### AIMS Mathematics

2023, Issue 1: 1329-1344. doi: 10.3934/math.2023066
Research article

# Stationary distribution of an SIR epidemic model with three correlated Brownian motions and general Lévy measure

• Received: 21 August 2022 Revised: 01 October 2022 Accepted: 11 October 2022 Published: 19 October 2022
• MSC : 37A50

• Exhaustive surveys have been previously done on the long-time behavior of illness systems with Lévy motion. All of these works have considered a Lévy–Itô decomposition associated with independent white noises and a specific Lévy measure. This setting is very particular and ignores an important class of dependent Lévy noises with a general infinite measure (finite or infinite). In this paper, we adopt this general framework and we treat a novel correlated stochastic $SIR_p$ system. By presuming some assumptions, we demonstrate the ergodic characteristic of our system. To numerically probe the advantage of our proposed framework, we implement Rosinski's algorithm for tempered stable distributions. We conclude that tempered tails have a strong effect on the long-term dynamics of the system and abruptly alter its behavior.

Citation: Yassine Sabbar, Anwar Zeb, Nadia Gul, Driss Kiouach, S. P. Rajasekar, Nasim Ullah, Alsharef Mohammad. Stationary distribution of an SIR epidemic model with three correlated Brownian motions and general Lévy measure[J]. AIMS Mathematics, 2023, 8(1): 1329-1344. doi: 10.3934/math.2023066

### Related Papers:

• Exhaustive surveys have been previously done on the long-time behavior of illness systems with Lévy motion. All of these works have considered a Lévy–Itô decomposition associated with independent white noises and a specific Lévy measure. This setting is very particular and ignores an important class of dependent Lévy noises with a general infinite measure (finite or infinite). In this paper, we adopt this general framework and we treat a novel correlated stochastic $SIR_p$ system. By presuming some assumptions, we demonstrate the ergodic characteristic of our system. To numerically probe the advantage of our proposed framework, we implement Rosinski's algorithm for tempered stable distributions. We conclude that tempered tails have a strong effect on the long-term dynamics of the system and abruptly alter its behavior.

