Mathematical models are used to determine if infection wave fronts could occur by traveling geographically in a loop around a region or continent. These infection wave fronts arise by Hopf bifurcation for some spatial models for infectious disease transmission with distributed-contacts. Periodic traveling waves are shown to exist for the spatial analog of the SIRS endemic model, in which the temporary immunity is described by a delay, but they do not exist in a similar spatial SIRS endemic model without a delay. Specifically, we found that the ratio of the delay ω in the recovered class and the average infectious period1/γ must be sufficiently large for Hopf bifurcation to occur.
& Hethcote, H. W.
(2009). Periodic Traveling Waves in SIRS Endemic Models. Mathematical and Computer Modelling, 49 (1-2), 393-401.