Threshold dynamics of stochastic cholera epidemic model with direct transmission

Roshan Ara; Saeed Ahmad; Zareen A. Khan; Mostafa Zahri; Roshan Ara; Saeed Ahmad; Zareen A. Khan; Mostafa Zahri

doi:10.3934/math.20231375

AIMS Mathematics

2023, Volume 8, Issue 11: 26863-26881. doi: 10.3934/math.20231375

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Research article

Threshold dynamics of stochastic cholera epidemic model with direct transmission

1.
Department of Mathematics University of Malakand Chakdara Dir (L), Pakistan
2.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
3.
Department of Mathematics, Research Groups MASEP and Bioinformatics FG, University of Sharjah, United Arab Emirates

Received: 10 May 2023 Revised: 03 September 2023 Accepted: 11 September 2023 Published: 21 September 2023
MSC : 65C30

This paper extends the cholera human-to-human direct transmission model from a deterministic to a stochastic framework. This is expressed as mixed system of stochastic and deterministic differential equations. A Lyapunov function is created to investigate the global stability of the stochastic cholera epidemic, which shows the existence of global positivity of the solution using the theory of stopping time. We then find the threshold quantity of the extended stochastic cholera epidemic model. We derive a parametric condition $ \widetilde{R}_0 $, and for additive white noise, we establish sufficient conditions for the extinction and the persistence of the cholera infection. Finally, for a suitable choice of the parameter of the system for $ \widetilde{R}_0 $, we perform numerical simulations for both scenarios of extinction and persistence of the dynamic of the cholera infection.
- stochastic cholera epidemic system,
- extinction,
- persistence,
- global positivity,
- Lyapunov function
Citation: Roshan Ara, Saeed Ahmad, Zareen A. Khan, Mostafa Zahri. Threshold dynamics of stochastic cholera epidemic model with direct transmission[J]. AIMS Mathematics, 2023, 8(11): 26863-26881. doi: 10.3934/math.20231375

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Abstract

This paper extends the cholera human-to-human direct transmission model from a deterministic to a stochastic framework. This is expressed as mixed system of stochastic and deterministic differential equations. A Lyapunov function is created to investigate the global stability of the stochastic cholera epidemic, which shows the existence of global positivity of the solution using the theory of stopping time. We then find the threshold quantity of the extended stochastic cholera epidemic model. We derive a parametric condition $ \widetilde{R}_0 $, and for additive white noise, we establish sufficient conditions for the extinction and the persistence of the cholera infection. Finally, for a suitable choice of the parameter of the system for $ \widetilde{R}_0 $, we perform numerical simulations for both scenarios of extinction and persistence of the dynamic of the cholera infection.

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