A novel numerical dynamics of fractional derivatives involving singular and nonsingular kernels: designing a stochastic cholera epidemic model

Saima Rashid; Fahd Jarad; Hajid Alsubaie; Ayman A. Aly; Ahmed Alotaibi; Saima Rashid; Fahd Jarad; Hajid Alsubaie; Ayman A. Aly; Ahmed Alotaibi

doi:10.3934/math.2023178

AIMS Mathematics

2023, Volume 8, Issue 2: 3484-3522. doi: 10.3934/math.2023178

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A novel numerical dynamics of fractional derivatives involving singular and nonsingular kernels: designing a stochastic cholera epidemic model

1.
Department of Mathematics, Government College university, Faislabad, Pakistan
2.
Department of Mathematics, Çankaya University, Ankara, Turkey
3.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
4.
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
5.
Department of Mechnical Enginnering, College of Engineering, Taif University, Taif, Saudi Arabia

Received: 16 August 2022 Revised: 30 October 2022 Accepted: 10 November 2022 Published: 22 November 2022
MSC : 46S40, 47H10, 54H25

In this research, we investigate the direct interaction acquisition method to create a stochastic computational formula of cholera infection evolution via the fractional calculus theory. Susceptible people, infected individuals, medicated individuals, and restored individuals are all included in the framework. Besides that, we transformed the mathematical approach into a stochastic model since it neglected the randomization mechanism and external influences. The descriptive behaviours of systems are then investigated, including the global positivity of the solution, ergodicity and stationary distribution are carried out. Furthermore, the stochastic reproductive number for the system is determined while for the case $ \mathbb{R}_{0}^{s} > 1, $ some sufficient condition for the existence of stationary distribution is obtained. To test the complexity of the proposed scheme, various fractional derivative operators such as power law, exponential decay law and the generalized Mittag-Leffler kernel were used. We included a stochastic factor in every case and employed linear growth and Lipschitz criteria to illustrate the existence and uniqueness of solutions. So every case was numerically investigated, utilizing the newest numerical technique. According to simulation data, the main significant aspects of eradicating cholera infection from society are reduced interaction incidence, improved therapeutic rate, and hygiene facilities.
- cholera epidemic model,
- fractional derivative operator,
- numerical solutions,
- Itô derivative,
- ergodic and stationary distribution
Citation: Saima Rashid, Fahd Jarad, Hajid Alsubaie, Ayman A. Aly, Ahmed Alotaibi. A novel numerical dynamics of fractional derivatives involving singular and nonsingular kernels: designing a stochastic cholera epidemic model[J]. AIMS Mathematics, 2023, 8(2): 3484-3522. doi: 10.3934/math.2023178

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Abstract

In this research, we investigate the direct interaction acquisition method to create a stochastic computational formula of cholera infection evolution via the fractional calculus theory. Susceptible people, infected individuals, medicated individuals, and restored individuals are all included in the framework. Besides that, we transformed the mathematical approach into a stochastic model since it neglected the randomization mechanism and external influences. The descriptive behaviours of systems are then investigated, including the global positivity of the solution, ergodicity and stationary distribution are carried out. Furthermore, the stochastic reproductive number for the system is determined while for the case $ \mathbb{R}_{0}^{s} > 1, $ some sufficient condition for the existence of stationary distribution is obtained. To test the complexity of the proposed scheme, various fractional derivative operators such as power law, exponential decay law and the generalized Mittag-Leffler kernel were used. We included a stochastic factor in every case and employed linear growth and Lipschitz criteria to illustrate the existence and uniqueness of solutions. So every case was numerically investigated, utilizing the newest numerical technique. According to simulation data, the main significant aspects of eradicating cholera infection from society are reduced interaction incidence, improved therapeutic rate, and hygiene facilities.

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