
-
Previous Article
Bifurcation analysis of a general activator-inhibitor model with nonlocal dispersal
- DCDS-B Home
- This Issue
-
Next Article
Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem
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 |
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.
References:
[1] | D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edition, Cambridge University Press, 2009. Google Scholar |
[2] |
J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616. Google Scholar |
[3] |
J. Bao and J. Shao, Asymptotic behavior of SIRS models in state-dependent random environments, arXiv: 1802.02309. Google Scholar |
[4] |
J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise, J. Math. Anal. Appl., 391 (2012), 363-375. Google Scholar |
[5] |
I. Barbalat, Systemes déquations différentielles doscillations non linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[16] |
Y. Guo, Stochastic regime switching SIR model driven by Lévy noise, Physica A: Statistical Mechanics and its Applications, 479 (2017), 1-11. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[21] |
Q. Liu, The threshold of a stochastic Susceptible-Infective epidemic model under regime switching, Nonlinear Analysis: Hybrid Systems, 21 (2016), 49-58. Google Scholar |
[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. Google Scholar |
[23] |
X. Mao, G. Marion and E. Renshaw, Environmental noise suppresses explosion in population dynamics, Stochastic Process. Appl., 97 (2002), 95-110. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[28] |
M. Slatkin, The dynamics of a population in a Markovian environment, Ecology, 59 (1978), 249-256. Google Scholar |
[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. Google Scholar |
[32] |
C. Xu, Global threshold dynamics of a stochastic differential equation SIS model, Journal of Mathematical Analysis and Applications, 447 (2017), 736-757. Google Scholar |
[33] |
G. G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Stoch. Model. Appl. Probab. 63, Springer, New York, 2010. Google Scholar |
[34] |
X. Zhang and K. Wang, Stochastic SIR model with jumps, Appl. Math. Lett., 26 (2013), 867-874. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[2] |
J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616. Google Scholar |
[3] |
J. Bao and J. Shao, Asymptotic behavior of SIRS models in state-dependent random environments, arXiv: 1802.02309. Google Scholar |
[4] |
J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise, J. Math. Anal. Appl., 391 (2012), 363-375. Google Scholar |
[5] |
I. Barbalat, Systemes déquations différentielles doscillations non linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[16] |
Y. Guo, Stochastic regime switching SIR model driven by Lévy noise, Physica A: Statistical Mechanics and its Applications, 479 (2017), 1-11. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[21] |
Q. Liu, The threshold of a stochastic Susceptible-Infective epidemic model under regime switching, Nonlinear Analysis: Hybrid Systems, 21 (2016), 49-58. Google Scholar |
[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. Google Scholar |
[23] |
X. Mao, G. Marion and E. Renshaw, Environmental noise suppresses explosion in population dynamics, Stochastic Process. Appl., 97 (2002), 95-110. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[28] |
M. Slatkin, The dynamics of a population in a Markovian environment, Ecology, 59 (1978), 249-256. Google Scholar |
[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. Google Scholar |
[32] |
C. Xu, Global threshold dynamics of a stochastic differential equation SIS model, Journal of Mathematical Analysis and Applications, 447 (2017), 736-757. Google Scholar |
[33] |
G. G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Stoch. Model. Appl. Probab. 63, Springer, New York, 2010. Google Scholar |
[34] |
X. Zhang and K. Wang, Stochastic SIR model with jumps, Appl. Math. Lett., 26 (2013), 867-874. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. |





















[1] |
Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201 |
[2] |
Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223 |
[3] |
Braxton Osting, Jérôme Darbon, Stanley Osher. Statistical ranking using the $l^{1}$-norm on graphs. Inverse Problems & Imaging, 2013, 7 (3) : 907-926. doi: 10.3934/ipi.2013.7.907 |
[4] |
Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113 |
[5] |
Pascal Noble, Sebastien Travadel. Non-persistence of roll-waves under viscous perturbations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 61-70. doi: 10.3934/dcdsb.2001.1.61 |
[6] |
Akio Matsumot, Ferenc Szidarovszky. Stability switching and its directions in cournot duopoly game with three delays. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021069 |
[7] |
Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199 |
[8] |
J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 |
[9] |
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 |
[10] |
Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208 |
[11] |
María J. Garrido-Atienza, Bohdan Maslowski, Jana Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088 |
[12] |
Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023 |
[13] |
Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192 |
[14] |
Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021066 |
[15] |
Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225 |
[16] |
Johannes Kellendonk, Lorenzo Sadun. Conjugacies of model sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3805-3830. doi: 10.3934/dcds.2017161 |
[17] |
Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic model for avascular tumor growth. Conference Publications, 2009, 2009 (Special) : 101-108. doi: 10.3934/proc.2009.2009.101 |
[18] |
Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427 |
[19] |
Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 |
[20] |
Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021004 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]