An ideal control objective is that the removedbyimmunity 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 staticstate 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:
\{\begin{array}{c}\dot{x}(t)=f(x(t))+g(x(t))u(t),\hfill \\ y(t)=h(x(t)),\hfill \end{array}
(3.1)
where x(t)={[I(t)\phantom{\rule{0.25em}{0ex}}E(t)\phantom{\rule{0.25em}{0ex}}S(t)]}^{T}\in {\mathbb{R}}_{0+}^{3}, y(t)=I(t)\in {\mathbb{R}}_{0+} and u(t)=V(t)\in {\mathbb{R}}_{0+} are considered as the state vector, the measurable output signal (i.e. the infectious population) and the input signal of the system \mathrm{\forall}t\in {\mathbb{R}}_{0+} , respectively, and R(t)=NS(t)E(t)I(t) is used with
\begin{array}{r}f(x(t))=\left[\begin{array}{c}(\mu +\gamma )I(t)+\sigma E(t)\\ (\mu +\sigma )E(t)+{\beta}_{1}I(t)S(t)\\ \omega (I(t)+E(t))+(\mu +\omega )(NS(t)){\beta}_{1}I(t)S(t)\end{array}\right];\\ g(x(t))=\left[\begin{array}{c}0\\ 0\\ \mu N\end{array}\right];\phantom{\rule{2em}{0ex}}h(x(t))=I(t),\end{array}
(3.2)
where {\beta}_{1}=\beta /N. 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 thorder Lie derivative of h(x(t)) along f(x(t)) is {L}_{f}^{k}h(x(t))\stackrel{\mathrm{\Delta}}{=}\frac{\partial ({L}_{f}^{k1}h(x(t)))}{\partial x}f(x(t)) with {L}_{f}^{0}h(x(t))\stackrel{\mathrm{\Delta}}{=}h(x(t)) 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 {L}_{g}{L}_{f}^{k}h(x(t))=0 for k<r1 and {L}_{g}{L}_{f}^{r1}h(x(t))\ne 0.
From (3.2), {L}_{g}h(x(t))={L}_{g}{L}_{f}h(x(t))=0, while {L}_{g}{L}_{f}^{2}h(x(t))=\mu \sigma \beta I(t), so the relative degree of the system is 3 in D\stackrel{\mathrm{\Delta}}{=}\{x={[I\phantom{\rule{0.25em}{0ex}}E\phantom{\rule{0.25em}{0ex}}S]}^{T}\in {\mathbb{R}}_{0+}^{3}I\ne 0\}, i.e. \mathrm{\forall}x={[I\phantom{\rule{0.25em}{0ex}}E\phantom{\rule{0.25em}{0ex}}S]}^{T}\in {\mathbb{R}}_{0+}^{3} except in the singular surface I=0 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 x\in D, the nonlinear coordinate change defined as follows:
\begin{array}{r}\overline{I}(t)={L}_{f}^{0}h(x(t))=I(t),\\ \overline{E}(t)={L}_{f}h(x(t))=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]f(x(t))=(\mu +\gamma )I(t)+\sigma E(t),\\ \overline{S}(t)={L}_{f}^{2}h(x(t))=\left[\begin{array}{ccc}(\mu +\gamma )& \sigma & 0\end{array}\right]f(x(t))\\ \phantom{\overline{S}(t)}={(\mu +\gamma )}^{2}I(t)\sigma (2\mu +\sigma +\gamma )E(t)+\sigma {\beta}_{1}I(t)S(t),\end{array}
(3.3)
allows representing the SEIR model in the socalled normal form in a neighbourhood of any x\in D. Namely
\{\begin{array}{c}\dot{\overline{x}}(t)=\overline{f}(\overline{x}(t))+\overline{g}(\overline{x}(t))u(t),\hfill \\ y(t)=h(\overline{x}(t)),\hfill \end{array}
(3.4)
where \overline{x}(t)={[\overline{I}(t)\phantom{\rule{0.25em}{0ex}}\overline{E}(t)\phantom{\rule{0.25em}{0ex}}\overline{S}(t)]}^{T} and
The equations in (3.3) define a mapping \mathrm{\Phi}:{[I\phantom{\rule{0.25em}{0ex}}E\phantom{\rule{0.25em}{0ex}}S]}^{T}\to {[\overline{I}\phantom{\rule{0.25em}{0ex}}\overline{E}\phantom{\rule{0.25em}{0ex}}\overline{S}]}^{T} whose Jacobian matrix J(x(t))\stackrel{\mathrm{\Delta}}{=}[{J}_{i,j}(x(t))]\in {\mathbb{R}}^{3\times 3}, with {J}_{i,j}(x(t))\stackrel{\mathrm{\Delta}}{=}[\frac{\partial {\overline{x}}_{i}(t)}{\partial {x}_{j}(t)}] for i,j\in \{1,2,3\}, is nonsingular \mathrm{\forall}x\in D since Det[J(x(t))]={\sigma}^{2}{\beta}_{1}I(t)\ne 0 if I(t)\ne 0. Then the reverse transformation, namely {\mathrm{\Phi}}^{1}:{[\overline{I}\phantom{\rule{0.25em}{0ex}}\overline{E}\phantom{\rule{0.25em}{0ex}}\overline{S}]}^{T}\to {[I\phantom{\rule{0.25em}{0ex}}E\phantom{\rule{0.25em}{0ex}}S]}^{T}, is available in order to obtain the original state vector x(t) from the new one \overline{x}(t) whenever I=\overline{I}\ne 0. By direct calculations, such a reverse transformation is given by
\begin{array}{r}I(t)=\overline{I}(t);\phantom{\rule{2em}{0ex}}E(t)=\frac{1}{\sigma}[(\mu +\gamma )\overline{I}(t)+\overline{E}(t)];\\ S(t)=\frac{1}{\sigma {\beta}_{1}\overline{I}(t)}[(\mu +\gamma )(\mu +\sigma )\overline{I}(t)+(2\mu +\sigma +\gamma )\overline{E}(t)+\overline{S}(t)].\end{array}
(3.6)
Both transformations \mathrm{\Phi}(x(t)) and {\mathrm{\Phi}}^{1}(\overline{x}(t)) are smooth mappings, i.e. they have continuous partial derivatives of any order. Then \mathrm{\Phi}(x(t)) defines a diffeomorphism on D. The feature that the relative degree of the system is equal to the system order \mathrm{\forall}x\in D allows to change it into a linear and controllable one around any point x\in D via the coordinate transformation (3.3) and an exact linearization feedback control [23, 28]. The following result being relative to the inputoutput linearization of the system is established.
