# Disease extinction in finite time for a continuous-time SIRS model

Abderrahman Iggidr, Mohamed Ouzahra, Gauthier Sallet

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Abderrahman Iggidr, Mohamed Ouzahra, Gauthier Sallet. Disease extinction in finite time for a continuous-time SIRS model. 2026. hal-05484414

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Disease extinction in finite time for a
continuous-time SIRS model

Abderrahman Iggidr<fnref n="1" />, Mohamed Ouzahra<fnref n="2" />*, Gauthier Sallet<fnref n="3" />

*Corresponding author(s). E-mail(s): mohamed.ouzahra@usmba.ac.ma;

Contributing authors: abderrahman.Iggidr@inria.fr;

gauthier.sallet@univ-lorraine.fr;

**Abstract**

This paper addresses the problem of controlling the spread of an epidemic through a multiplicatively controlled SIRS model, in which the control input modulates the infection rate in a bilinear manner. Classical approaches based on asymptotic stability only guarantee disease eradication as time tends to infinity, which may be inadequate for timely and effective intervention. To overcome this limitation, we investigate the finite-time behavior of the infected population, ensuring its convergence to zero either in finite time (FTS) or within a fixed time independent of the initial conditions (FxTS). Moreover, through an appropriate choice of the control gain, this convergence time can be arbitrarily prescribed, leading to prescribed-time stability (PrTS). We propose explicit feedback control laws under which the infected population vanishes in finite, fixed, or prescribed time, while the susceptible and recovered populations converge exponentially to their equilibrium values as time tends to infinity. Numerical simulations are provided to validate the theoretical results, illustrating rapid disease eradication and global system stability. These findings demonstrate the effectiveness of finite, fixed, and prescribed-time control strategies for bilinear epidemic models and offer practical guidelines for the design of responsive and robust public health interventions.

**Keywords:** Bilinear dynamical systems, SIRS model, Stability in finite time, Lyapunov method

<footnote n="1">¹Univ. de Lorraine, CNRS, Inria, IECL, F-57000 Metz, France.</footnote>

<footnote n="2">²*, Univ. de Fez, ENS, Bensouda, Fez, 30030, Fez-Meknes, Morocco.</footnote>

<footnote n="3">³Univ. de Lorraine, CNRS, IECL, F-57000 Metz, France.</footnote>
# 1 Introduction

The concept of stability is one of the foundational notions in control theory. It describes the system's response to perturbations. In this framework, asymptotic stability ensures that system trajectories tend to an equilibrium point as time progresses to infinity, but it does not specify how fast this convergence occurs. However, many real-world applications require convergence to the desired state within a finite time, which may or may not depend on the initial conditions. This requirement has motivated the introduction of stronger notions of stability, namely finite-time stability (FTS), fixed-time stability (FxTS) and prescribed-time stability (PrTS). Finite-time stability guarantees that the trajectories reach the equilibrium in a finite time T, which may depend on the initial conditions. Fixed-time stability strengthens this property by ensuring convergence within a uniformly bounded time, independent of the initial conditions. Moreover, when fixed-time stability can be achieved for any a priori chosen time, the property is referred to as prescribed-time stability. These concepts, introduced by Roxin [10] and developed further by Bhat and Bernstein [2], have been widely studied in recent years (e.g., [3–6, 11]). Extensions to infinite-dimensional controlled systems can be found in [1, 7, 9].

In many control problems, the objective is not necessarily the finite-time stabilization of the entire state vector, but rather the finite-time regulation of a critical subset of the state variables. This leads to the concept of partial finite-time stability, where only selected components, often those associated with available output measurements, are required to converge in finite time. Such situations naturally arise in epidemic modeling, where the primary control objective is to eliminate the infectious compartment as quickly as possible, without imposing the same convergence requirement on the remaining compartments (such as the susceptible or recovered populations). This framework is particularly relevant for the design of intervention strategies during outbreaks, where rapid suppression of the disease spread is essential. In this context, partial finite-time stabilization ensures that the infected population vanishes in finite (or fixed) time, while the other compartments are allowed to converge asymptotically.

This paper investigates a multiplicatively controlled SIRS epidemic model, where the control is the treatment rate of infected individuals. The proposed feedback law is designed to drive the number of infected individuals $I(t)$ to zero in finite or fixed/pre-scribed time, while the susceptible and recovered populations $S(t)$ and $R(t)$ decay exponentially to their respective equilibrium values. Numerical simulations illustrate the effectiveness of the proposed strategy in driving the epidemic to extinction rapidly, demonstrating the practical potential of finite/fixed-time control techniques in public health applications.