 [1] S. P. Rajasekar, M. Pitchaimani, Q. Zhu, Higher order stochastically perturbed SIRS epidemic model with relapse and media impact, Math. Method. Appl. Sci., 45 (2022), 843–863. http://doi.org/10.1002/mma.7817 doi: 10.1002/mma.7817 [2] D. Kiouach, Y. Sabbar, S. E. A. El-idrissi, New results on the asymptotic behavior of an SIS epidemiological model with quarantine strategy, stochastic transmission, and Levy disturbance, Math. Method. Appl. Sci., 44 (2021), 13468–13492. http://doi.org/10.1002/mma.7638 doi: 10.1002/mma.7638 [3] Z. Wang, K. Tang, Combating COVID-19: health equity matters, Nat. Med., 26 (2020), 458. http://doi.org/10.1038/s41591-020-0823-6 doi: 10.1038/s41591-020-0823-6 [4] Z. Neufeld, H. Khataee, A. Czirok, Targeted adaptive isolation strategy for COVID-19 pandemic, Infectious Disease Modelling, 5 (2020), 357–361. http://doi.org/10.1016/j.idm.2020.04.003 doi: 10.1016/j.idm.2020.04.003 [5] Y. Sabbar, A. Din, D. Kiouach, Predicting potential scenarios for wastewater treatment under unstable physical and chemical laboratory conditions: A mathematical study, Results Phys., 39 (2022), 105717. https://doi.org/10.1016/j.rinp.2022.105717 doi: 10.1016/j.rinp.2022.105717 [6] Y. Sabbar, A. Zeb, D. Kiouach, N. Gul, T. Sitthiwirattham, D. Baleanu, et al., Dynamical bifurcation of a sewage treatment model with general higher-order perturbation, Results Phys., 39 (2022), 105799. https://doi.org/10.1016/j.rinp.2022.105799 doi: 10.1016/j.rinp.2022.105799 [7] Y. Sabbar, A. Khan, A. Din, Probabilistic analysis of a marine ecological system with intense variability, Mathematics, 10 (2022), 2262. https://doi.org/10.3390/math10132262 doi: 10.3390/math10132262 [8] Y. Sabbar, D. Kiouach, New method to obtain the acute sill of an ecological model with complex polynomial perturbation, Math. Method. Appl. Sci., in press. https://doi.org/10.1002/mma.8654 [9] A. Khan, Y. Sabbar, A. Din, Stochastic modeling of the Monkeypox 2022 epidemic with cross-infection hypothesis in a highly disturbed environment, Math. Biosci. Eng., 19 (2022), 13560–13581. http://doi.org/10.3934/mbe.2022633 doi: 10.3934/mbe.2022633 [10] Y. Sabbar, A. Khan, A. Din, D. Kiouach, S. P. Rajasekar, Determining the global threshold of an epidemic model with general interference function and high-order perturbation, AIMS Mathematics, 7 (2022), 19865–19890. http://doi.org/10.3934/math.20221088 doi: 10.3934/math.20221088 [11] Y. Sabbar, D. Kiouach, S. Rajasekar, S. E. A. El-idrissi, 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 Soliton. Fract., 159 (2022), 112110. http://doi.org/10.1016/j.chaos.2022.112110 doi: 10.1016/j.chaos.2022.112110 [12] D. Kiouach, Y. Sabbar, Developing new techniques for obtaining the threshold of a stochastic SIR epidemic model with 3-dimensional Levy process, Journal of Applied Nonlinear Dynamics, 11 (2022), 401–414. http://doi.org/10.5890/JAND.2022.06.010 doi: 10.5890/JAND.2022.06.010 [13] D. Kiouach, Y. Sabbar, The long-time behaviour of a stochastic SIR epidemic model with distributed delay and multidimensional Levy jumps, Int. J. Biomath., 15 (2022), 2250004. http://doi.org/10.1142/S1793524522500048 doi: 10.1142/S1793524522500048 [14] D. Kiouach, Y. Sabbar, Dynamic characterization of a stochastic sir infectious disease model with dual perturbation, Int. J. Biomath., 14 (2021), 2150016. https://doi.org/10.1142/S1793524521500169 doi: 10.1142/S1793524521500169 [15] D. Kiouach, Y. Sabbar, Ergodic stationary distribution of a stochastic hepatitis B epidemic model with interval-valued parameters and compensated poisson process, Comput. Math. Meth. Med., 2020 (2020), 9676501. http://doi.org/10.1155/2020/9676501 doi: 10.1155/2020/9676501 [16] R. Ikram, A. Khan, M. Zahri, A. Saeed, M. Yavuz, P. Kumam, Extinction and stationary distribution of a stochastic COVID-19 epidemic model with time-delay, Comput. Biol. Med., 141 (2022), 105115. http://doi.org/10.1016/j.compbiomed.2021.105115 doi: 10.1016/j.compbiomed.2021.105115 [17] B. Buonomo, Effects of information-dependent vaccination behavior on coronavirus outbreak: insights from a SIRI model, Ricerche di Matematica, 69 (2020), 483–499. http://doi.org/10.1007/s11587-020-00506-8 doi: 10.1007/s11587-020-00506-8 [18] N. T. Dieu, T. Fugo, N. H. Du, Asymptotic behaviors of stochastic epidemic models with jump-diffusion, Appl. Math. Model., 86 (2020), 259–270. http://doi.org/10.1016/j.apm.2020.05.003 doi: 10.1016/j.apm.2020.05.003 [19] I. I. Gihman, A. V. Skorohod, Stochastic