A Vaccination Strategy Based on a State Feedback Control Law for Linearizing SEIR Epidemic Models - Biomedical Engineering Systems and Technologies - page 174

taking into account the inhibition effect from the behavioral change of susceptible
individuals when the infectious individual number increases [12].
The analysis of the existence of equilibrium points, relative to either the persis-
tence or extinction of the epidemics, the conditions for the existence of a backward
bifurcation where both equilibrium points co-exist and the constraints for guarantee-
ing the positivity and the boundedness of the solutions of such models have been
some of the main objectives in the aforementioned papers. Also, the conditions that
generate an oscillatory behavior in such solutions have been dealt with in the litera-
ture about epidemic models [13]. Other important aim is that relative to the design of
control strategies in order to eradicate the persistence of the infection within the host
population [1], [4], [6], [9]. In this context, an explicit vaccination function of many
different kinds may be considered, namely: constant, continuous-time, impulsive,
mixed constant/impulsive, mixed continuous-time/impulsive, discrete-time and so on.
In this paper, a SEIR epidemic model is considered. The dynamics of susceptible
(S) and immune (R) populations are directly affected by a vaccination function V(t) ,
which also has indirectly influence in the time evolution of infected or exposed (E)
and infectious (I) populations. In fact, such a vaccination function has to be suitably
designed in order to eradicate the infection from the population. This model has been
already studied in [1] from the viewpoint of equilibrium points in the controlled and
free-vaccination cases. A vaccination auxiliary control law being proportional to the
susceptible population was proposed in order to achieve the whole population be
asymptotically immune. Such an approach assumed that the parameters of the model
were known and the illness transmission was not critical. Moreover, some important
issues of positivity, stability and tracking of the SEIR model were discussed. The
present paper proposes an alternative method to obtain the vaccination control law to
asymptotically eradicate the epidemic disease. Concretely, the vaccination function is
synthesized by means of an input-output exact feedback linearization technique. Such
a linearization control strategy constitutes the main contribution of the paper. More-
over, mathematical proofs about the disease eradication based on such a controlled
SEIR epidemic model while maintaining the non-negativity of all the partial popula-
tions for all time are issued. The exact feedback linearization can be implemented by
using a proper nonlinear coordinate transformation and a static-state feedback control.
The use of such a linearization strategy is motivated by three main facts, namely: (i) it
is a power tool for controlling nonlinear systems which is based on well-established
technical principles [14], (ii) the given SEIR model is highly nonlinear and (iii) such a
control strategy has not been yet applied in epidemic models.

2 SEIR Epidemic Model

Let S(t) , E(t) , I(t) and R(t) be, respectively, the susceptible, infected (or ex-
posed), infectious and removed-by-immunity populations at time t . Consider a time-
invariant true-mass action type SEIR epidemic model given by:

S(t)I(t)



[ ] S(t) =−μ S(t) +ω R(t) −β +μ N 1 − V(t) (1)

N