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

Yassine Sabbar; Anwar Zeb; Nadia Gul; Driss Kiouach; S. P. Rajasekar; Nasim Ullah; Alsharef Mohammad; Yassine Sabbar; Anwar Zeb; Nadia Gul; Driss Kiouach; S. P. Rajasekar; Nasim Ullah; Alsharef Mohammad

doi:10.3934/math.2023066

AIMS Mathematics

2023, Volume 8, Issue 1: 1329-1344. doi: 10.3934/math.2023066

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Research article

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

1.
LPAIS Laboratory, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, Fez 30000, Morocco
2.
Department of Mathematics, COMSATS University of Islamabad, Abbottabad Campus, Abbottabad, Khyber Pakhtunkhwa, Pakistan
3.
Department of Mathematics, Shaheed Benazir Bhutto Women University, Peshawar, 25000, Khyber Pakhtunkhwa, Pakistan
4.
Department of Mathematics, Government Arts College for Women, Nilakottai - 624202, Tamilnadu, India
5.
Department of Electrical Engineering College of Engineering, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

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.
- mathematical modeling,
- Lévy motion,
- Lévy measure,
- tempered distribution,
- stationary distribution
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

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Abstract

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.

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