# American Institute of Mathematical Sciences

August  2021, 26(8): 4359-4373. doi: 10.3934/dcdsb.2020291

## Density function analysis for a stochastic SEIS epidemic model with non-degenerate diffusion

 1 School of Mathematics and Statistics, Key Laboratory of Applied Statistics of MOE, Northeast Normal University, Changchun 130024, Jilin Province, China 2 School of Continuing Education, Northeast Normal University, Changchun 130024, Jilin Province, China

* Correspondence should be addressed to Qingmei Chen, E-mail: chenqingmei.2007@163.com, Tel.:+8613617755207; fax:+8613617755207

Received  April 2020 Revised  July 2020 Published  August 2021 Early access  October 2020

Fund Project: The authors were supported by the National Natural Science Foundation of China (No.12001090) and the Fundamental Research Funds for the Central Universities of China (No.2412020QD024)

In this paper, we construct a stochastic SEIS epidemic model that incorporates constant recruitment, non-degenerate diffusion and infectious force in the latent period and infected period. By solving the corresponding Fokker-Planck equation, we obtain the exact expression of the density function around the endemic equilibrium of the deterministic system provided that the basic reproduction number is greater than one. Our work greatly improves the result of Chen [A new idea on density function and covariance matrix analysis of a stochastic SEIS epidemic model with degenerate diffusion, Appl. Math. Lett., 2020, 106200].

Citation: Qun Liu, Qingmei Chen. Density function analysis for a stochastic SEIS epidemic model with non-degenerate diffusion. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4359-4373. doi: 10.3934/dcdsb.2020291
##### References:
 [1] Y. Cai, Y. Kang and W. Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Appl. Math. Comput., 305 (2017), 221-240.  doi: 10.1016/j.amc.2017.02.003. [2] Z. Cao, W. Feng, X. Wen and L. Zu, Dynamical behavior of a stochastic SEI epidemic model with saturation incidence and logistic growth, Physica A, 523 (2019), 894-907.  doi: 10.1016/j.physa.2019.04.228. [3] T. Caraballo, M. J. Garrido-Atienza and J. L. de-la Cruz, Dynamics of some stochastic chemostat models with multiplicative noise, Commun. Pure Appl. Anal., 16 (2017), 1893-1914.  doi: 10.3934/cpaa.2017092. [4] Z. Chang, X. Meng and T. Zhang, A new way of investigating the asymptotic behaviour of a stochastic SIS system with multiplicative noise, Appl. Math. Lett., 87 (2019), 80-86.  doi: 10.1016/j.aml.2018.07.014. [5] Q. Chen, A new idea on density function and covariance matrix analysis of a stochastic SEIS epidemic model with degenerate diffusion, Appl. Math. Lett., 103 (2020), 106200, 6 pp. doi: 10.1016/j.aml.2019.106200. [6] M. Fan, M. Y. Li and K. Wang, Global stability of an SEIS epidemic model with recruitment and a varying total population size, Math. Biosci., 170 (2001), 199-208.  doi: 10.1016/S0025-5564(00)00067-5. [7] T. Feng, Z. Qiu, X. Meng and L. Rong, Analysis of a stochastic HIV-1 infection model with degenerate diffusion, Appl. Math. Comput., 348 (2019), 437-455.  doi: 10.1016/j.amc.2018.12.007. [8] J. Grasman, Stochastic epidemics: The expected duration of the endemic period in higher dimensional models, Math. Biosci., 152 (1998), 13-27.  doi: 10.1016/S0025-5564(98)10020-2. [9] S. Han and C. Lei, Global stability of equilibria of a diffusive SEIR epidemic model with nonlinear incidence, Appl. Math. Lett., 98 (2019), 114-120.  doi: 10.1016/j.aml.2019.05.045. [10] H.-F. Huo, P. Yang and H. Xiang, Stability and bifurcation for an SEIS epidemic model with the impact of media, Phys. A, 490 (2018), 702-720.  doi: 10.1016/j.physa.2017.08.139. [11] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics (Part I), Proc. Soc. Lond. Ser. A, 115 (1927), 700-721. [12] G. Li and Z. Jin, Global stability of an SEI epidemic model, Chaos, Soliton. Fract., 21 (2004), 925-931. [13] H. Li, R. Peng and Z. Wang, On a diffusive susceptible-infected-susceptible epidemic model with mass action mechanism and birth-death effect: Analysis, simulations, and comparison with other mechanisms, SIAM J. Appl. Math., 78 (2018), 2129-2153.  doi: 10.1137/18M1167863. [14] G. Li and J. Zhen, Global stability of an SEI epidemic model with general contact rate, Chaos Solitons Fractals, 23 (2005), 997-1004.  doi: 10.1016/j.chaos.2004.06.012. [15] Q. Liu, D. Jiang, T. Hayat and A. Alsaedi, Dynamics of a stochastic SIR epidemic model with distributed delay and degenerate diffusion, J. Franklin Inst., 356 (2019), 7347-7370.  doi: 10.1016/j.jfranklin.2019.06.030. [16] S. Liu, Y. Pei, C. Li and L. Chen, Three kinds of TVS in a SIR epidemic model with saturated infectious force and vertical transmission, Appl. Math. Model., 33 (2009), 1923-1932.  doi: 10.1016/j.apm.2008.05.001. [17] J. Liu and F. Wei, Dynamics of stochastic SEIS epidemic model with varying population size, Phys. A, 464 (2016), 241-250.  doi: 10.1016/j.physa.2016.06.120. [18] X. Mao, Stationary distribution of stochastic population systems, Systems Control Lett., 60 (2011), 398-405.  doi: 10.1016/j.sysconle.2011.02.013. [19] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Limited, Chichester, 1997. [20] B. Mukhopadhyay and R. Bhattacharyya, Analysis of a spatially extended nonlinear SEIS epidemic model with distinct incidence for exposed and infectives, Nonlinear Anal. Real World Appl., 9 (2008), 585-598.  doi: 10.1016/j.nonrwa.2006.12.003. [21] R. Xu, Global dynamics of an SEIS epidemic model with saturation incidence and latent period, Appl. Math. Comput., 218 (2012), 7927-7938.  doi: 10.1016/j.amc.2012.01.076.

