Dynamics of a Cholera Transmission Model with Immunological Threshold and Natural Phage Control in Reservoir

Abstract

Cholera remains epidemic and endemic in the world, causing thousands of deaths annually in locations lacking adequate sanitation and water infrastructure. Yet, its dynamics are still not fully understood. In this paper, we simplify and improve Jensen et al.’s model (PNAS 103:4652–4657, 2006) by incorporating a Minimum Infection Dose (MID) into the incidence term. We perform local stability analysis and provide bifurcation diagrams of the bacterial carrying capacity with or without shedding. Choosing parameters such that the endemic or epidemic equilibrium is unstable (as it is the case in reality), we observe numerically that for the bacterial carrying capacity (K) less than the MID (c), oscillating trajectories exist only in the microbial scale, whereas for $K>c$, they exist in both the microbial and population scales. In both cases, increasing pathogen shed rate $\xi$ increases the amplitude of the trajectories and the period of the trajectories for those that are periodic. Our findings highlight the importance of the relationship among the shedding rates, K, MID, the maximum bacterial growth rate (r) and the features of the disease outbreak. In addition, we identified a region in the parameter space of our model that leads to chaotic behaviour. This could be used to explain the irregularity in the seasonal patterns of outbreaks amongst different countries, especially if the positive relationship between bacterial proliferation and temperature is considered.