In the next section, we recall the notions of finite-time, fixed-time and prescribed-time stability, and then present a lemma that will be used to prove fixed-time and prescribed finite-time stability. In the third section, we provide a necessary condition for finite-time stability and state and prove the main results on finite-time, fixed-time, and prescribed-time stability. In Section 4, we investigate finite-time stabilization using output control. In Section 5, we provide extensions to some class of nonlinear systems with bilinear control. Section 6 is devoted to simulations for specific parameters and initial conditions. The conclusion is given in the last section.
## 2 Preliminary

### 2.1 Notions of stability in finite time

Let us recall the notions of finite-time (FTS), fixed-time (FxTS) and prescribed time stability (PrTS).

**Definition 1** A control system $\dot{x} = f(x, u)$, with $u$ being the control input and $x$ the corresponding state, is said to be finite-time stabilizable toward an equilibrium $E_0$ if:

* $E_0$ is a Lyapunov stable equilibrium;

* there exists a feedback control such that the closed-loop system admits a unique solution; and there exists $T = T(x_0) > 0$ such that $x(t) = E_0$ for all $t \ge T$.

In this case, the time $T_*(x_0) := \inf\{t > 0 : x(t) = E_0\}$ is called the settling time.

If in addition $\sup_{x_0} T_*(x_0) < +\infty$, we say that the system is fixed-time stable (FxTS).

In the case where the settling time can be chosen a priori and arbitrarily, we refer to this property as prescribed-time stability (PrTS).

**Remark 1** • In the case $E_0 = 0$, we also speak of finite-time (or fixed-time) stability at the origin.

* If, in the above definition, the relation $x(t) = E_0 \quad \forall t \ge T$ is replaced by $Cx(t) = CE_0 \quad \forall t \ge T$ for some output operator $C$, then we speak of partial finite-time (or partial fixed/prescribed-time) stability. Moreover, if $x_1 := Cx$ is a state variable (i.e., a trajectory of the system in closed-loop), we also say that $x_1$ is FTS (or FxTS, PrTS).

From Lemma 2 of [8], we have the following result regarding FxTS:

**Lemma 1** Consider the following system:

$$ \dot{x}(t) = g(t, x(t)), \quad (1) $$

where $g: \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{R}^n$ is a possibly nonlinear function such that $g(t, 0) = 0$, $t \ge 0$. Let $V: \mathbb{R}^n \to \mathbb{R}$ be a continuous radially unbounded function and positive defined.

If $x: t \in \mathbb{R}^+ \to x(t) \in \mathbb{R}^n$ is a solution $x(t)$ of (1) satisfying the inequality:

$$ \frac{d}{dt}V(x(t)) \le -aV^{1-\nu}(x(t)) - bV^{1+\nu}(x(t)), $$

for some $a, b > 0$ and $0 < \nu < 1/2$, then we have

$$ x(t) = 0, \ t \ge T_{\max} := \frac{\pi}{2\nu\sqrt{ab}}. $$

**Remark 2** If the main inequality of the above lemma holds for the value $b = 0$, then the system exhibits finite-time stability, with a settling time that depends on the initial state.
## 2.2 Considered SIRS-model

Consider the following SIRS epidemic model with control:

$$ egin{cases} \dot{S} = \Lambda - \mu S + \eta R - \beta SI + u p I, \\ \dot{I} = \beta SI - (\mu + \gamma)I - u I, \\ \dot{R} = \gamma I - (\mu + \eta)R + u q I, \end{cases} \qquad (2) $$

where $p, q \ge 0$ satisfy $p + q = 1$. The parameters $\mu, \eta, \beta$, and $\gamma$ denote, respectively, the natural death rate, the rate of loss of immunity, the transmission rate, and the recovery rate, while $\Lambda$ represents the recruitment (birth) rate into the susceptible class. The control input $u \ge 0$ corresponds to the treatment or intervention rate applied to infected individuals.

This system can be written in the following form:

$$ \dot{x} = X(x) + uY(x) \qquad (3) $$

where

$$ x = \begin{pmatrix} S \\ I \\ R \end{pmatrix}, \quad X(x) = \begin{pmatrix} \Lambda - \mu S + \eta R - \beta SI \\ \beta SI - \mu I - \gamma I \\ \gamma I - (\mu + \eta)R \end{pmatrix} \quad \text{and} \quad Y(x) = \begin{pmatrix} pI \\ -I \\ qI \end{pmatrix}. $$

The system (3) possesses the Disease Free Equilibrium point (DFE) $E_0 = (\frac{\Lambda}{\mu}, 0, 0)$.

