# American Institute of Mathematical Sciences

October  2019, 39(10): 5683-5706. doi: 10.3934/dcds.2019249

## Dynamical behavior of a multigroup SIRS epidemic model with standard incidence rates and Markovian switching

 a. School of Mathematics and Statistics, Key Laboratory of Applied Statistics of MOE, Northeast Normal Universit, Changchun 130024, Jilin Province, China b. Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, King Abdulaziz University, Jeddah, Saudi Arabia c. College of Science, China University of Petroleum, Qingdao 266580, Shandong Province, China d. Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan

* Corresponding author at School of Mathematics and Statistics, Key Laboratory of Applied Statistics of MOE, Northeast Normal University, Changchun, Jilin 130024, China

Received  June 2018 Published  July 2019

In this paper, we consider a multigroup SIRS epidemic model with standard incidence rates and Markovian switching. Firstly, we obtain sufficient conditions for extinction of the diseases. Then we establish sufficient conditions for persistence in the mean of the diseases. Moreover, in the case of persistence, we derive sufficient conditions for the existence of positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching.

Citation: Qun Liu, Daqing Jiang, Tasawar Hayat, Ahmed Alsaedi. Dynamical behavior of a multigroup SIRS epidemic model with standard incidence rates and Markovian switching. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5683-5706. doi: 10.3934/dcds.2019249
##### References:
 [1] N. Bacaër and M. Khaladi, On the basic reproduction number in a random environment, J. Math. Biol., 67 (2013), 1729-1739.  doi: 10.1007/s00285-012-0611-0. [2] N. Bacaër and A. Ed-Darraz, On linear birth-and-death processes in a random environment, J. Math. Biol., 69 (2014), 73-90.  doi: 10.1007/s00285-013-0696-0. [3] D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman, Harlow, 1993. [4] J. Bao and J. Shao, Permanence and extinction of regime-switching predator-prey models, SIAM J. Math. Anal., 48 (2016), 725-739.  doi: 10.1137/15M1024512. [5] E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model, in: T.G. Hallam, L.J. Gross, S.A. Levin (Eds.), Mathematical Ecology, World Scientific, Teaneck, NJ, 1988,317–342. [6] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, New York, 1979. [7] N. H. Dang, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise, J. Differential Equations, 257 (2014), 2078-2101.  doi: 10.1016/j.jde.2014.05.029. [8] N. H. Du and N. H. Dang, Dynamics of Kolmogorov systems of competitive type under the telegraph noise, J. Differential Equations, 250 (2011), 386-409.  doi: 10.1016/j.jde.2010.08.023. [9] Z. Feng, W. Huang and C. Castillo-Chavez, Global behavior of a multi-group SIS epidemic model with age structure, J. Differential Equations, 218 (2005), 292-324.  doi: 10.1016/j.jde.2004.10.009. [10] A. Gray, D. Greenhalgh, X. Mao and J. Pan, The SIS epidemic model with Markovian switching, J. Math. Anal. Appl., 394 (2012), 496-516.  doi: 10.1016/j.jmaa.2012.05.029. [11] D. Greenhalgh, Y. Liang and X. Mao, Modelling the effect of telegraph noise in the SIRS epidemic model using Markovian switching, Physica A, 462 (2016), 684-704.  