Dynamics of an SEIR Epidemic Model with Vaccination, Treatment and Saturated Incidence: Threshold Analysis and Numerical Study
TL;DR
A deterministic SEIR model in which susceptible people are vaccinated at a constant rate, infectious people are treated at a constant per-capita rate, and the force of infection saturates at high prevalence to reflect behavioural change and limited contact is studied.
Abstract
Vaccination and treatment are the two main tools used to bring an outbreak under control, yet they act at different stages of the infection and are rarely compared inside a single, simple mathematical framework. In this paper we study a deterministic SEIR model in which susceptible people are vaccinated at a constant rate, infectious people are treated at a constant per-capita rate, and the force of infection saturates at high prevalence to reflect behavioural change and limited contact. We derive the control reproduction number Rc with the next-generation method and show that the disease-free equilibrium is globally asymptotically stable when Rc < 1, while a unique endemic equilibrium exists, with an explicit closed form, whenever Rc > 1. The model therefore has no backward bifurcation. A normalised sensitivity analysis ranks the transmission rate, vaccination rate and recovery-related rates as the most influential parameters. Numerical simulations with illustrative parameters show that the combined strategy lowers the epidemic peak by about 85% compared with no intervention, and that treatment is the faster-acting measure while vaccination is the more effective long-run one. The results give a compact, analytically tractable basis for comparing intervention mixes