Mathematical analysis and optimal control of a cholera epidemic in different human communities with individuals' migration. (August 2020) - Kokomo, Eric; Danhrée, Bongor; Emvudu, Yves
Abstract: We propound a deterministic, nonlinear model for the transmission dynamics of cholera in different human communities with individuals' migration. The considered different human communities are crossed by a running water which is contaminated by the vibrio cholerae bacterium. The formulated model for each community which is an initial/boundary-value problem constituted of four parabolic partial differential equations, integrates antibiotic treatment, hydration therapy and contaminated water treatment as control mechanisms of the disease. Using semigroup theory, we prove that this model has a unique bounded positive solution. Also under a given condition, the existence of a trivial equilibrium and of a nontrivial equilibrium of each community is established and their local and global stabilities are studied. In analysis of Turing's instability, we determine sufficient conditions allowing the formation of a spatially stationary and periodic heterogeneous pattern. Analytically the existence of a unique optimal control is established by the use of functional analysis techniques and an optimal control θ ̄ is determined to eradicate the epidemic in each community. In order to confirm our theoretical results, we finish with a real-world application concerning the cholera epidemic that took place in Cameroon in 2011.