Can pathogen spread keep pace with its host invasion?

We consider the Fisher-KPP equation in a wavelike shifting environment for which the wave profile of the environment is given by a monotonically decreasing function changing signs (shifting from favorable to unfavorable environment). This type of equation arises naturally from the consideration of pathogen spread in a classical susceptible-infected-susceptible epidemiological model of a host population where the disease impact on host mobility and mortality is negligible. We conclude that there are three different ranges of the disease transmission rate where the disease spread has distinguished spatiotemporal patterns: extinction; spread in pace with the host invasion; spread not in a wave format and slower than the host invasion. We calculate the disease propagation speed when disease does spread. Our analysis for a related elliptic operator provides closed form expressions for two generalized eigenvalues in an unbounded domain. The obtained closed forms yield unsolvability of the related elliptic equation in the critical case, which relates to the open problem 4.6 in.