Theorem 3.1
The state feedback control law defined as
u(t)=\frac{1}{{L}_{g}{L}_{f}^{2}h(x(t))}[{L}_{f}^{3}h(x(t)){\lambda}_{0}h(x(t)){\lambda}_{1}{L}_{f}h(x(t)){\lambda}_{2}{L}_{f}^{2}h(x(t))],
(3.7)
where {\lambda}_{i} for i\in \{0,1,2\} are the controller tuning parameters, induces the linear closedloop dynamics given by
\stackrel{\u20db}{y}(t)+{\lambda}_{2}\ddot{y}(t)+{\lambda}_{1}\dot{y}(t)+{\lambda}_{0}y(t)=0
(3.8)
around any point x\in D.
Proof The following state equation for the closedloop system is obtained:
\left[\begin{array}{c}\dot{\overline{I}}(t)\\ \dot{\overline{E}}(t)\\ \dot{\overline{S}}(t)\end{array}\right]=\left[\begin{array}{c}\overline{E}(t)\\ \overline{S}(t)\\ \phi (\overline{x}(t)){L}_{f}^{3}h(x(t)){\lambda}_{0}\overline{I}(t){\lambda}_{1}\overline{E}(t){\lambda}_{2}\overline{S}(t)\end{array}\right]
(3.9)
by introducing the control law (3.7) in (3.4) and taking into account the coordinate transformation (3.3) and the fact that {L}_{g}{L}_{f}^{2}h(x(t))=\mu \sigma \beta I(t)=\mu \sigma \beta \overline{I}(t)\ne 0 \mathrm{\forall}x\in D. Moreover, it follows by direct calculations that
\begin{array}{rcl}{L}_{f}^{3}h(x(t))& =& [\sigma \beta (\mu +\omega ){(\mu +\gamma )}^{3}]I(t)+\sigma [{(\mu +\gamma )}^{2}+(2\mu +\sigma +\gamma )(\mu +\sigma )]E(t)\\ \sigma {\beta}_{1}\omega I(t)[I(t)+E(t)]\\ \sigma {\beta}_{1}(4\mu +\sigma +2\gamma +\omega )I(t)S(t)+{\sigma}^{2}{\beta}_{1}E(t)S(t)\sigma {\beta}_{1}^{2}{I}^{2}(t)S(t).\end{array}
(3.10)
One may express {L}_{f}^{3}h(x(t)) in the state space defined by \overline{x}(t) via the application of the coordinate transformation in (3.6). Then it follows directly that {L}_{f}^{3}h(x(t))=\phi (\overline{x}(t)). Thus, the state equation of the closedloop system in the state space defined by \overline{x}(t) can be written as
\dot{\overline{x}}(t)=A\overline{x}(t)\phantom{\rule{1em}{0ex}}\text{with}A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ {\lambda}_{0}& {\lambda}_{1}& {\lambda}_{2}\end{array}\right].
