### 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 ${\lambda}_{i}$ 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 closed*-*loop dynamics* (3.8) *are of strictly negative real part via an appropriate choice of the free*-*design 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 left-half 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 closed-loop dynamics have been prefixed, the constants

${c}_{j}$ for

$j\in \{1,2,3\}$ of the time-evolution of

$I(t)$ are obtained from

$K={R}_{p}^{-1}M$ since

${R}_{p}$ is a non-singular 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 non-negative 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 non-negative and the roots

$(-{r}_{j})$ for

$j\in \{1,2,3\}$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

$\overline{E}(t)=\dot{\overline{I}}(t)$ 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 non-negativity 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 non-negativity of

$S(t)$ has been proven. Note that the function

$H(v)$ defined by (3.28) is an upper-open parabola zero-valued 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 non-negativity 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 non-negativity 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 non-negativity 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 closed-loop 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 non-negativity 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 non-negativity 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.