Bifurcation and Sensitivity Analysis of Immunity Duration in an Epidemic Model

Most disease transmission models assume no immunity or permanent immunity for simplicity, however, hosts have temporary immunity for most diseases. In this note we find that the immunity duration is actually the most sensitive parameter for dynamics of disease transmission. We provide numerical schemes to sketch Hopf bifurcations (forward or backward), sensitivity surfaces, periodicity diagrams (one dimensional or two dimensional parameter space) for a disease transmission model with immunity delay. The methods introduced here can be easily modified for a specific disease transmission model. We also test how different incidence functions change dynamics via bifurcation diagrams.