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May  2020, 19(5): 2513-2531. doi: 10.3934/cpaa.2020110

A stochastic threshold for an epidemic model with isolation and a non linear incidence

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla, Spain

2. 

Département de Mathématiques, Faculté des Sciences, Université Ibn Tofail, BP 133, Kénitra, Morocco

3. 

Université Chouaib Doukkali, EST Sidi Benour, El Jadida, Morocco

4. 

Université Sultan Moulay Slimane, Faculté Polydisciplinaire, Beni Mellal, Morocco

* Corresponding author

Received  November 2018 Revised  October 2019 Published  March 2020

Fund Project: The first author is supported by Ministerio de Ciencia. Innovación y Universidades (Spain), FEDER (European Community) under grant PGC2018-096540-B-I00, and Consejer\'\i a de Innovación Ciencia y Empresa de la Junta de Andaluc\'\i a (Spain) under grant P12-FQM-1492. The second and the third authors are supported by Ibn Tofail University of Kénitra (Morocco)

In this paper, we study a stochastic epidemic model with isolation and nonlinear incidence. In particular, we propose a stochastic threshold for the model without any sharp sufficient assumptions on model parameters as compared to existing works. Firstly, we establish the uniqueness of the global positive solution according to Lyapunov function method. Secondly, we prove stochastic permanence of the solutions. Then, we establish sufficient condition for the extinction. Thirdly, we investigate necessary and sufficient conditions for persistence in mean of the disease. Finally, we provide some numerical simulations to illustrate our theoretical results.

Citation: Tomás Caraballo, Mohamed El Fatini, Idriss Sekkak, Regragui Taki, Aziz Laaribi. A stochastic threshold for an epidemic model with isolation and a non linear incidence. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2513-2531. doi: 10.3934/cpaa.2020110
References:
[1]

O. AdebimpeL. M. Erinle-Ibrahim and A. F. Adebisi, Stability analysis of SIQS epidemic model with saturated incidence rate, Appl. Math., 7 (2016), 1082-1086.  doi: 10.1016/j.amc.2014.06.026.

[2] N. J. T. Bailey, The Mathematical Theory of Infectious Diseases and Its Applications, 3, edition, Oxford University Press, Oxford, 1975. 
[3]

T. CaraballoM. El FatiniR. 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.

[4]

J. Chen, Local stability and global stability of SIQS models for disease, J. Biomath., 19 (2004), 57-64. 

[5]

Y. ChenB. Wen and Z. Teng, The global dynamics for a stochastic SIS epidemic model with isolation, Physica A, 492 (2018), 1604-1624.  doi: 10.1016/j.physa.2017.11.085.

[6]

M. El FatiniA. LaaribiR. Pettersson and R. Taki, L$\acute{e}$vy noise perturbation for an epidemic model with impact of media coverage, Stochastics, 91 (2019), 998-1019.  doi: 10.1080/17442508.2019.1595622.

[7]

M. El FatiniA. LahrouzR. PetterssonA. 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.

[8]

H. W. Hethcote, The Basic Epidemiology Models: Models, Expressions for R0, Parameter Estimation, and Applications, World Scientific Publishing Co. Pte. Ltd, 2009. doi: 10.1142/9789812834836_0001.

[9]

H. W. HethcoteM Zhien and L Shengbing, Effects of quarantine in six endemic models for infectious diseases, Math. Biosci., 180 (2002), 141-160.  doi: 10.1016/S0025-5564(02)00111-6.

[10]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.

[11]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edition, North-Holland Mathematical, 1989.

[12]

C. Ji and D. Jiang, Threshold behaviour of a stochastic SIR model, Appl. Math. Model., 38 (2014), 5067-5079.  doi: 10.1016/j.apm.2014.03.037.

[13]

D. JiangN. Shi and X. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 340 (2008), 588-597.  doi: 10.1016/j.jmaa.2007.08.014.

[14]

R. Khasminskii, Stochastic Stability of Differential Equations, Springer, 2012. doi: 10.1007/978-3-642-23280-0.

[15]

P. E. Kloeden, E. Platen and H. Schurz, Numerical Solution of SDE Through Computer Experiments, Springer, 1992. doi: 10.1007/978-3-642-57913-4.

[16]

A. Lahrouz, L. Omari and D. Kiouach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model,Nonlinear Anal. Model. Control., 16 (2011), 59-76. doi: 10.15388/NA.16.1.14115.

[17]

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.

[18]

Q. Liu and Q. Chen, Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence, Physica A, 13 (2015), 140-153.  doi: 10.1016/j.physa.2015.01.075.

