# Epidemiology: The SEIR model

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For many important infections there is a significant period of time during which the individual has been infected but is not yet infectious himself. During this latent period the individual is in compartment E (for exposed).

Assuming that the period of staying in the latent state is a random variable with exponential distribution with parameter a (i.e. the average latent period is $\displaystyle{ a^{-1} }$), and also assuming the presence of vital dynamics with birth rate equal to death rate, we have the model:

$\displaystyle{ \frac{dS}{dt} = \mu N - \mu S - \beta \frac{I}{N} S }$
$\displaystyle{ \frac{dE}{dt} = \beta \frac{I}{N} S - (\mu +a ) E }$
$\displaystyle{ \frac{dI}{dt} = a E - (\gamma +\mu ) I }$
$\displaystyle{ \frac{dR}{dt} = \gamma I - \mu R. }$

Of course, we have that $\displaystyle{ S+E+I+R=N }$.

The lines in the JSXGraph-simulation below have the following meaning:

* Blue: Rate of susceptible population
* Black: Rate of exposed population
* Red: Rate of infectious population
* Green: Rate of recovered population (which means: immune, isolated or dead)