Diffusion - Epidemiology PDE Model

[retrofit81]

Homework Statement

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.

Homework 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).

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

 

[mighty2000]

Dear Michael,

Thank you for your inquiry. I am a scientist specializing in epidemiology and I would be happy to assist you with your question.

Based on the information you have provided, it is possible to find an explicit traveling wave solution to the simple epidemiology diffusion model you have described. This type of solution is known as a traveling wave solution because it describes the spread of a disease over time and space, much like a wave moving through a medium.

To find this solution, you can use the method of characteristics, which involves rewriting the system of equations in terms of a new variable, called the characteristic variable. This variable is defined as x - ct, where c is the speed of the wave. By substituting this variable into the equations, you can reduce the system to a single partial differential equation, which can then be solved using standard techniques.

I would recommend starting by looking at the characteristics of the system, which can be found by setting the right-hand sides of the equations equal to zero. This will give you two equations, one for the characteristic variable and one for the speed of the wave. By solving these equations, you can determine the form of the traveling wave solution.

I hope this helps. Let me know if you have any further questions or need any clarification.