Towards Disease Eradication: Long-Term Control with Constant Vaccination Rates in the Normalized SIR Model

Abstract

In this paper, we investigate a normalized SIR model incorporating exponential natural birth and death rates, as well as disease-induced mortality and a constant vaccination control parameter denoted as u. This entails vaccinating a fixed percentage of susceptibles in each campaign, a pragmatic approach considering that available economic and human resources often correlate with population size at any given time. Then, we identify a bifurcation value, u bv , determined by other parameters and show that, for u in the interval [0, u bv ), the system converges to a steady state with a positive proportion of infected individuals, while for u in (u bv , 1], this proportion approaches zero asymptotically. Notably, the threshold u bv serves as the minimum percentage of the population that should be vaccinated in each campaign to effectively pursue disease eradication. Additionally, we explore the cost implications of a two-phase control strategy. Initially, we employ optimal control techniques to expedite the system's transition to a state where the infected population proportion stabilizes. Subsequently, we implement a constant-rate vaccination policy to drive the proportion of infected individuals to zero. Our analysis utilizes generic parameters, as we do not focus on a specific disease or population.