Compartmental Models Driven by Renewal Processes: Survival Analysis and Applications to SVIS Epidemic Models

Abstract

Compartmental models with exponentially distributed lifetime stages assume a constant hazard rate, limiting their scope. This study develops a theoretical framework for systems with general lifetime distributions, modeled as transition rates in a renewal process. Applications are provided for the SVIS (Susceptible-Vaccinated-Infected-Susceptible) disease epidemic model to investigate the impacts of hazard rate functions (HRFs) on disease control. The novel SVIS model is formulated as a non-autonomous nonlinear system (NANLS) of ordinary differential equations (ODEs), with coefficients defined by HRFs. Key statistical properties and the basic reproduction number () are derived, and conditions for the system’s asymptotic autonomy are established for specific lifetime distributions. Four HRF behaviors—monotonic, bathtub, reverse bathtub, and constant—are analyzed to determine conditions for disease eradication and the asymptotic population under these scenarios. Sensitivity analysis examines how HRF behaviors shape system trajectories. Numerical simulations illustrate the influence of diverse lifetime models on vaccine efficacy and immunity, offering insights for effective disease management.