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 , 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 , and are considered as the state vector, the measurable output signal (i.e. the infectious population) and the input signal of the system , respectively, and is used with
where . The first step to apply a coordinate transformation based on the Lie derivation is to determine the relative degree of the system. For such a purpose, the following definitions are taken into account: (i) The k th-order Lie derivative of along is with and (ii) the relative degree 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 for and .
From (3.2), , while , so the relative degree of the system is 3 in , i.e. except in the singular surface of the state space where the relative degree is not well defined. Since the relative degree of the system is exactly equal to the dimension of the state space for any , the nonlinear coordinate change defined as follows:
allows representing the SEIR model in the so-called normal form in a neighbourhood of any . Namely
The equations in (3.3) define a mapping whose Jacobian matrix , with for , is non-singular since if . Then the reverse transformation, namely , is available in order to obtain the original state vector from the new one whenever . By direct calculations, such a reverse transformation is given by
Both transformations and are smooth mappings, i.e. they have continuous partial derivatives of any order. Then defines a diffeomorphism on D. The feature that the relative degree of the system is equal to the system order allows to change it into a linear and controllable one around any point 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.
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 . Moreover, it follows by direct calculations that
One may express in the state space defined by via the application of the coordinate transformation in (3.6). Then it follows directly that . Thus, the state equation of the closed-loop system in the state space defined by can be written as
Furthermore, the output equation of the closed-loop system is with since . From (3.11) and the closed-loop output equation, it follows that
with ℓ denoting the order of the differentiation of . Finally, the dynamics of the closed-loop system (3.8) is obtained by direct calculations from (3.12). □
Remarks 3.1 (i) The controller parameters , for , will be adjusted so that the roots of the closed-loop system characteristic polynomial are located at prescribed positions, i.e. for and , with denoting the desired roots of . If one of the control objectives is to guarantee the exponential stability of the closed-loop system, then all roots of have to be in the open left-half plane, i.e. for all . Then the values , and for the controller parameters have to be chosen in order to achieve such a stability result. It implies that the strict positivity of the controller parameters is a necessary condition for the exponential stability of the closed-loop system.
The control (3.7) may be rewritten as follows:
by using (3.3) and (3.10), or
where is the control parameters vector, by using (3.3) and the facts that and .
The control law (3.7) is well defined for all except in the surface . However, the infection may be considered eradicated from the population once the infectious population strictly exceeds zero while it is smaller than one individual. So the vaccination strategy may be switched off when . This fact implies that the singularity in the control law is not going to be reached, 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 (3.15) is given by the linearizing control law (3.7) while denotes the eventual time instant after which the infection propagation may be assumed ended. Formally, such a time instant is defined as follows:
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.
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.
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 for . Such parameters have to be appropriately chosen in order to guarantee the following suitable properties: (i) the stability of the controlled SEIR model, (ii) the eradication of the infection, i.e. the asymptotic convergence of and to zero as time tends to infinity and (iii) the positivity property of the controlled SEIR model under a vaccination based on such a control strategy. The following theorems related to the choice of the controller tuning parameter values for are proven, in order to meet such properties under an eventual vaccination effort.
