Solitons and Fractals, 40 (2009), 145-158. doi: 10.1016/j.chaos.2007.07.047. [27] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-Distributed or discrete, Nonlinear Analysis RWA, 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014. [28] J. Mena-Lorca and H. W. Hethcote, Dynamic models of infectious diseases as regulators of population sizes, J. Math. Biol., 30 (1992), 693-716. doi: 10.1007/BF00173264. [29] G. Mulone and B. Straughan, A note on heroin epidemics, Math. Biosci., 218 (2009), 138-141. doi: 10.1016/j.mbs.2009.01.006. [30] Y. Muroya, Practical monotonous iterations for nonlinear equations, Memoirs of the Faculty of Science, Kyushu University Ser A, 22 (1968), 56-73. doi: 10.2206/kyushumfs.22.56. [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), 999-1008. doi: 10.3934/dcdss.2015.8.999. [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), 1693-1704. doi: 10.1016/j.nonrwa.2012.11.005. [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), 341-361. doi: 10.1016/S0252-9602(13)60003-X. [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), 1897-1910. doi: 10.1016/j.nonrwa.2010.12.002. [35] Y. Muroya and T. Kuniya, Global stability of nonresident computer virus models, Math. Methods Appl. Sciences, 38 (2015), 281-295. doi: 10.1002/mma.3068. [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), 73-78. doi: 10.1016/j.aml.2014.07.005. [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), 1550048. doi: 10.1142/S1793524515500485. [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), 415-439. doi: 10.1016/j.jmaa.2014.12.019. [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), 719-732. doi: 10.1016/j.jmaa.2013.08.024. [40] Y. Muroya, H. Li and T. Kuniya, On global stability of a nonresident computer virus model, Acta. Math. Scientia., 34 (2014), 1427-1445. doi: 10.1016/S0252-9602(14)60094-1. [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), 205-245. [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), 1119-1128. [43] Y. Nakata and G. Röst, Global analysis for spread of infectious diseases via transportation networks, J. Math. Biol., 70 (2015), 1411-1456. doi: 10.1007/s00285-014-0801-z. [44] J. Ortega and W. Rheinboldt, Monotone iterations for nonlinear equations with application to Gauss-Seidel methods, SIAM J. Numer. Anal., 4 (1967), 171-190. doi: 10.1137/0704017. [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), 1581-1592. doi: 10.1016/j.nonrwa.2011.11.016. [46] R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear distributed incidence, Comput. Math. Appl., 60 (2010), 2286-2291. doi: 10.1016/j.camwa.2010.08.020. [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), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [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), 578-599. [49] J. Wang, Y. Takeuchi and S. Liu, A multi-group SVEIR epidemic model with distributed delay and vaccination, Inter. J. Biomath., 5 (2012), 1260001(18 pages). doi: 10.1142/S1793524512600017. [50] E. White and C. Comiskey, Heroin epidemics, treatment and ODE modelling, Mathematical Biosciences, 208 (2007), 312-324. doi: 10.1016/j.mbs.2006.10.008. [51] Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Analysis RWA., 11 (2010), 995-1004. doi: 10.1016/j.nonrwa.2009.01.040. [52] 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), 3479-3490. doi: 10.1016/j.nonrwa.2009.12.008. [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), 421-444.

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##### References:
 [1] R. M. Anderson and R. M. May, Population biology of infectious diseases: Part I, Nature, 280 (1979), 361-367. doi: 10.1038/280361a0. [2] J. Arino, Disease in metapopulations, Modeling and Dynamics of Infectious Diseases, Higher Education Press, Beijing, 11 (2009), 64-122. doi: 10.1142/7223. [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), 4107-4115. doi: 10.1016/S0362-546X(01)00528-4. [4] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. [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), 4391-4400. doi: 10.1016/j.amc.2011.10.015. [6] Y. Chen, J. Yang and F. Zhang, The global stability of an SIRS model with infection age, Math. Bios. Eng., 11 (2014), 449-469. doi: 10.3934/mbe.2014.11.449. [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), 365-382. doi: 10.1007/BF00178324. [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), 2120-2133. doi: 10.1016/j.nonrwa.2012.01.007. [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), 715-733. doi: 10.3934/dcdsb.2014.19.715. [10] T. Faria, Global dynamics for Lotka-Volterra systems with infinite delay and patch structure, Appl. Math. Comput., 245 (2014), 575-590. doi: 10.1016/j.amc.2014.08.009. [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), 301-330. doi: 10.1017/S0308210513001194. [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), 414-430. doi: 10.1007/s00285-005-0331-9. [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), 259-284. [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), 