Compartmental models with underlying renewal process: survival analysis and applications to SVIS epidemic models

Abstract Compartmental models assuming exponentially distributed lifetime stages are limited because of a constant hazard rate. Here, a theoretical compartmental model for a system with general lifetime distributions is studied. The model represents the transition rates between jumps of a renewal process in the system. Applications are given for the SVIS disease epidemic model, to investigate the impacts of the hazard rate functions (HRFs) on disease control. The new SVIS model is a non-autonomous nonlinear system (NANLS) of ordinary differential equations (ODEs), with coefficients that are HRFs. Moreover, for a class of lifetime distributions, the NANLS of ODEs is asymptotically autonomous. Four asymptotic behaviors of the HRFs: a monotonic, a bathtub, a reverse bathtub and a constant shape are explored to determine the asymptotic population for disease eradication. Also, analysis is conducted to determine the sensitivity of the epidemic system to the hazard rate behaviors over time. Numerical simulation results are given for different lifetime models representing hazard behaviors for vaccine efficacy and immunity. 2000 MSC: 92B15, 62N05, 60E05, 92D25