Dynamics and simulations of stochastic COVID-19 epidemic model using Legendre spectral collocation method

Ishtiaq Ali; Sami Ullah Khan; Ishtiaq Ali; Sami Ullah Khan

doi:10.3934/math.2023210

AIMS Mathematics

2023, Volume 8, Issue 2: 4220-4236. doi: 10.3934/math.2023210

Previous Article Next Article

Research article Special Issues

Dynamics and simulations of stochastic COVID-19 epidemic model using Legendre spectral collocation method

Ishtiaq Ali ^{1
,
,},
Sami Ullah Khan ²

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
2.
Department of Mathematics, City University of Science and Information Technology Peshawar, KP 2500, Pakistan

Received: 02 October 2022 Revised: 22 November 2022 Accepted: 28 November 2022 Published: 02 December 2022
MSC : 34K50, 37H30, 65M70

The aim of this study is to investigate the dynamics of epidemic transmission of COVID-19 SEIR stochastic model with generalized saturated incidence rate. We assume that the random perturbations depends on white noises, which implies that it is directly proportional to the steady states. The existence and uniqueness of the positive solution along with the stability analysis is provided under disease-free and endemic equilibrium conditions for asymptotically stable transmission dynamics of the model. An epidemiological metric based on the ratio of basic reproduction is used to describe the transmission of an infectious disease using different parameters values involve in the proposed model. A higher order scheme based on Legendre spectral collocation method is used for the numerical simulations. For the better understanding of the proposed scheme, a comparison is made with the deterministic counterpart. In order to confirm the theoretical analysis, we provide a number of numerical examples.
- stochastic COVID-19 model,
- reproduction ratio,
- stability analysis,
- Legendre spectral method
Citation: Ishtiaq Ali, Sami Ullah Khan. Dynamics and simulations of stochastic COVID-19 epidemic model using Legendre spectral collocation method[J]. AIMS Mathematics, 2023, 8(2): 4220-4236. doi: 10.3934/math.2023210

Related Papers:

Abstract

The aim of this study is to investigate the dynamics of epidemic transmission of COVID-19 SEIR stochastic model with generalized saturated incidence rate. We assume that the random perturbations depends on white noises, which implies that it is directly proportional to the steady states. The existence and uniqueness of the positive solution along with the stability analysis is provided under disease-free and endemic equilibrium conditions for asymptotically stable transmission dynamics of the model. An epidemiological metric based on the ratio of basic reproduction is used to describe the transmission of an infectious disease using different parameters values involve in the proposed model. A higher order scheme based on Legendre spectral collocation method is used for the numerical simulations. For the better understanding of the proposed scheme, a comparison is made with the deterministic counterpart. In order to confirm the theoretical analysis, we provide a number of numerical examples.