2793-2802. doi: 10.1090/S0002-9939-08-09341-6. [15] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Vol. 99, Springer, New York, 1993. [16] G. Huang and A. Liu, A note on global stability for a heroin epidemic model with distributed delay, Appl. Math. Lett., 26 (2013), 687-691. doi: 10.1016/j.aml.2013.01.010. [17] W. Kermack and A. McKendrick, Contributions to the mathematical theory of epidemics I, II and III, Bulletin of Mathematical Biology, 53 (1991), 33-55, 57-87 and 89-118. [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), 1105-1118. doi: 10.3934/dcdsb.2014.19.1105. [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), 785-798. doi: 10.1016/j.amc.2015.05.124. [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), 1375-1393. doi: 10.3934/mbe.2014.11.1375. [21] A. Lajmanovich and J. A. Yorke, A deterministic model for Gonorrhea in a nonhomogeneous population, Math. Biosci, 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5. [22] J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976. [23] M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Diff. Equat., 248 (2010), 1-20. doi: 10.1016/j.jde.2009.09.003. [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), 38-47. doi: 10.1016/j.jmaa.2009.09.017. [25] J. Liu and T. Zhang, Global behaviour of a heroin epidemic model with distributed delays, Appl. Math. Lett., 24 (2011), 1685-1692. doi: 10.1016/j.aml.2011.04.019. [26] J. Liu and Y. Zhou, Global stability of an SIRS epidemic model with transport-related infection, Chaos Solitons and Fractals, 40 (2009), 145-158. doi: 10.1016/j.chaos.2007.07.047. [27] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-Distributed or discrete, Nonlinear Analysis RWA, 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014. [28] J. Mena-Lorca and H. W. Hethcote, Dynamic models of infectious diseases as regulators of population sizes, J. Math. Biol., 30 (1992), 693-716. doi: 10.1007/BF00173264. [29] G. Mulone and B. Straughan, A note on heroin epidemics, Math. Biosci., 218 (2009), 138-141. doi: 10.1016/j.mbs.2009.01.006. [30] Y. Muroya, Practical monotonous iterations for nonlinear equations, Memoirs of the Faculty of Science, Kyushu University Ser A, 22 (1968), 56-73. doi: 10.2206/kyushumfs.22.56. [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), 999-1008. doi: 10.3934/dcdss.2015.8.999. [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), 1693-1704. doi: 10.1016/j.nonrwa.2012.11.005. [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), 341-361. doi: 10.1016/S0252-9602(13)60003-X. [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), 1897-1910. doi: 10.1016/j.nonrwa.2010.12.002. [35] Y. Muroya and T. Kuniya, Global stability of nonresident computer virus models, Math. Methods Appl. Sciences, 38 (2015), 281-295. doi: 10.1002/mma.3068. [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), 73-78. doi: 10.1016/j.aml.2014.07.005. [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), 1550048. doi: 10.1142/S1793524515500485. [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), 415-439. doi: 10.1016/j.jmaa.2014.12.019. [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), 719-732. doi: 10.1016/j.jmaa.2013.08.024. [40] Y. Muroya, H. Li and T. Kuniya, On global stability of a nonresident computer virus model, Acta. Math. Scientia., 34 (2014), 1427-1445. doi: 10.1016/S0252-9602(14)60094-1. [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), 205-245. [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), 1119-1128. [43] Y. Nakata and G. Röst, Global analysis for spread of infectious diseases via transportation networks, J. Math. Biol., 70 (2015), 1411-1456. doi: 10.1007/s00285-014-0801-z. [44] J. Ortega and W. Rheinboldt, Monotone iterations for nonlinear equations with application to Gauss-Seidel methods, SIAM J. Numer. Anal., 4 (1967), 171-190. doi: 10.1137/0704017. [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), 1581-1592. doi: 10.1016/j.nonrwa.2011.11.016. [46] R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear distributed incidence, Comput. Math. Appl., 60 (2010), 2286-2291. doi: 10.1016/j.camwa.2010.08.020. [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), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [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), 578-599. [49] J. Wang, Y. Takeuchi and S. Liu, A multi-group SVEIR epidemic model with distributed delay and vaccination, Inter. J. Biomath., 5 (2012), 1260001(18 pages). doi: 10.1142/S1793524512600017. [50] E. White and C. Comiskey, Heroin epidemics, treatment and ODE modelling, Mathematical Biosciences, 208 (2007), 312-324. doi: 10.1016/j.mbs.2006.10.008. [51] Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Analysis RWA., 11 (2010), 995-1004. doi: 10.1016/j.nonrwa.2009.01.040. [52] 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), 3479-3490. doi: 10.1016/j.nonrwa.2009.12.008. [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), 421-444.
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