Optimal control problem of a quarantine model in multi region with spatial dynamics

Firstly we define a multi-region discrete SIR model describing the spread of an epidemic that can emerge in one region and spread to regions that are connected to its neighbors by any type of anthropological movement. We assume that the infected who leave the main source of epidemic or other areas that were partially affected later, in order to enter safer areas, represent the population category to be controlled, and we consider a surface colored cell grid is presented to illustrate the entire domain affected by the epidemic, each cell may represent a subdomain or region. Secondly we introduce a control variable into our model to show the effectiveness of movement restrictions of infected individuals from the primary source we target by a quarantine optimal control strategy. The characterization of the sought optimal control is derived from Pontryagin’s Maximum Principle. Numerical illustrations are given to the obtained results.