# American Institute of Mathematical Sciences

September  2021, 26(9): 5101-5134. doi: 10.3934/dcdsb.2020335

## Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps

 School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

* Corresponding author: Shangjiang Guo

Received  March 2020 Revised  July 2020 Published  September 2021 Early access  November 2020

Fund Project: The first author is supported by China Postdoctoral Science Foundation, and the Post-doctoral Innovative Research Positions in Hubei Province (Grant No. 1232037), the second author is supported by NSFC (Grant Nos. 11671123, 12071446)

This paper is devoted to a stochastic regime-switching susceptible-infected-susceptible epidemic model with nonlinear incidence rate and Lévy jumps. A threshold $\lambda$ in terms of the invariant measure, different from the usual basic reproduction number, is obtained to completely determine the extinction and prevalence of the disease: if $\lambda>0$, the disease is persistent and there is a stationary distribution; if $\lambda<0$, the disease goes to extinction and the susceptible population converges weakly to a boundary distribution. Moreover, some numerical simulations are performed to illustrate our theoretical results. It is very interesting to notice that random fluctuations (including the white noise and Lévy noise) acting the infected individuals can prevent the outbreak of disease, that the disease of a regime-switching model may have the opportunity to persist eventually even if it is extinct in one regime, and that the prevalence of the disease can also be controlled by reducing the value of transmission rate of disease.