(3.11)
Furthermore, the output equation of the closedloop system is y(t)=C\overline{x}(t) with C=[1\phantom{\rule{0.25em}{0ex}}0\phantom{\rule{0.25em}{0ex}}0] since y(t)=I(t)=\overline{I}(t). From (3.11) and the closedloop output equation, it follows that
{y}^{(\ell )}(t)=C{A}^{\ell}{e}^{At}\overline{x}(0)\phantom{\rule{1em}{0ex}}\text{for}\ell \in \{0,1,2,3\}
(3.12)
with ℓ denoting the order of the differentiation of y(t). Finally, the dynamics of the closedloop system (3.8) is obtained by direct calculations from (3.12). □
Remarks 3.1 (i) The controller parameters {\lambda}_{i}, for i\in \{0,1,2\}, will be adjusted so that the roots of the closedloop system characteristic polynomial P(s)=Det(s{I}_{3}A) are located at prescribed positions, i.e. {\lambda}_{i}={\lambda}_{i}({r}_{j}) for i\in \{0,1,2\} and j\in \{1,\mathrm{2},\mathrm{3}\}, with ({r}_{j}) denoting the desired roots of P(s). If one of the control objectives is to guarantee the exponential stability of the closedloop system, then all roots of P(s)=(s+{r}_{1})(s+{r}_{2})(s+{r}_{3}) have to be in the open lefthalf plane, i.e. Re\{{r}_{j}\}>0 for all j\in \{1,2,3\}. Then the values {\lambda}_{0}={r}_{1}{r}_{2}{r}_{3}>0, {\lambda}_{1}={r}_{1}{r}_{2}+{r}_{1}{r}_{3}+{r}_{2}{r}_{3}>0 and {\lambda}_{2}={r}_{1}+{r}_{2}+{r}_{3}>0 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 closedloop system.

(ii)
The control (3.7) may be rewritten as follows:
\begin{array}{rcl}u(t)& =& \frac{(\mu +\omega )\sigma \beta {(\mu +\gamma )}^{3}+{\lambda}_{0}{\lambda}_{1}(\mu +\gamma )+{\lambda}_{2}{(\mu +\gamma )}^{2}}{\mu \sigma \beta}\\ \frac{\omega}{\mu N}[I(t)+E(t)]\frac{(3\mu +\sigma +2\gamma {\lambda}_{2})}{\mu N}S(t)\\ +\frac{{(\mu +\gamma )}^{2}+(2\mu +\sigma +\gamma )(\mu +\sigma )+{\lambda}_{1}{\lambda}_{2}(2\mu +\sigma +\gamma )}{\mu \beta}\frac{E(t)}{I(t)}\\ +\frac{\sigma}{\mu N}\frac{E(t)S(t)}{I(t)}\frac{\beta}{\mu {N}^{2}}I(t)S(t)\end{array}
(3.13)
by using (3.3) and (3.10), or
\begin{array}{rcl}u(t)& =& \frac{1}{\mu \sigma \beta \overline{I}(t)}[\phi (\overline{x}(t))+{\lambda}_{0}\overline{I}(t)+{\lambda}_{1}\overline{E}(t)+{\lambda}_{2}\overline{S}(t)]\\ =& \frac{1}{\mu \sigma \beta \overline{I}(t)}[\phi (\overline{x}(t))+{\mathrm{\Lambda}}^{T}\overline{x}(t)],\end{array}
(3.14)
where \mathrm{\Lambda}\stackrel{\mathrm{\Delta}}{=}{[{\lambda}_{0}\phantom{\rule{0.25em}{0ex}}{\lambda}_{1}\phantom{\rule{0.25em}{0ex}}{\lambda}_{2}]}^{T} is the control parameters vector, by using (3.3) and the facts that {L}_{f}^{3}h(x(t))=\phi (\overline{x}(t)) and {L}_{g}{L}_{f}^{2}h(x(t))=\mu \sigma \beta \overline{I}(t).

(iii)
The control law (3.7) is well defined for all x\in {\mathbb{R}}_{0+}^{3} except in the surface I=0. 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 0<{\delta}^{\prime}\le I(t)\le \delta <1. 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
{u}_{p}(t)=\{\begin{array}{cc}u(t)\hfill & \text{for}0\le t\le {t}_{f},\hfill \\ 0\hfill & \text{for}t{t}_{f}\hfill \end{array}
(3.15)
may be used instead of (3.7) in a practical situation. The signal u(t) in (3.15) is given by the linearizing control law (3.7) while {t}_{f} denotes the eventual time instant after which the infection propagation may be assumed ended. Formally, such a time instant is defined as follows:
{t}_{f}\stackrel{\mathrm{\Delta}}{=}min\{t\in {\mathbb{R}}_{0}^{+}\mid I(t)<\delta \text{for some}0\delta 1\}.
(3.16)
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 0\le t\le {t}_{f}, 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 inputoutput linearization strategy, makes the closedloop dynamics of the infectious population be given by (3.8). Such a dynamics depends on the control parameters {\lambda}_{i} for i\in \{0,1,2\}. 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 I(t) and E(t) 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 {\lambda}_{i} for i\in \{0,1,2\} are proven, in order to meet such properties under an eventual vaccination effort.
Theorem 3.2 Assume that the initial condition x(0)={[I(0)\phantom{\rule{0.25em}{0ex}}E(0)\phantom{\rule{0.25em}{0ex}}S(0)]}^{T}\in {\mathbb{R}}_{0+}^{3} is bounded, and all roots ({r}_{j}) for j\in \{1,2,3\} of the characteristic polynomial P(s) associated with the closedloop dynamics (3.8) are of strictly negative real part via an appropriate choice of the freedesign controller parameters {\lambda}_{i}>0 for i\in \{0,1,2\}. 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: E(t), I(t), S(t)I(t) and S(t)+R(t)=N[E(t)+I(t)] are bounded for all time, E(t)\to 0, I(t)\to 0, S(t)+R(t)\to N and S(t)I(t)\to 0 exponentially as t\to \mathrm{\infty}, and I(t)=o(1/S(t)).
