In this paper, we propose a multi-cell discrete-time SIRI epidemic model that describes the spatial-temporal spread of an infectious disease in a geographical domain represented as a grid of cells (regions) divided into \(p^{2}\) cells. We assume that people can move between these cells due to their connectivity. To investigate the efficacy of travel restrictions and vaccination in preventing epidemic spread, we include two control variables in our model. We aim to reduce the number of infected individuals in each cell designated as infectious by the health authority, while also minimizing the expenses associated with implementing the travel ban and vaccination efforts. We develop an optimal control approach using the definition of a supplementary function that identifies the automatic activation of travel restrictions and vaccination in every cell according to health authority decisions. We present numerical results testing the spread of epidemics without any intervention, applying travel restrictions only, implementing vaccination only, and combining the two strategies.
