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February  2020, 19(2): 1089-1110. doi: 10.3934/cpaa.2020050

## Dynamics of a multigroup SIRS epidemic model with random perturbations and varying total population size

 1 School of Mathematics and Statistics, Key Laboratory of Applied Statistics of MOE, Northeast, Normal University, Changchun 130024, Jilin Province, China 2 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 121589, Saudi Arabia 3 College of Science, China University of Petroleum (East China), Qingdao 266580, Shandong Province, China

*Corresponding author

Received  March 2019 Revised  May 2019 Published  October 2019

Fund Project: The authors were supported by the National Natural Science Foundation of P.R. China (No. 11871473) and Natural Science Foundation of Guangxi Province (No. 2016GXNSFBA380006).

In this paper, we analyze a multigroup SIRS epidemic model with random perturbations and varying total population size. By utilizing the stochastic Lyapunov function method, we establish sufficient conditions for the existence of a stationary distribution of the positive solutions to the model. Since our model is multidimensional, it is extremely difficult to construct an appropriate stochastic Lyapunov function to prove the existence of the stationary distribution, which implies stochastic weak stability. Then we establish sufficient conditions for extinction of the diseases. These conditions are related to the basic reproduction number in its corresponding deterministic system.

Citation: Qun Liu, Daqing Jiang. Dynamics of a multigroup SIRS epidemic model with random perturbations and varying total population size. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1089-1110. doi: 10.3934/cpaa.2020050
##### References:
 [1] M. S. Bartlett, Deterministic and stochastic models for recurrent epidemics, in Neyman, J. (Ed.), Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 4., University of California Press, Berkeley, 1956.  doi: 10.2307/1401576. [2] E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model, Mathematical Ecology, World Scientific, Teaneck, NJ (1986), 317–342. doi: 10.1007/978-3-642-69888-0. [3] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.  doi: 10.1137/1.9781611971262. [4] Y. Chen, B. Wen and Z. Teng, The global dynamics for a stochastic SIS epidemic model with isolation, Physica A, 492 (2018), 1604-1624.  doi: 10.1016/j.physa.2017.11.085. [5] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use, J. Math. Anal. Appl., 325 (2007), 36-53.  doi: 10.1016/j.jmaa.2006.01.055. [6] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics, J. Math. Anal. Appl., 341 (2008), 1084-1101.  doi: 10.1016/j.jmaa.2007.11.005. [7] 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. [8] 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. [9] 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: 10.2307/20535481. [10] 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. [11] 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: 10.2307/2102404. [12] 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. [13] L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differential Equations, 217 (2005), 26-53.  doi: 10.1016/j.jde.2005.06.017. [14] 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. [15] C. Ji, D. Jiang and N. Shi, Multigroup SIR epidemic model with stochastic perturbation, Physica A, 390 (2011), 1747-1762.  doi: 10.1016/j.physa.2010.12.042. [16] C. Ji, D. Jiang, Q. Yang and N. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica, 48 (2012), 121-131.  doi: 10.1016/j.automatica.2011.09.044. [17] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008.  doi: 10.1086/591197. [18] R. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1980. doi: 10.1007/978-94-009-9121-7. [19] C. Koide and H. Seno, Sex ratio features of two-group SIR model for asymmetrie transmission of heterosexual disease, Math. Comput. Model., 23 (1996), 67-91.  doi: 10.1016/0895-7177(96)00004-0. [20] 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. [21] 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. [22] 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. [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] Q. Liu and D. Jiang, Stationary distribution of a stochastic SIS epidemic model with double diseases and the Beddington-DeAngelis incidence, Chaos, 27 (2017), 083126. doi: 10.1063/1.4986838. [25] Q. Liu, D. Jiang, T. Hayat and A. Alsaedi, Dynamics of a stochastic multigroup SIQR epidemic model with standard incidence rates, J. Franklin Inst., 356 (2019), 2960-2993.  doi: 10.1016/j.jfranklin.2019.01.038. [26] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, 1997. [27] 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. [28] L. Rass and J. Radcliffe, Global asymptotic convergence results for multitype models, Int. J. Appl. Math. Comput. Sci., 10 (2000), 63-79. [29] E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Physica A, 354 (2005), 111-126.  doi: 10.1016/j.physa.2005.02.057. [30] L. Wang, Z. Teng, C. Ji, X. Feng and K. Wang, Dynamical behaviors of a stochastic malaria model: A case study for Yunnan, China, Physica A, 521 (2019), 435-454.  doi: 10.1016/j.physa.2018.12.030. [31] B. Wen, Z. Teng and Z. Li, The threshold of a periodic stochastic SIVS epidemic model with nonlinear incidence, Physica A, 508 (2018), 532-549.  doi: 10.1016/j.physa.2018.05.056. [32] D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996. doi: 10.1007/978-3-8348-9329-1_2. [33] D. Xu, Y. Huang and Z. Yang, Existence theorems for periodic Markov process and stochastic functional differential equations, Discrete Contin. Dyn. Syst., 24 (2009), 1005-1023.  doi: 10.3934/dcds.2009.24.1005. [34] J. Yu, D. Jiang and N. Shi, Global stability of two-group SIR model with random perturbation, J. Math. Anal. Appl., 360 (2009), 235-244.  doi: 10.1016/j.jmaa.2009.06.050.

