S. Alonso-Quesada

A Control Theory point of view on Beverton–Holt equation in population dynamics and some of its generalizations

Abstract This paper is devoted to develop some “ad hoc” Control Theory formalism useful for the famous Beverton–Holt equation arising in population dynamics. In particular, the inverse equation is redefined for a finite set of consecutive samples under the equivalent form of a discrete linear dynamic system whose input sequence is defined by the sequence of carrying capacity gains and the unforced dynamics is directly related to the intrinsic growth rate. For that purpose, the environment carrying capacity gains are allowed to be time-varying and designed for control purposes. The controllability property is also investigated on this dynamic extended system as well as the stability, equilibrium points and attractor oscillating trajectories. The properties of the dynamic system associated with the Beverton–Holt inverse equation allow extrapolate in a simple dual way the above properties to the standard Beverton–Holt equation. Some generalizations are given for the case when there are extra parameters in the equation or when the system is subject to the presence of additive disturbances. In all cases, a reference model being also of Beverton–Holt type is proposed to be followed by the control system.

On a Generalized Time-Varying SEIR Epidemic Model with Mixed Point and Distributed Time-Varying Delays and Combined Regular and Impulsive Vaccination Controls

This paper discusses a generalized time-varying SEIR propagation disease model subject to delays which potentially involves mixed regular and impulsive vaccination rules. The model takes also into account the natural population growing and the mortality associated to the disease, and the potential presence of disease endemic thresholds for both the infected and infectious population dynamics as well as the lost of immunity of newborns. The presence of outsider infectious is also considered. It is assumed that there is a finite number of time-varying distributed delays in the susceptible-infected coupling dynamics influencing the susceptible and infected differential equations. It is also assumed that there are time-varying point delays for the susceptible-infected coupled dynamics influencing the infected, infectious, and removed-by-immunity differential equations. The proposed regular vaccination control objective is the tracking of a prescribed suited infectious trajectory for a set of given initial conditions. The impulsive vaccination can be used to improve discrepancies between the SEIR model and its suitable reference one.

Robustly stable multiestimation scheme for adaptive control and identification with model reduction issues.

A discrete pole-placement-based and multiestimation-based adaptive control scheme involving a relative adaptation dead zone is presented for a plant with known poles and unknown zeros. The basic usefulness of the proposed multiestimation scheme is related to the use of a set of models of reduced order associated with the multiestimation scheme instead of a high-order one. Depending on the frequency spectrum characteristics of the input and on the estimates evolution, the multiestimation scheme selects on-line the most appropriate model and its related estimation scheme in order to improve the identification and control performances. Robust closed-loop stability is proved even in the presence of unmodeled dynamics of sufficiently small sizes as it has been confirmed by simulation results. The scheme chooses in real time the estimator/controller associated with a particular reduced model possessing the best performance according to an identification performance index by implementing a switching rule between estimators. The switching rule is subject to a minimum residence time at each identifier/adaptive controller parameterization for closed-loop stabilization purposes. A conceptually simple higher-level supervisor, based on heuristic updating rules which estimate on-line the weights of the switching rule between estimation schemes, is discussed.

On the Existence of Equilibrium Points, Boundedness, Oscillating Behavior and Positivity of a SVEIRS Epidemic Model under Constant and Impulsive Vaccination

This paper discusses the disease-free and endemic equilibrium points of a SVEIRS propagation disease model which potentially involves a regular constant vaccination. The positivity of such a model is also discussed as well as the boundedness of the total and partial populations. The model takes also into consideration the natural population growing and the mortality associated to the disease as well as the lost of immunity of newborns. It is assumed that there are two finite delays affecting the susceptible, recovered, exposed, and infected population dynamics. Some extensions are given for the case when impulsive nonconstant vaccination is incorporated at, in general, an aperiodic sequence of time instants. Such an impulsive vaccination consists of a culling or a partial removal action on the susceptible population which is transferred to the vaccinated one. The oscillatory behavior under impulsive vaccination, performed in general, at nonperiodic time intervals, is also discussed.

