# American Institute of Mathematical Sciences

May  2020, 25(5): 1985-1997. doi: 10.3934/dcdsb.2020012

## On the threshold dynamics of the stochastic SIRS epidemic model using adequate stopping times

 1 Department of Mathematics, Faculty of Sciences and Technics, Abdelmalek Essaâdi University, BP 416, Tangier, Morocco 2 Department of Mathematics, Faculty of Sciences, Ibn Tofail University, BP 133 Kénitra, Morocco 3 Department of Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, China

*Correponding author (melfatini@gmail.com)

Received  April 2019 Revised  July 2019 Published  December 2019

As it is well known, the dynamics of the stochastic SIRS epidemic model with mass action is governed by a threshold $\mathcal{R_S}$. If $\mathcal{R_S}<1$ the disease dies out from the population, while if $\mathcal{R_S}> 1$ the disease persists. However, when $\mathcal{R_S} = 1$, classical techniques used to study the asymptotic behaviour do not work any more. In this paper, we give answer to this open problem by using a new approach involving some adequate stopping times. Our results show that if $\mathcal{R_S} = 1$ then, small noises promote extinction while the large one promote persistence. So, it is exactly the opposite role of the noises in case when $\mathcal{R_S}\neq 1$.

Citation: Adel Settati, Aadil Lahrouz, Mustapha El Jarroudi, Mohamed El Fatini, Kai Wang. On the threshold dynamics of the stochastic SIRS epidemic model using adequate stopping times. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1985-1997. doi: 10.3934/dcdsb.2020012
##### References:
 [1] E. Allen, Modeling with Itô Stochastic Differential Equations, Springer, The Netherlands, 2007. Google Scholar [2] B. Berrhazi, M. El Fatini, T. Caraballo and R. Pettersson, A stochastic SIRI epidemic model with Lévy noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2415-2431.  doi: 10.3934/dcdsb.2018057.  Google Scholar [3] B. Berrhazi, M. El Fatini, A. Laaribi, R. Pettersson and R. Taki, A stochastic SIRS epidemic model incorporating media coverage and driven by Lévy noise, Chaos Solitons Fractals, 105 (2017), 60-68.  doi: 10.1016/j.chaos.2017.10.007.  Google Scholar [4] B. Berrhazi, M. El Fatini, A. Lahrouz, A. Settati and R. Taki, A stochastic SIRS epidemic model with a general awareness-induced incidence, Phys. A, 512 (2018), 968-980.  doi: 10.1016/j.physa.2018.08.150.  Google Scholar [5] V. Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomathematics, 97. Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-540-70514-7.  Google Scholar [6] T. Caraballo, M. El Fatini, R. Pettersson and R. Taki, A stochastic SIRI epidemic model with relapse and media coverage, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3483-3501.  doi: 10.3934/dcdsb.2018250.  Google Scholar [7] T. Caraballo and R. Colucci, A comparison between random and stochastic modeling for a SIR model, Commun. Pure Appl. Anal., 16 (2017), 151-162.  doi: 10.3934/cpaa.2017007.  