Proof The dynamics of the controlled SEIR model (3.8) can be equivalently rewritten with the state equation (3.11) and the output equation y(t)=C\overline{x}(t), where C=[1\phantom{\rule{0.25em}{0ex}}0\phantom{\rule{0.25em}{0ex}}0], by taking into account that y(t)=\overline{I}(t), \dot{y}(t)=\overline{E}(t) and \ddot{y}(t)=\overline{S}(t). The initial condition \overline{x}(0)={[\overline{I}(0)\phantom{\rule{0.25em}{0ex}}\overline{E}(0)\phantom{\rule{0.25em}{0ex}}\overline{S}(0)]}^{T} in such a realization is bounded since it is related to x(0) via the coordinate transformation (3.3), and x(0) is assumed to be bounded. The controlled SEIR model is exponentially stable since the eigenvalues of the matrix A are the roots ({r}_{j}) for j\in \{1,2,3\} of P(s) which are assumed to be in the open lefthalf plane. Then the state vector \overline{x}(t) exponentially converges to zero as time tends to infinity while being bounded for all time. Moreover, I(t) and E(t) are also bounded and converge exponentially to zero as t\to \mathrm{\infty} from the boundedness and exponential convergence to zero of \overline{x}(t) as t\to \mathrm{\infty} 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 S(t)+R(t) follows from that of E(t) and I(t), and the fact that the total population is constant for all time. Also, the exponential convergence of S(t)+R(t) to the total population as t\to \mathrm{\infty} is derived from the exponential convergence to zero of I(t) and E(t) as t\to \mathrm{\infty}, and the fact that S(t)+E(t)+I(t)+R(t)=N \mathrm{\forall}t\in {\mathbb{R}}_{0+}. Finally, from the third equation of (3.3), it follows that S(t)I(t) is bounded and it converges exponentially to zero as t\to \mathrm{\infty} from the boundedness and convergence to zero of I(t), E(t) and \overline{x}(t) as t\to \mathrm{\infty}. The facts that I(t)\to 0 and S(t)I(t)\to 0 as t\to \mathrm{\infty} imply directly that I(t)=o(1/S(t)). □
Remark 3.2 Theorem 3.2 implies the existence of a finite time instant {t}_{f} 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 I(t) to zero as t\to \mathrm{\infty} via the application of the control law (3.7).
Theorem 3.3 Assume that an initial condition for the SEIR model satisfies R(0)\ge 0, x(0)\in {\mathbb{R}}_{0+}^{3}, i.e. I(0)\ge 0, E(0)\ge 0 and S(0)\ge 0, and the constraint S(0)+E(0)+I(0)+R(0)=N. Assume also that some strictly positive real numbers {r}_{j} for j\in \{1,2,3\} are chosen such that

(a)
0<{r}_{1}<\mu +min\{\sigma ,\gamma \}, {r}_{2}=\mu +\gamma and {r}_{3}>\mu +max\{\sigma ,\gamma \}, so that {r}_{3}>{r}_{2}>{r}_{1}>0,

(b)
{r}_{1} and {r}_{3} satisfy the inequalities:
\{\begin{array}{c}{r}_{1}+{r}_{3}\ge 2\mu +\sigma +\gamma +\beta \omega ,\hfill \\ {r}_{1}{r}_{3}\ge (\mu +\sigma )({r}_{1}+{r}_{3})+(\gamma \sigma )(2\mu +\sigma +\gamma ){(\mu +\gamma )}^{2},\hfill \\ ({r}_{3}{r}_{1})({r}_{3}\mu \gamma )\ge \sigma \beta .\hfill \end{array}
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 I(t)\ge 0, E(t)\ge 0 and S(t)\ge 0 \mathrm{\forall}t\in {\mathbb{R}}_{0+}, and

(ii)
the application of the control law (3.15) guarantees the epidemics eradication after a finite time {t}_{f}, the positivity of the controlled SEIR epidemic model \mathrm{\forall}t\in {\mathbb{R}}_{0}^{+} and that u(t)=V(t)\ge 1 \mathrm{\forall}t\in [0,{t}_{f}) so that u(t)\ge 0 \mathrm{\forall}t\in {\mathbb{R}}_{0+},
provided that the controller tuning parameters {\lambda}_{i} for i\in \{0,1,2\} are chosen such that ({r}_{j}) for j\in \{1,2,3\} are the roots of the characteristic polynomial P(s) 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 \overline{x}(t)={[\overline{I}(t)\phantom{\rule{0.25em}{0ex}}\overline{E}(t)\phantom{\rule{0.25em}{0ex}}\overline{S}(t)]}^{T} as in (3.11). From such a realization, taking into account the first equation in (3.3) and the fact that ({r}_{j}) for j\in \{1,2,3\} are the eigenvalues of A, it follows that