Moreover, by performing the change of variable $\tilde{S} := S^* - S$ with $S^* = \frac{\Lambda}{\mu}$, the system is transformed into one with an equilibrium at the origin.

For the free uncontrolled system ($u = 0$), the basic reproduction number is given by

$$ \mathcal{R}_0 = \frac{\beta\Lambda}{\mu(\mu + \gamma)}. $$

It is well known that if $\mathcal{R}_0 < 1$, then the DFE is a globally asymptotically stable equilibrium for the uncontrolled system $\dot{x} = X(x)$. When $\mathcal{R}_0 > 1$, the uncontrolled system admits an endemic equilibrium which is globally asymptotically stable equilibrium.

# 3 Stabilization in finite-time of the SIRS model

## 3.1 A necessary condition of FTS

Let us consider the uncontrolled version of the system (3):

$$ \dot{\varphi} = X(\varphi), \quad \varphi(0) = x_0 \qquad (4) $$

We have the following result.

**Theorem 2** If the system (3) is FTS toward an equilibrium $E_0$, then for all $x_0 \in \mathbb{R}^3$, we have:
$$Y(\varphi_{x_0}(t)) = 0, \forall t \ge 0 \Rightarrow x_0 = E_0$$

where $\varphi_{x_0}$ is the solution of (4).

*Proof* Let $x_0 \in \mathbb{R}^3$ be such that $Y(\varphi_{x_0}(t)) = 0, \forall t \ge 0$. Then $\varphi_{x_0}(t)$ solves the system (3) regardless the choice of the control $u$. Hence the FTS assumption implies that $\varphi_{x_0}(t) = E_0, \forall t \ge T$ for some $T = T(x_0) > 0$.

But for all $t \ge 0$, we have $E_0 = \varphi_{E_0}(t)$. It follows that $\varphi_{x_0}(t) = \varphi_{E_0}(t), \forall t \ge T$, and hence $x_0 = E_0$.

□

We deduce the following non global stability in finite time.

**Corollary 3** The system (2) is not globally FTS.

*Proof* Letting $x_0 = (\frac{\Lambda}{\mu}, 0, R_0)$ with $R_0 \neq 0$, we can see that $Y(\varphi_{x_0}(t)) = 0, \forall t \ge 0$, while $x_0 \neq E_0$. Hence the global FTS fails to hold according to the previous theorem.

□

## 3.2 Finite-time stabilization

**Theorem 4** Assume that $\frac{\beta\Lambda}{\mu} \ge \mu + \gamma$ (i.e., $R_0 \ge 1$).

Then the component $I$ of the system (2) can be stabilized in finite time using the following positive feedback law, involving only the variable $I$:

$$u = \left( \frac{\beta\Lambda}{\mu} - (\mu + \gamma) \right) + \mathbf{1}_{(I \neq 0)} I^{-\rho}, \quad 0 < \rho < 1, \qquad (5)$$

where $\mathbf{1}_{(I \neq 0)}$ denotes the indicator function of the set $\{I \neq 0\}$.

Furthermore, this control guarantees the exponential convergence of the components $S$ and $R$ towards $\frac{\Lambda}{\mu}$ and $0$ respectively.

*Proof* First let us point out that, according to the nature of the model, we deal with positive solutions $(S, R, I)$, in particular only positive initial solutions $(S_0, I_0, R_0)$ are allowed. Moreover, one can show that the following compact set is positively invariant:

$$\Omega = \{(S, I, R) / S, I, R \ge 0, S + I + R \le \frac{\Lambda}{\mu}\}$$

Let us set $z = (S, I, R)$. Taking into account the positivity of $I$, we consider the following candidate Lyapunov function that involves only the component $I$ of the full state $z: V(z) = I \ge 0$.

Then, using the second equation of (2), we have for any solution $(S, I, R)$:

$$\dot{I} \le \left( \frac{\beta\Lambda}{\mu} - (\mu + \gamma) \right) I - uI.$$
Now the control

$$u = \left( \frac{\beta\Lambda}{\mu} - (\mu + \gamma) \right) + I^{-\rho} \mathbf{1}_{(I \neq 0)}, \quad 0 < \rho < 1$$

is positive if

$$\frac{\beta\Lambda}{\mu} \geq \mu + \gamma \text{ i.e., } \mathcal{R}_0 \geq 1.$$

Thanks to the positively invariance property of $\Omega$, the solution $(S, I, R)$ lies in $\Omega$ whenever the initial condition is chosen in $\Omega$.

Let $(S_0, I_0, R_0) \in \Omega$. We may assume that $I_0 \neq 0$, since if $I_0 = 0$, then the second equation of (2) admits the trivial solution.

