Stochastic dynamical behavior of COVID-19 model based on secondary vaccination

Xinyu Bai; Shaojuan Ma; Xinyu Bai; Shaojuan Ma

doi:10.3934/mbe.2023141

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 2: 2980-2997. doi: 10.3934/mbe.2023141

Previous Article Next Article

Research article Special Issues

Stochastic dynamical behavior of COVID-19 model based on secondary vaccination

Xinyu Bai ¹,
Shaojuan Ma ^{1,2
,
,}

1.
School of Mathematics and Information Science, North Minzu University, YinChuan 750021, China
2.
Ningxia Key Laboratory of Intelligent Information and Big Data Processing Yinchuan, YinChuan 750021, China

Academic Editor: Pierre Magal

Received: 13 August 2022 Revised: 29 October 2022 Accepted: 08 November 2022 Published: 01 December 2022

This paper mainly studies the dynamical behavior of a stochastic COVID-19 model. First, the stochastic COVID-19 model is built based on random perturbations, secondary vaccination and bilinear incidence. Second, in the proposed model, we prove the existence and uniqueness of the global positive solution using random Lyapunov function theory, and the sufficient conditions for disease extinction are obtained. It is analyzed that secondary vaccination can effectively control the spread of COVID-19 and the intensity of the random disturbance can promote the extinction of the infected population. Finally, the theoretical results are verified by numerical simulations.
- stochastic model of COVID-19,
- brownian motion,
- global positive solution,
- disease extinction,
- secondary vaccination
Citation: Xinyu Bai, Shaojuan Ma. Stochastic dynamical behavior of COVID-19 model based on secondary vaccination[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 2980-2997. doi: 10.3934/mbe.2023141

Related Papers:

Abstract

This paper mainly studies the dynamical behavior of a stochastic COVID-19 model. First, the stochastic COVID-19 model is built based on random perturbations, secondary vaccination and bilinear incidence. Second, in the proposed model, we prove the existence and uniqueness of the global positive solution using random Lyapunov function theory, and the sufficient conditions for disease extinction are obtained. It is analyzed that secondary vaccination can effectively control the spread of COVID-19 and the intensity of the random disturbance can promote the extinction of the infected population. Finally, the theoretical results are verified by numerical simulations.

