Optimal Theory Applied in Integrodifference Equation Models and in a Cholera Differential Equation Model

Integrodifference equations are discrete in time and continuous in space, and are used to model the spread of populations that are growing in discrete generations, or at discrete times, and dispersing spatially. We investigate optimal harvesting strategies, in order to maximize the profit and minimize the cost of harvesting. Theoretical results on the existence, uniqueness and characterization, as well as numerical results of optimized harvesting rates are obtained. The order of how the three events, growth, dispersal and harvesting, are arranged also affects the harvesting behavior.

Cholera remains a public health threat in many parts of the world and improved intervention strategies are needed. We investigate a key intervention strategy, vaccination, with optimal control applied to a cholera model. This system of differential equations has human compartments with susceptibles with different levels of immunity, symptomatic and asymptomatic infecteds, and two cholera vibrio compartments, hyperinfectious and non-hyperinfectious. The spread of the infection in the model is shown to be most sensitive to certain parameters, and the effect of varying these parameters on the optimal vaccination strategy is shown in numerical simulations. Our simulations also show the importance of the infection rate under various parameter cases.