[19]

K. E. LambD. Greenhalgh and C. Robertson, A simple mathematical model for genetic effects in pneumococcal carriage and transmission, J. Comput. Appl. Math., 235 (2011), 1812-1818.  doi: 10.1016/j.cam.2010.03.019.

[20]

Y. Lin and D. Jiang, Long-time behaviour of a perturbed SIR model by white noise, Discrete Contin. Dyn. Syst., 18 (2013), 1873-1887.  doi: 10.3934/dcdsb.2013.18.1873.

[21]

M. Liu and M. Fan, Permanence of stochastic Lotka-Volterra systems, J. Nonlinear Sci., 27 (2017), 425-452.  doi: 10.1007/s00332-016-9337-2.

[22]

Q. LiuD. Jiang and N. Shi, Threshold behavior in a stochastic SIQR epidemic model with standard incidence and regime switching, Appl. Math. Comput., 316 (2018), 310-325.  doi: 10.1016/j.amc.2017.08.042.

[23]

X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Horwood, 2007. doi: 10.1533/9780857099402.

[24]

X. MaoG. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stoch. Process. Their Appl., 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0.

[25]

F. Wei and F. Chen, Stochastic permanence of an SIQS epidemic model with saturated incidence and independant random perturbations, Commun. Nonlinear Sci. Numer. Simul., 453 (2016), 99-107.  doi: 10.1016/j.physa.2016.01.059.

[26]

X. Yang and X. Chunrong, An SIQS infection model with nonlinear and isolation, Int. J. Biomath., 1 (2008), 239-245.  doi: 10.1142/S1793524508000199.

[27]

X. Yang, F. Li and Y. Cheng, Global stability analysis on the dynamics of an SIQ model with nonlinear incidence rate, Advances in Future Computer and Control Systems, 2 (2012), 561–565.

[28]

D. Zhao, Study on the threshold of a stochastic SIR epidemic model and its extensions, Commun. Nonlinear Sci. Numer. Simul., 38 (2016), 172-177.  doi: 10.1016/j.cnsns.2016.02.014.

[29]

X. B ZhangH. HuoH Xiang and X. Meng, Dynamics of the deterministic and stochastic SIQS epidemic model with nonlinear incidence, Appl. Math. Comput., 243 (2014), 546-558.  doi: 10.1016/j.amc.2014.05.136.

[30]

X. B. ZhangH. F. HuoH. XiangQ. Shi and D. Li, The threshold of a stochastic SIQS epidemic model, Physica A, 482 (2017), 362-374.  doi: 10.1016/j.physa.2017.04.100.

[31]

X. B. ZhangQ. ShiS. H. MaH. F. Huo and D. Li, Dynamic behavior of a stochastic SIQS epidemic model with Lévy jumps, Nonlinear Dyn., 93 (2018), 1481-1493.  doi: 10.1007/s11071-018-4272-4.

show all references

References:
[1]

O. AdebimpeL. M. Erinle-Ibrahim and A. F. Adebisi, Stability analysis of SIQS epidemic model with saturated incidence rate, Appl. Math., 7 (2016), 1082-1086.  doi: 10.1016/j.amc.2014.06.026.

[2] N. J. T. Bailey, The Mathematical Theory of Infectious Diseases and Its Applications, 3, edition, Oxford University Press, Oxford, 1975. 
[3]

T. CaraballoM. El FatiniR. 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.

[4]

J. Chen, Local stability and global stability of SIQS models for disease, J. Biomath., 19 (2004), 57-64. 

[5]

Y. ChenB. Wen and Z. Teng, The global dynamics for a stochastic SIS epidemic model with isolation, Physica A, 492 (2018), 1604-1624.  doi: 10.1016/j.physa.2017.11.085.

[6]

M. El FatiniA. LaaribiR. Pettersson and R. Taki, L$\acute{e}$vy noise perturbation for an epidemic model with impact of media coverage, Stochastics, 91 (2019), 998-1019.  doi: 10.1080/17442508.2019.1595622.

[7]

M. El FatiniA. LahrouzR. PetterssonA. 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.

[8]

H. W. Hethcote, The Basic Epidemiology Models: Models, Expressions for R0, Parameter Estimation, and Applications, World Scientific Publishing Co. Pte. Ltd, 2009. doi: 10.1142/9789812834836_0001.

[9]

H. W. HethcoteM Zhien and L Shengbing, Effects of quarantine in six endemic models for infectious diseases, Math. Biosci., 180 (2002), 141-160.  doi: 10.1016/S0025-5564(02)00111-6.

[10]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.

[11]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edition, North-Holland Mathematical, 1989.

[12]

C. Ji and D. Jiang, Threshold behaviour of a stochastic SIR model, Appl. Math. Model., 38 (2014), 5067-5079.  doi: 10.1016/j.apm.2014.03.037.