References

[1]	M. Mandal, S. Jana, S. K. Nandi, A. Khatua, S. Adak, T. K. Kar, A model based study on the dynamics of COVID-19: Prediction and control, Chaos Solitons Fractals, 136 (2020), 109889. https://doi.org/10.1016/j.chaos.2020.109889 doi: 10.1016/j.chaos.2020.109889
[2]	S. Muhammad, M. A. Z. Raja, M. T. Sabir, A. H. Bukhari, H. Alrabaiah, Z. Shah, et al., A stochastic numerical analysis based on hybrid NAR-RBFs networks nonlinear SITR model for novel COVID-19 dynamics, Comput. Methods Programs Biomed., 202 (2021), 105973. https://doi.org/10.1016/j.cmpb.2021.105973 doi: 10.1016/j.cmpb.2021.105973
[3]	O. M. Otunuga, Estimation of epidemiological parameters for COVID-19 cases using a stochastic SEIRS epidemic model with vital dynamics, Results Phys., 28 (2021), 104664. https://doi.org/10.1016/j.rinp.2021.104664 doi: 10.1016/j.rinp.2021.104664
[4]	X. Zhu, B. Gao, Y. Zhong, C. Gu, K. Choi, Extended Kalman filter based on stochastic epidemiological model for COVID-19 modelling, Comput. Biol. Med., 137 (2021), 104810. https://doi.org/10.1016/j.compbiomed.2021.104810 doi: 10.1016/j.compbiomed.2021.104810
[5]	J. P. Hespanha, C. Raphael, R. R. Costa, M. K. Erdal, G. Yang, Forecasting COVID-19 cases based on a parameter-varying stochastic SIR model, Annu. Rev. Control, 51 (2021), 460–476. https://doi.org/10.1016/j.arcontrol.2021.03.008 doi: 10.1016/j.arcontrol.2021.03.008
[6]	O. E. Deeb, M. Jalloul, The dynamics of COVID-19 spread: evidence from Lebanon, Math. Biosci. Eng., 17 (2020), 5618–5632. https://doi.org/10.3934/mbe.2020302 doi: 10.3934/mbe.2020302
[7]	B. Machado, L. Antunes, C. Caetano, et al., The impact of vaccination on the evolution of COVID-19 in Portugal, Math. Biosci. Eng., 19 (2022), 936–952. https://doi.org/10.3934/mbe.2022043 doi: 10.3934/mbe.2022043
[8]	S. He, S. Tang, L. Rong, A discrete stochastic model of the COVID-19 outbreak: Forecast and control, Math. Biosci. Eng., 17 (2020), 2792–2804. https://doi.org/10.3934/mbe.2020153 doi: 10.3934/mbe.2020153
[9]	W. O. Kermack, A. G. McKendrick, Contributions to the mathematical theory of epidemics-III. Further studies of the problem of endemicity, Bull. Math. Biol., 53 (1991), 89–118. https://doi.org/10.1016/S0092-8240(05)80042-4 doi: 10.1016/S0092-8240(05)80042-4
[10]	S. Djilali, A. Zeb, T. Saeed, Effect of Occasional heroin consumers on the spread of heroin addiction, Fractals, 5 (2022), 2240164. https://doi.org/10.1142/S0218348X22401648 doi: 10.1142/S0218348X22401648
[11]	E. A. Iboi, O. Sharomi, C. N. Ngonghala, A. B. Gumel, Mathematical modeling and analysis of COVID-19 pandemic in Nigeria, Math. Biosci. Eng., 17 (2020), 7192–7220. https://doi.org/10.3934/mbe.2020369 doi: 10.3934/mbe.2020369
[12]	S. Djilali, S. Bentout, T. M. Touaoula, A. Tridanee, S. Kumarf, Global behavior of Heroin epidemic model with time distributed delay and nonlinear incidence function, Results Phys., 31 (2021), 104953. https://doi.org/10.1016/j.rinp.2021.104953 doi: 10.1016/j.rinp.2021.104953
[13]	V. Piccirillo, COVID-19 pandemic control using restrictions and vaccination, Math. Biosci. Eng., 19 (2022), 1355–1372. https://doi.org/10.3934/mbe.2022062 doi: 10.3934/mbe.2022062
[14]	V. Piccirillo, Nonlinear control of infection spread based on a deterministic SEIR model, Chaos Solitons Fractals, 149 (2021), 111051. https://doi.org/10.1016/j.chaos.2021.111051 doi: 10.1016/j.chaos.2021.111051
[15]	S. Djilali, S. Bentout, T. M. Touaoula, A. Tridane, Global dynamics of alcoholism epidemic model with distributed delays, Math. Biosci. Eng., 18 (2021), 8245–8256. https://doi.org/10.3934/mbe.2021409 doi: 10.3934/mbe.2021409
[16]	S. Batabyal, COVID-19: Perturbation dynamics resulting chaos to stable with seasonality transmission, Chaos Solitons Fractals, 145 (2021), 110772. https://doi.org/10.1016/j.chaos.2021.110772 doi: 10.1016/j.chaos.2021.110772
[17]	A. B. Gumel, E. A. Iboi, C. N. Ngonghala, E. H. Elbasha, A primer on using mathematics to understand COVID-19 dynamics: Modeling, analysis and simulations, Infect. Dis. Model., 6 (2021), 148–168. https://doi.org/10.1016/j.idm.2020.11.005 doi: 10.1016/j.idm.2020.11.005
[18]	A. Zeb, S. Djilali, T. Saeed, M. Sh. Alhodalyd, N. Gule, Global proprieties of an SIR epidemic model with nonlocal diffusion and immigration, Results Phys., 39 (2022), 105758. https://doi.org/10.1016/j.rinp.2022.105758 doi: 10.1016/j.rinp.2022.105758
[19]	F. A. Rihan, H. J. Alsakaji, Dynamics of a stochastic delay differential model for COVID-19 infection with asymptomatic infected and interacting people: Case study in the UAE, Results Phys., 28 (2021), 104658. https://doi.org/10.1016/j.rinp.2021.104658 doi: 10.1016/j.rinp.2021.104658
[20]	Z. Zhang, A. Zeb, S. Hussain, E. Alzahrani, Dynamics of COVID-19 mathematical model with stochastic perturbation, Adv. Differ. Equations, 1 (2020), 451. https://doi.org/10.1186/s13662-020-02909 doi: 10.1186/s13662-020-02909
[21]	J. Danane, K. Allali, Z. Hammouch, K. S. Nisar, Mathematical analysis and simulation of a stochastic COVID-19 L$\acute{e}$vy jump model with isolation strategy, Results Phys., 23 (2021), 103994. https://doi.org/10.1016/j.rinp.2021.103994 doi: 10.1016/j.rinp.2021.103994