I(t)=\overline{I}(t)=y(t)={c}_{1}{e}^{{r}_{1}t}+{c}_{2}{e}^{{r}_{2}t}+{c}_{3}{e}^{{r}_{3}t}\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in {\mathbb{R}}_{0+}
(3.17)
for some constants {c}_{j} for j\in \{1,2,3\} being dependent on the initial conditions y(0), \dot{y}(0) and \ddot{y}(0). 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 x(t)={[I(t)\phantom{\rule{0.25em}{0ex}}E(t)\phantom{\rule{0.25em}{0ex}}S(t)]}^{T} via (3.3). The constants {c}_{j} for j\in \{1,2,3\} can be obtained by solving the following set of linear equations:
\begin{array}{r}\overline{I}(0)=y(0)={c}_{1}+{c}_{2}+{c}_{3}=I(0),\\ \overline{E}(0)=\dot{y}(0)=({c}_{1}{r}_{1}+{c}_{2}{r}_{2}+{c}_{3}{r}_{3})=(\mu +\gamma )I(0)+\sigma E(0),\\ \overline{S}(0)=\ddot{y}(0)={c}_{1}{r}_{1}^{2}+{c}_{2}{r}_{2}^{2}+{c}_{3}{r}_{3}^{2}\\ \phantom{\overline{S}(0)}={(\mu +\gamma )}^{2}I(0)\sigma (2\mu +\sigma +\gamma )E(0)+\sigma {\beta}_{1}I(0)S(0),\end{array}
(3.18)
where (3.3) and (3.17) have been used. Such equations can be more compactly written as {R}_{p}\cdot K=M, where
\begin{array}{r}{R}_{p}=\left[\begin{array}{ccc}1& 1& 1\\ {r}_{1}& {r}_{2}& {r}_{3}\\ {r}_{1}^{2}& {r}_{2}^{2}& {r}_{3}^{2}\end{array}\right],\phantom{\rule{2em}{0ex}}K=\left[\begin{array}{c}{c}_{1}\\ {c}_{2}\\ {c}_{3}\end{array}\right]\phantom{\rule{1em}{0ex}}\text{and}\\ M=\left[\begin{array}{c}I(0)\\ (\mu +\gamma )I(0)\sigma E(0)\\ {(\mu +\gamma )}^{2}I(0)\sigma (2\mu +\sigma +\gamma )E(0)+\sigma {\beta}_{1}I(0)S(0)\end{array}\right].\end{array}
(3.19)
Once the desired roots of the characteristic equation of the closedloop dynamics have been prefixed, the constants {c}_{j} for j\in \{1,2,3\} of the timeevolution of I(t) are obtained from K={R}_{p}^{1}M since {R}_{p} is a nonsingular matrix, i.e. an invertible matrix. In this sense, note that Det({R}_{p})=({r}_{2}{r}_{1})({r}_{3}{r}_{1})({r}_{3}{r}_{2})\ne 0 since {R}_{p} is the Vandermonde matrix [31] and the roots ({r}_{j}) for j\in \{1,2,3\} have been chosen different among them. Namely
K=\left[\begin{array}{c}{c}_{1}\\ {c}_{2}\\ {c}_{3}\end{array}\right]=\left[\begin{array}{c}\frac{F({r}_{2},{r}_{3})I(0)+\sigma G({r}_{2},{r}_{3})E(0)+\sigma {\beta}_{1}I(0)S(0)}{({r}_{2}{r}_{1})({r}_{3}{r}_{1})}\\ \frac{F({r}_{1},{r}_{3})I(0)+\sigma G({r}_{1},{r}_{3})E(0)+\sigma {\beta}_{1}I(0)S(0)}{({r}_{2}{r}_{1})({r}_{3}{r}_{2})}\\ \frac{F({r}_{1},{r}_{2})I(0)+\sigma G({r}_{1},{r}_{2})E(0)+\sigma {\beta}_{1}I(0)S(0)}{({r}_{3}{r}_{1})({r}_{3}{r}_{2})}\end{array}\right],
(3.20)
where the functions F:{\mathbb{R}}_{+}^{2}\to \mathbb{R} and G:{\mathbb{R}}_{+}^{2}\to \mathbb{R} are defined as follows:
\begin{array}{r}F(v,w)=vw(\mu +\gamma )(v+w)+{(\mu +\gamma )}^{2}\phantom{\rule{1em}{0ex}}\text{and}\\ G(v,w)=v+w(2\mu +\sigma +\gamma ).\end{array}
(3.21)
In particular, {c}_{1}=\frac{\sigma ({r}_{3}\mu \gamma )E(0)+\sigma {\beta}_{1}I(0)S(0)}{(\mu +\gamma {r}_{1})({r}_{3}{r}_{1})}>0 since I(0)\ge 0, S(0)\ge 0, E(0)\ge 0, F({r}_{2},{r}_{3})=0, G({r}_{2},{r}_{3})={r}_{3}\mu \gamma >0, \mu +\gamma {r}_{1}>0 and {r}_{3}{r}_{1}>0 by taking into account the constraints in (a). On the one hand, I(t)\ge 0 \mathrm{\forall}t\in {\mathbb{R}}_{0+} is proven directly from (3.17) as follows. One ‘a priori’ knows that {c}_{1}>0. However, the sign of both {c}_{2} and {c}_{3} may not be ‘a priori’ determined from the initial conditions and constraints in (a). The following four cases may be possible: (i) {c}_{2}\ge 0 and {c}_{3}\ge 0, (ii) {c}_{2}\ge 0 and {c}_{3}<0, (iii) {c}_{2}<0 and {c}_{3}\ge 0, and (iv) {c}_{2}<0 and {c}_{3}<0. For the cases (i) and (ii), i.e. if {c}_{2}\ge 0, it follows from (3.17) that
