= Solution
The susceptible equation may be written
$$
\dot S
=\underbrace{be^{-aS}S}_{\text{birth of susceptibles}}
-\underbrace{\beta SI}_{\text{infection}}
-\underbrace{dS}_{\text{natural death}}.
$$
The factor $be^{-aS}$ is a density-dependent per-capita birth rate. The <law of mass action> gives the infection incidence $\beta SI$, which transfers individuals out of the susceptible compartment.
Likewise,
$$
\dot I
=\underbrace{\beta SI}_{\text{new infections}}
-\underbrace{dI}_{\text{natural death}}
-\underbrace{\delta I}_{\text{disease-induced death}}.
$$
There is no birth term in the infective equation, so only susceptible individuals reproduce in this model. Infectives die at total per-capita rate $d+\delta$. These terms define the <susceptible-infective model with exponential density-dependent birth>.
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