Mathematical modeling of the migration's effect and analysis of the spreading of a cholera epidemic

We propound a mathematical modeling of the migration's effect on the size of any population dynamic from a site of a heterogeneous space \Omega\subset \textbf{R}^{d}, d=۱,۲,\ldots. The  obtained model is afterwards added at SIR model including the dynamics of the bacteria and some control parameters to model the spreading of a cholera epidemic which occurs in \Omega. The formulated model is given by a system of four parabolic partial differential equations. Existence and stability of equilibria, Turing's instability and optimal control problem of this model are studied. We finish with a real-world application in which we apply the model specifically to the cholera epidemic that took place in Cameroon in ۲۰۱۱.

Keywords:

Cholera epidemic , Semigroup , partial differential equation , Dirac distribution , optimal control