\begin{array}{rcl}I(t)& =& {c}_{1}{e}^{{r}_{1}t}+{c}_{2}{e}^{{r}_{2}t}+[I(0){c}_{1}{c}_{2}]{e}^{{r}_{3}t}\\ =& {c}_{1}({e}^{{r}_{1}t}{e}^{{r}_{3}t})+{c}_{2}({e}^{{r}_{2}t}{e}^{{r}_{3}t})+I(0){e}^{{r}_{3}t}\ge 0\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in {\mathbb{R}}_{0+},\end{array}
(3.22)
where the facts that I(0)={c}_{1}+{c}_{2}+{c}_{3}\ge 0 and, {e}^{{r}_{1}t}{e}^{{r}_{3}t}\ge 0 and {e}^{{r}_{2}t}{e}^{{r}_{3}t}\ge 0 \mathrm{\forall}t\in {\mathbb{R}}_{0+} since {r}_{1}<{r}_{2}<{r}_{3}, have been taken into account. For the case (iii), i.e. if {c}_{2}<0 and {c}_{3}\ge 0, it follows from (3.17) that
\begin{array}{rcl}I(t)& =& [I(0){c}_{2}{c}_{3}]{e}^{{r}_{1}t}+{c}_{2}{e}^{{r}_{2}t}+{c}_{3}{e}^{{r}_{3}t}\\ =& [I(0){c}_{3}]{e}^{{r}_{1}t}+{c}_{2}({e}^{{r}_{2}t}{e}^{{r}_{1}t})+{c}_{3}{e}^{{r}_{3}t}\ge 0\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in {\mathbb{R}}_{0+},\end{array}
(3.23)
by taking into account that I(0)={c}_{1}+{c}_{2}+{c}_{3}, {e}^{{r}_{2}t}{e}^{{r}_{1}t}\le 0 \mathrm{\forall}t\in {\mathbb{R}}_{0+} since {r}_{1}<{r}_{2} and the fact that
I(0){c}_{3}=\frac{[({r}_{3}{r}_{1})({r}_{3}\mu \gamma )\sigma {\beta}_{1}S(0)]I(0)+\sigma (\mu +\gamma {r}_{1})E(0)}{({r}_{3}{r}_{1})({r}_{3}\mu \gamma )}\ge 0,
(3.24)
where (3.20), (3.21), F({r}_{1},{r}_{2})=0, G({r}_{1},{r}_{2})={r}_{1}\mu \gamma <0 and the constraints in (a) and (b) have been used. In particular, the coefficient multiplying to I(0) in (3.24) is nonnegative if {r}_{1} and {r}_{3} satisfy the third inequality of the constraints (b) by taking into account \sigma {\beta}_{1}S(0)=\sigma \beta \frac{S(0)}{N}\le \sigma \beta and S(0)\le N. This later inequality is directly implied by I(0)\ge 0, E(0)\ge 0, S(0)\ge 0, R(0)\ge 0 and N=I(0)+E(0)+S(0)+R(0). Finally, for the case (iv), i.e. if {c}_{2}<0 and {c}_{3}<0, it follows from (3.17) that
\begin{array}{rcl}I(t)& =& [I(0){c}_{2}{c}_{3}]{e}^{{r}_{1}t}+{c}_{2}{e}^{{r}_{2}t}+{c}_{3}{e}^{{r}_{3}t}\\ =& I(0){e}^{{r}_{1}t}+{c}_{2}({e}^{{r}_{2}t}{e}^{{r}_{1}t})+{c}_{3}({e}^{{r}_{3}t}{e}^{{r}_{1}t})\ge 0\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in {\mathbb{R}}_{0+},\end{array}
(3.25)
where the constraints I(0)={c}_{1}+{c}_{2}+{c}_{3}\ge 0, {e}^{{r}_{2}t}{e}^{{r}_{1}t}\le 0 and {e}^{{r}_{3}t}{e}^{{r}_{1}t}\le 0 \mathrm{\forall}t\in {\mathbb{R}}_{0+}, since {r}_{1}<{r}_{2}<{r}_{3}, have been taken into account. In summary, I(t)\ge 0 \mathrm{\forall}t\in {\mathbb{R}}_{0+} if all partial populations are initially nonnegative and the roots ({r}_{j}) for j\in \{1,2,3\}of the closedloop 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 \overline{E}(t)=\dot{\overline{I}}(t) and \overline{S}(t)=\ddot{\overline{I}}(t). If one fixes the parameter {r}_{2}=\mu +\gamma then
where the fact that the function H:{\mathbb{R}}_{+}\to \mathbb{R} defined by
H(v)={v}^{2}(2\mu +\sigma +\gamma )v+(\mu +\sigma )(\mu +\gamma )
(3.28)
is zero for v={r}_{2}=\mu +\gamma has been used. From the first equation in (3.27), it follows that {c}_{3}(\mu +\gamma {r}_{3})=\sigma E(0){c}_{1}(\mu +\gamma {r}_{1}) and then
E(t)=\frac{1}{\sigma}[{c}_{1}(\mu +\gamma {r}_{1})({e}^{{r}_{1}t}{e}^{{r}_{3}t})+\sigma E(0){e}^{{r}_{3}t}]\ge 0\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in {\mathbb{R}}_{0+}
(3.29)
