Abstract

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 the susceptible and infected. The control objective is 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. A state observer is used to estimate the true various partial populations of the susceptible, infected, infectious, and immune which are assumed to be unknown. The model parameters are also assumed to be, in general, unknown. In this case, the parameters are replaced by available estimates to implement the vaccination action.

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 investigated at the levels of stability, cycle-oscillatory behavior, permanence, and control through the manipulation of the carrying capacity (see, e.g., [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 mathematical models is exhaustive in many books and papers. A nonexhaustive list of references is given in this paper; compare [6–14] (see also the references listed therein). The sets of models include the most basic ones, [6, 7].(i)SI-models where not removed-by-immunity population is assumed. In other words, only susceptible and infected populations are assumed.(ii)SIR models, which include susceptible plus infected plus removed-by—immunity populations. (iii)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 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. There are many variants of the above models, for instance, including vaccination of different kinds as follows: constant [8, 9], impulsive [9, 12], discrete-time, and so forth, incorporating point or distributed delays [12, 13], oscillatory behaviours [14], and so forth. On the other hand, variants of such models become considerably simpler for the illness transmission among plants [6, 7]. Recent work on the influence of latent and infective periods and studiers of persistence of infection has been performed in [15, 16]. Other mathematical models involving coupled partial populations of the whole ones require very close analysis of equilibrium points, stability, and positivity to that required in epidemic models (see, e.g., [17, 18]).

In this paper, a continuous-time vaccination observer-based control strategy is given for a SEIR epidemic model which takes directly the estimated susceptible, infected, infectious, and immune populations to design the vaccination strategy. It is not required either the knowledge through time of the true partial populations of the susceptible, infected, infectious, and immune, since this task is performed by the observer, nor the knowledge of the true parameters. The incorporation of the observer to the vaccination control rule is the main contribution of this paper. On the other hand, it is assumed that the total population is known and equal to the total estimated population and also that it remains constant through time, so that the illness transmission is not critical, and the SEIR model is of the above-mentioned true-mass action type. Some important issues of positivity, stability, and model reference tracking of a suitable population evolution of the combined SEIR model and its observer are discussed.

1.1. Notation

is the first open -real orthant, and is the first closed -real orthant.

is a positive real -vector in the usual sense that all its components are nonnegative. This can be also denoted by if . If all the components are strictly positive, this is denoted by or . In the same way, is a positive real -matrix in the usual sense that all its entries are nonnegative and we can use the notations and (if ). A square real matrix is a Metzler matrix if and only if all its off-diagonal entries are nonnegative and then its associate exponential matrix function is positive; that is, none of its entries takes a negative value at any time.

is the set of real functions of class of domain Do and image Im. is the set of real functions of class () of domain Do and image Im whose th derivative exits but it is not necessarily everywhere continuous on its definition domain.

2. SEIR Epidemic Model

Let be the “susceptible” population of infection at time , the “infected” (i.e., those which incubate the illness but do not still have any symptoms) at time , is the “infectious” (or “infective”) population at time , and is the “removed-by-immunity” (or “immune”) population at time . Consider the true-mass action SEIR-type epidemic model: subject to initial conditions , , , and under the constraint ; for all and the vaccination function . In the above SEIR model, is the total population, is the rate of deaths from causes unrelated to the infection, is the rate of losing immunity, is the transmission constant (with the total number of infections per unity of time at time being ), and and are finite and, respectively, the average durations of the latent and infective periods. All the above parameters are assumed to be nonnegative. Note that , , , if . However, if then , , , , their time-derivatives exist but they are not everywhere continuous on , in general. This model has been studied in [19] from the point of view of equilibrium points in the free-vaccination case and control. Two vaccination auxiliary controls (grouping the last three terms of the right-hand side of (2.1)) being, respectively, proportional to the susceptible or to the whole population so that the whole population is asymptotically immune have been proposed. Note that controllability is a very suitable property of practical control problems since each state variable is allowed to track a prescribed value, in a finite time (see, e.g., [20]). However, we have to point out that epidemic models are not, in general, controllable since all the partial populations cannot be separately controlled through vaccination since there is always an inherent interchange among the various populations [19]. The SEIR model (2.1)–(2.4) is uncontrollable according to the above principle so that there are potential final values of some of the partial populations which are not within the set of reachable states [21]. The epidemic models proposed in previous works assume that the parameters of the SEIR model are known. However, the research in this paper does not require such a perfect parametrical knowledge since an observer with parametrical estimation is incorporated. It has to be pointed out that the model following and tracking of desired references as well as the optimization of certain well-posed criteria are very important objectives in control theory problems including robotics and adaptive control [22–24]. They are also relevant in predator-prey processes or in epidemics as, for instance, the removal or asymptotic removal of diseases through vaccination strategies [25–28]. In this paper, the vaccination rules are based on the estimation of the partial populations provided by an “ad hoc” synthesized observer which is potentially useful in the common real case that not all partial populations which are integrated in the whole dynamic epidemic model are precisely known or measurable for all time.

