Threshold dynamics in a periodic epidemic model with imperfect quarantine, isolation and vaccination

Abstract

A nonautonomous mathematical model was presented to explore the complex dynamics of disease spread over time, incorporating a time-periodic transmission parameter and imperfections in quarantine, isolation and vaccination strategies. Through a detailed examination of threshold dynamics, it was revealed that the global dynamics of disease transmission are influenced by the basic reproduction number ($ \mathcal{R}_0 $), a critical threshold that determines extinction, persistence, and the presence of periodic solutions. It was shown that the disease-free equilibrium is globally asymptotically stable if $ \mathcal{R}_0 < 1 $, while the disease persists if $ \mathcal{R}_0 > 1 $. To support and validate our analytical results, the basic reproduction number and the dynamics of the disease were estimated by fitting monthly data from two Asian countries, namely Saudi Arabia and Pakistan. Furthermore, a sensitivity analysis of the time-averaged reproduction number ($ \langle \mathcal{R}_0 \rangle $) of the associated time-varying model showed a significant sensitivity to key parameters such as infection rates, quarantine rate, vaccine coverage rate, and recovery rates, supported by numerical simulations. These simulations validated theoretical findings and explored the impact of seasonal contact rate, imperfect quarantine, isolation, imperfect vaccination, and other parameters on the dynamics of measles transmission. The results showed that increasing the rate of immunization, improving vaccine management, and raising public awareness can reduce the incidence of the epidemic. The study highlighted the importance of understanding these patterns to prevent future periodic epidemics.

Keywords

• compartmental model
• time-periodic transmission rates
• basic reproduction number ($ \mathcal{r}_0 $)
• global dynamics
• periodic solutions
• sensitivity analysis