by applying such a relation between {c}_{1} and {c}_{3} in (3.27) and by taking into account that {c}_{1}(\mu +\gamma {r}_{1})>0, E(0)\ge 0 and {e}^{{r}_{1}t}{e}^{{r}_{3}t}\ge 0 \mathrm{\forall}t\in {\mathbb{R}}_{0+} since {r}_{1}<{r}_{3}. In this way, the nonnegativity of E(t) has been proven. From the second equation in (3.27), it follows that {c}_{3}H({r}_{3})=\sigma {\beta}_{1}I(0)S(0){c}_{1}H({r}_{1}) and then
S(t)=\frac{1}{\sigma {\beta}_{1}I(t)}[{c}_{1}H({r}_{1})({e}^{{r}_{1}t}{e}^{{r}_{3}t})+\sigma {\beta}_{1}I(0)S(0){e}^{{r}_{3}t}]\ge 0\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in {\mathbb{R}}_{0+}
(3.30)
by applying such a relation between {c}_{1} and {c}_{3} in (3.27) and by taking into account that {c}_{1}H({r}_{1})>0 since {r}_{1}<\mu +min\{\sigma ,\gamma \}, I(0)\ge 0, S(0)\ge 0, and I(t)\ge 0 and {e}^{{r}_{1}t}{e}^{{r}_{3}t}\ge 0 \mathrm{\forall}t\in {\mathbb{R}}_{0+} since {r}_{1}<{r}_{3}. In this way, the nonnegativity of S(t) has been proven. Note that the function H(v) defined by (3.28) is an upperopen parabola zerovalued for {v}_{1}=\mu +\sigma and {v}_{2}=\mu +\gamma so H({r}_{1})>0 from the assumption that {r}_{1}<\mu +min\{\sigma ,\gamma \}.

(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(t) to zero as t\to \mathrm{\infty} in (3.17) implies directly the existence of a finite time instant {t}_{f} at which the control (3.15) switches off. Obviously, the nonnegativity of I(t), E(t) and S(t) \mathrm{\forall}t\in [0,{t}_{f}] is proven by following the same reasoning used in the part (i) of the current theorem. The nonnegativity of R(t) \mathrm{\forall}t\in [0,{t}_{f}] is proven by using continuity arguments. In this sense, if R(t) reaches negative values for some t\in [0,{t}_{f}] starting from an initial condition R(0)\ge 0, then R(t) passes through zero, i.e. there exists at least a time instant {t}_{0}\in [0,{t}_{f}) such that R({t}_{0})=0. Then it follows from (2.4) that
\begin{array}{rcl}\dot{R}({t}_{0})& =& \gamma I({t}_{0})+\mu NV({t}_{0})\\ =& \gamma I({t}_{0})+\frac{\mu \sigma \beta +{\lambda}_{0}{\lambda}_{1}(\mu +\gamma )+{\lambda}_{2}{(\mu +\gamma )}^{2}{(\mu +\gamma )}^{3}}{\sigma \beta}N\\ +({\lambda}_{2}+\omega 3\mu \sigma 2\gamma )S({t}_{0})\\ +\frac{{(\mu +\gamma )}^{2}+(2\mu +\sigma +\gamma )(\mu +\sigma )+{\lambda}_{1}{\lambda}_{2}(2\mu +\sigma +\gamma )}{\beta}N\frac{E({t}_{0})}{I({t}_{0})}\\ +\sigma \frac{E({t}_{0})S({t}_{0})}{I({t}_{0})}\frac{\beta}{N}I({t}_{0})S({t}_{0})\end{array}
(3.31)
by introducing the control law (3.15) and taking into account the facts that V(t)=u(t) and I({t}_{0})+E({t}_{0})+S({t}_{0})=N since R({t}_{0})=0 has been used. Moreover, the nonnegativity of I(t), E(t) and S(t) \mathrm{\forall}t\in [0,{t}_{f}] as it has been previously proven, implies that I({t}_{0})\le N, E({t}_{0})\le N and S({t}_{0})\le N. Also, I({t}_{0})\ge \delta >0 since {t}_{0}<{t}_{f} and from the definition of {t}_{f} in (3.16). Then one obtains
\begin{array}{rcl}\dot{R}({t}_{0})& \ge & \gamma I({t}_{0})+\frac{\mu \sigma \beta +{\lambda}_{0}{\lambda}_{1}(\mu +\gamma )+{\lambda}_{2}{(\mu +\gamma )}^{2}{(\mu +\gamma )}^{3}}{\sigma \beta}N\\ +({\lambda}_{2}+\omega 3\mu \sigma 2\gamma \beta )S({t}_{0})\\ +\frac{{(\mu +\gamma )}^{2}+(2\mu +\sigma +\gamma )(\mu +\sigma )+{\lambda}_{1}{\lambda}_{2}(2\mu +\sigma +\gamma )}{\beta}N\frac{E({t}_{0})}{I({t}_{0})}\\ +\sigma \frac{E({t}_{0})S({t}_{0})}{I({t}_{0})}\end{array}
(3.32)
from (3.31). The controller tuning parameters {\lambda}_{i} for i\in \{0,1,2\} are related to the roots ({r}_{j}) for j\in \{1,2,3\} of the closedloop characteristic polynomial P(s), see Remark 3.1 (i), by
{\lambda}_{0}={r}_{1}{r}_{2}{r}_{3};\phantom{\rule{2em}{0ex}}{\lambda}_{1}={r}_{1}{r}_{2}+{r}_{1}{r}_{3}+{r}_{2}{r}_{3};\phantom{\rule{2em}{0ex}}{\lambda}_{2}={r}_{1}+{r}_{2}+{r}_{3}.
