A Vaccination Strategy Based on a State Feedback Control Law for Linearizing SEIR Epidemic Models - Biomedical Engineering Systems and Technologies - page 173

A Vaccination Strategy Based on a State Feedback
Control Law for Linearizing SEIR Epidemic Models

S. Alonso-Quesada 1 , M. De la Sen 1 , and A. Ibeas 2

1 Department of Electricity and Electronics, Faculty of Science and Technology,
University of the Basque Country, UPV/EHU, Campus of Leioa, 48940-Leioa, Bizkaia, Spain
{santiago.alonso,manuel.delasen}@ehu.es
2 Departamento de Telecomunicación e Ingeniería de Sistemas,
Escuela Técnica Superior de Ingeniería, Universitat Autònoma, Barcelona, Spain
[e-mail]

Abstract. A vaccination strategy for fighting against the propagation of epi-
demic diseases within a host population is purposed. A SEIR epidemic model is
used to describe the propagation of the illness. This compartmental model
divides the population in four classes by taking into account their status related
to the infection. In this way, susceptible, exposed, infectious and recovered
populations are included in the model. The vaccination strategy is based on a
continuous-time nonlinear control law synthesized via an exact feedback input-
output linearization approach. The asymptotic eradication of the infection from
the host population under such a vaccination is proved. Moreover, the positivity
and stability properties of the controlled system are investigated.

Keywords: SEIR epidemic models, Vaccination, State feedback control,
Stability, Positivity.

1 Introduction

A relevant area in the mathematical theory of epidemiology is the development of
models for studying the propagation of epidemic diseases within a host population.
The epidemic mathematical models include the most basic ones [1-5], namely: (i) SI
models where only susceptible and infected populations are assumed to be present in
the model, (ii) SIR models which include susceptible plus infected plus removed-by-
immunity populations and (iii) SEIR models where the infected population is split
into two ones, namely, the “infected” (or “exposed”) which incubate the disease but
they do not still have any disease symptoms and the “infectious” (or “infective”)
which do have the external disease symptoms. There are many variants of the above
models as, for instance, the SVEIR epidemic models which incorporate the dynamics
of a vaccinated population [6], [7], the SEIQR-SIS model which adds a quarantine
population [8] and the model proposed in [9] which incorporates vaccinated, quaran-
tine and hospitalized populations. Other variant consists of the generalization of such
models by incorporating point and/or distributed delays [10], [11]. Another one is
concerned with the inclusion of a saturated disease transmission incidence rate for