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.... More >

Releasing amount of natural enemy generally depends on its population data, while impulsive releasing natural enemy usually brings about a time delay after the data is observed in the practical pest management. Therefore, it is very important for pest managers to assess the impact of the time delay in pest management. In this paper, we construct a pest-natural enemy model with time delay in impulsive releasing natural enemy. We prove that the pest-free periodic solution of model $(2.1)$ is globally attractive with $\eta \tau<\frac{\zeta y^{\ast}(1-e^{-\lambda \tau})}{\lambda}$. We also prove that model $(2.1)$ is permanent with $\eta \tau>\frac{\zeta y^{\ast}(1-e^{-\lambda \tau})}{\lambda}$.... More >