### 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.

##### Keywords:

SEIR epidemic models; vaccination; nonlinear control; stability; positivity; nonlinear observers design### 1 Introduction

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 [1-20]. The epidemic mathematical models analysed in such an exhaustive list of books and papers include the most basic ones [1-9], 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’ (or ‘exposed’) 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 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. There are many variants of the above models as, for instance, the SVEIR epidemic models which incorporate the dynamics of a vaccinated population in comparison with the SEIR models [10-12] and the SEIQR-SIS model which adds a quarantine population [13]. Other variant consists of the generalisation of such models by incorporating point and/or distributed delays [8,10-12,14]. All of the aforementioned models are so-called compartmental models since host individuals are classified depending on their status in relation to the infectious disease. However, there are diseases where some factors such as the disease transmission, the mortality rate and so on are the functions of age. Then such diseases are described more precisely by means of the so-called compartmental models with age structure [15]. Moreover, although the dynamics of infectious diseases transmission through a host population is continuous-time, some researchers have proposed models composed of difference equations to describe the dynamics of epidemics and develop treatments to minimise its effects within the population [16]. On the one hand, a key point in such research works is the choice of an optimal-time step in order to obtain an acceptable discrete-time model from the discretisation of the continuous-time ones. On the other hand, an advantage is that discrete-time models are easier to analyse than continuous-time ones, and then the effectiveness of a potential treatment to eradicate the disease from the host population can be easier to derive.

The analysis of the existence of equilibrium points, relative to either the persistence (endemic equilibrium point) or extinction (disease-free equilibrium point) of the epidemics in the host population [6,9,11-14], the constraints for guaranteeing the positivity and the boundedness of the solutions of such models [11,12,17] and the conditions that generate an oscillatory behaviour in such solutions [11,18] have been some of the main objectives in the literature about epidemic mathematical models. Other important aim is that relative to the design of control strategies in order to eradicate the persistence of the infection in the host population [2,5,11,12,17]. In this context, an explicit vaccination function of many different kinds may be added to all aforementioned epidemic models, namely constant [5,12], continuous-time [2,17], impulsive [10], mixed constant/impulsive [11], mixed continuous-time/impulsive [14], discrete-time and so on. Concretely, the research in [17] exhaustively analyses the equilibrium points of an SEIR epidemic model under a vaccination strategy based on a state feedback control law with respect to the model parameters and/or the controller gains. The conditions for the eradication of the diseases from the host population, the extinction of the host population or the persistence of the disease in a non-extinguished host population are derived form such a study. Other alternative approaches, as those based on fuzzy rules [19] or networks framework [13,20], have been also proposed for modelling the epidemics transmission through a host population. In this way, the influence of certain social network parameters such as visiting probability, hub radius and contact radius on the epidemics propagation has been investigated [13]. Moreover, there are studies about the influence of the immigration on the persistence or extinction of the epidemics in a population subject to immigration from other regions [9]. Also, the influence of epidemic diseases on the dynamics of prey-predator models has been considered in ecoepidemic models [21].

In this paper, an SEIR epidemic model which includes susceptible (S), infected or
exposed (E), infectious (I) and removed-by-immunity (R) populations is considered.
The dynamics of susceptible and immune populations are directly affected by a vaccination
function
*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*. *Such a law only switches once and in this way the control process is divided into
two stages. In the first stage, the so-called observation stage, the control function
is identically zero and only the observer is working to reduce the initial difference
between the true infectious population measure and the estimated one provided by the
own observer below a prescribe threshold. 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. Such a combination of a linearization control strategy with
a nonlinear observer to online estimate all the partial populations constitutes the
main contribution of the paper. Moreover, mathematical proofs about the epidemics
eradication based on such a controlled SEIR model coupled with the nonlinear observer
are presented while maintaining the non-negativity of all the partial populations
for all time.* The exact feedback linearization can be implemented by using a proper nonlinear coordinate
transformation and a static-state feedback control. The use of such a linearization
strategy is motivated by three main facts, namely (i) it is a power tool for controlling
nonlinear systems which is based on well-established technical principles [22,23], (ii) the given SEIR model is highly nonlinear and (iii) such a control strategy
has not been yet applied in epidemic models.

On the one hand, approaches based on switching control laws have been broadly dealt with in the control theory and its applications [24]. On the other hand, the combination of exact feedback linearization techniques with state observers has been widely used in many control applications, for instance, in biological systems and chemical engineering [25,26]. The exact linearization technique requires the system to satisfy some structural and regularity conditions, like the existence of relative degree, the minimum phase property and the integrability condition [27,28]. The SEIR epidemic model satisfies such assumptions, and the aforementioned linearization technique can be applied without any modification. Otherwise, alternative approaches developed to approximately linearize nonlinear systems violating one or more of such assumptions could be used [29,30].

