September  2016, 21(7): 2363-2378. doi: 10.3934/dcdsb.2016051

Stationary distribution of stochastic SIRS epidemic model with standard incidence

1. 

College of Mathematic, Jilin University, Changchun 130012, Jilin, China, China

2. 

College of Mathematics, Beihua University, Jilin 132013, Jilin, China

3. 

College of Science, China University of Petroleum(East China), Qingdao 266580, Shandong, China

4. 

Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26, Richmond Street, Glasgow G1 1XH

Received  December 2014 Revised  September 2015 Published  August 2016

We study stochastic versions of a deterministic SIRS(Susceptible, Infective, Recovered, Susceptible) epidemic model with standard incidence. We study the existence of a stationary distribution of stochastic system by the theory of integral Markov semigroup. We prove the distribution densities of the solutions can converge to an invariant density in $L^1$. This shows the system is ergodic. The presented results are demonstrated by numerical simulations.
Citation: Yanan Zhao, Yuguo Lin, Daqing Jiang, Xuerong Mao, Yong Li. Stationary distribution of stochastic SIRS epidemic model with standard incidence. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2363-2378. doi: 10.3934/dcdsb.2016051
References:
[1]

R. M. Anderson and R. M. May, Population biology of infectious diseases I,, Nature, 280 (1979), 361. doi: 10.1007/978-3-642-68635-1. Google Scholar

[2]

R. M. Anderson and R. M. May, Population biology of infectious diseases II,, Nature, 280 (1979), 455. Google Scholar

[3]

R. M. Anderson and R. M. May, Population Biology of Infectious Diseases,, Berlin, (1982). doi: 10.1007/978-3-642-68635-1. Google Scholar

[4]

R. M. Anderson and R. M. May, Infectious Diseases of Human: Dynamics and Control,, Oxford: Oxford University Press. 1991., (1991). Google Scholar

[5]

G.B. Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale (II),, Probab. Theory Relat. Fields, 90 (1991), 377. doi: 10.1007/BF01193751. Google Scholar

[6]

D. R. Bell, The Malliavin Calculus,, Dover publications, (2006). Google Scholar

[7]

S. Busenberg and P. van den Driessche, Analysis of a disease transmission model in a population with varying size,, J. Math. Biol., 28 (1990), 257. doi: 10.1007/BF00178776. Google Scholar

[8]

N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use,, J. Math. Anal. Appl., 325 (2007), 36. doi: 10.1016/j.jmaa.2006.01.055. Google Scholar

[9]

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. doi: 10.1137/10081856X. Google Scholar

[10]

R. Z. Hasminskii, Stochastic Stability of Differential Equations,, Sijthoff & Noordhoff, (1980). Google Scholar

[11]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Rev., 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar

[12]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations,, SIAM Rev., 43 (2001), 525. doi: 10.1137/S0036144500378302. Google Scholar

[13]

C. Ji, D. Jiang, Q. Yang and N. Z. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation,, Automatica, 48 (2012), 121. doi: 10.1016/j.automatica.2011.09.044. Google Scholar

[14]

W. O. Kermack and A. G. McKendrick, Contribution to mathematical theory of epidemics,, P. Roy. Soc. Lond. A Math., 115 (1927), 700. Google Scholar

[15]

A. Lahrouz, L. Omari and D. Kiouach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model,, Nonlinear Anal. Model. Control, 16 (2011), 59. doi: 10.1515/rose-2016-0005. Google Scholar

[16]

A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence,, Statistics & Probability Letters, 83 (2013), 960. doi: 10.1016/j.spl.2012.12.021. Google Scholar

[17]

M. Li, J. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size,, Mathematical Biosciences, 160 (1999), 191. doi: 10.1016/S0025-5564(99)00030-9. Google Scholar

[18]

