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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

Adel Settati 1, , Aadil Lahrouz 1, , Mustapha El Jarroudi 1, , Mohamed El Fatini 2,, and  Kai Wang 3,

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  May 2020 Early access  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 $.

Keywords: Stochastic threshold, SIRS epidemic model, Wiener process.
Mathematics Subject Classification: 92B05, 60G51, 60H30, 60G57.
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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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]

H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356.  doi: 10.1016/0025-5564(76)90132-2.

[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.

[16]

C. Y. Ji, D. Q. Jiang and N. Z. Shi, Multigroup SIR epidemic model with stochastic perturbation, Physica A, 390 (2011), 1747-1762. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Physica A, 35 (2005), 4111-4126. 

[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.

[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.

show all references

References:
[1]

E. Allen, Modeling with Itô Stochastic Differential Equations, Springer, The Netherlands, 2007.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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]

H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356.  doi: 10.1016/0025-5564(76)90132-2.

[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.

[16]

C. Y. Ji, D. Q. Jiang and N. Z. Shi, Multigroup SIR epidemic model with stochastic perturbation, Physica A, 390 (2011), 1747-1762. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Physica A, 35 (2005), 4111-4126. 

[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.

[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.

Figure 1.  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
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Figure 2.  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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