differential equations, Berlin, Heidelberg: Springer, 1972. [20] J. Rosinski, Tempering stable processes, Stoch. Proc. Appl., 117 (2007), 677–707. http://doi.org/10.1016/j.spa.2006.10.003 [21] Y. Cheng, F. Zhang, M. Zhao, A stochastic model of HIV infection incorporating combined therapy of haart driven by Levy jumps, Adv. Differ. Equ., 2019 (2019), 321. http://doi.org/10.1186/s13662-019-2108-2 doi: 10.1186/s13662-019-2108-2 [22] Y. Cheng, M. Li, F. Zhang, A dynamics stochastic model with HIV infection of CD4 T cells driven by Levy noise, Chaos Soliton. Fract., 129 (2019), 62–70. http://doi.org/10.1016/j.chaos.2019.07.054 doi: 10.1016/j.chaos.2019.07.054 [23] S. Cai, Y. Cai, X. Mao, A stochastic differential equation sis epidemic model with two correlated brownian motions, Nonlinear Dyn., 97 (2019), 2175–2187. http://doi.org/10.1007/s11071-019-05114-2 doi: 10.1007/s11071-019-05114-2 [24] N. Privault, L. Wang, Stochastic SIR Levy jump model with heavy tailed increments, J. Nonlinear Sci., 31 (2021), 15. http://doi.org/10.1007/s00332-020-09670-5 doi: 10.1007/s00332-020-09670-5 [25] Y. Zhou, W. Zhang, Threshold of a stochastic SIR epidemic model with Levy jumps, Physica A, 446 (2016), 204–216. http://doi.org/10.1016/j.physa.2015.11.023 doi: 10.1016/j.physa.2015.11.023 [26] J. Tong, Z. Zhang, J. Bao, The stationary distribution of the facultative population model with a degenerate noise, Stat. Probabil. Lett., 83 (2013), 655–664. http://doi.org/10.1016/j.spl.2012.11.003 doi: 10.1016/j.spl.2012.11.003 [27] D. Zhao, S. Yuan, Sharp conditions for the existence of a stationary distribution in one classical stochastic chemostat, Appl. Math. Comput., 339 (2018), 199–205. http://doi.org/10.1016/j.amc.2018.07.020 doi: 10.1016/j.amc.2018.07.020 [28] M. Gholami, R. K. Ghaziani, Z. Eskandari, Three-dimensional fractional system with the stability condition and chaos control, Mathematical Modelling and Numerical Simulation with Applications, 2 (2022), 41–47. http://doi.org/10.53391/mmnsa.2022.01.004 doi: 10.53391/mmnsa.2022.01.004 [29] A. Zahid, S. Masood, S. Mubarik, A. Din, An efficient application of scrambled response approach to estimate the population mean of the sensitive variables, Mathematical Modelling and Numerical Simulation with Applications, 2 (2022), 127–146. http://doi.org/10.53391/mmnsa.2022.011 doi: 10.53391/mmnsa.2022.011 [30] A. Din, M. Z. Abidin, Analysis of fractional-order vaccinated Hepatitis-B epidemic model with Mittag-Leffler kernels, Mathematical Modelling and Numerical Simulation with Applications, 2 (2022), 59–72. http://doi.org/10.53391/mmnsa.2022.006 doi: 10.53391/mmnsa.2022.006 [31] N. Sene, Second-grade fluid with Newtonian heating under Caputo fractional derivative: Analytical investigations via Laplace transforms, Mathematical Modelling and Numerical Simulation with Applications, 2 (2022), 13–25. http://doi.org/10.53391/mmnsa.2022.01.002 doi: 10.53391/mmnsa.2022.01.002 [32] P. Kumar, V. S. Erturk, Dynamics of cholera disease by using two recent fractional numerical methods, Mathematical Modelling and Numerical Simulation with Applications, 1 (2021), 102–111. http://doi.org/10.53391/mmnsa.2021.01.010 doi: 10.53391/mmnsa.2021.01.010 [33] Z. Hammouch, M. Yavuz, N. Özdemir, Numerical solutions and synchronization of a variable-order fractional chaotic system, Mathematical Modelling and Numerical Simulation with Applications, 1 (2021), 11–23. http://doi.org/10.53391/mmnsa.2021.01.002 doi: 10.53391/mmnsa.2021.01.002 [34] B. Dasbasi, Stability analysis of an incommensurate fractional-order SIR model, Mathematical Modelling and Numerical Simulation with Applications, 1 (2021), 44–55. http://doi.org/10.53391/mmnsa.2021.01.005 doi: 10.53391/mmnsa.2021.01.005 [35] P. Veeresha, A numerical approach to the coupled atmospheric ocean model using a fractional operator, Mathematical Modelling and Numerical Simulation with Applications, 1 (2021), 1–10. http://doi.org/10.53391/mmnsa.2021.01.001 doi: 10.53391/mmnsa.2021.01.001 [36] M. Naim, Y. Sabbar, A. Zeb, Stability characterization of a fractional-order viral system with the non-cytolytic immune assumption. Mathematical Modelling and Numerical Simulation with Applications, 2 (2022), 164–176. http://doi.org/10.53391/mmnsa.2022.013
• © 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)
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

2.739 2.4

Article outline

Figures(3)

## Other Articles By Authors

• On This Site
• On Google Scholar

/