Dynamics of An SIR Model with Isolation and Pulse Vaccinated Susceptible Population for Delay Acquiring Immunity

Abstract

Vaccination is one of the most cost-effective and simplest interventions for protecting against infectious-disease epidemics. The COVID-19 (coronavirus disease 2019) outbreak in China demonstrated that physical protection and social isolation are critical to controlling an epidemic in the absence of vaccines or antiviral drugs. Considering isolation and pulse vaccination of the susceptible population subject to a delay in acquiring immunity, we propose a susceptible-infectious-recovered (SIR) model incorporating these features. Using the theory of impulsive differential equations and comparison theorems, we prove that the infection-free periodic solution $(\widetilde{S(t)},0)$ of system $(3.1)$ is globally asymptotically stable if $H<0$, and that the disease tends to become endemic if $H>0$. The expression of $H$ and the numerical simulations indicate that isolation and the delay in acquiring immunity of the vaccinated susceptible population play important roles in achieving the infection-free state, and that the maximum enrolling amount from the exterior region also affects disease elimination.