The paper is organised as follows. Section 2 describes the set of differential equations which compound the SEIR model for the propagation of an epidemic disease through a host population. A result related to the positivity property of such a model is proven. Section 3 presents a control action based on an input-output linearization technique, guaranteeing the positivity and stability properties of the system while asymptotically achieving the eradication of the infection from the host population and, simultaneously, the whole population becoming immune. The positivity property is required from the own nature of the system which forbids the existence of negative populations at any time instant. The control strategy requires the knowledge of the susceptible, infected, infectious and whole population for all time. In this context, the knowledge of the infectious and whole population for all time is feasible, but the knowledge of the susceptible and infected population for all time is not a realistic assumption. As a consequence, such partial populations have to be estimated by means of an observer dynamic system. Then a control action based on such estimates, instead of the corresponding true partial populations, is carried out in Section 4. These theoretical results and the effectiveness of the feedback input-output linearizing controller combined with the observer are illustrated by means of some simulation results in Section 5.

**Notation**
*n*th real orthant and
*n*th real orthant.
*n-*vector in the usual sense that all its components are non-negative. Also,
*M*.

### 2 SEIR epidemic model

Let
*t*. Consider a time-invariant true-mass action type SEIR epidemic model given by the
following equations:

subject to initial conditions
*μ* is the rate of deaths and births from causes unrelated to the infection, *ω* is the rate of losing immunity, *β* is the transmission constant (with the total number of infections per unity of time
at time *t* being

so that the total population

**Lemma 2.1***Assume the SEIR model* (2.1)-(2.4) *with an initial condition subject to*
*and under no vaccination action before a finite time instant*
*i*.*e*.
*Then*

*Proof* Let eventually existing finite time instants

• If

• If

• If

• If

Note that either

(a) Proceed by contradiction by assuming that there exists a finite

(b) Proceed by contradiction by assuming that there exists a finite

(c) Proceed by contradiction by assuming that there exists a finite

(d) Proceed by contradiction by assuming that there exists a finite

As a result, if

**Remark 2.1** The result
*Lemma 2.1*, combined with the equation (2.5), provided that

### 3 Vaccination strategy

*An ideal control objective is that the removed-by-immunity population asymptotically
tracks the whole population*. In this way, the joint infected plus infectious population asymptotically tends
to zero as time tends to infinity, so the infection is eradicated from the population.
A vaccination control law based on a static-state feedback linearization strategy
is developed for achieving such a control objective. This technique requires a nonlinear
coordinate transformation, based on the theory of Lie derivatives [23], in the system representation.

The dynamics equations (2.1)-(2.3) of the SEIR model can be equivalently written as the following nonlinear control affine system:

where
*i.e.* the infectious population) and the input signal of the system

where
*k*th-order Lie derivative of
*r* of the system is the number of times that the system output (*i.e.* the infectious population) must be differentiated in order to obtain the input explicitly,
*i.e.* the number *r* such that

From (3.2),
*i.e.*

allows representing the SEIR model in the so-called normal form in a neighbourhood
of any

where

The equations in (3.3) define a mapping

Both transformations
*i.e.* they have continuous partial derivatives of any order. Then
*D*. The feature that the relative degree of the system is equal to the system order
*via* the coordinate transformation (3.3) and an exact linearization feedback control [23,28]. The following result being relative to the input-output linearization of the system
is established.

**Theorem 3.1***The state feedback control law defined as*

*where*
*for*
*are the controller tuning parameters*, *induces the linear closed*-*loop dynamics given by*

*around any point*

*Proof* The following state equation for the closed-loop system is obtained:

by introducing the control law (3.7) in (3.4) and taking into account the coordinate
transformation (3.3) and the fact that

One may express
*via* the application of the coordinate transformation in (3.6). Then it follows directly
that

Furthermore, the output equation of the closed-loop system is

with *ℓ* denoting the order of the differentiation of

**Remarks 3.1** (i) The controller parameters
*i.e.*
*i.e.*

(ii) The control (3.7) may be rewritten as follows:

by using (3.3) and (3.10), or

where

(iii) The control law (3.7) is well defined for all
*i.e.* such a control law is well defined by the nature of the system. In this sense, the
control law given by

may be used instead of (3.7) in a practical situation. The signal

In this way, the control action is maintained active while the infection persists within the host population and it is switched off once the epidemics is eradicated.

(iv) The linear system (3.8) is strictly identical to the SEIR model (2.1)-(2.4) under
the transformation (3.3) and the control law (3.15) for
*i.e.* until the time instant at which the epidemics is eradicated.

(v) The implementation of the control law (3.15) requires online measurement of the susceptible, infected and infectious population. In a practical situation, only online measures of the infectious and whole populations may be feasible, so the populations of susceptible and infected can only be estimated. In this context, a complete state observer is going to be designed for such a purpose in Section 4.

