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Permanence of a delayed SIR epidemic model with density dependent birth rate. (English) Zbl 1117.34310
The authors consider the following SIR epidemic model with time delay \[ \frac{dS(t)}{dt}=-\beta S(t)I(t-h)-\mu _1S(t)+b\left( 1-\beta _1\frac{N(t)}{ 1+N(t)}\right), \]
\[ \frac{dI(t)}{dt}=\beta S(t)I(t-h)-\mu _2I(t)-\lambda I(t),\quad \frac{dR(t)}{dt}=\lambda I(t)-\mu _3R(t), \] where \(S(t),\) \(I(t)\) and \(R(t)\) denote the numbers of susceptible members to the disease, of infective members and of members who have been removed from the possibility of infection through full immunity, respectively. Some permanence results are established when the positive equilibrium exists.

MSC:
34K25 Asymptotic theory of functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
92D30 Epidemiology
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