Stability and Hopf bifurcation in an epidemic model with post-infection mortality

Post-infection mortality (PIM) plays a critical role in shaping infectious disease dynamics, yet its influence remains understudied in epidemiological modeling. A recent study introduced an ODE model incorporating PIM and demonstrated its potential to generate epidemic cycling. In this work, we conduct a rigorous theoretical and numerical analysis of the model and uncover new dynamical behaviors not previously identified. We establish global stability results for both the disease-free equilibrium and the unique endemic equilibrium and prove the existence of a transcritical bifurcation at the disease-free equilibrium and a supercritical Hopf bifurcation at the endemic equilibrium. These findings provide a more complete understanding of the model's stability landscape. Numerical simulations validate our theoretical results and confirm complex dynamical behaviors suggested in previous numerical studies. Our results highlight the fundamental role of PIM in shaping long-term epidemic dynamics and underscore the need for its careful consideration in infectious disease modeling.