Citation: Shangzhi Li, Shangjiang Guo. Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5101-5134. doi: 10.3934/dcdsb.2020335
##### References:
 [1] D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edition, Cambridge University Press, 2009. [2] J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616. [3] J. Bao and J. Shao, Asymptotic behavior of SIRS models in state-dependent random environments, arXiv: 1802.02309. [4] J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise, J. Math. Anal. Appl., 391 (2012), 363-375. [5] I. Barbalat, Systemes déquations différentielles doscillations non linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270. [6] S. Cai, Y. Cai and X. Mao, A stochastic differential equation SIS epidemic model with two independent Brownian motions, Journal of Mathematical Analysis and Applications, 474 (2019), 1536-1550.  doi: 10.1016/j.jmaa.2019.02.039. [7] M.-F. Chen, From Markov Chains to Non-equilibrium Particle Systems, 2nd ed., World Scientific, River Edge, NJ, 2004. [8] O. Diekmann, J. A. P. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382. [9] N. H. Du, R. Kon, K. Sato and Y. Takeuchi, Dynamical behaviour of Lotka-Volterra competition systems: Nonautonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math., 170 (2004), 399-422. [10] J. Gao and S. Guo, Effect of prey-taxis and diffusion on positive steady states for a predator-prey system, Math Meth Appl Sci., 41 (2018), 3570-3587.  doi: 10.1002/mma.4847. [11] J. Gao and S. Guo, Patterns in a modified Leslie-Gower model with Beddington-DeAngelis functional response and nonlocal prey competition, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050074, 28pp. doi: 10.1142/S0218127420500741. [12] Q. Ge, G. Ji, J. Xu and et al., Extinction and persistence of a stochastic nonlinear SIS epidemic model with jumps, Physica A: Statistical Mechanics and its Applications, 462 (2016), 1120-1127. [13] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.  doi: 10.1137/10081856X. [14] A. Gray, D. Greenhalgh, X. Mao and J. Pan, The SIS epidemic model with Markovian switchiing, J. Math. Anal. Appl., 394 (2012), 496-516.  doi: 10.1016/j.jmaa.2012.05.029. [15] S. Guo, Bifurcation and spatio-temporal patterns in a diffusive predator-prey system, Nonlinear Analysis: Real World Applications, 42 (2018), 448-477. [16] Y. Guo, Stochastic regime switching SIR model driven by Lévy noise, Physica A: Statistical Mechanics and its Applications, 479 (2017), 1-11. [17] H.J. Li and S. Guo, Dynamics of a SIRC epidemiological model, Electronic Journal of Differential Equations, 2017 (2017), Paper No. 121, 18 pp. [18] S. Li and S. Guo, Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions, Discrete & Continuous Dynamical Systems-B, 2020. doi: 10.3934/dcdsb.2020201. [19] S. Li and S. Guo, Random attractors for stochastic semilinear degenerate parabolic equations with delay, Physica A, 550 (2020), 124164, 24pp. doi: 10.1016/j.physa.2020.124164. [20] Y. Lin and Y. Zhao, Exponential ergodicity of a regime-switching SIS epidemic model with jumps, Applied Mathematics Letters, 94 (2019), 133-139. [21] Q. Liu, The threshold of a stochastic Susceptible-Infective epidemic model under regime switching, Nonlinear Analysis: Hybrid Systems, 21 (2016), 49-58. [22] Q. Liu, D. Jiang, T. Hayat and A. Alsaedi, Dynamical behavior of a hybrid switching SIS epidemic model with vaccination and Lévy jumps, Stochastic Analysis and Applications, 37 (2019), 388-411. [23] X. Mao, G. Marion and E. Renshaw, Environmental noise suppresses explosion in population dynamics, Stochastic Process. Appl., 97 (2002), 95-110. [24] D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Equations, 262 (2017), 1192-1225.  doi: 10.1016/j.jde.2016.10.005. [25] H. Qiu and S. Guo, Steady-states of a Leslie-Gower model with diffusion and advection, Applied Mathematics and Computation, 346 (2019), 695-709. [26] H. Qiu, S. Guo and S. Li, Stability and bifurcation in a predator-prey system with prey-taxis, Int. J. Bifur. Chaos, 30 (2020), 2050022, 25pp. doi: 10.1142/S0218127420500224. [27] R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering, Springer, 2005. [28] M. Slatkin, The dynamics of a population in a Markovian environment, Ecology, 59 (1978), 249-256. [29] B. Sounvoravong, S. Guo and Y. Bai, Bifurcation and stability of a diffusive SIRS epidemic model with time delay, Electronic Journal of Differential Equations, 2019 (2019), Paper No. 45, 16 pp. [30] Z. Teng and L. Wang, Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate, Physic A, 451 (2016), 507-518.  doi: 10.1016/j.physa.2016.01.084. [31] K. Wang, Stochastic Biomathematics Models, Science Press, Beijing, 2010. [32] C. Xu, Global threshold dynamics of a stochastic differential equation SIS model, Journal of Mathematical Analysis and Applications, 447 (2017), 736-757. [33] G. G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Stoch. Model. Appl. Probab. 63, Springer, New York, 2010. [34] X. Zhang and K. Wang, Stochastic SIR model with jumps, Appl. Math. Lett., 26 (2013), 867-874. [35] X. Zhong, S. Guo and M. Peng, Stability of stochastic SIRS epidemic models with saturated incidence rates and delay, Stochastic Analysis and Applications, 35 (2017), 1-26.  doi: 10.1080/07362994.2016.1244644. [36] J. Zhou and H. W. Hethcote, Populations size dependent incidence in models for diseases without immunity, J. Math. Biol., 32 (1994), 809-834. [37] Y. Zhou, S. Yuan and D. Zhao, Threshold behavior of a stochastic SIS model with Lévy jumps, Applied Mathematics and Computation, 275 (2016), 255-267. [38] Y. Zhou and W. Zhang, Threshold of a stochastic SIR epidemic model with Lévy jumps, Physica A: Statistical Mechanics and its Applications, 446 (2016), 204-216. [39] C. Zhu, Critical result on the threshold of a stochastic SIS model with saturated incidence rate, Physica A, 523 (2019), 426-437.  doi: 10.1016/j.physa.2019.02.012. [40] R. Zou and S. Guo, Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment, Discrete & Continuous Dynamical Systems-B, 25 (2020), 4189-4210.  doi: 10.3934/dcdsb.2020093. [41] R. Zou and S. Guo, Dynamics in a diffusive predator-prey system with ratio-dependent predator influence, Computers and Mathematics with Applications, 75 (2018), 1237-1258.  doi: 10.1016/j.camwa.2017.11.002.