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##### References:
 [1] Y. Cai, Y. Kang and W. Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Appl. Math. Comput., 305 (2017), 221-240.  doi: 10.1016/j.amc.2017.02.003. [2] Z. Cao, W. Feng, X. Wen and L. Zu, Dynamical behavior of a stochastic SEI epidemic model with saturation incidence and logistic growth, Physica A, 523 (2019), 894-907.  doi: 10.1016/j.physa.2019.04.228. [3] T. Caraballo, M. J. Garrido-Atienza and J. L. de-la Cruz, Dynamics of some stochastic chemostat models with multiplicative noise, Commun. Pure Appl. Anal., 16 (2017), 1893-1914.  doi: 10.3934/cpaa.2017092. [4] Z. Chang, X. Meng and T. Zhang, A new way of investigating the asymptotic behaviour of a stochastic SIS system with multiplicative noise, Appl. Math. Lett., 87 (2019), 80-86.  doi: 10.1016/j.aml.2018.07.014. [5] Q. Chen, A new idea on density function and covariance matrix analysis of a stochastic SEIS epidemic model with degenerate diffusion, Appl. Math. Lett., 103 (2020), 106200, 6 pp. doi: 10.1016/j.aml.2019.106200. [6] M. Fan, M. Y. Li and K. Wang, Global stability of an SEIS epidemic model with recruitment and a varying total population size, Math. Biosci., 170 (2001), 199-208.  doi: 10.1016/S0025-5564(00)00067-5. [7] T. Feng, Z. Qiu, X. Meng and L. Rong, Analysis of a stochastic HIV-1 infection model with degenerate diffusion, Appl. Math. Comput., 348 (2019), 437-455.  doi: 10.1016/j.amc.2018.12.007. [8] J. Grasman, Stochastic epidemics: The expected duration of the endemic period in higher dimensional models, Math. Biosci., 152 (1998), 13-27.  doi: 10.1016/S0025-5564(98)10020-2. [9] S. Han and C. Lei, Global stability of equilibria of a diffusive SEIR epidemic model with nonlinear incidence, Appl. Math. Lett., 98 (2019), 114-120.  doi: 10.1016/j.aml.2019.05.045. [10] H.-F. Huo, P. Yang and H. Xiang, Stability and bifurcation for an SEIS epidemic model with the impact of media, Phys. A, 490 (2018), 702-720.  doi: 10.1016/j.physa.2017.08.139. [11] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics (Part I), Proc. Soc. Lond. Ser. A, 115 (1927), 700-721. [12] G. Li and Z. Jin, Global stability of an SEI epidemic model, Chaos, Soliton. Fract., 21 (2004), 925-931. [13] H. Li, R. Peng and Z. Wang, On a diffusive susceptible-infected-susceptible epidemic model with mass action mechanism and birth-death effect: Analysis, simulations, and comparison with other mechanisms, SIAM J. Appl. Math., 78 (2018), 2129-2153.  doi: 10.1137/18M1167863. [14] G. Li and J. Zhen, Global stability of an SEI epidemic model with general contact rate, Chaos Solitons Fractals, 23 (2005), 997-1004.  doi: 10.1016/j.chaos.2004.06.012. [15] Q. Liu, D. Jiang, T. Hayat and A. Alsaedi, Dynamics of a stochastic SIR epidemic model with distributed delay and degenerate diffusion, J. Franklin Inst., 356 (2019), 7347-7370.  doi: 10.1016/j.jfranklin.2019.06.030. [16] S. Liu, Y. Pei, C. Li and L. Chen, Three kinds of TVS in a SIR epidemic model with saturated infectious force and vertical transmission, Appl. Math. Model., 33 (2009), 1923-1932.  doi: 10.1016/j.apm.2008.05.001. [17] J. Liu and F. Wei, Dynamics of stochastic SEIS epidemic model with varying population size, Phys. A, 464 (2016), 241-250.  doi: 10.1016/j.physa.2016.06.120. [18] X. Mao, Stationary distribution of stochastic population systems, Systems Control Lett., 60 (2011), 398-405.  doi: 10.1016/j.sysconle.2011.02.013. [19] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Limited, Chichester, 1997. [20] B. Mukhopadhyay and R. Bhattacharyya, Analysis of a spatially extended nonlinear SEIS epidemic model with distinct incidence for exposed and infectives, Nonlinear Anal. Real World Appl., 9 (2008), 585-598.  doi: 10.1016/j.nonrwa.2006.12.003. [21] R. Xu, Global dynamics of an SEIS epidemic model with saturation incidence and latent period, Appl. Math. Comput., 218 (2012), 7927-7938.  doi: 10.1016/j.amc.2012.01.076.
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