#### 3.1 Controller tuning parameters choice

The application of the control law (3.7), obtained from the exact input-output linearization
strategy, makes the closed-loop dynamics of the infectious population be given by
(3.8). Such a dynamics depends on the control parameters
*i.e.* the asymptotic convergence of

**Theorem 3.2***Assume that the initial condition*
*is bounded*, *and all roots*
*for*
*of the characteristic polynomial*
*associated with the closed*-*loop dynamics* (3.8) *are of strictly negative real part via an appropriate choice of the free*-*design controller parameters*
*for*
*Then the control law* (3.7) *guarantees the exponential stability of the transformed controlled SEIR model* (3.1)-(3.6) *while achieving the eradication of the infection from the host population as time
tends to infinity*. *Moreover*, *the SEIR model* (2.1)-(2.4) *has the following properties*:
*and*
*are bounded for all time*,
*and*
*exponentially as*
*and*

*Proof* The dynamics of the controlled SEIR model (3.8) can be equivalently rewritten with
the state equation (3.11) and the output equation
*via* the coordinate transformation (3.3), and
*A* are the roots

**Remark 3.2***Theorem 3.2* implies the existence of a finite time instant
*via* the application of the control law (3.7).

**Theorem 3.3***Assume that an initial condition for the SEIR model satisfies*
*i*.*e*.
*and*
*and the constraint*
*Assume also that some strictly positive real numbers*
*for*
*are chosen such that*

(a)
*and*
*so that*

(b)
*and*
*satisfy the inequalities*:

*Then*

(i) *the application of the control law* (3.7) *to the SEIR model guarantees that the epidemics is asymptotically eradicated from
the host population while*
*and*
*and*

(ii) *the application of the control law* (3.15) *guarantees the epidemics eradication after a finite time*
*the positivity of the controlled SEIR epidemic model*
*and that*
*so that*

*provided that the controller tuning parameters*
*for*
*are chosen such that*
*for*
*are the roots of the characteristic polynomial*
*associated with the closed loop dynamics* (3.8).

*Proof* (i) On the one hand, the epidemics asymptotic eradication is proven by following
the same reasoning as in *Theorem 3.2*. On the other hand, the dynamics of the controlled SEIR model (3.8) can be written
in the state space defined by
*A*, it follows that

for some constants
*i.e.* in the state space defined by
*via* (3.3). The constants

where (3.3) and (3.17) have been used. Such equations can be more compactly written
as

Once the desired roots of the characteristic equation of the closed-loop dynamics
have been prefixed, the constants
*i.e.* an invertible matrix. In this sense, note that

where the functions

In particular,
*a priori*’ knows that
*a priori*’ determined from the initial conditions and constraints in (a). The following four
cases may be possible: (i)
*i.e.* if

where the facts that
*i.e.* if

by taking into account that

where (3.20), (3.21),
*i.e.* if

where the constraints

by taking into account that

where the fact that the function

is zero for

by applying such a relation between

by applying such a relation between

(ii) On the one hand, if the control law (3.15) is used instead of that in (3.7),
then the time evolution of the infectious population is also given by (3.17) while
the control action is active. Thus, the exponential convergence of
*i.e.* there exists at least a time instant

by introducing the control law (3.15) and taking into account the facts that

from (3.31). The controller tuning parameters
*Remark 3.1 (i)*, by

The assignment of

Then
*via* complete induction. Finally, the positivity of the controlled SEIR model
*Lemma 2.1*.

On the other hand, it follows from (3.13) and (3.15) that

where the facts that

In summary, this section has dealt with a vaccination strategy based on linearization
control techniques for nonlinear systems. The proposed control law satisfies the main
objectives required in the field of epidemics models, namely the stability, the positivity
and the eradication of the infection from the population. Such results are proven
formally in *Theorems 3.2* and *3.3*. In Section 5, some simulation results illustrate the effectiveness of such a vaccination
strategy. However, such a strategy has a main drawback, namely the control law needs
the knowledge of the true values of the susceptible, infected and infectious populations
at all time instants which are not available in certain real situations. An alternative
approach useful to overcome such a drawback is dealt with in the following section
where an observer to estimate all the partial populations is proposed.

### 4 Vaccination control strategy based on the use of a state observer

The control laws (3.7), or equivalently (3.13) or (3.14), and (3.15) require the online
measurement of all the state variables, namely
*This observer provides online estimates*

with

where
*i.e.*

for some real constants
*i.e.* the deviation between the infectious population estimated by the observer and the
true one. Note that

*The SEIR model (**3.1**)-(**3.2**) is diffeomorphic on D to the system (**3.4**)-(**3.5**) by applying the nonlinear coordinate transformation (**3.3**)*. In the system representation (3.4)-(3.5) the functions

with
*uniformly observable on D for any input* in view of *Theorem 2* of [25]. This property allows constructing an observer in the coordinates corresponding to
the state representation (3.4)-(3.5). The state equation of such an observer is as
follows:

with an initial condition

where
*C* are the following matrices:

The following result relative to the existence of a finite time instant

**Lemma 4.1***Assume that*

(i) *The SEIR model parameters are such that*

(ii)
*in the definition of the switching time instant*
*satisfies*

(iii) *the control parameter*
*in* (4.2) *and the constant*
*in the definition of*
*are such that*
*and*

(iv) *the observer gain*
*is large enough for the estimation error*
*with*
*and*
*to converge asymptotically to zero as*

*Then a finite time instant*
*at which the control law* (4.1) *switches for the first time exists*.

*Proof* On the one hand, note that the functions

for some definite positive function
*θ* in the sense that such a convergence rate is increased as the observer gain increases.

The existence of the finite time instant

since the fact that the SEIR parameters fulfil the condition