Then, with the above choice of $u$, we obtain the following Lyapunov differential inequality (LDI):

$$\dot{I} \leq -I^{1-\rho}.$$

Then, over the maximal time interval $[0, \tau)$ in which $I(t)$ does not vanish, the above inequality (LDI) can be written in the following form:

$$\dot{I}I^{\rho-1} \leq -1, \forall t \in [0, \tau),$$

which by integrating gives

$$I^{\rho}(t) \leq I^{\rho}(0) - \rho t, \forall t \in [0, \tau)$$

In particular, this requires

$$0 < I^{\rho}(0) - \rho t, \forall t \in [0, \tau),$$

from which, we deduce that $\tau \leq \frac{I^{\rho}(0)}{\rho}$ and $I(t) = 0$ for all $t \geq T_* := \frac{I^{\rho}(0)}{\rho}$. In other words, the settling time for $I$ admits the upper bound

$$T^* := \inf\{t \geq 0 : I(t) = 0\} \leq \frac{I^{\rho}(0)}{\rho}.$$

Furthermore, we have the exponential stabilization of the remaining components $S(t)$ and $R(t)$. More precisely, substituting back into the full system (2), we deduce from the third equation that :

$$R(t) = R(T_*)e^{-(\mu+\eta)(t-T_*)}, \quad \forall t \geq T_* \quad (6)$$

Moreover, summing up the first and the last equation, we get

$$\frac{d}{dt}(S + R) = \Lambda - \mu(S + R), \forall t \geq T_*,$$

which, together with the above estimate on $R$, gives:

$$S(t) = \frac{\Lambda}{\mu} + \left( S(T_*) + R(T_*) - \frac{\Lambda}{\mu} \right) e^{-\mu(t-T_*)} - R(T_*)e^{-(\mu+\eta)(t-T_*)}, \forall t \geq T_* \quad (7)$$

**Remark 3** • Notice that in the case $\frac{\beta\Lambda}{\mu} < \mu + \gamma$, one may just use the control $u = \mathbf{1}_{(I \neq 0)} I^{-\rho}$, $0 < \rho < 1$, to get the result of the above theorem.

• We have the following estimates for the upper bound of the settling time $T_\rho := \frac{I^{\rho}(0)}{\rho}$: $T_\rho \to +\infty$ as $\rho \to 0^+$, and $T_\rho \to I(0)$ as $\rho \to 1^-$.

We observe that the closer $\rho$ is to $1^-$, the smaller the settling time, providing the best possible estimate of the settling time for the considered strategy of control.
### 3.3 Fixed and prescribed time stabilization

In the previous subsection, we proved finite-time stability (FTS) with a settling time that depends on the initial state. Here, we aim to prove FTS with a uniform settling time valid for all initial states. This requires a more accurate estimate of the Lyapunov function, which can be obtained using Lemma 1.

In the next theorem, we prove that the component $I$ of system (2) can be stabilized in fixed time using a positive feedback law. Moreover, this control law also guarantees the uniform exponential stability of the components $S$ and $R$.

**Theorem 5** Assume that $\frac{\beta\Lambda}{\mu} \geq \mu + \gamma$.

Then the component $I$ of the system (2) is stabilizable in fixed time using a positive feedback law involving only the variable $I$. Furthermore, this control guarantees the uniform exponential stability of the components $S$ and $R$.

*Proof* Under the assumption $\frac{\beta\Lambda}{\mu} \geq \mu + \gamma$, we consider the control

$$u = \left( \frac{\beta\Lambda}{\mu} - (\mu + \gamma) \right) + \mathbf{1}_{(I \neq 0)} I^{-\rho} + I^{\rho}, \quad 0 < \rho < 1/2, \quad (8)$$

which leads to

$$\dot{I} \leq -(I^{1-\rho} + I^{1+\rho})$$

giving rise to fixed-time stability at $T \leq T^* = \frac{\pi}{2\rho}$, according to Lemma 1 which we can apply

by taking $V = I$ and $g(t, I) = \beta SI - \mu I - \gamma I - uI$ for any given continuous function $S$ such that $0 \leq S \leq \frac{\Lambda}{\mu}$.

Following the same techniques as for finite-time stabilization, we further obtain the uniform exponential stability of $S$ and $R$ (i.e., the exponential estimate do not depend on the initial state) $R(t)$ and $S(t)$ are given by formulas analogous to (6) and (7).

□

**Corollary 6** Assume that $\frac{\beta\Lambda}{\mu} \geq \mu + \gamma$.

Then the component $I$ of the system (2) is stabilizable in prescribed time using a positive feedback law involving only the variable $I$. Furthermore, this control guarantees the uniform exponential stability of the components $S$ and $R$.

