An impulsive delayed SEIRS epidemic model with saturation incidence

Abstract

A delayed SEIRS epidemic model with pulse vaccination and saturation incidence rate is investigated. Using Krasnoselskii's fixed-point theorem, we obtain the existence of infection-free periodic solution of the impulsive delayed epidemic system. We define some new threshold values R 1 , R 2 and R 3 . Further, using the comparison theorem, we obtain the explicit formulae of R 1 and R 2 . Under the condition R 1 < 1, the infection-free periodic solution is globally attractive, and that R 2 > 1 implies that the disease is permanent. Theoretical results show that the disease will be extinct if the vaccination rate is larger than θ * and the disease is uniformly persistent if the vaccination rate is less than θ * . Our results indicate that a long latent period of the disease or a large pulse vaccination rate will lead to eradication of the disease. Moreover, we prove that the disease will be permanent as R 3 > 1.

A delayed epidemic model of diseases through droplet infection and direct contact with saturation incidence and pulse vaccination

“…Stone, Shulgin, and Agur (2000) presented a theoretical examination of the PVS in the SIR epidemic model and d 'Onofrio (2002a'Onofrio ( , 2002b analysed the use of pulse vaccination policy to eradicate infectious disease for SIR and SEIR epidemic models. Different types of vaccination policies and strategies combining pulse vaccination policy, treatment, pre-outbreak vaccination or isolation have already been introduced by several researchers (Babiuk, Babiuk, & Baca-Estrada, 2002;d'Onofrio, 2005;Gao, Chen, Nieto, & Torres, 2006;Gao, Chen, & Teng, 2007;Gjorrgjieva et al, 2005;Tang, Xiao, & Clancy, 2005;Wei & Chen, 2008;Zhang & Teng, 2008).…”