Prove convergence for the original Kermack–McKendrick variables

Prove that the original susceptible and infected variables \(S(t,x)\) and \(I(t,i,x)\) of the nonlocal Kermack–McKendrick model converge, for large times, to a traveling wave with the logarithmic spatial shift identified for the transformed transport problem.

Background

The transport problem studied in the paper is motivated by a nonlocal Kermack–McKendrick epidemic model, and the analysis is carried out primarily for the transformed density $\varrho=\rho/\pi$. The paper establishes a logarithmic delay relative to the minimal traveling wave for this transformed formulation.

The authors suggest transferring the sharp asymptotic analysis back to the original epidemic variables $S(t,x)$ and $I(t,i,x)$. The unresolved issue is to prove their large-time convergence to a traveling wave while retaining the logarithmic shift in the wave position.