Abstract
This paper studies some basic properties of an SEIR (Susceptible-Exposed-Infectious-Recovered) epidemic model subject to vaccination and treatment controls. Firstly, the basic stability, boundedness, and nonnegativity of the state trajectory solution are investigated. Then, the problem of partial state reachability from a certain state value to a targeted one in finite time is focused on since it turns out that epidemic models are, because of their nature, neither (state) controllable from a given state to the origin nor reachable from a given initial condition. The particular formal statement of the partial reachability is focused on as a problem of output-reachability by defining a measurable output or lower dimension than that of the state. A special case of interest is that when the output is defined as the infectious subpopulation to be step-to-step tracked under suitable amounts being compatible with the required constraints. As a result, and provided that the output-controllability Gramian is nonsingular on a certain time interval of interest, a feedback control effort might be designed so that a prescribed value of the output can be approximately tracked. A linearization approximation is performed to simplify and facilitate the above task which is based on a point-to-point linearization of the solution trajectory. To this end, an “ad hoc” sampled approximate output trajectory is defined as control objective to be targeted through a point-wise calculated Jacobian matrix. A supervised appropriate restatement of the targeted suited sampled output values is redefined, if necessary, to make the initial proposed sampled trajectory compatible with the various needed constraints on nonnegativity and control boundedness. The design can be optionally performed under constant or adaptive sampling rates. Finally, some numerical examples are given to test the theoretical aspects and the design efficiency of the model.
Highlights
Epidemic mathematical models are receiving very significant attention in the last decades due to their inherent interest to predict the progression of the force intensity of the infections through time so as to take appropriate decisions about how to mitigate them and how to conduct in an adequate way the hospital means requirements to fight against them
A modified SEIR model which adds a new infectious population is considered, so-called the SEIADR model, and, apart from the vaccination and treatment controls, an impulsive control is applied to this population which is interpreted as the recovery from the streets of the lying infective corpses by ad hoc organizes brigades or volunteers when necessary
For instance, [17,18,19] and references therein. e approximate controllability and reachability problems in time-varying systems have been considered in [20], and some ad hoc examples related to epidemic models have been given, while in [21] the classical hyperstability theory of control systems has been adapted to be used in derivation of very wide structure types for nonlinear feedback rules of vaccination and treatment controls in some epidemic models. e design of control laws for certain ecology models, like Beverton–Holt equation or epidemic models, has received significant attention in the literature, in particular, the monitoring of the carrying capacity in aquaculture or the updated design of the feedback gains of vaccination and treatment control
Summary
Epidemic mathematical models are receiving very significant attention in the last decades due to their inherent interest to predict the progression of the force intensity of the infections through time so as to take appropriate decisions about how to mitigate them and how to conduct in an adequate way the hospital means requirements to fight against them In this context, it is very important to study the influence in the disease evolution of potential control actions like typically the vaccination and treatment controls and to accommodate their convenient application to items like their availability and economic costs through time and their demand in view of the disease progression.
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