Stochastic dynamics of influenza infection: Qualitative analysis and numerical results

Jehad Alzabut; Ghada Alobaidi; Shah Hussain; Elissa Nadia Madi; Hasib Khan; Jehad Alzabut; Ghada Alobaidi; Shah Hussain; Elissa Nadia Madi; Hasib Khan

doi:10.3934/mbe.2022482

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 10: 10316-10331. doi: 10.3934/mbe.2022482

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Stochastic dynamics of influenza infection: Qualitative analysis and numerical results

1.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
2.
Department of Industrial Engineering, OSTİM Technical University, Ankara 06374, Türkiye
3.
Department of Mathematics and Statistics, American University of Sharjah, PO Box 26666, Sharjah, United Arab Emirates
4.
Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin (UniSZA), Besut Campus, Terengganu, Malaysia
5.
Department of Mathematics, Shaheed Benazir Bhutto University, Sheringal, Dir Upper, Khyber Pakhtunkhwa, Pakistan

Academic Editor: Sanling Yuan

Received: 11 April 2022 Revised: 26 June 2022 Accepted: 05 July 2022 Published: 21 July 2022

In this paper, a novel influenza $ \mathcal{S}\mathcal{I}_N\mathcal{I}_R\mathcal{R} $ model with white noise is investigated. According to the research, white noise has a significant impact on the disease. First, we explain that there is global existence and positivity to the solution. Then we show that the stochastic basic reproduction $ {{\underset{\scriptscriptstyle\centerdot}{\text{R}}}} {_r} $ is a threshold that determines whether the disease is cured or persists. When the noise intensity is high, we get $ {{\underset{\scriptscriptstyle\centerdot}{\text{R}}}}{_r} < 1 $ and the disease goes away; when the white noise intensity is low, we get $ {{\underset{\scriptscriptstyle\centerdot}{\text{R}}}}{_r} > 1 $, and a sufficient condition for the existence of a stationary distribution is obtained, which suggests that the disease is still there. However, the main objective of the study is to produce a stochastic analogue of the deterministic model that we analyze using numerical simulations to get views on the infection dynamics in a stochastic environment that we can relate to the deterministic context.
- influenza,
- stochastic modeling,
- white noise,
- persistence and extinction of disease
Citation: Jehad Alzabut, Ghada Alobaidi, Shah Hussain, Elissa Nadia Madi, Hasib Khan. Stochastic dynamics of influenza infection: Qualitative analysis and numerical results[J]. Mathematical Biosciences and Engineering, 2022, 19(10): 10316-10331. doi: 10.3934/mbe.2022482

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Abstract

In this paper, a novel influenza $ \mathcal{S}\mathcal{I}_N\mathcal{I}_R\mathcal{R} $ model with white noise is investigated. According to the research, white noise has a significant impact on the disease. First, we explain that there is global existence and positivity to the solution. Then we show that the stochastic basic reproduction $ {{\underset{\scriptscriptstyle\centerdot}{\text{R}}}} {_r} $ is a threshold that determines whether the disease is cured or persists. When the noise intensity is high, we get $ {{\underset{\scriptscriptstyle\centerdot}{\text{R}}}}{_r} < 1 $ and the disease goes away; when the white noise intensity is low, we get $ {{\underset{\scriptscriptstyle\centerdot}{\text{R}}}}{_r} > 1 $, and a sufficient condition for the existence of a stationary distribution is obtained, which suggests that the disease is still there. However, the main objective of the study is to produce a stochastic analogue of the deterministic model that we analyze using numerical simulations to get views on the infection dynamics in a stochastic environment that we can relate to the deterministic context.

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