Epidemic on a geographic network: analysis and vaccination strategy

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Speakers: Jean-Guy Caputo Laboratoire de Mathematiques, INSA de Rouen Normandie, France

Title: Epidemic on a geographic network: analysis and vaccination strategy

Abstract: We analyze an epidemic model on a network consisting of susceptible-infected-recovered equations at the nodes coupled by diffusion using a graph Laplacian. The model has few parameters enabling fitting to real data. It also has the essential ingredient of importation of infected people. These two features were particularly important for the COVID-19 epidemic. Using a complete graph describing the main airports on the planet, the model allowed us to predict the arrival of the COVID epidemic in Mexico in March 2020. It was also useful to evaluate deconfinement scenarios and prevent a so-called second wave.

The size of the outbreak is governed by the initial growth rate of the disease given by the maximal eigenvalue of the epidemic matrix formed by the susceptibles and the graph Laplacian representing the mobility. We use matrix perturbation theory to analyze the epidemic matrix and define a vaccination strategy, assuming vaccination reduces the susceptibles. When mobility and local disease dynamics have similar time scales, it is most efficient to vaccinate the whole network because the disease grows uniformly. If only a few vertices can be vaccinated, then which ones do we choose? We answer this question, and show that it is most efficient to vaccinate along an eigenvector corresponding to the largest eigenvalue of the Laplacian.

When mobility is slower than local disease dynamics the epidemic grows on the vertex with largest susceptibles. The epidemic growth rate is more reduced when vaccinating a larger degree vertex; it also depends on the neighboring vertices. These results are illustrated by numerical calculations on several examples.