[22]	D. Adak, A. Majumder, N. Bairagi, Mathematical perspective of Covid-19 pandemic: Disease extinction criteria in deterministic and stochastic models, Chaos Solitons Fractals, 142 (2020), 110381. https://doi.org/10.1016/j.chaos.2020.110381 doi: 10.1016/j.chaos.2020.110381
[23]	B. Boukanjime, T. Caraballo, M. El Fatini, M. El Khalifi, Dynamics of a stochastic coronavirus (COVID-19) epidemic model with Markovian switching, Chaos Solitons Fractals, 141 (2020), 110361. https://doi.org/10.1016/j.chaos.2020.110361 doi: 10.1016/j.chaos.2020.110361
[24]	N. H. Sweilam, S. M. AL-Mekhlafi, D. Baleanu, A hybrid stochastic fractional order Coronavirus (2019-nCov) mathematical model, Chaos Solitons Fractals, 145 (2021), 110762. https://doi.org/10.1016/j.chaos.2021.110762 doi: 10.1016/j.chaos.2021.110762
[25]	A. Din, A. Khan, D. Baleanuc, Stationary distribution and extinction of stochastic coronavirus (COVID-19) epidemic model, Chaos Solitons Fractals, 139 (2020), 110036. https://doi.org/10.1016/j.chaos.2020.110036 doi: 10.1016/j.chaos.2020.110036
[26]	T. Khan, G. Zaman, Y. El-Khatib, Modeling the dynamics of novel coronavirus (COVID-19) via stochastic epidemic model, Results Phys., 24 (2020), 104004. https://doi.org/10.1016/j.rinp.2021.104004 doi: 10.1016/j.rinp.2021.104004
[27]	A. Tesfay, T. Saeed, A. Zeb, D. Tesfay, A. Khalaf, J. Brannan, Dynamics of a stochastic COVID-19 epidemic model with jump-diffusion, Adv. Differ. Equations, 1 (2021), 228. https://doi.org/10.1186/s13662-021-03396-8 doi: 10.1186/s13662-021-03396-8
[28]	K. Zhao, S. Ma, Qualitative analysis of a two-group SVIR epidemic model with random effect, Adv. Differ. Equations, 1 (2021), 172. https://doi.org/10.1186/s13662-021-03332-w doi: 10.1186/s13662-021-03332-w
[29]	Q. Liu, D. Jiang, T. Hayat, A. Alsaedi, Stationary distribution of a stochastic delayed SVEIR epidemic model with vaccination and saturation incidence, Phys. A, 512 (2018), 849–863. https://doi.org/10.1016/j.physa.2018.08.054 doi: 10.1016/j.physa.2018.08.054
[30]	Z. Zhang, R. K. Upadhyay, Dynamical analysis for a deterministic SVIRS epidemic model with Holling type II incidence rate and multiple delays, Results Phys., 24 (2021), 104181. https://doi.org/10.1016/j.rinp.2021.104181 doi: 10.1016/j.rinp.2021.104181
[31]	Y. Xing, H. Li, Almost periodic solutions for a SVIR epidemic model with relapse, Math. Biosci. Eng., 18 (2021), 7191–7217. https://doi.org/10.3934/mbe.2021356 doi: 10.3934/mbe.2021356
[32]	S. Djilali, S. Bentout, Global dynamics of SVIR epidemic model with distributed delay and imperfect vaccine, Results Phys., 25 (2021), 104245. https://doi.org/10.1016/j.rinp.2021.104245 doi: 10.1016/j.rinp.2021.104245
[33]	X. Zhang, D. Jiang, T. Hayat, B. Ahmad, Dynamical behavior of a stochastic SVIR epidemic model with vaccination, Phys. A, 483 (2017), 94–108. https://doi.org/10.1016/j.physa.2017.04.173 doi: 10.1016/j.physa.2017.04.173
[34]	A. W. Tesfaye, T. S. Satana, Stochastic model of the transmission dynamics of COVID-19 pandemic, Adv. Differ. Equations, 1 (2021), 457–457. https://doi.org/10.1186/s13662-021-03597-1 doi: 10.1186/s13662-021-03597-1
[35]	F. Wang, L. Cao, X. Song, Mathematical modeling of mutated COVID-19 transmission with quarantine, isolation and vaccination, Math. Biosci. Eng., 19 (2022), 8035–8056. https://doi.org/10.3934/mbe.2022376 doi: 10.3934/mbe.2022376
[36]	O. A. M. Omar, Y. Alnafisah, R. A. Elbarkouky, H. M. Ahmed, COVID-19 deterministic and stochastic modelling with optimized daily vaccinations in Saudi Arabia, Results Phys., 28 (2021), 104629. https://doi.org/10.1016/j.rinp.2021.104629 doi: 10.1016/j.rinp.2021.104629
[37]	M. A. Alshaikh, Stability of discrete-time delayed influenza model with two-strain and two vaccinations, Results Phys., 28 (2021), 104563. https://doi.org/10.1016/j.rinp.2021.104629 doi: 10.1016/j.rinp.2021.104629
[38]	D. J. Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Rev., 43 (2001), 525–546. https://doi.org/10.1137/S0036144500378302 doi: 10.1137/S0036144500378302
[39]	P. Agarwal, R. Singh, A. ul Rehmanc, Numerical solution of hybrid mathematical model of dengue transmission with relapse and memory via Adam-Bashforth-Moulton predictor-corrector scheme, Chaos Solitons Fractals, 143 (2021), 110564. https://doi.org/10.1016/j.chaos.2020.110564 doi: 10.1016/j.chaos.2020.110564
[40]	N. Anggriani, M.Z. Ndii, R. Ameliaa, W. Suryaningrata, M. A. AjiPratamaa, A mathematical COVID-19 model considering asymptomatic and symptomatic classes with waning immunity, Alexandria Eng. J., 61(2022), 113–124. https://doi.org/10.1016/j.aej.2021.04.104 doi: 10.1016/j.aej.2021.04.104
[41]	X. Zhang, X. Zhang, The threshold of a deterministic and a stochastic SIQS epidemic model with varying total population size, Appl. Math. Model., 91 (2021), 749–767. https://doi.org/10.1016/j.apm.2020.09.050 doi: 10.1016/j.apm.2020.09.050

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)