Global analysis of a diffusive Cholera model with multiple transmission pathways, general incidence and incomplete immunity in a heterogeneous environment

Shengfu Wang; Linfei Nie; Shengfu Wang; Linfei Nie

doi:10.3934/mbe.2024218

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 4: 4927-4955. doi: 10.3934/mbe.2024218

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Global analysis of a diffusive Cholera model with multiple transmission pathways, general incidence and incomplete immunity in a heterogeneous environment

Shengfu Wang ,
Linfei Nie ^,

College of Mathematics and System Science, Xinjiang University, Urumqi 830017, China

Academic Editor: Xiangsheng Wang

Received: 15 December 2023 Revised: 17 January 2024 Accepted: 01 February 2024 Published: 01 March 2024

With the consideration of the complexity of the transmission of Cholera, a partially degenerated reaction-diffusion model with multiple transmission pathways, incorporating the spatial heterogeneity, general incidence, incomplete immunity, and Holling type Ⅱ treatment was proposed. First, the existence, boundedness, uniqueness, and global attractiveness of solutions for this model were investigated. Second, one obtained the threshold condition $ \mathcal{R}_{0} $ and gave its expression, which described global asymptotic stability of disease-free steady state when $ \mathcal{R}_{0} < 1 $, as well as the maximum treatment rate as zero. Further, we obtained the disease was uniformly persistent when $ \mathcal{R}_{0} > 1 $. Moreover, one used the mortality due to disease as a branching parameter for the steady state, and the results showed that the model undergoes a forward bifurcation at $ \mathcal{R}_{0} $ and completely excludes the presence of endemic steady state when $ \mathcal{R}_{0} < 1 $. Finally, the theoretical results were explained through examples of numerical simulations.
- reaction-diffusion model,
- vaccination,
- multiple routes of transmission,
- bifurcation analysis,
- Holling type Ⅱ treatment
Citation: Shengfu Wang, Linfei Nie. Global analysis of a diffusive Cholera model with multiple transmission pathways, general incidence and incomplete immunity in a heterogeneous environment[J]. Mathematical Biosciences and Engineering, 2024, 21(4): 4927-4955. doi: 10.3934/mbe.2024218

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Abstract

With the consideration of the complexity of the transmission of Cholera, a partially degenerated reaction-diffusion model with multiple transmission pathways, incorporating the spatial heterogeneity, general incidence, incomplete immunity, and Holling type Ⅱ treatment was proposed. First, the existence, boundedness, uniqueness, and global attractiveness of solutions for this model were investigated. Second, one obtained the threshold condition $ \mathcal{R}_{0} $ and gave its expression, which described global asymptotic stability of disease-free steady state when $ \mathcal{R}_{0} < 1 $, as well as the maximum treatment rate as zero. Further, we obtained the disease was uniformly persistent when $ \mathcal{R}_{0} > 1 $. Moreover, one used the mortality due to disease as a branching parameter for the steady state, and the results showed that the model undergoes a forward bifurcation at $ \mathcal{R}_{0} $ and completely excludes the presence of endemic steady state when $ \mathcal{R}_{0} < 1 $. Finally, the theoretical results were explained through examples of numerical simulations.

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