Susceptible-infective model with exponential density-dependent birth

The equationscombine mass-action infection with an exponential density-dependent birth model. Susceptibles reproduce at per-capita rate $b{e}^{-aS}$, while infectives have no reproductive term and die at total per-capita rate $d+\delta$.

For $b>d$, the disease-free equilibrium isIt is linearly unstable precisely when

When the disease-free equilibrium is unstable, the positive equilibrium satisfiesIts Jacobian has negative trace and positive determinant, so the equilibrium is locally asymptotically stable.