An SIS epidemic model with variable population size and a delay

The SIS epidemiological model has births, natural deaths, disease-related deaths and a delay corresponding to the infectious period. The thresholds for persistence, equilibria and stability are determined. The persistence of the disease combined with the disease-related deaths can cause the population size to decrease to zero, to remain finite, or to grow exponentially with a smaller growth rate constant. For some parameter values, the endemic infective-fraction equilibrium is asymptotically stable, but for other parameter values, it is unstable and a surrounding periodic solution appears by Hopf bifurcation.