Stochastic dynamics of a syphilis epidemic model with perturbation: Stationary distribution and extinction thresholds

Amir Khan; Inayat Khan; Awatif J. Alqarni; Amir Khan; Inayat Khan; Awatif J. Alqarni

doi:10.3934/math.20261114

AIMS Mathematics

2026, Volume 11, Issue 9: 27890-27920. doi: 10.3934/math.20261114

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Stochastic dynamics of a syphilis epidemic model with perturbation: Stationary distribution and extinction thresholds

1.
Department of Mathematics and Statistics, University of Swat, KPK, Pakistan
2.
Department of Mathematics, College of Science, University of Bisha, P.O. Box 551, Bisha, 61922, Saudi Arabia

Received: 08 June 2026 Revised: 21 July 2026 Accepted: 07 August 2026 Published: 02 September 2026
MSC : 92D30, 60H10, 34D20

In this study, a novel epidemiological model which incorporates stochastic perturbations is discussed. The perturbations are introduced through Brownian motion, which accounts for the inherent randomness and environmental fluctuations that affect the disease transmission dynamics. The main goal is to find the existence of a unique non-negative nonlocal solution. We show that the stochastic model has a stationary distribution using the fundamental reproduction number $ R_0 $ that was obtained from the related deterministic model. Furthermore, under specific circumstances, we examine the fluctuation of the unique solution of the deterministic issue around the disease-free equilibrium. Additionally, the stochastic stability of the system is examined, and sufficient criteria for both extinction and persistence are established in the presence of noise. Our findings are further validated by numerical simulations, which supplement the theoretical insights.
- stochastic model,
- stochastic perturbation,
- Brownian motion,
- stochastic stability
Citation: Amir Khan, Inayat Khan, Awatif J. Alqarni. Stochastic dynamics of a syphilis epidemic model with perturbation: Stationary distribution and extinction thresholds[J]. AIMS Mathematics, 2026, 11(9): 27890-27920. doi: 10.3934/math.20261114

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Abstract

In this study, a novel epidemiological model which incorporates stochastic perturbations is discussed. The perturbations are introduced through Brownian motion, which accounts for the inherent randomness and environmental fluctuations that affect the disease transmission dynamics. The main goal is to find the existence of a unique non-negative nonlocal solution. We show that the stochastic model has a stationary distribution using the fundamental reproduction number $ R_0 $ that was obtained from the related deterministic model. Furthermore, under specific circumstances, we examine the fluctuation of the unique solution of the deterministic issue around the disease-free equilibrium. Additionally, the stochastic stability of the system is examined, and sufficient criteria for both extinction and persistence are established in the presence of noise. Our findings are further validated by numerical simulations, which supplement the theoretical insights.

References

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