Analysis of an SVIR Epidemic Model with Vaccination and Nonlinear Incidence

Abstract

This paper studies an SVIR epidemic model that incorporates vaccination and a nonlinear incidence term to represent behavioral responses and saturation in disease transmission.
The population was divided into four classes—susceptible, vaccinated, infectious, and recovered states —and both vaccine‑induced and naturally acquired immunity were allowed to wane, so that individuals could return to the susceptible class. The original model is an SIR model with nonmonotone incidence. Then we added a vaccinated compartment and assumed imperfect vaccination, meaning that vaccinated individuals may still become infected at a reduced rate.
We derived the disease‑free and endemic equilibria and computed the basic reproduction number  by the next‑generation method, obtaining a threshold that separated disease elimination from persistence. Stabilities of the equilibria were investigated under appropriate parameter conditions using Jacobian and Routh–Hurwitz criteria. Numerical simulations then illustrated how changes in the vaccination rate and other key parameters shaped the long‑term dynamics, including convergence to a disease‑free equilibrium or approach to an endemic equilibrium with a nonzero infectious population. The findings indicated that maintaining vaccination coverage above a critical level was essential to keep   and prevented sustained endemic transmission in the presence of nonlinear incidence and waning immunity.