Nonlocal modeling of insect borne diseases

We construct and analyze a nonlinear reaction-diffusion epidemiology model consisting of two integral-differential equations and an ordinary differential equation, which is suggested by various insect borne diseases like Yellow Fever. We first define a nonlinear auxiliary problem and establish the existence and uniqueness of its solution via a priori estimates and a fixed point argument. This leads to the existence and uniqueness of the classical solution to the analytic problem. We then develop a finite-difference method to approximate our model and conduct some numerical experiments, which demonstrate the biological applicability of the model. A large portion of this analysis originally appeared in: Cannon, J.R. and Galiffa, D.J. An Epidemiology Model Suggested by Yellow Fever. Math. Methods Appl. Sci. 2012, 35, 196-206. We supplement the analysis of the aforesaid paper by discussing ways to enhance the model therein and describe an open problem. We then conclude this chapter with an extension that yields a nonlocal global pandemic model for insect borne diseases, which is the first of its kind, and also give some preliminary results and future considerations.