Extinction and Stationary Distribution of a Stochastic SIRS Epidemic Model Incorporating Media Coverage

Main Article Content

Eric N'zi
Modeste N'zi


In this paper, we include stochastic perturbation into SIRS epidemic model incorporating media coverage and study their dynamics. Our model is obtained by taking into account both for demographic stochasticity and environmental fluctuations on contact rate before alert media β1. First, we show that the model is biologically well-posed by proving the global existence, positivity and boundedness of solution . Then, sufficient conditions for the extinction of infectious disease
is proved. We also established sufficient conditions for the existence of an ergodic stationary distribution to the model. Finally, the theoretical results are illustrated by numerical simulations; in addition we show that the media coverage can reduce the peak of infective individuals via numerical simulations.

Stochastic SIRS model, media coverage, stationary distribution, extinction.

Article Details

How to Cite
N’zi, E., & N’zi, M. (2019). Extinction and Stationary Distribution of a Stochastic SIRS Epidemic Model Incorporating Media Coverage. Journal of Advances in Mathematics and Computer Science, 34(3), 1-19. https://doi.org/10.9734/jamcs/2019/v34i3-430205
Original Research Article


Kermack WO, McKendrick AG. A contribution to mathematical theory of epidemics.
Proceedings of the Royal Society of London. 1927;115(772):700-721.

Enatsu Y, Nakata Y, Muroya Y. Global stability of SIRS epidemic models with a class of nonlinear rates and distributed delays. Acta Mathematica Scientia. 2012;32(3):851-865.

Enatsu Y, Nakata Y, Muroya Y. Global stability of a delayed SIRS epidemic model with nonmonotonic incidence rate. Journal of Mathematical Analysis and Applications. 2011;377(1):1-14.

Liu W, Levin SA, Iwasa Y. Influence of nonlinear incidence rates upon the behavior of sirs epidemiological models. Journal of Mathematical Biology. 1986;23(2):187-204.

Cui JA, Tao X, Zhu H. An SIS infection model incorporating media coverage. Rocky Mountain Journal of Mathematics. 2008;38(5):1323-1334.

Brinn MP, Carson KV, Esterman AJ, et al. Mass media interventions for preventing smoking in young people. Cochrane Database of Systematic Reviews. 2010;11:CD001006.

Xiao Y, Zhao T, Tang S. Dynamics of an infectious diseases with media/psychology induced non-smooth incidence. Mathematical Biosciences and Engineering. 2013;10(2):445-461.

Xiao Y, Tang S, Wu J. Media impact switching surface during an infectious disease outbreak. Scientific Reports. 2015;5:7838.

Liu Y, Cui JA. The impact of media coverage on the dynamics of infectious disease. International Journal of Biomathematics. 2008;1(1):65-74.

Sun C, Yang W, Arino J, et al. Effect of media-induced social distancing on disease transmission in a two patch setting. Mathematical Biosciences. 2011;230(2):87-95.

Li Y, Cui J. The effect of constant and pulse vaccination on sis epidemic models incorporating media coverage. Communications in Nonlinear Science and Numerical Simulation. 2009;14(5):2353-2365.

Spencer S. Stochastic epidemic models for emerging diseases [Ph D Thesis]. University of Nottingham; 2008.

Rand DA, Wilson HB. Chaotic stochasticity: A ubiquitos source of unpredictability in epidemics. Proceedings: Biological Sciences. 1991;246(1316):179-184.

Imhof L, Walcher S. Exclusion and persistence in deterministic and stochastic chemostat models. Journal of Differential Equations. 2005;217(1):26-53.

Miaochan Z , Huitao Z. Asymptotic behavior of global positive solution to a stochastic SIR model incorporating media coverage. Advances in Difference Equations. 2016;2016. DOI: 10.1186/s13662-0884-4

Sarkar RR. A stochastic model for autotroph-herbivore system with nutrient reclycing.Ecological Modelling. 2004;178(3-4):429-440.

Tornatore E, Buccellato SM, Vetro P. Stability of a stochastic SIR system. Physica A: Statistical Mechanics and Its Applications. 2005;354(C):111-126.

Liu W. A SIRS epidemic model incorporating media coverage with random perturbation. Abstract and Applied Analysis. 2013;9. Article ID 792308.

Jiang D, Yu J, Ji C, et al. Asymptotic behavior of global positive solution to a stochastic SIR model. Mathematical and Computer Modelling. 2011;54(1-2):221-232.

Lahrouz A, Settati A. Qualitative study of a nonlinear stochastic SIRS epidemic system. Stochastic Analysis and Applications. 2014;32(6):992-1008.

Has’minskii R. Stochastic stability of differential equations sijthoff and noordhoff. 2nd ed. Alphen aan den Rijn, The Netherlands; 1980.

Higham DJ. An algorithmic introduction to numerical simulation of stochastic differentiel equations. SIAM Review. 2001;43(3):525-546.

Tchuenche J M, Dube N, Bhunu CP, et al. The impact of media coverage on the transmission dynamics of human influenza. BMC Public Health. 2011;11(Suppl 1):S5.

Xiao D, Ruan S. Global analysis of an epidemic model with nonmonotone incidence rate. Mathematical Biosciences. 2007;208(2):419-429.

Allen LJS. An Introduction to stochastic processes with applications to biology. Pearson prentice Hall, Upper Saddle River, N J; 2003.

Mao X, Marion G, Renshaw E. Environmental brownian noise suppresses explosions in population dynamics. Stochastic Processes and their Applications. 2002;97(1):95-110.

Allen LJS, Allen EJ. A comparison of three different stochastic population models with regard to persistence time. Theoretical Population Biology. 2003;64(4):439-449.

Keeling MJ, Rohani P. Modeling infectious diseases in human and animals. New Jersey: Princeton University Press; 2008.

Weiming W, Wang L, Huang H, et al. Stochastic extinction in an sirs epidemic model incorporating media coverage. Abstract and Applied Analysis. 2013;2013:8. Article ID 891765.

Liu W, Zheng Q. A stochastic SIS epidemic model incorporating media coverage in a two patch setting. Applied Mathematics and Computation. 2015;262(C):160-168.

Lu Q. Stability of SIRS system with random perturbations. Physica A: Statistical Mechanics and Its Applications. 2009;388(18):3677-3686.

Foster D, Young P. Stochastic evolutionary game dynamics. Theoretical Population Biology. 1990;38(2):219-232.

Yan Z, Kuangang F, Shujing G, et al. Ergodic stationary distribution of a stochastic SIRS epidemic model incorporating media coverage and saturated incidence rate. Physica A. 2019;514. DOI: 10.1016/j.physa.2018.09.124):671-685

Mao X. Stochastic differential equations and applications. 2nd ed. Chichester: Horwood Publishing; 1997.

Revuz D, Yor M. Continuous martingales and brownian motion. 3rd ed. Berlin: Springer; 2005.

Zhu C, Yin G. Asymptotic properties of hybrid diffusion systems. SIAM Journal on Control and Optimization. 2007;46(4):1155-1179.

Strang G. Linear algebra and its applications. Fourth ed. Thomson Learning Inc.; 1988.