show all references

##### References:
 [1] D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edition, Cambridge University Press, 2009. [2] J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616. [3] J. Bao and J. Shao, Asymptotic behavior of SIRS models in state-dependent random environments, arXiv: 1802.02309. [4] J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise, J. Math. Anal. Appl., 391 (2012), 363-375. [5] I. Barbalat, Systemes déquations différentielles doscillations non linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270. [6] S. Cai, Y. Cai and X. Mao, A stochastic differential equation SIS epidemic model with two independent Brownian motions, Journal of Mathematical Analysis and Applications, 474 (2019), 1536-1550.  doi: 10.1016/j.jmaa.2019.02.039. [7] M.-F. Chen, From Markov Chains to Non-equilibrium Particle Systems, 2nd ed., World Scientific, River Edge, NJ, 2004. [8] O. Diekmann, J. A. P. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382. [9] N. H. Du, R. Kon, K. Sato and Y. Takeuchi, Dynamical behaviour of Lotka-Volterra competition systems: Nonautonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math., 170 (2004), 399-422. [10] J. Gao and S. Guo, Effect of prey-taxis and diffusion on positive steady states for a predator-prey system, Math Meth Appl Sci., 41 (2018), 3570-3587.  doi: 10.1002/mma.4847. [11] J. Gao and S. Guo, Patterns in a modified Leslie-Gower model with Beddington-DeAngelis functional response and nonlocal prey competition, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050074, 28pp. doi: 10.1142/S0218127420500741. [12] Q. Ge, G. Ji, J. Xu and et al., Extinction and persistence of a stochastic nonlinear SIS epidemic model with jumps, Physica A: Statistical Mechanics and its Applications, 462 (2016), 1120-1127. [13] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.  doi: 10.1137/10081856X. [14] A. Gray, D. Greenhalgh, X. Mao and J. Pan, The SIS epidemic model with Markovian switchiing, J. Math. Anal. Appl., 394 (2012), 496-516.  doi: 10.1016/j.jmaa.2012.05.029. [15] S. Guo, Bifurcation and spatio-temporal patterns in a diffusive predator-prey system, Nonlinear Analysis: Real World Applications, 42 (2018), 448-477. [16] Y. Guo, Stochastic regime switching SIR model driven by Lévy noise, Physica A: Statistical Mechanics and its Applications, 479 (2017), 1-11. [17] H.J. Li and S. Guo, Dynamics of a SIRC epidemiological model, Electronic Journal of Differential Equations, 2017 (2017), Paper No. 121, 18 pp. [18] S. Li and S. Guo, Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions, Discrete & Continuous Dynamical Systems-B, 2020. doi: 10.3934/dcdsb.2020201. [19] S. Li and S. Guo, Random attractors for stochastic semilinear degenerate parabolic equations with delay, Physica A, 550 (2020), 124164, 24pp. doi: 10.1016/j.physa.2020.124164. [20] Y. Lin and Y. Zhao, Exponential ergodicity of a regime-switching SIS epidemic model with jumps, Applied Mathematics Letters, 94 (2019), 133-139. [21] Q. Liu, The threshold of a stochastic Susceptible-Infective epidemic model under regime switching, Nonlinear Analysis: Hybrid Systems, 21 (2016), 49-58. [22] Q. Liu, D. Jiang, T. Hayat and A. Alsaedi, Dynamical behavior of a hybrid switching SIS epidemic model with vaccination and Lévy jumps, Stochastic Analysis and Applications, 37 (2019), 388-411. [23] X. Mao, G. Marion and E. Renshaw, Environmental noise suppresses explosion in population dynamics, Stochastic Process. Appl., 97 (2002), 95-110. [24] D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka-Volterra models, J. Differential Equations, 262 (2017), 1192-1225.  doi: 10.1016/j.jde.2016.10.005. [25] H. Qiu and S. Guo, Steady-states of a Leslie-Gower model with diffusion and advection, Applied Mathematics and Computation, 346 (2019), 695-709. [26] H. Qiu, S. Guo and S. Li, Stability and bifurcation in a predator-prey system with prey-taxis, Int. J. Bifur. Chaos, 30 (2020), 2050022, 25pp. doi: 10.1142/S0218127420500224. [27] R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering, Springer, 2005. [28] M. Slatkin, The dynamics of a population in a Markovian environment, Ecology, 59 (1978), 249-256. [29] B. Sounvoravong, S. Guo and Y. Bai, Bifurcation and stability of a diffusive SIRS epidemic model with time delay, Electronic Journal of Differential Equations, 2019 (2019), Paper No. 45, 16 pp. [30] Z. Teng and L. Wang, Persistence and extinction for a class of stochastic SIS epidemic models with nonlinear incidence rate, Physic A, 451 (2016), 507-518.  