show all references

##### References:
 [1] M. S. Bartlett, Deterministic and stochastic models for recurrent epidemics, in Neyman, J. (Ed.), Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 4., University of California Press, Berkeley, 1956.  doi: 10.2307/1401576. [2] E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model, Mathematical Ecology, World Scientific, Teaneck, NJ (1986), 317–342. doi: 10.1007/978-3-642-69888-0. [3] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.  doi: 10.1137/1.9781611971262. [4] Y. Chen, B. Wen and Z. Teng, The global dynamics for a stochastic SIS epidemic model with isolation, Physica A, 492 (2018), 1604-1624.  doi: 10.1016/j.physa.2017.11.085. [5] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use, J. Math. Anal. Appl., 325 (2007), 36-53.  doi: 10.1016/j.jmaa.2006.01.055. [6] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics, J. Math. Anal. Appl., 341 (2008), 1084-1101.  doi: 10.1016/j.jmaa.2007.11.005. [7] 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. [8] 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. [9] 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: 10.2307/20535481. [10] 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. [11] 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: 10.2307/2102404. [12] 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. [13] L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differential Equations, 217 (2005), 26-53.  doi: 10.1016/j.jde.2005.06.017. [14] 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. [15] C. Ji, D. Jiang and N. Shi, Multigroup SIR epidemic model with stochastic perturbation, Physica A, 390 (2011), 1747-1762.  doi: 10.1016/j.physa.2010.12.042. [16] C. Ji, D. Jiang, Q. Yang and N. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica, 48 (2012), 121-131.  doi: 10.1016/j.automatica.2011.09.044. [17] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008.  doi: 10.1086/591197. [18] R. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1980. doi: 10.1007/978-94-009-9121-7. [19] C. Koide and H. Seno, Sex ratio features of two-group SIR model for asymmetrie transmission of heterosexual disease, Math. Comput. Model., 23 (1996), 67-91.  doi: 10.1016/0895-7177(96)00004-0. [20] 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. [21] 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. [22] 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. [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] Q. Liu and D. Jiang, Stationary distribution of a stochastic SIS epidemic model with double diseases and the Beddington-DeAngelis incidence, Chaos, 27 (2017), 083126. doi: 10.1063/1.4986838. [25] Q. Liu, D. Jiang, T. Hayat and A. Alsaedi, Dynamics of a stochastic multigroup SIQR epidemic model with standard incidence rates, J. Franklin Inst., 356 (2019), 2960-2993.  doi: 10.1016/j.jfranklin.2019.01.038. [26] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, 1997. [27] 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. [28] L. Rass and J. Radcliffe, Global asymptotic convergence results for multitype models, Int. J. Appl. Math. Comput. Sci., 10 (2000), 63-79. [29] E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Physica A, 354 (2005), 111-126.  doi: 10.1016/j.physa.2005.02.057. [30] L. Wang, Z. Teng, C. Ji, X. Feng and K. Wang, Dynamical behaviors of a stochastic malaria model: A case study for Yunnan, China, Physica A, 521 (2019), 435-454.  doi: 10.1016/j.physa.2018.12.030. [31] B. Wen, Z. Teng and Z. Li, The threshold of a periodic stochastic SIVS epidemic model with nonlinear incidence, Physica A, 508 (2018), 532-549.  doi: 10.1016/j.physa.2018.05.056. [32] D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996. doi: 10.1007/978-3-8348-9329-1_2. [33] D. Xu, Y. Huang and Z. Yang, Existence theorems for periodic Markov process and stochastic functional differential equations, Discrete Contin. Dyn. Syst., 24 (2009), 1005-1023.  doi: 10.3934/dcds.2009.24.1005. [34] J. Yu, D. Jiang and N. Shi, Global stability of two-group SIR model with random perturbation, J. Math. Anal. Appl., 360 (2009), 235-244.  doi: 10.1016/j.jmaa.2009.06.050.
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