# Diffusion - Epidemiology PDE Model

1. Nov 30, 2006

### retrofit81

1. The problem statement, all variables and given/known data

I'm wondering if there is an explicit traveling wave solution to a simple epidemiology diffusion model. This model is a basic representation of rabies spread among organisms. Rabies causes its victims to become delirious; hence the diffusion there.

Here x is the spatial variable and t is the temporal variable. r, a, and D are all positive parameters. S represents susceptible fox not yet infected with rabies and I represents infected fox. S_0 in particular is the initial number of susceptible fox.

2. Relevant equations

The system is

partialS/partialt = -rIS
partialI/partialt = rIS - aI + D*(partial^2 I/partial x^2)

with boundary conditions S(inf)=1, I(inf)=0, S'(-inf)=0, I(-inf)=0 and initial condition (S,I) = (S_0, 0).

3. The attempt at a solution

I have a DVI solution file I wrote showing that a solution exists (I find a heterocline between the two relevant equlibria in the SI-plane). It uses linear stability/nullcline analysis, not terribly complicated. Email me (retrofit81@aol.com) or respond here and I can send it to you.

I'd really just like to know if a closed-form expression exists, and if so, maybe a hint on how I can start finding it...?

Regards and thank you in advance,

Michael