Beyond R0: How Vaccination and Post-Infection Mortality Drive Epidemic Persistence and Oscillations.

The long-term consequences of infectious diseases, particularly post-infection mortality (PIM), represent a significant yet understudied aspect of epidemiology. Furthermore, the interaction between PIM and vaccination, a key public health intervention, remains poorly understood from a dynamical systems perspective. In this paper, we develop a six-dimensional compartmental model that incorporates both PIM and vaccination. Analytical results show that the model admits a rich bifurcation structure, including a backward bifurcation where a stable endemic equilibrium coexists with a stable disease-free state even when the basic reproduction number R0<1. We further prove the existence of Hopf bifurcations, demonstrating the potential for sustained epidemic oscillations driven by PIM. Numerical continuation in the two-dimensional parameter space of vaccination and PIM rates reveals organizing centers, specifically Bogdanov-Takens and cusp points, that govern transitions among these dynamical regimes. Our central finding is that the interplay between vaccination and PIM can undermine control strategies based solely on reducing R0 below unity. This highlights the importance of incorporating post-infection health outcomes and their interaction with vaccination into epidemiological models to design robust and effective disease-control policies.