A delay mathematical model for cholera transmission: Stability, chaos and bifurcation

In this study, a mathematical model was developed and analyzed to describe the dynamics of cholera transmission in the presence of a time delay. The model incorporated both direct (human-to-human) and indirect (environmental) transmission pathways, with houseflies assumed to facilitate the transfer of Vibrio cholerae from contaminated to uncontaminated environments. The existence of disease-free and endemic equilibria was established, and stability conditions were examined. Analytical results demonstrated that, in the absence of time delay and under certain conditions, both equilibria remained stable. To account for the effects of time delay, the Lyapunov–Krasovskii approach was employed to establish global stability conditions. Numerical simulations were conducted to analyze the time-series behavior and the oscillatory dynamics induced by the time delay, also three-dimensional bifurcation and chaos patterns were presented. The occurrence of periodic cholera outbreaks was investigated, and a criterion for Hopf bifurcation was established by treating time delay as a bifurcation parameter. Finally, Hopf bifurcation analysis was conducted to identify the critical conditions under which the system transitions from a stable state to sustained oscillations, providing insights into the long-term dynamics of cholera epidemics.