Integral equation modeling of an SEIQHR epidemic system with non-exponentially distributed disease stages

Many infectious diseases, such as COVID-19 and tuberculosis, are characterized by latent infectious stages, substantial transmission potential, and a non-negligible risk of severe outcomes, which necessitates integrated control strategies including early detection, quarantine, and hospitalization. Compared with exponential distributions, non-exponential distributions provide a more realistic description of disease transmission processes. In this paper, we incorporate general (non-exponential) distributions to describe stage durations associated with infection, recovery, quarantine, and hospital isolation, including Gamma distributions as a special case. Under suitable assumptions, we formulate an SEIQHR epidemic model as a system of integral equations, incorporating natural mortality, disease-induced mortality, and hospital-related mortality. We establish the existence and uniqueness of solutions. An explicit expression of the control reproduction number is derived and used as a threshold quantity for disease control. If the control reproduction number is less than one, the disease-free equilibrium is globally asymptotically stable, otherwise, the endemic equilibrium is globally asymptotically stable. Under the assumption of Gamma-distributed stage durations, we derive an equivalent system of ODEs. Numerical simulations indicate that the shape parameters of the Gamma distributions can significantly affect the control reproduction number.