# A Multiple-Strain Susceptible-Infected Model with Diffusion Formulated on the Space of Radon Measures

We formulate a multiple strain susceptible-infected model with diffusion on the space of Radon measures which has the advantage of unifying discrete and continuous strain spaces under one framework. We first establish the well-posedness of this model. Then we study the long-time behavior for the case of discrete strain spaces. We define the basic reproduction number for each strain. We establish the existence of a disease-free equilibrium and a strain-specific endemic equilibrium including a competitive exclusion equilibrium where the density of individuals at one strain is positive and the density at the remaining n-1 is zero and a coexistence equilibrium where the density of individuals at more than one strain is positive. We study the global asymptotic stability of these equilibria. In particular, for the endemic equilibrium, we establish conditions guaranteeing its global asymptotic stability under the assumption that the diffusion rate of the susceptible individuals is equal to the diffusion rate of the infected individuals. We then extend some of the long-time behavior results from the discrete strain space case to the continuous strain space case.