The risk matrix of vector-borne diseases in metapopulation networks and its relation with local and global R0

The basic reproduction number R0 is an index worldwide commonly used by public health organizations as a key estimator of the severity of a given epidemic. In this work we use a Lagrangian approach to model vector-borne diseases (SIR-SI) into a metapopulation network in order to derive an expression of the basic reproduction number and we analyze its dependency on human mobility. We prove that this index can be computed by evaluating the spectral radius of the risk matrix W, whose entries Wij are the number of secondary cases in patch j produced by the inclusion of a single infected human in patch i. Based on the risk matrix, we propose a risk index which locally describes the epidemic vulnerability, while R0 give us an estimation of the global vulnerability. Further, we numerically analyze the effect of human mobility over the values of R0 in a system composed of two and three patches, and for a network connected in a star topology configuration.