Disease-free equilibrium of the susceptible-infective model with exponential birth

For

b > d

, the disease-free equilibrium is

(S, I) = (\frac{1}{a} lo g \frac{b}{d}, 0) .

It is linearly unstable precisely when

\frac{1}{a} lo g \frac{b}{d} > \frac{d + δ}{β} .

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