A partially degenerate reaction–diffusion cholera model with temporal and spatial heterogeneity

Abstract

Cholera remains a significant threat to public health and economics. Although various mathematical models published in recent years have made an essential contribution to the study of cholera outbreaks, their dynamics are still not fully understood. To study the temporal-spatial dynamics of cholera spread, we propose a partially degenerate reaction–diffusion cholera model, which allows for the transmission rate and the shedding rate of cholera bacteria varying explicitly with both time and space. We define the basic reproduction number ${\mathcal{R}}_0$ for the model and then show that ${\mathcal{R}}_0$ acts as a threshold parameter determining whether or not the disease can invade a population. Numerically, we investigate the influences of diffusion rate, seasonality, and heterogeneity on ${\mathcal{R}}_0$. It is found that ${\mathcal{R}}_0$ is not monotone concerning seasonality for indirect transmission.

Keywords:

• Cholera
• reaction–diffusion model
• basic reproduction number
• threshold dynamics

2020 Mathematics Subject Classifications:

• 37N25
• 92D30
• 35K57

Acknowledgments

We sincerely thank Dr. Lei Zhang for helpful discussions on the definition of ${\mathcal{R}}_0$. We are also grateful to two anonymous referees for careful reading and insightful comments which led to improvements of our original manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by the NSF of China [grant number 11971369], the Fundamental Research Funds for the Central Universities [grant number JB210711] and the NSF of Shaanxi Province of China [grant number 2019JM-241].