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, ubv, determined by other parameters and show that, for u in the interval [0, ubv), the system converges to a steady state with a positive proportion of infected individuals, while for u in (ubv, 1], this proportion approaches zero asymptotically. Notably, the threshold ubv 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.

Keywords

• normalized sir model
• infectious diseases
• epidemiology
• vaccination policies
• steady states
• optimal control