Global stability of a multi-group delayed epidemic model with logistic growth

B. M. Almuqati; F. M. Allehiany; B. M. Almuqati; F. M. Allehiany

doi:10.3934/math.20231173

AIMS Mathematics

2023, Volume 8, Issue 10: 23046-23061. doi: 10.3934/math.20231173

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Global stability of a multi-group delayed epidemic model with logistic growth

B. M. Almuqati ^,,
F. M. Allehiany

Department of Mathematical Sciences, College of Applied Sciences, Umm Al-Qura University, Makkah, Saudi Arabia

Received: 07 February 2023 Revised: 29 May 2023 Accepted: 02 July 2023 Published: 19 July 2023
MSC : 37N25, 92C42

In this work, we aim to investigate the mechanism of a multi-group epidemic model taking into account the influences of logistic growth and delay time distribution. Despite the importance of the logistic growth effect in such models, its consideration remains rare. We show that $ \mathcal{R}_0 $ has a crusher role in the global stability of a disease-free and endemic equilibria. That is, if $ \mathcal{R}_0 $ is less than or equal to one, then the disease-free equilibrium is globally asymptotically stable, whereas, if $ \mathcal{R}_0 $ is greater than one, then a unique endemic equilibrium exists and is globally asymptotically stable. In addition, we construct suitable Lyapunov functions to investigate the global stability of disease-free and endemic equilibria. Finally, we introduce numerical simulations of the model.
- multi-group epidemic model,
- delay distribution,
- global stability,
- Lyapunov functions,
- logistic growth
Citation: B. M. Almuqati, F. M. Allehiany. Global stability of a multi-group delayed epidemic model with logistic growth[J]. AIMS Mathematics, 2023, 8(10): 23046-23061. doi: 10.3934/math.20231173

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Abstract

In this work, we aim to investigate the mechanism of a multi-group epidemic model taking into account the influences of logistic growth and delay time distribution. Despite the importance of the logistic growth effect in such models, its consideration remains rare. We show that $ \mathcal{R}_0 $ has a crusher role in the global stability of a disease-free and endemic equilibria. That is, if $ \mathcal{R}_0 $ is less than or equal to one, then the disease-free equilibrium is globally asymptotically stable, whereas, if $ \mathcal{R}_0 $ is greater than one, then a unique endemic equilibrium exists and is globally asymptotically stable. In addition, we construct suitable Lyapunov functions to investigate the global stability of disease-free and endemic equilibria. Finally, we introduce numerical simulations of the model.

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