Y. Lin and D. Jiang, Long-time behaviour of a perturbed SIR model by white noise,, Discrete and Continuous Dynamical Systems-Series B, 18 (2013), 1873. doi: 10.3934/dcdsb.2013.18.1873. Google Scholar

[19]

H. Liu, Q. Yang and D. Jiang, The asymptotic behavior of stochastically perturbed DI SIR epidemic models with saturated incidences,, Automatica, 48 (2012), 820. doi: 10.1016/j.automatica.2012.02.010. Google Scholar

[20]

Z. Liu, Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates,, Nonlinear Analysis: Real World Applications, 14 (2013), 1286. doi: 10.1016/j.nonrwa.2012.09.016. Google Scholar

[21]

R. M. May, Stability and Complexity in Model Ecosystems,, Princeton University, (1973). Google Scholar

[22]

K. Pichór and R. Rudnicki, Stability of Markov semigroups and applications to parabolic systems,, J. Math. Anal. Appl., 215 (1997), 56. doi: 10.1006/jmaa.1997.5609. Google Scholar

[23]

R. Rudnicki, Long-time behaviour of a stochastic prey-predator model,, Stochastic Process. Appl., 108 (2003), 93. doi: 10.1016/S0304-4149(03)00090-5. Google Scholar

[24]

R. Rudnicki and K. Pichór, Influence of stochastic perturbation on prey-predator systems,, Math. Biosci., 206 (2007), 108. doi: 10.1016/j.mbs.2006.03.006. Google Scholar

[25]

D. W. Stroock and S. R. S. Varadhan, On the Support of Diffusion Processes with Applications to the Strong Maximum Principle,, Proc. Sixth Berkeley Symposium on Mathematical Statistics and Probability, (1972). Google Scholar

[26]

S. Aida, S. Kusuoka and D. Strook, On the Support of Diffusion Processes with Applications to the Strong Maximum Principle,, Proc. Sixth Berkeley Symposium on Mathematical Statistics and Probability, (1972). Google Scholar

[27]

Q. Yang, D. Jiang and N. Shi, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence,, Journal of Mathematical Analysis and Applications, 388 (2012), 248. doi: 10.1016/j.jmaa.2011.11.072. Google Scholar

[28]

Y. Zhao and D. Jiang, The threshold of a stochastic SIRS epidemic model with saturated incidence,, Applied Mathematical Letter, 34 (2014), 90. doi: 10.1016/j.aml.2013.11.002. Google Scholar

[29]

Y. Zhao, D. Jiang, X. Mao and A. Gray, The threshold of a stochastic SIRS epidemic model in a population with varying size,, Discrete Continuous Dynam. Systems - B, 20 (2015), 1277. doi: 10.3934/dcdsb.2015.20.1277. Google Scholar

[30]

Y. Zhao, D. Jiang and D. O'Regan, The extinction and persistence of the stochastic SIS epidemic model with vaccination,, Physica A, 392 (2013), 4916. doi: 10.1016/j.physa.2013.06.009. Google Scholar

show all references

References:
[1]

R. M. Anderson and R. M. May, Population biology of infectious diseases I,, Nature, 280 (1979), 361. doi: 10.1007/978-3-642-68635-1. Google Scholar

[2]

R. M. Anderson and R. M. May, Population biology of infectious diseases II,, Nature, 280 (1979), 455. Google Scholar

[3]

R. M. Anderson and R. M. May, Population Biology of Infectious Diseases,, Berlin, (1982). doi: 10.1007/978-3-642-68635-1. Google Scholar

[4]

R. M. Anderson and R. M. May, Infectious Diseases of Human: Dynamics and Control,, Oxford: Oxford University Press. 1991., (1991). Google Scholar

[5]

G.B. Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale (II),, Probab. Theory Relat. Fields, 90 (1991), 377. doi: 10.1007/BF01193751. Google Scholar

[6]

D. R. Bell, The Malliavin Calculus,, Dover publications, (2006). Google Scholar

[7]