References

[1]	P. K. Anderson, A. A. Cunningham, N. G. Patel, F. J. Morales, P. R. Epstein, P. Daszak, Emerging infectious diseases of plants: pathogen pollution, climate change and agrotechnology drivers, Trends Ecol. Evol., 19 (2004), 535–544. https://doi.org/10.1016/j.tree.2004.07.021 doi: 10.1016/j.tree.2004.07.021
[2]	D. Wang, B. Hu, C. Hu, F. Zhu, X. Liu, J. Zhang, et al., Clinical characteristics of 138 hospitalized patients with 2019 novel coronavirus-infected pneumonia in Wuhan, China, JAMA, 323 (2020), 1061–1069. https://doi.org/10.1001/jama.2020.1585 doi: 10.1001/jama.2020.1585
[3]	Y. G. Sanchez, Z. Sabir, J. L. Guirao, Design of a nonlinear SITR fractal model based on the dynamics of a novel coronavirus (COVID-19), Fractals, 28 (2020), 2040026. https://doi.org/10.1142/S0218348X20400265 doi: 10.1142/S0218348X20400265
[4]	B. C. Baumann, K. M. MacArthur, J. C. Baumann, Emotional support animals on commercial flights: a risk to allergic patients, Lancet Resp. Med., 4 (2016), 544–545. https://doi.org/10.1016/S2213-2600(16)30143-6 doi: 10.1016/S2213-2600(16)30143-6
[5]	World Health Organization, Coronavirus disease 2019 (COVID-19): situation report, 2020. Available from: https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports.
[6]	F. Evirgen, S. Uçar, N. Özdemir, System analysis of HIV infection model with CD4+ T under non-singular kernel derivative, Appl. Math. Nonlinear Sci., 5 (2020), 139–146. https://doi.org/10.2478/amns.2020.1.00013 doi: 10.2478/amns.2020.1.00013
[7]	N. H. Sweilam, S. M. Al-Mekhlafi, D. Baleanu, Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains, J. Adv. Res., 17 (2019), 125–137. https://doi.org/10.1016/j.jare.2019.01.007 doi: 10.1016/j.jare.2019.01.007
[8]	W. Gao, P. Veeresha, D. G. Prakasha, H. M. Baskonus, Novel dynamic structures of 2019-nCoV with nonlocal operator via powerful computational technique, Biology, 9 (2020), 107. https://doi.org/10.3390/biology9050107 doi: 10.3390/biology9050107
[9]	A. Atangana, S. İğret Araz, Mathematical model of COVID-19 spread in Turkey and South Africa: theory, methods, and applications, Adv. Differ. Equ., 2020 (2020), 659. https://doi.org/10.1186/s13662-020-03095-w doi: 10.1186/s13662-020-03095-w
[10]	J. Tanimoto, Sociophysics approach to epidemics, Springer, 2021.
[11]	A. Din, A. Khan, Y. Sabbar, Long-term bifurcation and stochastic optimal control of a triple delayed Ebola virus model with vaccination and quarantine strategies, Fractal Fract., 6 (2022), 578. https://doi.org/10.3390/fractalfract6100578 doi: 10.3390/fractalfract6100578
[12]	A. Khan, Y. Sabbar, A. Din, Stochastic modeling of the Monkey pox 2022 epidemic with cross infection hypothesis in a highly disturbed environment, Math. Biosci. Eng., 19 (2022), 13560–13581.
[13]	Y. Sabbar, D. Kiouacha, S. P. Rajasekarb, S. El AzamiEl-idrissia, The influence of quadratic Lévy noise on the dynamic of an SIC contagious illness model: new framework, critical comparison and an application to COVID-19 (SARS-CoV-2) case, Chaos, Solitons Fract., 159 (2022), 112110. https://doi.org/10.1016/j.chaos.2022.112110 doi: 10.1016/j.chaos.2022.112110
[14]	D. Lehotzky, T. Insperger, G. Stepan, Extension of the spectral element method for stability analysis of time-periodic delay-differential equations with multiple and distributed delays, Commun. Nonlinear Sci. Numer. Simul., 35 (2016), 177–189. https://doi.org/10.1016/j.cnsns.2015.11.007 doi: 10.1016/j.cnsns.2015.11.007
[15]	S. U. Khan, I. Ali, Application of Legendre spectral-collocation method to delay differential and stochastic delay differential equation, AIP Adv., 8 (2018), 035301. https://doi.org/10.1063/1.5016680 doi: 10.1063/1.5016680