Control issues for the Beverton-Holt equation in ecology by locally monitoring the environment carrying capacity: Non-adaptive and adaptive cases

This paper proposes a control algorithm to govern the solution of the Beverton-Holt equation (BHE) under the potentially presence of additive disturbances. The BHE to be controlled is defined by certain intrinsic growth rate and environment carrying capacity sequences, the last one being susceptible of local modifications around nominal values. In fact, the control action provides the carrying capacity which makes that the solution of the current BHE tracks a reference sequence given by another BHE defined by appropriate intrinsic growth rate and environment carrying capacity sequences. In this context, the fact that the inverse of the BHE is a discrete time-varying linear system is taken into account where the inverse of the carrying capacity sequence plays the role of control sequence. The current and the reference BHEs have to be close enough to each other in order that local modifications of the carrying capacity be able to meet the tracking objective. A feedback control law is designed to achieve such an objective with a zero tracking-error in the ideal case of known intrinsic growth rate sequence and no presence of disturbances. An adaptive control law, with the associated parameter estimation algorithm, is considered when the intrinsic growth rate is fully or partially unknown and disturbances are present. Such a control strategy guarantees a bounded tracking-error with the error converging asymptotically to zero in case that additive disturbances also converge to zero. Some results obtained from a simulation example illustrate the effectiveness of this control strategy.

Model-Matching-Based Control of the Beverton-Holt Equation in Ecology

This paper discusses the generation of a carrying capacity of the environment so that the famous Beverton-Holt equation of Ecology has a prescribed solution. The way used to achieve the tracking objective is the design of a carrying capacity through a feedback law so that the prescribed reference sequence, which defines the suitable behavior, is achieved. The advantage that the inverse of the Beverton-Holt equation is a linear time-varying discrete dynamic system whose external input is the inverse of the environment carrying capacity is taken in mind. In the case when the intrinsic growth rate is not perfectly known, an adaptive law implying parametrical estimation is incorporated to the scheme so that the tracking property of the reference sequence becomes an asymptotic objective in the absence of additive disturbances. The main advantage of the proposal is that the population evolution might behave as a prescribed one either for all time or asymptotically, which defines the desired population evolution. The technique might be of interest in some industrial exploitation problems like, for instance, in aquaculture management.

Stable multi-estimation model for single-input single-output discrete adaptive control systems

A pole-placement-based adaptive controller synthesized from a multi-estimation scheme is designed for linear single-input single-output time-invariant plants. A higher level switching structure between the various estimation schemes is used to supervise the reparameterization of the adaptive controller in real time. The basic usefulness of the proposed scheme is to improve the transient behaviour while guaranteeing closed loop stability. The scheme becomes specifically useful when extended to linear plants whose parameters are piecewise constants while changing abruptly to new constant parameterizations or in the case when the parameters are slowly time varying rather than constant. Thus, the scheme becomes attractive from a modelling point of view since the plant, while being potentially time varying, or in particular, possessing several operation points, is modelled as a set of time-invariant plant unknown parameterizations each possessing its own estimation scheme. In that way, the model description becomes conceptually simple and easy to implement concerned with both estimation and control issues. A description of the controller architecture with multiple parameterizations, together with its associated multi-estimation scheme is given. In addition, the proofs of boundedness of all the relevant signals are given so that the closed-loop system is proved to be stable.

Vaccination strategies based on feedback control techniques for a general SEIR-epidemic model

Abstract This paper presents several simple linear vaccination-based control strategies 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 vaccination control objective is the asymptotically tracking of the removed-by-immunity population to the total population while achieving simultaneously that the remaining populations (i.e. susceptible plus infected plus infectious) tend asymptotically to zero.

On vaccination control tools for a general SEIR-epidemic model

This paper presents a simple continuous-time linear vaccination-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 the asymptotically tracking of the removed-by-immunity population to the total population while achieving simultaneously that the remaining populations tend asymptotically to zero.

On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules

This paper discusses and formulates a continuous-time SEIR -type epidemic model of pseudo-mass action type with finitely distributed delays under a very general, potentially time-varying, vaccination control rule which eventually generates feedback actions on the susceptible, infectious and recovered subpopulations. A lot of particular vaccination laws can be got from the proposed general one. The disease-free and endemic equilibrium points are characterized and their local stability properties discussed depending on the limits of the vaccination control gains provided that they converge asymptotically. Then, the global asymptotic stability to the disease-free equilibrium point is studied under an infective transmission rate below a certain maximum threshold. Later on, an extended SEIR epidemic model is discussed through simulated examples with stochastic Wiener-type perturbations around the equilibrium points.