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Reproduction matrix for an epidemic and lockdowns in a city

We consider an epidemic spreading in a city which is divided geographically into different districts. We introduce the reproduction matrix $\R=\bigl(R(i,j)\bigr)$ between districts, where $R(i,j)$ is the mean number of individuals in district $j$ infected by an individual from district $i$. We analyse policies of partial lockdowns of the city, that is of a set of districts, based on the study of matrix $\R$, where rows and columns corresponding to districts in lockdown are set to zero. This schema can also be applied to a country divided into regions or other appropriate units, provided the relevant information is available. We conclude by analyzing a matrix~$\R$ which was constructed for the spread of COVID-19 in Santiago, Chile, with the aid of an agent-based simulator for generating surrogate district data.