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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 |
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 $.
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. |


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