doi: 10.1016/j.physa.2016.01.084. [31] K. Wang, Stochastic Biomathematics Models, Science Press, Beijing, 2010. [32] C. Xu, Global threshold dynamics of a stochastic differential equation SIS model, Journal of Mathematical Analysis and Applications, 447 (2017), 736-757. [33] G. G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Stoch. Model. Appl. Probab. 63, Springer, New York, 2010. [34] X. Zhang and K. Wang, Stochastic SIR model with jumps, Appl. Math. Lett., 26 (2013), 867-874. [35] X. Zhong, S. Guo and M. Peng, Stability of stochastic SIRS epidemic models with saturated incidence rates and delay, Stochastic Analysis and Applications, 35 (2017), 1-26.  doi: 10.1080/07362994.2016.1244644. [36] J. Zhou and H. W. Hethcote, Populations size dependent incidence in models for diseases without immunity, J. Math. Biol., 32 (1994), 809-834. [37] Y. Zhou, S. Yuan and D. Zhao, Threshold behavior of a stochastic SIS model with Lévy jumps, Applied Mathematics and Computation, 275 (2016), 255-267. [38] Y. Zhou and W. Zhang, Threshold of a stochastic SIR epidemic model with Lévy jumps, Physica A: Statistical Mechanics and its Applications, 446 (2016), 204-216. [39] C. Zhu, Critical result on the threshold of a stochastic SIS model with saturated incidence rate, Physica A, 523 (2019), 426-437.  doi: 10.1016/j.physa.2019.02.012. [40] R. Zou and S. Guo, Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment, Discrete & Continuous Dynamical Systems-B, 25 (2020), 4189-4210.  doi: 10.3934/dcdsb.2020093. [41] R. Zou and S. Guo, Dynamics in a diffusive predator-prey system with ratio-dependent predator influence, Computers and Mathematics with Applications, 75 (2018), 1237-1258.  doi: 10.1016/j.camwa.2017.11.002.
Trajectories of solutions to model (2) with parameters (29) in regimes 1 and 2
Trajectories of susceptible and infected populations of model (2) with parameters (29)
The joint density distribution of $(S,I)$ of model (2) with parameters (29). (a) The case without jumps; (b) The case with jumps
Trajectories of solutions to model (2) with parameters (30) in regimes 1 and 2
Trajectories of susceptible and infected populations of model (2) with parameters (30)
The weak convergence of $S$ to the stationary solution $\varphi$ of (10) with parameters (30)
The joint density distribution of $(S,I)$ of model (2) with parameters (30). (a) The case without jumps; (b) The case with jumps
Trajectories of solutions to model (2) with parameters (31) in regimes 1 and 2
Trajectories of susceptible and infected populations of model (2) with parameters (31)
The weak convergence of $S$ to the stationary solution $\varphi$ of (10) with parameters (31)
The joint density distribution of $(S,I)$ of model (2) with parameters (31). (a) The case without jumps; (b) The case with jumps
Trajectories of solutions to model (2) with parameters (32) in regimes 1 and 2
Trajectories of susceptible and infected populations of model (2) with parameters (32)
The joint density distribution of $(S,I)$ of model (2) with parameters (32). (a) The case without jumps; (b) The case with jumps
Trajectories of solutions to model (2) with parameters (33) in regimes 1 and 2
Trajectories of susceptible and infected populations of model (2) with parameters (33)
The joint density distribution of $(S,I)$ of model (2) with parameters (33). (a) The case without jumps; (b) The case with jumps
Trajectories of solutions to model (2) with parameters (34) in regimes 1 and 2
Trajectories of susceptible and infected populations of model (2) with parameters (34) in regimes 1 and 2
The weak convergence of $S$ to the stationary solution $\varphi$ of (10) with parameters (34)
The joint density distribution of $(S,I)$ of model (2) with parameters (34). (a) The case without jumps; (b) The case with jumps
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