Compartmental Models in Epidemiology - The SEIR Model

The SEIR Model

For many important infections there is a significant period of time during which the individual has been infected but is not yet infectious themselves. 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 ), and also assuming the presence of vital dynamics with birth rate equal to death rate, we have the model:

Of course, we have that .

For this model, the basic reproduction number is:

Similarly to the SIR model, also in this case we have a Disease-Free-Equilibrium (N,0,0,0) and an Endemic Equilibrium EE, and one can show that, independently form biologically meaningful initial conditions

it holds that:

In case of periodically varying contact rate the condition for the global attractiveness of DFE is that the following linear system with periodic coefficients:

is stable (i.e. it has its Floquet's eigenvalues inside the unit circle in the complex plane).