  We formulate a general SEIR epidemic model in a heterogenous population
characterized by some trait in a discrete or continuous subset of a space R d.
The incubation and recovery rates governing the evolution of each homogenous
subpopulation depend upon this trait, and no restriction is assumed on the
contact matrix that defines the probability for an individual of a given trait
to be infected by an individual with another trait. Our goal is to derive and
study the final size equation fulfilled by the limit distribution of the
population. We show that this limit exists and satisfies the final size
equation. The main contribution is to prove the uniqueness of this solution
among the distributions smaller than the initial condition. We also establish
that the dominant eigenvalue of the next-generation operator (whose initial
value is equal to the basic reproduction number) decreases along every
trajectory until a limit smaller than 1. The results are shown to remain valid
in presence of diffusion term. They generalize previous works corresponding to
finite number of traits (including metapopulation models) or to rank 1 contact
matrix (modeling e.g. susceptibility or infectivity presenting heterogeneity
independently of one another).
