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.