 Solution

The susceptible equation may be writtenThe factor $b{e}^{-aS}$ 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,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.