Abstract
This paper presents a feedback linearization-based control strategy for a SEIR (susceptible plus infected plus infectious plus removed populations) propagation disease 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 novel in the sense that the asymptotically tracking of the removed-by-immunity population to the total population while achieving simultaneously the remaining population (i.e. susceptible plus infected plus infectious) to asymptotically converge to zero. The vaccination policy is firstly designed on the above proposed tracking objective. Then, it is proven that identical vaccination rules might be found based on a general feedback linearization technique. Such a formal technique is very useful in control theory which provides a general method to generate families of vaccination policies with sound technical background which include those proposed in the former sections of the paper. The output zero dynamics of the normal canonical form in the theoretical feedback linearization analysis is identified with that of the removed-by-immunity population. The various proposed vaccination feedback rules involved one of more of the partial populations and there is a certain flexibility in their designs since some control parameters being multiplicative coefficients of the various populations may be zeroed. The basic properties of stability and positivity of the solutions are investigated in a joint way. The equilibrium points and their stability properties as well as the positivity of the solutions are also investigated.
Highlights
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
Those models have two major variants, 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)
Note that in the same way that the stability property is a minimum requirement for control problems, positivity is a minimum requirement for properly dealing with models involving populations so as to have a better chance to adjust the model solutions to the real and foreseen evolution of the partial populations
Summary
Rn+ is the first open n-real orthant and Rn0+ is the first closed n-real orthant. m ∈ Rn0+ is a positive real n-vector in the usual sense that all its components are nonnegative. Rn+ is the first open n-real orthant and Rn0+ is the first closed n-real orthant. M ∈ Rn0+ is a positive real n-vector in the usual sense that all its components are nonnegative. M ∈ Rn0+×n is a positive real n-matrix in the usual sense that all its entries are nonnegative. The notations Rn+ and R+n×n refer to the stronger properties that all the respective components or entries are positive. C(q)(Do; Im) is the set of real functions of class q of domain Do and image Im. P C(q)(Do; Im) is the set of real functions of class (q − 1) of domain Do and image Im www.mii.lt/NA whose q-th derivative exits but it is not necessarily everywhere continuous on its definition domain
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