Abstract
We first propose a new epidemic disease model governed by system of impulsive delay differential equations. Then, based on theories for impulsive delay differential equations, we skillfully solve the difficulty in analyzing the global dynamical behavior of the model with pulse vaccination and impulsive population input effects at two different periodic moments. We prove the existence and global attractivity of the “infection-free” periodic solution and also the permanence of the model. We then carry out numerical simulations to illustrate our theoretical results, showing us that time delay, pulse vaccination, and pulse population input can exert a significant influence on the dynamics of the system which confirms the availability of pulse vaccination strategy for the practical epidemic prevention. Moreover, it is worth pointing out that we obtained an epidemic control strategy for controlling the number of population input.
Highlights
IntroductionSusceptible-infectious-recovered type of models is well known [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] such models very often ignore the incubation period in the development of mathematical models for some diseases
We can prove that it is the unique globally asymptotically stable positive periodic solution of system (4). We summarize this conclusion in the following lemma
We claim that there exists an m2 > 0 such that I(t) > m2 for t is large enough. We prove this claim in two steps
Summary
Susceptible-infectious-recovered type of models is well known [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] such models very often ignore the incubation period in the development of mathematical models for some diseases. It is of great significance to investigate epidemic models with time delay and impulsive effects due to the incubation period and vaccination period [26,27,28,29]. Motivated by Jiang et al [30] and Song et al [19], we built a new mathematical model: susceptible, vaccinated, exposed, Abstract and Applied Analysis infectious, recovered, and susceptible epidemic model with two time delays and two nonlinear incidences with pulse vaccination and a constant periodic population input at two different moments as follows: dS (t) = −bS (t) − βSp (t) I (t) + γI (t − ω) e−bω, dt dV (t) dt. Terms βSpI and VqI are the nonlinear incidence rates, and in our paper we only discuss the case 1 ≤ q ≤ p
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