Abstract
An SIR epidemic model with saturated treatment function and nonlinear pulse vaccination is studied. The existence and stability of the disease-free periodic solution are investigated. The sufficient conditions for the persistence of the disease are obtained. The existence of the transcritical and flip bifurcations is considered by means of the bifurcation theory. The stability of epidemic periodic solutions is discussed. Furthermore, some numerical simulations are given to illustrate our results.
Highlights
The SIR epidemic models have attracted much attention in recent years
Impulsive differential equations [6, 7] are suitable for the mathematical simulation of evolutionary processes in which the parameters state variables undergo relatively long periods of smooth variation followed by a short-term rapid change in their values
Many results have been obtained for SIR epidemic models described by impulsive differential equations [8,9,10,11,12,13]
Summary
The SIR epidemic models have attracted much attention in recent years. In most cases, ordinary differential equations are used to build SIR epidemic models [1,2,3,4,5]. Where α ≥ 0 represents the maximal medical resources supplied per unit time and ω > 0 is half-saturation constant, Discrete Dynamics in Nature and Society which measures the efficiency of the medical resource supply in the sense that if ω is smaller, the efficiency is higher They investigated the following SIR model: S. Θ is the half-saturation constant, that is, the number of susceptible individuals when the vaccination rate is half the largest vaccination rate They established the following SIR epidemic model: S (t) = aA − βSI − dS, 1 + kI σ) I, R (t) = (1 − a) A + γI − dR, t ≠ nT,. Motivated by [16, 18], the following SIR epidemic model with saturated treatment function and nonlinear pulse vaccination is considered: dR, t ≠ nT, (7).
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