S. Busenberg and P. van den Driessche, Analysis of a disease transmission model in a population with varying size,, J. Math. Biol., 28 (1990), 257. doi: 10.1007/BF00178776. Google Scholar

[8]

N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use,, J. Math. Anal. Appl., 325 (2007), 36. doi: 10.1016/j.jmaa.2006.01.055. Google Scholar

[9]

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. doi: 10.1137/10081856X. Google Scholar

[10]

R. Z. Hasminskii, Stochastic Stability of Differential Equations,, Sijthoff & Noordhoff, (1980). Google Scholar

[11]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Rev., 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar

[12]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations,, SIAM Rev., 43 (2001), 525. doi: 10.1137/S0036144500378302. Google Scholar

[13]

C. Ji, D. Jiang, Q. Yang and N. Z. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation,, Automatica, 48 (2012), 121. doi: 10.1016/j.automatica.2011.09.044. Google Scholar

[14]

W. O. Kermack and A. G. McKendrick, Contribution to mathematical theory of epidemics,, P. Roy. Soc. Lond. A Math., 115 (1927), 700. Google Scholar

[15]

A. Lahrouz, L. Omari and D. Kiouach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model,, Nonlinear Anal. Model. Control, 16 (2011), 59. doi: 10.1515/rose-2016-0005. Google Scholar

[16]

A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence,, Statistics & Probability Letters, 83 (2013), 960. doi: 10.1016/j.spl.2012.12.021. Google Scholar

[17]

M. Li, J. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size,, Mathematical Biosciences, 160 (1999), 191. doi: 10.1016/S0025-5564(99)00030-9. Google Scholar

[18]

Y. Lin and D. Jiang, Long-time behaviour of a perturbed SIR model by white noise,, Discrete and Continuous Dynamical Systems-Series B, 18 (2013), 1873. doi: 10.3934/dcdsb.2013.18.1873. Google Scholar

[19]

H. Liu, Q. Yang and D. Jiang, The asymptotic behavior of stochastically perturbed DI SIR epidemic models with saturated incidences,, Automatica, 48 (2012), 820. doi: 10.1016/j.automatica.2012.02.010. Google Scholar

[20]

Z. Liu, Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates,, Nonlinear Analysis: Real World Applications, 14 (2013), 1286. doi: 10.1016/j.nonrwa.2012.09.016. Google Scholar

[21]

R. M. May, Stability and Complexity in Model Ecosystems,, Princeton University, (1973). Google Scholar

[22]

K. Pichór and R. Rudnicki, Stability of Markov semigroups and applications to parabolic systems,, J. Math. Anal. Appl., 215 (1997), 56. doi: 10.1006/jmaa.1997.5609. Google Scholar

[23]

R. Rudnicki, Long-time behaviour of a stochastic prey-predator model,, Stochastic Process. Appl., 108 (2003), 93. doi: 10.1016/S0304-4149(03)00090-5. Google Scholar

[24]

R. Rudnicki and K. Pichór, Influence of stochastic perturbation on prey-predator systems,, Math. Biosci., 206 (2007), 108. doi: 10.1016/j.mbs.2006.03.006. Google Scholar

[25]

D. W. Stroock and S. R. S. Varadhan, On the Support of Diffusion Processes with Applications to the Strong Maximum Principle,, Proc. Sixth Berkeley Symposium on Mathematical Statistics and Probability, (1972). Google Scholar

[26]

S. Aida, S. Kusuoka and D. Strook, On the Support of Diffusion Processes with Applications to the Strong Maximum Principle,, Proc. Sixth Berkeley Symposium on Mathematical Statistics and Probability, (1972). Google Scholar

[27]

Q. Yang, D. Jiang and N. Shi, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence,, Journal of Mathematical Analysis and Applications, 388 (2012), 248. doi: 10.1016/j.jmaa.2011.11.072. Google Scholar

[28]

