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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:

Note that the variables give the fraction of individuals - that is, we have normalized them so that
\begin{displaymath}
S+E+I+R = 1.
\end{displaymath} (9)

Furthermore, suppose that This leads us to consider the following model:
$\displaystyle \frac{dS}{dt}$ $\textstyle =$ $\displaystyle \mu - \beta(t) S I - \mu S$ (10)
$\displaystyle \frac{dE}{dt}$ $\textstyle =$ $\displaystyle \beta(t) S I - (\mu + \alpha) E$ (11)
$\displaystyle \frac{dI}{dt}$ $\textstyle =$ $\displaystyle \alpha E - (\mu + \gamma) I.$ (12)

The variable $R$ is determined from the other variables according to equation (9). When $\beta = \beta_0 = {\rm constant}$, this is a three-dimensional autonomous system of ordinary differential equations, and is well understood. Defining
\begin{displaymath}
R_0 = \frac{\beta_0 \alpha}{(\mu + \alpha) (\mu + \gamma)},
\end{displaymath} (13)

it can be shown that for $R_0>1$ the model has a fixed point with $I=0$ which is unstable, and a fixed point with $I>0$ which is stable, etc. See Aron and Schwartz for more detail and references.

If $\beta$ 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 up previous
Next: An SEIR model with Up: APC591 Tutorial 6: Seasonal Previous: An SIS model with
Jeffrey M. Moehlis 2001-11-07