Permanence of a delayed SIR epidemic model with density dependent birth rate

In this paper, we consider the permanence of a modified delayed SIR epidemic model with density dependent birth rate which is proposed in [M. Song, W. Ma, Asymptotic properties of a revised SIR epidemic model with density dependent birth rate and time delay, Dynamic of Continuous, Discrete and Impulsive Systems, 13 (2006) 199–208]. It is shown that global dynamic property of the modified delayed SIR epidemic model is very similar as that of the model in [W. Ma, Y. Takeuchi, T. Hara, E. Beretta, Permanence of an SIR epidemic model with distributed time delays, Tohoku Math. J. 54 (2002) 581–591; W. Ma, M. Song, Y. Takeuchi, Global stability of an SIR epidemic model with time delay, Appl. Math. Lett. 17 (2004) 1141–1145].