(3.33)
The assignment of {r}_{j} for j\in \{1,2,3\} such that the constraints in (a) and (b) are fulfilled implies that
Then \dot{R}({t}_{0})\ge 0 by taking into account (3.34) in (3.32). The facts that R(t)\ge 0 \mathrm{\forall}t\in [0,{t}_{0}), R({t}_{0})=0 and \dot{R}({t}_{0})\ge 0 imply that R(t)\ge 0 \mathrm{\forall}t\in [0,{t}_{f}] via complete induction. Finally, the positivity of the controlled SEIR model \mathrm{\forall}t\in {\mathbb{R}}_{0}^{+} follows from the nonnegativity of I(t), E(t), S(t) and R(t) \mathrm{\forall}t\in [0,{t}_{f}] and Lemma 2.1.
On the other hand, it follows from (3.13) and (3.15) that
\begin{array}{rcl}u(t)& =& \frac{\mu \sigma \beta {(\mu +\gamma )}^{3}+{\lambda}_{0}{\lambda}_{1}(\mu +\gamma )+{\lambda}_{2}{(\mu +\gamma )}^{2}}{\mu \sigma \beta}\\ +\frac{\omega}{\mu N}R(t)\frac{(3\mu +\sigma +2\gamma \omega {\lambda}_{2})}{\mu N}S(t)\\ +\frac{{(\mu +\gamma )}^{2}+(2\mu +\sigma +\gamma )(\mu +\sigma )+{\lambda}_{1}{\lambda}_{2}(2\mu +\sigma +\gamma )}{\mu \beta}\frac{E(t)}{I(t)}\\ +\frac{\sigma}{\mu N}\frac{E(t)S(t)}{I(t)}\frac{\beta}{\mu {N}^{2}}I(t)S(t)\end{array}
(3.35)
\mathrm{\forall}t\in [0,{t}_{f}] by taking into account that S(t)+E(t)+I(t)+R(t)=N. Moreover,
\begin{array}{rcl}u(t)& \ge & \frac{\mu \sigma \beta {(\mu +\gamma )}^{3}+{\lambda}_{0}{\lambda}_{1}(\mu +\gamma )+{\lambda}_{2}{(\mu +\gamma )}^{2}}{\mu \sigma \beta}+\frac{{\lambda}_{2}+\omega 3\mu \sigma 2\gamma \beta}{\mu N}S(t)\\ +\frac{{(\mu +\gamma )}^{2}+(2\mu +\sigma +\gamma )(\mu +\sigma )+{\lambda}_{1}{\lambda}_{2}(2\mu +\sigma +\gamma )}{\mu \beta}\frac{E(t)}{I(t)}\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in [0,{t}_{f}],\end{array}
(3.36)
where the facts that 0<\delta \le I(t)\le N, E(t)\ge 0, S(t)\ge 0 and R(t)\ge 0 \mathrm{\forall}t\in [0,{t}_{f}] have been used. If the roots of the polynomial P(s) satisfy the conditions in (a) and (b), it follows from (3.36) that
\begin{array}{rcl}u(t)& \ge & 1+\frac{{\lambda}_{2}+\omega 3\mu \sigma 2\gamma \beta}{\mu N}S(t)\\ +\frac{{(\mu +\gamma )}^{2}+(2\mu +\sigma +\gamma )(\mu +\sigma )+{\lambda}_{1}{\lambda}_{2}(2\mu +\sigma +\gamma )}{\mu \beta}\frac{E(t)}{I(t)}\ge 1\end{array}
(3.37)
\mathrm{\forall}t\in [0,{t}_{f}] by taking into account the third equation in (3.34) and the nonnegativity of S(t), E(t) and I(t) \mathrm{\forall}t\in [0,{t}_{f}]. Finally, it follows that u(t)\ge 0 \mathrm{\forall}t\in {\mathbb{R}}_{0}^{+} 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.