On vaccination controls for the SEIR epidemic model M. De la Sen a,⇑ , A. Ibeas b , S. Alonso-Quesada a a Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country, P.O. Box 644, Bilbao, Spain b Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain article info Article history: Received 29 January 2011 Received in revised form 15 September 2011 Accepted 15 October 2011 Available online 24 October 2011 Keywords: Epidemic models Control SEIR epidemic models Stability abstract This paper presents a simple continuous-time linear vaccination-based control strategy for a SEIR (susceptible plus infected plus infectious plus removed populations) disease propa- gation model. The model takes into account the total population amounts as a refrain for the illness transmission since its increase makes more difficult contacts among susceptible and infected. The control objective is the asymptotically tracking of the removed-by- immunity population to the total population while achieving simultaneously the remain- ing population (i.e. susceptible plus infected plus infectious) to asymptotically tend to zero. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction Important control problems nowadays related to Life Sciences are the control of ecological models like, for instance, those of population evolution (Beverton–Holt model, Hassell model, Ricker model, etc.) via the online adjustment of the species environment carrying capacity, that of the population growth or that of the regulated harvesting quota as well as the disease propagation via vaccination control. In a set of papers, several variants and generalizations of the Beverton–Holt model (standard time-invariant, time-varying parameterized, generalized model or modified generalized model) have been inves- tigated at the levels of stability, cycle-oscillatory behavior, permanence and control through the manipulation of the carrying capacity (see, for instance [1–5]). The design of related control actions has been proved to be important in those papers at the levels, for instance, of aquaculture exploitation or plague fighting. On the other hand, the literature about epidemic math- ematical models is exhaustive in many books and papers. A non-exhaustive list of references is given in this manuscript, cf. [6–27] including the consideration of eventual presence of delays in the various models, nonlinear incidence rates, non-monotonic incidence rates, nonlinear transmission effects and bifurcation problems. See also the references listed there- in. The sets of models include the most basic ones [6,7]: – SI-models where no removed-by – immunity population is assumed. In other words, only susceptible and infected pop- ulations are assumed. – SIR models, which include susceptible plus infected plus removed-by – immunity populations. – SEIR-models where the infected populations is split into two ones (namely, the ‘‘infected’’ which incubate the disease but do not still have any disease symptoms and the ‘‘infectious’’ or ‘‘infective’’ which do have the external disease symptoms). Those models have also two major variants, namely, the so-called ‘‘pseudo-mass action models’’, where the total popu- lation is not taken into account as a relevant disease contagious factor and the so-called ‘‘true-mass action models’’, where 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.10.012 ⇑ Corresponding author. E-mail address: wepdepam@lg.ehu.es (M. De la Sen). Commun Nonlinear Sci Numer Simulat 17 (2012) 2637–2658 Contents lists available at SciVerse ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns