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An SEIR model

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).
Note that the variables give the fraction of individuals - that is, we have normalized them so that
 (9)

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.
This leads us to consider the following model:
 (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)

it can be shown that for the model has a fixed point with which is unstable, and a fixed point with which is stable, etc. See Aron and Schwartz for more detail and references.

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.

Next: An SEIR model with Up: APC/EEB/MOL 514 Tutorial 4: Previous: An SIS model with
Jeffrey M. Moehlis 2002-10-14