[13]

D. JiangN. Shi and X. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 340 (2008), 588-597.  doi: 10.1016/j.jmaa.2007.08.014.

[14]

R. Khasminskii, Stochastic Stability of Differential Equations, Springer, 2012. doi: 10.1007/978-3-642-23280-0.

[15]

P. E. Kloeden, E. Platen and H. Schurz, Numerical Solution of SDE Through Computer Experiments, Springer, 1992. doi: 10.1007/978-3-642-57913-4.

[16]

A. Lahrouz, L. Omari and D. Kiouach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model,Nonlinear Anal. Model. Control., 16 (2011), 59-76. doi: 10.15388/NA.16.1.14115.

[17]

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.

[18]

Q. Liu and Q. Chen, Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence, Physica A, 13 (2015), 140-153.  doi: 10.1016/j.physa.2015.01.075.

[19]

K. E. LambD. Greenhalgh and C. Robertson, A simple mathematical model for genetic effects in pneumococcal carriage and transmission, J. Comput. Appl. Math., 235 (2011), 1812-1818.  doi: 10.1016/j.cam.2010.03.019.

[20]

Y. Lin and D. Jiang, Long-time behaviour of a perturbed SIR model by white noise, Discrete Contin. Dyn. Syst., 18 (2013), 1873-1887.  doi: 10.3934/dcdsb.2013.18.1873.

[21]

M. Liu and M. Fan, Permanence of stochastic Lotka-Volterra systems, J. Nonlinear Sci., 27 (2017), 425-452.  doi: 10.1007/s00332-016-9337-2.

[22]

Q. LiuD. Jiang and N. Shi, Threshold behavior in a stochastic SIQR epidemic model with standard incidence and regime switching, Appl. Math. Comput., 316 (2018), 310-325.  doi: 10.1016/j.amc.2017.08.042.

[23]

X. Mao, Stochastic Differential Equations and Applications, 2nd edition, Horwood, 2007. doi: 10.1533/9780857099402.

[24]

X. MaoG. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stoch. Process. Their Appl., 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0.

[25]

F. Wei and F. Chen, Stochastic permanence of an SIQS epidemic model with saturated incidence and independant random perturbations, Commun. Nonlinear Sci. Numer. Simul., 453 (2016), 99-107.  doi: 10.1016/j.physa.2016.01.059.

[26]

X. Yang and X. Chunrong, An SIQS infection model with nonlinear and isolation, Int. J. Biomath., 1 (2008), 239-245.  doi: 10.1142/S1793524508000199.

[27]

X. Yang, F. Li and Y. Cheng, Global stability analysis on the dynamics of an SIQ model with nonlinear incidence rate, Advances in Future Computer and Control Systems, 2 (2012), 561–565.

[28]

D. Zhao, Study on the threshold of a stochastic SIR epidemic model and its extensions, Commun. Nonlinear Sci. Numer. Simul., 38 (2016), 172-177.  doi: 10.1016/j.cnsns.2016.02.014.

[29]

X. B ZhangH. HuoH Xiang and X. Meng, Dynamics of the deterministic and stochastic SIQS epidemic model with nonlinear incidence, Appl. Math. Comput., 243 (2014), 546-558.  doi: 10.1016/j.amc.2014.05.136.

[30]

X. B. ZhangH. F. HuoH. XiangQ. Shi and D. Li, The threshold of a stochastic SIQS epidemic model, Physica A, 482 (2017), 362-374.  doi: 10.1016/j.physa.2017.04.100.

[31]

X. B. ZhangQ. ShiS. H. MaH. F. Huo and D. Li, Dynamic behavior of a stochastic SIQS epidemic model with Lévy jumps, Nonlinear Dyn., 93 (2018), 1481-1493.  doi: 10.1007/s11071-018-4272-4.

Figure 1.  Paths of stochastic and deterministic systems as given in Example 1
Figure 2.  Paths of stochastic and deterministic systems as given in Example 2
Figure 3.  Paths of stochastic and deterministic systems as given in Example 3
Figure 4.  Paths of stochastic and deterministic systems as given in Example 4
Figure 5.  Effect of quarantine
Table 1.  Table of parameter used in the numerical simulation
$ \quad A $ $ \qquad \beta $ $ \qquad\ \mu $ $ \quad \gamma $ $ \; \varepsilon $
206.04 $ 2.865\times 10^{-7} $ $ 1.3736\times 10^{-3} $ 0.02011 0.1
$ \quad A $ $ \qquad \beta $ $ \qquad\ \mu $ $ \quad \gamma $ $ \; \varepsilon $
206.04 $ 2.865\times 10^{-7} $ $ 1.3736\times 10^{-3} $ 0.02011 0.1
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