[16]	S. U. Khan, I. Ali, Convergence and error analysis of a spectral collocation method for solving system of nonlinear Fredholm integral equations of second kind, Comput. Appl. Math., 38 (2019), 125. https://doi.org/10.1007/s40314-019-0897-2 doi: 10.1007/s40314-019-0897-2
[17]	S. U. Khan, I. Ali, Applications of Legendre spectral collocation method for solving system of time delay differential equations, Adv. Mech. Eng., 12 (2020). https://doi.org/10.1177/1687814020922113
[18]	I. Ali, S. U. Khan, Analysis of stochastic delayed SIRS model with exponential birth and saturated incidence rate, Chaos, Solitons Fract., 138 (2020), 110008. https://doi.org/10.1016/j.chaos.2020.110008 doi: 10.1016/j.chaos.2020.110008
[19]	I. Ali, S. U. Khan, Threshold of stochastic SIRS epidemic model from infectious to susceptible class with saturated incidence rate Using spectral method, Symmetry, 14 (2022), 1838. https://doi.org/10.3390/sym14091838 doi: 10.3390/sym14091838
[20]	S. U. Khan, M. Ali, I. Ali, A spectral collocation method for stochastic Volterra integro-differential equations and its error analysis, Adv. Differ. Equ., 2019 (2019), 1–14. https://doi.org/10.1186/s13662-019-2096-2 doi: 10.1186/s13662-019-2096-2
[21]	N. Gul, S. U. Khan, I. Ali, F. U. Khan, Transmission dynamic of stochastic hepatitis C model by spectral collocation method, Comput. Methods Biomech. Biomed. Eng., 25 (2022), 578–592. https://doi.org/10.1080/10255842.2021.1970143 doi: 10.1080/10255842.2021.1970143
[22]	I. Ali, S. U. Khan, Asymptotic behavior of three connected stochastic delay neoclassical growth systems using spectral technique, Mathematics, 10 (2022), 3639. https://doi.org/10.3390/math10193639 doi: 10.3390/math10193639
[23]	S. U. Khan, I. Ali, Numerical analysis of stochastic SIR model by Legendre spectral collocation method, Adv. Mech. Eng., 11 (2019). https://doi.org/10.1177/1687814019862918
[24]	A. Ali, S. U. Khan, I. Ali, F. U. Khan, On dynamics of stochastic avian influenza model with asymptomatic carrier using spectral method, Math. Methods Appl. Sci., 45 (2022), 8230–8246. https://doi.org/10.1002/mma.8183 doi: 10.1002/mma.8183
[25]	I. Ali, S. U. Khan, Threshold of stochastic SIRS epidemic model from infectious to susceptible class with saturated incidence rate using spectral method, Symmetry, 9 (2022), 1838. https://doi.org/10.3390/sym14091838 doi: 10.3390/sym14091838
[26]	D. Wang, J. Zhou, Z. Wang, W. Wang, Random gradient-free optimization for multiagent systems with communication noises under a time-varying weight balanced digraph, IEEE Trans. Syst., Man, Cybern.: Syst., 50 (2020), 281–289, https://doi.org/10.1109/TSMC.2017.2757265 doi: 10.1109/TSMC.2017.2757265
[27]	C. Li, J. Lian, Y. Wang, Stability of switched memristive neural networks with impulse and stochastic disturbance, Neurocomputing, 275 (2018), 2565–2573. https://doi.org/10.1016/j.neucom.2017.11.031 doi: 10.1016/j.neucom.2017.11.031
[28]	Y. Cai, J. Jiao, Z. Gui, Y. Liu, W. Wang, Environmental variability in a stochastic epidemic model, Appl. Math. Comput., 329 (2018), 210–226. https://doi.org/10.1016/j.amc.2018.02.009 doi: 10.1016/j.amc.2018.02.009
[29]	Y. Cai, Y. Kang, W. Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Appl. Math. Comput., 305 (2017), 221–240. https://doi.org/10.1016/j.amc.2017.02.003 doi: 10.1016/j.amc.2017.02.003
[30]	Y. Song, A. Miao, T. Zhang, X. Wang, J. Liu, Extinction and persistence of a stochastic SIRS epidemic model with saturated incidence rate and transfer from infectious to susceptible, Adv. Differ. Equ., 2018 (2018), 293. https://doi.org/10.1186/s13662-018-1759-8 doi: 10.1186/s13662-018-1759-8
[31]	X. Meng, S. Zhao, T. Feng, T. Zhang, Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis, J. Math. Anal. Appl., 433, (2016), 227–242. https://doi.org/10.1016/j.jmaa.2015.07.056

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)