We'll now consider the epidemic model from ``Seasonality and
period-doubling bifurcations in an epidemic model'' by J.L. Aron and
I.B. Schwartz, *J. Theor. Biol.* **110**:665-679, 1984 in which
the population consists of four groups:

- is the fraction of susceptible individuals (those able to contract the disease),
- is the fraction of exposed individuals (those who have been infected but are not yet infectious),
- is the fraction of infective individuals (those capable of transmitting the disease),
- is the fraction of recovered individuals (those who have become immune).

Furthermore, suppose that

- There are equal birth and death rates ,
- is the mean latent period for the disease,
- is the mean infectious period,
- recovered individuals are permanently immune,
- the contact rate may be a function of time.

(10) | |||

(11) | |||

(12) |

The variable is determined from the other variables according to equation (9). When , this is a three-dimensional autonomous system of ordinary differential equations, and is well understood. Defining

(13) |

If depends on time, we have a three-dimensional nonautonomous system, which can be converted to a four-dimensional autonomous system as was done above for the SIS model.