A delayed SEIQR epidemic model with pulse vaccination and the quarantine measure

Abstract

A delayed SEIQR epidemic model with pulse vaccination and the quarantine measure is investigated. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact periodic disease-free solution. Using the comparison method, we prove that the disease-free periodic solution is globally attractive when the basic reproductive number (ℜ∗) is less than unity, and that the disease is permanent when another basic reproductive number (ℜ∗) is greater than unity. In other words, the disease will be extinct if the pulse vaccination rate is larger than a critical value θ∗ and the disease will be uniformly persistent if the vaccination rate is less than another critical value θ∗. Our results indicate that a longer latent period of the disease or a larger pulse vaccination rate will lead to the eradication of the disease, and whether the disease will be extinct or not is independent of the removal rate from the quarantined group. Furthermore, a larger fraction of susceptibles should be vaccinated against the disease unless the quarantine measure is taken. Finally, we find that the number of the infected decreases as the quarantine measure is taken. We carry out numerical simulations to verify our results.