Coupling the modeling of phage-bacteria interaction and cholera epidemiological model with and without optimal control.

In this work, we assess the impact of the phage-bacteria infection and optimal control on the indirectly transmitted cholera disease. The phage-bacteria interactions are described by predator-prey system using the Smith functional response, which takes into account the number of bacteria binding sites. The study is done in two steps, namely the model without control and the model with control. For the first scenario, we explicitly compute the basic reproduction number R which serves as stability threshold and bifurcation parameter. The proposed model exhibits a bi-stability phenomenon via the existence of backward bifurcation, which implies that the classical requirement of bringing the reproduction number under unity, while necessary, is no longer sufficient for cholera elimination from the population. We intuitively introduce a new threshold number N needed for the global stability of the disease free equilibrium point which is achieved when R⩽1 and N⩽1. It is further shown that the phage absorption is a possible cause of bi-stability, since in its absence, the condition R⩽1 is sufficient for cholera to die out. The existence of endemic equilibrium points depends on the range of both R and N. Regarding the model extended to an optimal control problem, which involves the use of virulent vibriophages to reduce or eliminate the bacteria population, we use optimal control theory techniques. We establish the conditions under which the spread of cholera can be stopped, and examine the impact of control measures on the transmission dynamic of cholera. The Pontryagin's maximum principle is used to characterize the optimal control. Numerical simulations suggest that, the release of lytic vibriophages can significantly reduce the spread of the disease. We discuss opportunities for phage therapy as treatment of some bacterial-borne diseases without side effects.