Abstract
In this paper, we are interested in finding the periodic oscillation of seasonally forced SEIR models with pulse vaccination. Many infectious diseases show seasonal patterns of incidence. Pulse vaccination strategy is an effective tool to control the spread of these infectious diseases. Assuming that the seasonally dependent transmission rate is a T-periodic forcing, we obtain the existence of positive T-periodic solutions of seasonally forced SEIR models with pulse vaccination by Mawhin’s coincidence degree method. Some relevant numerical simulations are presented to illustrate the effectiveness of such pulse vaccination strategy.
Highlights
It is a common phenomenon that the incidence of many infectious diseases often changes periodically with the seasonal cycle, such as measles, chickenpox, mumps, rubella, pertussis, and influenza [1,2,3]
Pulse vaccination strategy (PVS) is an effective tool to control the spread of epidemics, for example, control of poliomyelitis and measles in Central [11] and South American [12] and the UK vaccination campaigns against measles in 1994 [13]. e theoretical study on pulse vaccination strategy was firstly presented by Agur et al [14]
We focus on the existence of periodic solution of seasonally forced SEIR models with pulse vaccination; we consider models of the form dS(t) dt dE(t) dt dI(t)
Summary
It is a common phenomenon that the incidence of many infectious diseases often changes periodically with the seasonal cycle, such as measles, chickenpox, mumps, rubella, pertussis, and influenza [1,2,3]. D’Onofrio applied the pulse vaccination method for SIR and SEIR epidemic models [18, 19]. Using Leray–Schauder degree theory, Zu and the author [28] established new results on the existence of at least one positive periodic solution for a seasonally forced SIR model with impact of media coverage. E author [29] proved the existence of positive periodic solutions of seasonally forced SIR models with impulse vaccination at fixed time by Mawhin’s coincidence degree method if the basic reproductive number. Coincidence degree theory has been applied to prove the existence of multiple periodic solutions of the epidemic model with seasonal periodic rate [30, 31]. E aim of this paper is to study the existence of periodic solution of seasonally forced SEIR models with pulse vaccination.
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