Stationary distribution and extinction of a stochastic SEIQ epidemic model with a general incidence function and temporary immunity

Yuhuai Zhang; Xinsheng Ma; Anwarud Din; Yuhuai Zhang; Xinsheng Ma; Anwarud Din

doi:10.3934/math.2021715

AIMS Mathematics

2021, Volume 6, Issue 11: 12359-12378. doi: 10.3934/math.2021715

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Research article

Stationary distribution and extinction of a stochastic SEIQ epidemic model with a general incidence function and temporary immunity

1.
College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 211106, China
2.
Department of Mathematics, Zhejiang International Studies University, Hangzhou 310012, China
3.
Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

Received: 25 May 2021 Accepted: 16 August 2021 Published: 27 August 2021
MSC : 34F05, 60H10, 92D30

In this paper, we propose a novel stochastic SEIQ model of a disease with the general incidence rate and temporary immunity. We first investigate the existence and uniqueness of a global positive solution for the model by constructing a suitable Lyapunov function. Then, we discuss the extinction of the SEIQ epidemic model. Furthermore, a stationary distribution for the model is obtained and the ergodic holds by using the method of Khasminskii. Finally, the theoretical results are verified by some numerical simulations. The simulation results show that the noise intensity has a strong influence on the epidemic spreading.
- stochastic SEIQ model,
- temporary immunity,
- extinction,
- stationary distribution,
- general incidence function
Citation: Yuhuai Zhang, Xinsheng Ma, Anwarud Din. Stationary distribution and extinction of a stochastic SEIQ epidemic model with a general incidence function and temporary immunity[J]. AIMS Mathematics, 2021, 6(11): 12359-12378. doi: 10.3934/math.2021715

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Abstract

In this paper, we propose a novel stochastic SEIQ model of a disease with the general incidence rate and temporary immunity. We first investigate the existence and uniqueness of a global positive solution for the model by constructing a suitable Lyapunov function. Then, we discuss the extinction of the SEIQ epidemic model. Furthermore, a stationary distribution for the model is obtained and the ergodic holds by using the method of Khasminskii. Finally, the theoretical results are verified by some numerical simulations. The simulation results show that the noise intensity has a strong influence on the epidemic spreading.

