## Preprint 2013-22

## Veronica Anaya, Mostafa Bendahmane, Michael Langlais, Mauricio Sepúlveda:

### A convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross-diffusion

### Abstract:

In this paper, we are concerned with a finite volume method for a model with cross-diffusion of the indirect transmission of an epidemic disease between two spatially distributed host populations having non-coincident spatial domains, transmission occurring through a contaminated environment. The mobility of each class is assumed to be influenced by the gradient of the other classes. We propose a Finite Volume scheme and proved the well-posedness, nonnegativity and convergence of the discrete solution. The convergence proof is based on deriving a series of a priori estimates and by using a general Lp compactness criterion. Additionally, we address the questions of existence of weak solutions and existence and uniqueness of classical solution by using, respectively, a regularization method and an interpolation result between Banach spaces. The proofs of these results are given in the Appendix. Finally, the numerical scheme is illustrated by some examples.

This preprint gave rise to the following definitive publication(s):

**Veronica ANAYA, Mostafa BENDAHMANE, Michael LANGLAIS, Mauricio SEPúLVEDA: ***A convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross-diffusion*. Computers & Mathematics with Applications, vol. 70, 2, pp. 132-157, (2015).