doi: 10.1016/j.physa.2016.06.125. [12] 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. [13] H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q., 14 (2006), 259-284. [14] H. W. Hethcote and H. R. Thieme, Stability of endemic equilibrium in epidemic models with subpopulations, Math. Biosci., 75 (1985), 205-227.  doi: 10.1016/0025-5564(85)90038-0. [15] N. T. Hieu, N. H. Du, P. Auger and N. H. Dang, Dynamical behavior of a stochastic SIRS epidemic model, Math. Model. Nat. Phenom., 10 (2015), 56-73.  doi: 10.1051/mmnp/201510205. [16] W. Huang, K. L. Cooke and C. Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math., 52 (1992), 835-854.  doi: 10.1137/0152047. [17] A. Iggidr, G. Sallet and M. O. Souza, On the dynamics of a class of multi-group models for vector-borne diseases, J. Math. Anal. Appl., 441 (2016), 723-743.  doi: 10.1016/j.jmaa.2016.04.003. [18] C. Ji and D. Jiang, The asymptotic behavior of a stochastic multigroup SIS model, Int. J. Biomath., 11 (2018), 1850037 (16 pages). doi: 10.1142/S1793524518500377. [19] C. Ji, D. Jiang and N. Shi, Multigroup SIR epidemic model with stochastic perturbation, Physica A, 390 (2011), 1747-1762. [20] C. Koide and H. Seno, Sex ratio features of two-group SIR model for asymmetric transmission of heterosexual disease, Math. Comput. Model., 23 (1996), 67-91. [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] V. Lakshmikantham, D. Bainov and P. Simeonov, Theory of Impulsive Differential Equations, World Scientific Press, ingapore, 1989.  doi: 10.1142/0906. [23] 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. [24] D. Li, S. Liu and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching, J. Differential Equations, 263 (2017), 8873-8915.  doi: 10.1016/j.jde.2017.08.066. [25] M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20.  doi: 10.1016/j.jde.2009.09.003. [26] M. Liu and K. Wang, Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment, J. Theor. Biol., 264 (2010), 934-944.  doi: 10.1016/j.jtbi.2010.03.008. [27] Q. Luo and X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69-84.  doi: 10.1016/j.jmaa.2006.12.032. [28] X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 79 (1999), 45-67.  doi: 10.1016/S0304-4149(98)00070-2. [29] X. Mao and C. Yuan, Stochastic Differential Equations With Markovian Switching, Imperial College Press, London, 2006.  doi: 10.1142/p473. [30] Y. Muroya, Y. Enatsu and T. Kuniya, Global stability for a multi-group SIRS epidemic model with varying population sizes, Nonlinear Anal. RWA, 14 (2013), 1693-1704.  doi: 10.1016/j.nonrwa.2012.11.005. [31] L. Rass and J. Radcliffe, Global asymptotic convergence results for multitype models, Int. J. Appl. Math. Comput. Sci., 10 (2000), 63-79. [32] D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996. [33] X. Zhang, D. Jiang, A. Alsaedi and T. Hayat, Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching, Appl. Math. Lett., 59 (2016), 87-93.  doi: 10.1016/j.aml.2016.03.010. [34] L. Zu, D. Jiang and D. O'Regan, Conditions for persistence and ergodicity of a stochastic Lotka-Volterra predator-prey model with regime switching, Commun. Nonlinear Sci. Numer. Simul., 29 (2015), 1-11.  doi: 10.1016/j.cnsns.2015.04.008.

