  In this work we analyze mathematically the consequences and effectiveness of
strategies to control an epidemic in the framework of classical SEIR models
with multiple parallel infectious stages. We define the mathematical concept of
a control strategy, showing that it implies turning classic epidemiological
models into systems of non-autonomous differential equations. The analysis of
these non-autonomous systems is based on the two main results obtained in this
work: the first establishes a condition that implies a dynamic without epidemic
outbreaks; the second establishes a maximum value for the susceptible
population associated to the fixed points that are attractors, moreover, we
proof that any trajectory converges to some of these attractors. An important
consequence of this last result is the existence of an insurmountable limit on
the number of infected individuals after the end of a given control strategy.
This restriction can only be mitigated by changing the maximum value of
susceptible population associated to the system attractors, which could only be
done with permanent control action, that is, without returning to normality.
Another interesting result of our work is to show how the moment to start and
the way how the control strategy ends strongly impacts the asymptotic value for
the total number of infected individuals. We illustrate our analysis and
results in a SEIR model (with two or three parallel stages) applied to describe
the COVID-19 epidemic.
