  We improve a recent mathematical model for cholera by adding a time delay
that represents the time between the instant at which an individual becomes
infected and the instant at which he begins to have symptoms of cholera
disease. We prove that the delayed cholera model is biologically meaningful and
analyze the local asymptotic stability of the equilibrium points for positive
time delays. An optimal control problem is proposed and analyzed, where the
goal is to obtain optimal treatment strategies, through quarantine, that
minimize the number of infective individuals and the bacterial concentration,
as well as treatment costs. Necessary optimality conditions are applied to the
delayed optimal control problem, with a $L^1$ type cost functional. We show
that the delayed cholera model fits better the cholera outbreak that occurred
in the Department of Artibonite -- Haiti, from 1 November 2010 to 1 May 2011,
than the non-delayed model. Considering the data of the cholera outbreak in
Haiti, we solve numerically the delayed optimal control problem and propose
solutions for the outbreak control and eradication.