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##### References:
 [1] N. Bacaër and M. Khaladi, On the basic reproduction number in a random environment, J. Math. Biol., 67 (2013), 1729-1739.  doi: 10.1007/s00285-012-0611-0. [2] N. Bacaër and A. Ed-Darraz, On linear birth-and-death processes in a random environment, J. Math. Biol., 69 (2014), 73-90.  doi: 10.1007/s00285-013-0696-0. [3] D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman, Harlow, 1993. [4] J. Bao and J. Shao, Permanence and extinction of regime-switching predator-prey models, SIAM J. Math. Anal., 48 (2016), 725-739.  doi: 10.1137/15M1024512. [5] E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model, in: T.G. Hallam, L.J. Gross, S.A. Levin (Eds.), Mathematical Ecology, World Scientific, Teaneck, NJ, 1988,317–342. [6] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, New York, 1979. [7] N. H. Dang, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise, J. Differential Equations, 257 (2014), 2078-2101.  doi: 10.1016/j.jde.2014.05.029. [8] N. H. Du and N. H. Dang, Dynamics of Kolmogorov systems of competitive type under the telegraph noise, J. Differential Equations, 250 (2011), 386-409.  doi: 10.1016/j.jde.2010.08.023. [9] Z. Feng, W. Huang and C. Castillo-Chavez, Global behavior of a multi-group SIS epidemic model with age structure, J. Differential Equations, 218 (2005), 292-324.  doi: 10.1016/j.jde.2004.10.009. [10] A. Gray, D. Greenhalgh, X. Mao and J. Pan, The SIS epidemic model with Markovian switching, J. Math. Anal. Appl., 394 (2012), 496-516.  doi: 10.1016/j.jmaa.2012.05.029. [11] D. Greenhalgh, Y. Liang and X. Mao, Modelling the effect of telegraph noise in the SIRS epidemic model using Markovian switching, Physica A, 462 (2016), 684-704.  doi: 10.1016/j.physa.2016.06.125. [12] 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. [13] H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q., 14 (2006), 259-284. [14] H. W. Hethcote and H. R. Thieme, Stability of endemic equilibrium in epidemic models with subpopulations, Math. Biosci., 75 (1985), 205-227.  doi: 10.1016/0025-5564(85)90038-0. [15] N. T. Hieu, N. H. Du, P. Auger and N. H. Dang, Dynamical behavior of a stochastic SIRS epidemic model, Math. Model. Nat. Phenom., 10 (2015), 56-73.  doi: 10.1051/mmnp/201510205. [16] W. Huang, K. L. Cooke and C. Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math., 52 (1992), 835-854.  doi: 10.1137/0152047. [17] A. Iggidr, G. Sallet and M. O. Souza, On the dynamics of a class of multi-group models for vector-borne diseases, J. Math. Anal. Appl., 441 (2016), 723-743.  doi: 10.1016/j.jmaa.2016.04.003. [18] C. Ji and D. Jiang, The asymptotic behavior of a stochastic multigroup SIS model, Int. J. Biomath., 11 (2018), 1850037 (16 pages). doi: 10.1142/S1793524518500377. [19] C. Ji, D. Jiang and N. Shi, Multigroup SIR epidemic model with stochastic perturbation, Physica A, 390 (2011), 1747-1762. [20] C. Koide and H. Seno, Sex ratio features of two-group SIR model for asymmetric transmission of heterosexual disease, Math. Comput. Model., 23 (1996), 67-91. [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] V. Lakshmikantham, D. Bainov and P. Simeonov, Theory of Impulsive Differential Equations, World Scientific Press, ingapore, 1989.  doi: 10.1142/0906. [23] 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. [24] D. Li, S. Liu and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching, J. Differential Equations, 263 (2017), 8873-8915.  doi: 10.1016/j.jde.2017.08.066. [25] M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20.  doi: 10.1016/j.jde.2009.09.003. [26] M. Liu and K. Wang, Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment, J. Theor. Biol., 264 (2010), 934-944.  doi: 10.1016/j.jtbi.2010.03.008. [27] Q. Luo and X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69-84.  doi: 10.1016/j.jmaa.2006.12.032. [28] X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 79 (1999), 45-67.  doi: 10.1016/S0304-4149(98)00070-2. [29] X. Mao and C. Yuan, Stochastic Differential Equations With Markovian Switching, Imperial College Press, London, 2006.  doi: 10.1142/p473. [30] Y. Muroya, Y. Enatsu and T. Kuniya, Global stability for a multi-group SIRS epidemic model with varying population sizes, Nonlinear Anal. RWA, 14 (2013), 1693-1704.  doi: 10.1016/j.nonrwa.2012.11.005. [31] L. Rass and J. Radcliffe, Global asymptotic convergence results for multitype models, Int. J. Appl. Math. Comput. Sci., 10 (2000), 63-79. [32] D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996. [33] X. Zhang, D. Jiang, A. Alsaedi and T. Hayat, Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching, Appl. Math. Lett., 59 (2016), 87-93.  doi: 10.1016/j.aml.2016.03.010. [34] L. Zu, D. Jiang and D. O'Regan, Conditions for persistence and ergodicity of a stochastic Lotka-Volterra predator-prey model with regime switching, Commun. Nonlinear Sci. Numer. Simul., 29 (2015), 1-11.  doi: 10.1016/j.cnsns.2015.04.008.
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