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 , where , by taking into account that , and . The initial condition in such a realization is bounded since it is related to via the coordinate transformation (3.3), and is assumed to be bounded. The controlled SEIR model is exponentially stable since the eigenvalues of the matrix A are the roots for of which are assumed to be in the open left-half plane. Then the state vector exponentially converges to zero as time tends to infinity while being bounded for all time. Moreover, and are also bounded and converge exponentially to zero as from the boundedness and exponential convergence to zero of as according to the first and second equations of the coordinate transformation (3.3). Then the infection is eradicated from the host population. Furthermore, the boundedness of follows from that of and , and the fact that the total population is constant for all time. Also, the exponential convergence of to the total population as is derived from the exponential convergence to zero of and as , and the fact that . Finally, from the third equation of (3.3), it follows that is bounded and it converges exponentially to zero as from the boundedness and convergence to zero of , and as . The facts that and as imply directly that . □
Remark 3.2 Theorem 3.2 implies the existence of a finite time instant after which the epidemics is eradicated if the vaccination control law (3.15) is used instead of that in (3.7). Concretely, such an existence derives from the convergence of to zero as 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
, and , so that ,
and satisfy the inequalities:
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
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 as in (3.11). From such a realization, taking into account the first equation in (3.3) and the fact that for are the eigenvalues of A, it follows that
for some constants for being dependent on the initial conditions , and . In turn, such initial conditions are related to the initial conditions of the SEIR model in its original realization, i.e. in the state space defined by via (3.3). The constants for can be obtained by solving the following set of linear equations:
where (3.3) and (3.17) have been used. Such equations can be more compactly written as , where
Once the desired roots of the characteristic equation of the closed-loop dynamics have been prefixed, the constants for of the time-evolution of are obtained from since is a non-singular matrix, i.e. an invertible matrix. In this sense, note that since is the Vandermonde matrix  and the roots for have been chosen different among them. Namely
where the functions and are defined as follows:
In particular, since , , , , , and by taking into account the constraints in (a). On the one hand, is proven directly from (3.17) as follows. One ‘a priori’ knows that . However, the sign of both and may not be ‘a priori’ determined from the initial conditions and constraints in (a). The following four cases may be possible: (i) and , (ii) and , (iii) and , and (iv) and . For the cases (i) and (ii), i.e. if , it follows from (3.17) that
where the facts that and, and since , have been taken into account. For the case (iii), i.e. if and , it follows from (3.17) that
by taking into account that , since and the fact that
where (3.20), (3.21), , and the constraints in (a) and (b) have been used. In particular, the coefficient multiplying to in (3.24) is non-negative if and satisfy the third inequality of the constraints (b) by taking into account and . This later inequality is directly implied by , , , and . Finally, for the case (iv), i.e. if and , it follows from (3.17) that
where the constraints , and , since , have been taken into account. In summary, if all partial populations are initially non-negative and the roots for of the closed-loop characteristic polynomial satisfy the constraints in (a) and (b). On the other hand, one obtains by direct calculations from (3.6) and (3.17) that
by taking into account that and . If one fixes the parameter then
where the fact that the function defined by
is zero for has been used. From the first equation in (3.27), it follows that and then
by applying such a relation between and in (3.27) and by taking into account that , and since . In this way, the non-negativity of has been proven. From the second equation in (3.27), it follows that and then
by applying such a relation between and in (3.27) and by taking into account that since , , , and and since . In this way, the non-negativity of has been proven. Note that the function defined by (3.28) is an upper-open parabola zero-valued for and so from the assumption that .
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 to zero as in (3.17) implies directly the existence of a finite time instant at which the control (3.15) switches off. Obviously, the non-negativity of , and is proven by following the same reasoning used in the part (i) of the current theorem. The non-negativity of is proven by using continuity arguments. In this sense, if reaches negative values for some starting from an initial condition , then passes through zero, i.e. there exists at least a time instant such that . Then it follows from (2.4) that
by introducing the control law (3.15) and taking into account the facts that and since has been used. Moreover, the non-negativity of , and as it has been previously proven, implies that , and . Also, since and from the definition of in (3.16). Then one obtains
from (3.31). The controller tuning parameters for are related to the roots for of the closed-loop characteristic polynomial , see Remark 3.1 (i), by
The assignment of for such that the constraints in (a) and (b) are fulfilled implies that
Then by taking into account (3.34) in (3.32). The facts that , and imply that via complete induction. Finally, the positivity of the controlled SEIR model follows from the non-negativity of , , and and Lemma 2.1.
On the other hand, it follows from (3.13) and (3.15) that
by taking into account that . Moreover,
where the facts that , , and have been used. If the roots of the polynomial satisfy the conditions in (a) and (b), it follows from (3.36) that
by taking into account the third equation in (3.34) and the non-negativity of , and . Finally, it follows that from (3.15) and (3.37). □
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.