We are interested in this paper in the modeling and analysis of the disease of COVID-19 applied to the capital of Niger: Niamey. The model we are presenting takes into account the strategy that the country has adopted to fight this pandemic. The spread of the infectious agent within the population is a dynamic phenomenon: the number of the healthy and sick individuals changes over time, depending on the contacts during which the pathogen passes from an infected individual to a healthy individual. We model this propagation phenomenon by a set of differential systems equations and determine its behavior through a numerical resolution.
We present in this paper a new technique based on Gelfand's triplet [1] and include differential theory to make a theoretical analysis of an optimal control problem with constraints governed by coupled partial differential equations. This technique allowed us to give some theoretical results of existence and uniqueness of the solution of constraints and characterize the optimal control.
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