# American Institute of Mathematical Sciences

October  2018, 23(8): 3483-3501. doi: 10.3934/dcdsb.2018250

## A stochastic SIRI epidemic model with relapse and media coverage

 1 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain 2 Ibn Tofail University, FS, Department of Mathematics, BP 133, Kénitra, Morocco 3 Linnaeus University, Department of Mathematics, 351 95 Växjö, Sweden

Received  January 2018 Published  August 2018

Fund Project: This work has been partially supported by FEDER and the Spanish Ministerio de Economía y Competitividad project MTM2015-63723-P, the Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under grant 2010/FQM314 and Proyecto de Excelencia P12-FQM-1492 and the Faculty of Sciences, Ibn Tofail University

This work is devoted to investigate the existence and uniqueness of a global positive solution for a stochastic epidemic model with relapse and media coverage. We also study the dynamical properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we show the existence of a stationary distribution. Numerical simulations are presented to confirm the theoretical results.

Citation: Tomás Caraballo, Mohamed El Fatini, Roger Pettersson, Regragui Taki. A stochastic SIRI epidemic model with relapse and media coverage. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3483-3501. doi: 10.3934/dcdsb.2018250
##### References:

show all references

##### References:
Trajectories of stochastic and deterministic systems with the parameters values given in Example 6.1.
Trajectories of stochastic and deterministic systems with the parameters values given in previous Example $2$
The kernel density function estimations of $S(t)$, $I(t)$ and $R(t)$ of stochastic system (1.2) at time $t$ = 9000, based on 10000 stochastic simulation with the parameters values given in Example 3
The kernel density function estimations of $S(t)$, $I(t)$ and $R(t)$ of stochastic system (1.2) at time $t$ = 9500, based on 10000 stochastic simulation with the parameters values given in example 3
Paths simulations of $I(t)$ for stochastic model with the parameters values as in Example 4 and $\beta_2 = 0.01, \; 0.1, \; 0.15$ respectively.
 [1] 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 [2] Jing Ge, Ling Lin, Lai Zhang. A diffusive SIS epidemic model incorporating the media coverage impact in the heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2763-2776. doi: 10.3934/dcdsb.2017134 [3] Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124 [4] 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 [5] Hongfu Yang, Xiaoyue Li, George Yin. Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3743-3766. doi: 10.3934/dcdsb.2016119 [6] Badr-eddine Berrhazi, Mohamed El Fatini, Tomás Caraballo, Roger Pettersson. A stochastic SIRI epidemic model with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2415-2431. doi: 10.3934/dcdsb.2018057 [7] Yan Wang, Lei Wang, Yanxiang Zhao, Aimin Song, Yanping Ma. A stochastic model for microbial fermentation process under Gaussian white noise environment. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 381-392. doi: 10.3934/naco.2015.5.381 [8] Hasan Alzubaidi, Tony Shardlow. Interaction of waves in a one dimensional stochastic PDE model of excitable media. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1735-1754. doi: 10.3934/dcdsb.2013.18.1735 [9] Yanan Zhao, Daqing Jiang, Xuerong Mao, Alison Gray. The threshold of a stochastic SIRS epidemic model in a population with varying size. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1277-1295. doi: 10.3934/dcdsb.2015.20.1277 [10] Yanzhao Cao, Dawit Denu. Analysis of stochastic vector-host epidemic model with direct transmission. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2109-2127. doi: 10.3934/dcdsb.2016039 [11] Julia Amador, Mariajesus Lopez-Herrero. Cumulative and maximum epidemic sizes for a nonlinear SEIR stochastic model with limited resources. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3137-3151. doi: 10.3934/dcdsb.2017211 [12] Victor Berdichevsky. Distribution of minimum values of stochastic functionals. Networks & Heterogeneous Media, 2008, 3 (3) : 437-460. doi: 10.3934/nhm.2008.3.437 [13] 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 [14] Youcef Amirat, Laurent Chupin, Rachid Touzani. Weak solutions to the equations of stationary magnetohydrodynamic flows in porous media. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2445-2464. doi: 10.3934/cpaa.2014.13.2445 [15] Stephen McDowall, Plamen Stefanov, Alexandru Tamasan. Gauge equivalence in stationary radiative transport through media with varying index of refraction. Inverse Problems & Imaging, 2010, 4 (1) : 151-167. doi: 10.3934/ipi.2010.4.151 [16] Deena Schmidt, Janet Best, Mark S. Blumberg. Random graph and stochastic process contributions to network dynamics. Conference Publications, 2011, 2011 (Special) : 1279-1288. doi: 10.3934/proc.2011.2011.1279 [17] Linda J. S. Allen, P. van den Driessche. Stochastic epidemic models with a backward bifurcation. Mathematical Biosciences & Engineering, 2006, 3 (3) : 445-458. doi: 10.3934/mbe.2006.3.445 [18] Ioana Ciotir. Stochastic porous media equations with divergence Itô noise. Evolution Equations & Control Theory, 2019, 0 (0) : 1-24. doi: 10.3934/eect.2020010 [19] Xiao-Qiang Zhao, Wendi Wang. Fisher waves in an epidemic model. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1117-1128. doi: 10.3934/dcdsb.2004.4.1117 [20] Qingshan Yang, Xuerong Mao. Stochastic dynamics of SIRS epidemic models with random perturbation. Mathematical Biosciences & Engineering, 2014, 11 (4) : 1003-1025. doi: 10.3934/mbe.2014.11.1003

2018 Impact Factor: 1.008