Google Scholar [8] C. Castillo-Chavez, Z. L. Feng and W. Z. Huang, On the computation of $\mathcal{R}_0$ and its role on global stability, Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction (Minneapolis, MN, 1999), IMA Vol. Math. Appl., Springer, New York, 125 (2002), 229-250.   Google Scholar [9] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcal{R}_0$ in models for infectious diseases in heterogeneous population, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar [10] M. El Fatini, B. Boukanjime, Stochastic analysis of a two delayed epidemic model incorporating Lévy processes with a general non-linear transmission, Stoch. Anal. Appl, (Published online: 26 Oct 2019) https://doi.org/10.1080/07362994.2019.1680295. doi: 10.1080/07362994.2019.1680295.  Google Scholar [11] M. El Fatini, A. Laaribi, R. Pettersson and R. Taki, Lévy noise perturbation for an epidemic model with impact of media coverage, Stochastics, 91 (2019), 998-1019.  doi: 10.1080/17442508.2019.1595622.  Google Scholar [12] M. El Fatini, A. Lahrouz, R. Pettersson, A. Settati and R. Taki, Stochastic stability and instability of an epidemic model with relapse, Appl. Math. Comput., 316 (2018), 326-341.  doi: 10.1016/j.amc.2017.08.037.  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.  Google Scholar [14] H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356.  doi: 10.1016/0025-5564(76)90132-2.  Google Scholar [15] C. Y. Ji and D. Q. Jiang, Threshold behaviour of a stochastic SIR model, Appl. Math. Model, 38 (2014), 5067-5079.  doi: 10.1016/j.apm.2014.03.037.  Google Scholar [16] C. Y. Ji, D. Q. Jiang and N. Z. Shi, Multigroup SIR epidemic model with stochastic perturbation, Physica A, 390 (2011), 1747-1762.   Google Scholar [17] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar [18] A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Statist. Probab. Lett., 83 (2013), 960-968.  doi: 10.1016/j.spl.2012.12.021.  Google Scholar [19] A. Lahrouz and A. Settati, Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation, Appl. Math. Comput., 233 (2014), 10-19.  doi: 10.1016/j.amc.2014.01.158.  Google Scholar [20] A. Lahrouz and A. Settati, Asymptotic properties of switching diffusion epidemic model with varying population size, Appl. Math. Comput., 219 (2013), 11134-11148.  doi: 10.1016/j.amc.2013.05.019.  Google Scholar [21] Y. G. Lin and D. Q. Jiang, Long-time behavior of a perturbed SIR model by white noise, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1873-1887.  doi: 10.3934/dcdsb.2013.18.1873.  Google Scholar [22] Q. Y. Lu, Stability of SIRS system with random perturbations, Physica A, 388 (2009), 3677-3686.  doi: 10.1016/j.physa.2009.05.036.  Google Scholar [23] A. Settati, A. Lahrouz, M. El Jarroudi and M. El Jarroudi, Dynamics of hybrid switching diffusions SIRS model, J. Appl. Math. Comput., 52 (2016), 101-123.  doi: 10.1007/s12190-015-0932-4.  Google Scholar [24] E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Physica A, 35 (2005), 4111-4126.   Google Scholar [25] P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar [26] Y. N. Zhao and D. Q. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Appl. Math. Comput., 243 (2014), 718-727.  doi: 10.1016/j.amc.2014.05.124.  Google Scholar

