Abstract
This paper presents a vaccination strategy for fighting against the propagation of epidemic diseases. The disease propagation is described by an SEIR (susceptible plus infected plus infectious plus removed populations) epidemic model. The model takes into account the total population amounts as a refrain for the illness transmission since its increase makes the contacts among susceptible and infected more difficult. The vaccination strategy is based on a continuous-time nonlinear control law synthesised via an exact feedback input-output linearization approach. An observer is incorporated into the control scheme to provide online estimates for the susceptible and infected populations in the case when their values are not available from online measurement but they are necessary to implement the control law. The vaccination control is generated based on the information provided by the observer. The control objective is to asymptotically eradicate the infection from the population so that the removed-by-immunity population asymptotically tracks the whole one without precise knowledge of the partial populations. The model positivity, the eradication of the infection under feedback vaccination laws and the stability properties as well as the asymptotic convergence of the estimation errors to zero as time tends to infinity are investigated.
Highlights
A relevant area in the mathematical theory of epidemiology is the development of models for studying the propagation of epidemic diseases in a host population [ – ]
The epidemic mathematical models analysed in such an exhaustive list of books and papers include the most basic ones [ – ], 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’ which incubate the disease but do not still have any disease symptoms and the ‘infectious’ which do have the external disease symptoms
Those models can be divided into two main classes, namely the so-called ‘pseudo-mass action models’, where the total population is not taken into account as a relevant disease contagious factor and the so-called ‘true-mass action models’, where the total population is more realistically considered as an inverse factor of the disease transmission rates
Summary
A relevant area in the mathematical theory of epidemiology is the development of models for studying the propagation of epidemic diseases in a host population [ – ]. The main motivation of the present paper is to provide a control solution to overcome such a drawback In this sense, the use of a switching control law coupled with a state observer to synthesise the vaccination function under no precise knowledge of the exact partial populations which are online estimated by the observer is proposed. In the second stage, related to a combined observation/control stage, the vaccination function is synthesised by means of an input-output exact feedback linearization technique while the observer is maintained active providing the estimates of the true partial populations In both stages, the state observer provides online estimations of susceptible and infected populations through time overcoming the unfeasibility of obtaining true measures of such partial populations.
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