An SIRS model with hospitalizations: economic impact by disease severity

We introduce a two-timescale SIRS-type model in which a fraction $\theta$ of infected individuals experiences a severe course of the disease, requiring hospitalization. During hospitalization, these individuals do not contribute to further infections. We analyze the model's equilibria, perform a bifurcation analysis, and explore its two-timescale nature (using techniques from Geometric Singular Perturbation Theory). Our main result provides an explicit expression for the value of $\theta$ that maximizes the total number of hospitalized individuals for long times, revealing that this fraction can be lower than 1. This highlights the interesting effect that a severe disease, by necessitating widespread hospitalization, can indirectly suppress contagions and, consequently, reduce hospitalizations. Numerical simulations illustrate the growth in the number of hospitalizations for short times. The model can also be interpreted as a scenario where only a fraction $\theta$ of infected individuals develops symptoms and self-quarantines.