show all references

##### References:
 [1] E. Allen, Modeling with Itô Stochastic Differential Equations, Springer, The Netherlands, 2007. Google Scholar [2] B. Berrhazi, M. El Fatini, T. Caraballo and R. Pettersson, A stochastic SIRI epidemic model with Lévy noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2415-2431.  doi: 10.3934/dcdsb.2018057.  Google Scholar [3] B. Berrhazi, M. El Fatini, A. Laaribi, R. Pettersson and R. Taki, A stochastic SIRS epidemic model incorporating media coverage and driven by Lévy noise, Chaos Solitons Fractals, 105 (2017), 60-68.  doi: 10.1016/j.chaos.2017.10.007.  Google Scholar [4] B. Berrhazi, M. El Fatini, A. Lahrouz, A. Settati and R. Taki, A stochastic SIRS epidemic model with a general awareness-induced incidence, Phys. A, 512 (2018), 968-980.  doi: 10.1016/j.physa.2018.08.150.  Google Scholar [5] V. Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomathematics, 97. Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-540-70514-7.  Google Scholar [6] T. Caraballo, M. El Fatini, R. Pettersson and R. Taki, A stochastic SIRI epidemic model with relapse and media coverage, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3483-3501.  doi: 10.3934/dcdsb.2018250.  Google Scholar [7] T. Caraballo and R. Colucci, A comparison between random and stochastic modeling for a SIR model, Commun. Pure Appl. Anal., 16 (2017), 151-162.  doi: 10.3934/cpaa.2017007.  Google Scholar [8] C. Castillo-Chavez, Z. L. Feng and W. Z. Huang, On the computation of $\mathcal{R}_0$ and its role on global stability, Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction (Minneapolis, MN, 1999), IMA Vol. Math. Appl., Springer, New York, 125 (2002), 229-250.   Google Scholar [9] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcal{R}_0$ in models for infectious diseases in heterogeneous population, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar [10] M. El Fatini, B. Boukanjime, Stochastic analysis of a two delayed epidemic model incorporating Lévy processes with a general non-linear transmission, Stoch. Anal. Appl, (Published online: 26 Oct 2019) https://doi.org/10.1080/07362994.2019.1680295. doi: 10.1080/07362994.2019.1680295.  Google Scholar [11] M. El Fatini, A. Laaribi, R. Pettersson and R. Taki, Lévy noise perturbation for an epidemic model with impact of media coverage, Stochastics, 91 (2019), 998-1019.  doi: 10.1080/17442508.2019.1595622.  Google Scholar [12] M. El Fatini, A. Lahrouz, R. Pettersson, A. Settati and R. Taki, Stochastic stability and instability of an epidemic model with relapse, Appl. Math. Comput., 316 (2018), 326-341.  doi: 10.1016/j.amc.2017.08.037.  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.  Google Scholar [14] H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356.  doi: 10.1016/0025-5564(76)90132-2.  Google Scholar [15] C. Y. Ji and D. Q. Jiang, Threshold behaviour of a stochastic SIR model, Appl. Math. Model, 38 (2014), 5067-5079.  doi: 10.1016/j.apm.2014.03.037.  Google Scholar [16] C. Y. Ji, D. Q. Jiang and N. Z. Shi, Multigroup SIR epidemic model with stochastic perturbation, Physica A, 390 (2011), 1747-1762.   Google Scholar [17] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar [18] A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Statist. Probab. Lett., 83 (2013), 960-968.  doi: 10.1016/j.spl.2012.12.021.  Google Scholar [19] A. Lahrouz and A. Settati, Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation, Appl. Math. Comput., 233 (2014), 10-19.  doi: 10.1016/j.amc.2014.01.158.  Google Scholar [20] A. Lahrouz and A. Settati, Asymptotic properties of switching diffusion epidemic model with varying population size, Appl. Math. Comput., 219 (2013), 11134-11148.  doi: 10.1016/j.amc.2013.05.019.  Google Scholar [21] Y. G. Lin and D. Q. Jiang, Long-time behavior of a perturbed SIR model by white noise, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1873-1887.  doi: 10.3934/dcdsb.2013.18.1873.  Google Scholar [22] Q. Y. Lu, Stability of SIRS system with random perturbations, Physica A, 388 (2009), 3677-3686.  doi: 10.1016/j.physa.2009.05.036.  Google Scholar [23] A. Settati, A. Lahrouz, M. El Jarroudi and M. El Jarroudi, Dynamics of hybrid switching diffusions SIRS model, J. Appl. Math. Comput., 52 (2016), 101-123.  doi: 10.1007/s12190-015-0932-4.  Google Scholar [24] E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Physica A, 35 (2005), 4111-4126.   Google Scholar [25] P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar [26] Y. N. Zhao and D. Q. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Appl. Math. Comput., 243 (2014), 718-727.  doi: 10.1016/j.amc.2014.05.124.  Google Scholar
simulation results of $S(t),I(t),R(t)$ respectively for the SDE model (1.1) with different initial conditions $(S_{0},I_{0},R_{0})$ and the parameters: $\mu = 0.1$, $\beta = 0.62$, $\lambda = 0.5$, $\gamma = 0.5$, $\sigma = 0.2$. Here $\mathcal{R_S} = 1$ and $\sigma^{2}<\beta$. For the different values of initial conditions the infection-free equilibrium $E_0(1,0,0)$ is asymptotically stable in probability
simulation results of $S(t),I(t),R(t)$ respectively for the SDE model (1.1) with different initial conditions $(S_{0},I_{0},R_{0})$ and the parameters: $\mu = 0.1$, $\beta = 0.7802$, $\lambda = 0.2$, $\gamma = 0.2$, $\sigma = 0.98$. Here $\mathcal{R_S} = 1$ and $\beta<\sigma^{2}$. For the different values of initial conditions, respectively, $S(t)$, $I(t)$ and $R(t)$ rises to or above the level $\xi_s = 0.6247$, $\xi_i = 0.2252$ and $\xi_r = 0.1501$ infinitely often with probability one
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