References

[1]	R. Anderson, R. May, Regulation and stability of host-parasite population interactions: I. Regulatory processes, J. Anim. Ecol., 47 (1978), 219–247. doi: 10.2307/3933
[2]	R. Anderson, R. May, Infectious disease of humans: Dynamics and control, Oxford University Press, 1992.
[3]	L. Allen, M. Langlais, C. Phillips, The dynamics of two viral infections in a single host population with applications to hantavirus, Math. Biosci., 186 (2003), 191–217. doi: 10.1016/j.mbs.2003.08.002
[4]	B. Buonomo, D. Lacitignola, C. Leon, Qualitative analysis and optimal control of an epidemic model with vaccination and treatment, Math. Comput. Simulat., 100 (2014), 88–102. doi: 10.1016/j.matcom.2013.11.005
[5]	S. Binder, A. Levitt, J. Sacks, J. Hughes, Emerging infectious diseases: Public health issues for the 21st century, Science, 284 (1999), 1311–1313. doi: 10.1126/science.284.5418.1311
[6]	S. Blower, A. McLean, Mixing ecology and epidemiology, Proc. R. Soc. Lond. B., 245 (1991), 187–192. doi: 10.1098/rspb.1991.0108
[7]	T. Caraballo, M. Fatini, R. Pettersson, R. Taki, A stochastic SIRI epidemic model with relapse and media coverage, Discrete Cont. Dyn. B, 23 (2018), 2483–3501.
[8]	X. Chen, J. Cao, J. Park, J. Qiu, Stability analysis and estimation of domain of attraction for the endemic equilibrium of an SEIQ epidemic model, Nonlinear Dynam., 87 (2017), 975–985. doi: 10.1007/s11071-016-3092-7
[9]	A. Din, Y. Li, T. Khan, K. Anwar, G. Zaman, Stochastic dynamics of hepatitis B epidemics, Results Phys., 20 (2021), 103730. doi: 10.1016/j.rinp.2020.103730
[10]	A. Din, Y. Li, A. Yusuf, Delayed hepatitis B epidemic model with stochastic analysis, Chaos Soliton. Fract., 146 (2021), 110839. doi: 10.1016/j.chaos.2021.110839
[11]	D. Ebert, C. Zschokke-Rohringer, H. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna, Oecologia, 122 (2000), 200–209. doi: 10.1007/PL00008847
[12]	K. Fushimi, Y. Enatsu, E. Ishiwata, Global stability of an SIS epidemic model with delays, Math. Method. Appl. Sci., 41 (2018), 5345–5354. doi: 10.1002/mma.5084
[13]	M. Fatini, M. Khalifi, R. Gerlach, A. Laaribi, R. Taki, Stationary distribution and threshold dynamics of a stochastic SIRS model with a general incidence, Physica A, 534 (2019), 120696. doi: 10.1016/j.physa.2019.03.061
[14]	X. Feng, L. Liu, S. Tang, X. Huo, Stability and bifurcation analysis of a two-patch SIS model on nosocomial infections, Appl. Math. Lett., 102 (2020), 106097. doi: 10.1016/j.aml.2019.106097
[15]	K. Fan, Y. Zhang, S. Gao, X. Wei, A class of stochastic delayed SIR epidemic models with generalized nonlinear incidence rate and temporary immunity, Physica A, 481 (2017), 198–208. doi: 10.1016/j.physa.2017.04.055
[16]	H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599–653. doi: 10.1137/S0036144500371907
[17]	H. Hethcote, P. Van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271–287. doi: 10.1007/BF00160539
[18]	D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525–546. doi: 10.1137/S0036144500378302
[19]	M. Jin, Classification of asymptotic behavior in a stochastic SIS epidemic model with vaccination, Physica A, 521 (2019), 661–666. doi: 10.1016/j.physa.2019.01.118
[20]	A. Kumar, M. Kumar, Nilam, A study on the stability behavior of an epidemic model with ratio-dependent incidence and saturated treatment, Theor. Biosci., 139 (2020), 225–234. doi: 10.1007/s12064-020-00314-6
[21]	A. Kumar, Nilam, Stability of a delayed SIR epidemic model by introducing two explicit treatment classes along with nonlinear incidence rate and Holling type treatment, Comput. Appl. Math., 38 (2019), 1–19. doi: 10.1007/s40314-019-0767-y
[22]	W. Kermack, A. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A., 115 (1927), 700–721. doi: 10.1098/rspa.1927.0118
[23]	R. Khasminskii, Stochastic stability of differential equations, Springer Science & Business Media, 2011.
[24]	Q. Liu, D. Jiang, T. Hayat, A. Alsaedi, B. Ahmad, A stochastic SIRS epidemic model with logistic growth and general nonlinear incidence rate, Physica A, 551 (2020), 124152. doi: 10.1016/j.physa.2020.124152
[25]	A. Lahrouz, L. Omari, D. Kiouach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Anal. Model., 16 (2011), 59–76. doi: 10.15388/NA.16.1.14115
[26]	A. Lahrouz, A. Settati, Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation, Appl. Math. Comput., 233 (2014), 10–19.
[27]	H. Li, R. Peng, Dynamics and asymptotic profiles of endemic equilibrium for SIS epidemic patch models, J. Math. Biol., 79 (2019), 1279–1317. doi: 10.1007/s00285-019-01395-8
[28]	J. Li, Y. Yang, Y. Zhou, Global stability of an epidemic model with latent stage and vaccination, Nonlinear Anal. Real, 12 (2011), 2163–2173. doi: 10.1016/j.nonrwa.2010.12.030
[29]	L. Li, Y. Bai, Z. Jin, Periodic solutions of an epidemic model with saturated treatment, Nonlinear Dynam., 76 (2014), 1099–1108. doi: 10.1007/s11071-013-1193-0
[30]	L. Liu, J. Wang, X. Liu, Global stability of an SEIR epidemic model with age-dependent latency and relapse, Nonlinear Anal. Real, 24 (2015), 18–35. doi: 10.1016/j.nonrwa.2015.01.001
[31]	C. Lv, L. Huang, Z. Yuan, Global stability for an HIV-1 infection model with Beddington-DeAngelis incidence rate and CTL immune response, Commun. Nonlinear Sci., 19 (2014), 121–127. doi: 10.1016/j.cnsns.2013.06.025
[32]	J. Mena-Lorca, H. Hethcote, Dynamic models of infectious disease as regulators of population size, J. Math. Biol., 30 (1992), 693–716.
[33]	H. Moreira, Y. Wang, Global stability in an SIRI model, SIAM Rev., 39 (1997), 496–502. doi: 10.1137/S0036144595295879
[34]	X. Mao, Stochastic differential equations and their applications, Horwood, Chichester, 1997.
[35]	S. Ruan, W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Equations, 188 (2003), 135–163. doi: 10.1016/S0022-0396(02)00089-X
[36]	X. Song, A. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281–297. doi: 10.1016/j.jmaa.2006.06.064
[37]	D. Tudor, A deterministic model for herpes infections in human and animal populations, SIAM Rev., 32 (1990), 136–139. doi: 10.1137/1032003
[38]	Y. Toshio, On a comparison theorem for solutions of stochastic differential equations and its applications, J. Math. Kyoto Univ., 13 (1973), 497–512.
[39]	R. Upadhyay, A. Pal, S. Kumari, P. Roy, Dynamics of an SEIR epidemic model with nonlinear incidence and treatment rates, Nonlinear Dynam., 96 (2019), 2351–2368. doi: 10.1007/s11071-019-04926-6
[40]	L. Wang, N. Huang, Ergodic stationary distribution of a stochastic nonlinear epidemic model with relapse and cure, Appl. Anal., 2020 (2020), 1–17.
[41]	World Health Organization, The World Health Report 1996: Fighting disease, Fostering development, World Health Organization, 1996.
[42]	World Health Organization, The world health report 2002: Reducing risks, promoting healthy life, World Health Organization, 2002.
[43]	Q. Yang, X. Mao, Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations, Nonlinear Anal. Real, 14 (2013), 1434–1456. doi: 10.1016/j.nonrwa.2012.10.007
[44]	Y. Yang, J. Zhou, C. H. Hsu, Threshold dynamics of a diffusive SIRI model with nonlinear incidence rate, J. Math. Anal. Appl., 478 (2019), 874–896. doi: 10.1016/j.jmaa.2019.05.059
[45]	Z. Zhang, Y. Suo, Qualitative analysis of a SIR epidemic model with saturated treatment rate, J. Appl. Math. Comput., 34 (2010), 177–194. doi: 10.1007/s12190-009-0315-9
[46]	Y. Zhao, D. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Appl. Math. Comput., 243 (2014), 718–727.

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