How to prove that there exist a travelling wave solution?

Question

Im currently studying the SIRS infectious disease model for modelling an epidemic. I have expanded the model to take spatial expansion into account, i.e. I have incorperated a diffusive term. The model is as follows $$ S_t(x,t) = -\beta S(x,t)I(x,t) + dS_{xx},\\ I_t(x,t) = \beta S(x,t)I(x,t)-\gamma I + dI_{xx},\\ R_t(x,t) = \gamma I + dR_{xx}, $$ where $\beta$ denotes the transmission rate, $\gamma$ denotes the recovery rate and $d$ denotes the diffusion rate. From studying it numerical, I've noticed that the infected compartment $I$ forms, what looks like, to be a travelling wave through time. But how do I prove that it exist?

Answer 1

Typically when one looks for traveling wave solutions, one imposes an ansatz for the unknown $U$ of the form $U(x,t)=u(x-tc)$ for some unknown velocity vector $c$. Then $\partial_t U(x,t) = -c \cdot \nabla u(x-ct)$, and the time-dependent PDE reduces to a time-idependent PDE with $c$ as a parameter. In your case you would then take $S(x,t) = s(x-tc), I(x,t) = i(x-tc), R(x,t) = r(x-tc)$, etc. The resulting stationary problem for $(s,i,r)$ is then a nonlinear elliptic system.