Vaccination strategy on a geographic network

Abstract

We considered a mathematical model describing the propagation of an epidemic on a geographical network. The initial growth rate of the disease is 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 the vaccination reduces the susceptibles. When the mobility is small compared to the local disease dynamics, it is best to vaccinate the vertex of least degree and not vaccinate neighboring vertices. Then the epidemic grows on the vertex corresponding to the largest eigenvalue. When the mobility is comparable to the local disease dynamics, the most efficient strategy is to vaccinate the whole network because the disease grows uniformly. However, 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 the eigenvector corresponding to the largest eigenvalue of the Laplacian. We illustrate these general results on a 7 vertex graph, a grid, and a realistic example of the french rail network