Y. Zhao and D. Jiang, The threshold of a stochastic SIRS epidemic model with saturated incidence,, Applied Mathematical Letter, 34 (2014), 90. doi: 10.1016/j.aml.2013.11.002. Google Scholar

[29]

Y. Zhao, D. Jiang, X. Mao and A. Gray, The threshold of a stochastic SIRS epidemic model in a population with varying size,, Discrete Continuous Dynam. Systems - B, 20 (2015), 1277. doi: 10.3934/dcdsb.2015.20.1277. Google Scholar

[30]

Y. Zhao, D. Jiang and D. O'Regan, The extinction and persistence of the stochastic SIS epidemic model with vaccination,, Physica A, 392 (2013), 4916. doi: 10.1016/j.physa.2013.06.009. Google Scholar

[1]

Zhong Tan, Leilei Tong. Asymptotic stability of stationary solutions for magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3435-3465. doi: 10.3934/dcds.2017146

[2]

Genni Fragnelli, A. Idrissi, L. Maniar. The asymptotic behavior of a population equation with diffusion and delayed birth process. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 735-754. doi: 10.3934/dcdsb.2007.7.735

[3]

Yuri Latushkin, Valerian Yurov. Stability estimates for semigroups on Banach spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5203-5216. doi: 10.3934/dcds.2013.33.5203

[4]

Masahiro Suzuki. Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics. Kinetic & Related Models, 2011, 4 (2) : 569-588. doi: 10.3934/krm.2011.4.569

[5]

Shuichi Kawashima, Shinya Nishibata, Masataka Nishikawa. Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane. Conference Publications, 2003, 2003 (Special) : 469-476. doi: 10.3934/proc.2003.2003.469

[6]

Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229

[7]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005

[8]

Françoise Pène. Asymptotic of the number of obstacles visited by the planar Lorentz process. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 567-587. doi: 10.3934/dcds.2009.24.567

[9]

Bara Kim, Jeongsim Kim. Explicit solution for the stationary distribution of a discrete-time finite buffer queue. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1121-1133. doi: 10.3934/jimo.2016.12.1121

[10]

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507

[11]

Guillaume Bal, Alexandre Jollivet. Stability estimates in stationary inverse transport. Inverse Problems & Imaging, 2008, 2 (4) : 427-454. doi: 10.3934/ipi.2008.2.427

[12]

Guangliang Zhao, Fuke Wu, George Yin. Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems. Mathematical Control & Related Fields, 2015, 5 (2) : 359-376. doi: 10.3934/mcrf.2015.5.359

[13]

Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147

[14]

Junping Shi, Jimin Zhang, Xiaoyan Zhang. Stability and asymptotic profile of steady state solutions to a reaction-diffusion pelagic-benthic algae growth model. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2325-2347. doi: 10.3934/cpaa.2019105

[15]

Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105

[16]

George Avalos. Strong stability of PDE semigroups via a generator resolvent criterion. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 207-218. doi: 10.3934/dcdss.2008.1.207

[17]

Katarzyna PichÓr, Ryszard Rudnicki. Stability of stochastic semigroups and applications to Stein's neuronal model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 377-385. doi: 10.3934/dcdsb.2018026

[18]

Kazuhiko Kuraya, Hiroyuki Masuyama, Shoji Kasahara. Load distribution performance of super-node based peer-to-peer communication networks: A nonstationary Markov chain approach. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 593-610. doi: 10.3934/naco.2011.1.593

[19]

Christopher E. Elmer. The stability of stationary fronts for a discrete nerve axon model. Mathematical Biosciences & Engineering, 2007, 4 (1) : 113-129. doi: 10.3934/mbe.2007.4.113

[20]

Dieter Schmidt, Lucas Valeriano. Nonlinear stability of stationary points in the problem of Robe. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1917-1936. doi: 10.3934/dcdsb.2016029

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (3)

[Back to Top]