3. Observer-Based Vaccination Control Strategy for the SEIR Model

It turns out that, while the assumption of the knowledge of the total population is not quite restrictive in practice, the knowledge of the individual various partial populations of the susceptible, infected, infectious, and immune may be considered severely restrictive. It is seen that if the partial initial populations are unknown then their evolution through time cannot be computed in a closed form from the differential system (2.1)–(2.4). A practical solution to circumvent the problem might be to estimate them based on percentages of the total population through time from experimental knowledge of the disease propagation. Another solution may be to estimate them online by using an online observer. The current paper focuses on this solution by using a SEIR-estimation algorithm (observer) of the SEIR model (2.1)–(2.4) which estimates through time the individual populations being involved. The vaccination strategy is obtained as a control strategy from the data supplied by the observer through time. Such a strategy does not require the knowledge of the partial populations to organize and perform the vaccination strategy. The estimates of the various individual populations are denoted by the same notations as the real populations with hat superscripts, namely, , , , , Thus, consider the SEIR-type observer for the SEIR-epidemic model (2.1)–(2.4) as follows: subject to initial conditions , , , and under the constant population constraint equalizing its total estimated value for all time, and the vaccination law generated by where , are constant real gains to be determined in order to achieve the control objective. In the above estimated SEIR model, is the estimate of , that is, the estimated rate of deaths from causes unrelated to the infection and also the estimated rate of births, is the estimate of , that is, the estimated rate of losing immunity, is the estimate of , that is, the estimated transmission constant (with the total number of infections per unity of time at time being ), and and are finite and, respectively, the estimates of and , that is, the estimated average durations of the latent and infective periods, respectively. The estimations of the above parameters can be done, through the use of available “a priori” knowledge, to be identical to the true values if those ones are known or estimated online from data measurements. Through this paper, we assume that those estimated parameters are fixed but not necessarily identical to the true parameters and all of them are nonnegative. The substitution of (3.3) in (3.1) yields the following combined observer-controller for the SEIR model: where The systems (3.4)–(3.6) and (3.7)–(3.10) may be compacted as an extended system as follows: with with being the observation error and is a parametrical error defined by It is direct to see that for any given real , with being the Euclidean norm of , if . Decompose where , , and are constant matrices being the nonunique decompositions (3.14) as follows: where denotes the term “” for any given prefixed constant values , and so that is decomposed as follows: If and are stability (or Hurwitz) matrices then the block triangular matrix is a stability matrix with stability abscissa which is subject to , where the first inequality is nonstrict if there is some multiple eigenvalue of .

4. About the Stability of the SEIR Extended Epidemic Model (3.11)–(3.16)

The SEIR epidemic model (2.1)–(2.4) and its observer (3.4)–(3.6) are both always globally stable if if the initial real and observed populations are bounded and equalize provided that both systems are positive as, on the other hand, the real problem at hand requires. This would imply that all the particular populations (susceptible, infected, infectious, and immune) are nonnegative and less than so that they are uniformly bounded for all time. Note that if any of the populations of the SEIR model is negative then boundedness of all the components of the associated system is not guaranteed. The boundedness of all the partial populations upperbounded by a bounded total population guarantees that the extended SEIR system (3.11)–(3.16), constructed with the estimated and observer error states, is stable with and Euclidean norm . This fact does not guarantee by itself that is a stability matrix function, that the observation error converges asymptotically to zero, or that, at least, it converges to a small residual set around zero. This feature merits further discussion. Note that simple inspection shows that is a stability matrix if is Hurwitz. Also, is a stability matrix if has all its zeros in Re , where

Assume that is a Hurwitz polynomial, that is, and define . Note that has all its solutions in for all of the form (4.1) from the small gain theorem if and only if being the imaginary complex unit) since is a Hurwitz polynomial and where is the -norm of the transfer function . Since is blocktriangular and constant then the following result is direct.

Assertion 4.1. is a stability matrix if and only if , ,, and (i.e., ; ; ; ) with .

From Assertion 4.1 and Gronwall’s Lemma [29] the following follows.

Assertion 4.2. The matrix function is stable if is a stability matrix, that is, if (4.3) holds and, furthermore, , where is the stability abscissa of the (stability) matrix .

Another alternative sufficiency type stability condition of, which replaces the constraint in Assertion 4.2 (see [30] for close related problems), is as follows.

Assertion 4.3. The matrix function is stable, so that the unforced extended system (3.11)–(3.16) is then globally exponentially stable, if is a stability matrix (see Assertion 4.1) and, furthermore, there exist norm-dependent real constants (being sufficiently small) and such that for any given fixed ; for all .

Note that the Euclidean norm of may be directly calculated from those of and using (3.5) and (3.13) leading to since and . Note that so that is larger than the maximum of the stability abscissas of for if Assertion 4.3 holds.

Assertion 4.4. If Assertion 4.3 holds then any solution of the forced system (3.11) to (3.16) satisfies the following inequality for some real constant : and the corresponding substates of satisfy

Note that as if (and then as ) and, if in addition, then as ) or if (and then as ). Since then as . However, this upperbound can be improved if a version of Assertion 4.3 applied to the matrix function leads to a smaller ratio of its corresponding constants than the ratio of the whole extended system.

5. About the Positivity of the SEIR Extended Epidemic Model (3.11)–(3.16)

Positive systems are those having nonnegative solutions in the sense that all the state components are nonnegative for all time [31, 32]. Because of the nature of the SEIR epidemic model (2.1)–(2.4), it is required that it be a positive system for the implemented vaccination law. The extended SEIR system has a unique solution for each initial state given by: From (3.4) and (3.7) the SEIR solution and its estimate through the observer are uniquely given by In principle, it is apparently nonnecessary to require in addition that the estimation algorithm or the extended system be positive. However, note the following features by direct inspection of (5.2)-(5.3).(1)If is a Metzler matrix, and , then , for all .(2)If , (i.e., if is a Metzler matrix, , and ), , , and is a Metzler matrix then , for all .(3) depends on the vaccination gain , does not depend on , but it depends on (see (3.15)), and in (3.10) depends on . Thus, if is chosen so that is not a Metzler matrix (because its (4.1) entry is negative), may be nonpositive at some time for some nonnegative initial condition . At the same time, the vector function may have some sufficiently negative component at some time “” so that some corresponding component of may be negative at that time. A close reasoning may be used for the case that is such that is nonpositive (even if is a Metzler matrix).