Vaccination strategy on a geographic network

We considered a simple model describing the propagation of an epidemic on a geographical network. The initial rate of growth of the epidemic is the maximal eigenvalue of a matrix formed by the susceptibles and the graph Laplacian. Assuming the vaccination reduces the susceptibles, we define different vaccination strategies: uniform, local, or following a given vector. Using perturbation theory and the special form of the graph Laplacian, we show that it is most efficient to vaccinate along with the eigenvector corresponding to the largest eigenvalue of the Laplacian. This result is illustrated on a 7 vertex graph, a grid, and a realistic example of the french rail network.