Delay equation formulation for an epidemic model with waning immunity: an application to mycoplasma pneumoniae

We formulate a general epidemic model with two arbitrary probability distributions for describing durations of infectivity and immunity. The model is given as a coupled system of a delay differential equation and a renewal equation for two dynamical variables: susceptible population and the force of infection. It is shown that there exists a unique endemic equilibrium if the basic reproduction number is greater than one. Assuming that a fixed duration of immunity we show that the endemic equilibrium becomes unstable via Hopf bifurcation. We briefly discuss that periodic outbreak of mycoplasma pneumoniae may be interpreted with the result of instability of the endemic equilibrium.