*Proof* Let us consider the following control law with gain

$$u = \left( \frac{\beta\Lambda}{\mu} - (\mu + \gamma) \right) + c(\mathbf{1}_{\{I \neq 0\}} I^{-\rho} + I^{\rho}), \quad 0 < \rho < 1/2, c > 0. \quad (9)$$

This choice yields

$$\dot{I} \leq -c(I^{1-\rho} + I^{1+\rho}).$$

Following the same arguments as in the proof of Theorem 5, and using again Lemma 1, we obtain that

$$I(t) = 0, \quad \forall t \geq T := \frac{\pi}{2c\rho}.$$
Hence, by choosing an appropriate value of the control gain c, finite-time stability (FTS) is achieved at any a priori fixed time T.

□

Remark 4 • In the case $\frac{\beta\Lambda}{\mu} \le \mu + \gamma$, the above result still holds with the use of the control $u = c(\mathbf{1}_{(I \ne 0)}I^{-\rho} + I^{\rho})$, $0 < \rho < 1/2$, $c > 0$.

• We observe that the larger the value of $\rho$ or $c$, the smaller the upper bound of the settling time.

# 4 Stabilization in finite time using output control

So far, we were using a feedback control $u(x(t))$ to stabilize system (2). However, in general, one does not have access to the actual state of the system and the only available measurement is some function of the state. So here, we are interested in stabilizing system (2) using an output feedback, i.e., a stabilizing control which depends only on the available output $y(t) = h(t, x(t))$.

## 4.1 Stability in finite time when measuring a proportion of the infected population

We assume that the only available measurement is a proportion of the infected population, i.e., only an output $y(t) = \alpha(t)I(t)$ is available. The measured proportion $0 < \alpha(t) < 1$, which we suppose continuous, is not known in general.

### 4.1.1 FTS with output control

Consider the system:

$$ 
\left\{ 
\begin{aligned}
\dot{S} &= \Lambda - \mu S + \eta R - \beta SI + (1-p)u, \\
\dot{I} &= \beta SI - \mu I - \gamma I - uI, \\
\dot{R} &= \gamma I - (\mu + \eta)R + puI, \\
y(t) &= \alpha(t)I(t)
\end{aligned}
\right.
\qquad (10) $$

**Theorem 7** The system (10) is FTS with the control law:

$$ u = \left( \frac{\beta\Lambda}{\mu} - (\mu + \gamma) \right) + \mathbf{1}_{(y \ne 0)} y^{-\rho}, \quad 0 < \rho < 1. $$

*Proof* With the control

$$ u = \left( \frac{\beta\Lambda}{\mu} - (\mu + \gamma) \right) + \mathbf{1}_{(y \ne 0)} y^{-\rho}, \quad 0 < \rho < 1, $$
we have

$$ \dot{I} = \beta SI - \mu I - \gamma I - u I = \left( \beta S - \frac{\beta \Lambda}{\mu} - \mathbf{1}_{(y \neq 0)} y^{-\rho} \right) I $$

Taking into account that $S \le \frac{\Lambda}{\mu}$, we obtain $\dot{I} \le -\mathbf{1}_{(y \neq 0)} y^{-\rho} I = -\alpha(t)^{-\rho} I^{1-\rho}$. Hence, we get

$$ \dot{I} \le -I^{1-\rho} $$

from which, we derive the FTS as in Subsection 3.2. □

### 4.1.2 FxTS and PrTS with output control

**Theorem 8** The system (10) is FxTS with the control law:

$$ u = \left( \frac{\beta \Lambda}{\mu} - \epsilon(\mu + \gamma) \right) + \mathbf{1}_{(y \neq 0)} y^{-\rho} + y^{\rho}, \quad 0 < \rho < 1/2, \ 0 < \epsilon < 1. $$

*Proof* We take the following output-feedback control that uses only the output y:

$$ u = \left( \frac{\beta \Lambda}{\mu} - \epsilon(\mu + \gamma) \right) + \mathbf{1}_{(y \neq 0)} y^{-\rho} + y^{\rho}, \quad 0 < \rho < 1, \ 0 < \epsilon < 1, $$

and let $x_0 \in \Omega$.

Taking into account that $S \le \frac{\Lambda}{\mu}$, we obtain

$$ \dot{I} \le -(\mathbf{1}_{(y \neq 0)} y^{-\rho} + y^{\rho}) I. $$

Assuming first that $I \le 1$, and using the monotony of $f: s \mapsto s^{-\rho} + s^{\rho}$, we can see that we have (since $y \le I$) $f(y) \ge f(I)$

$$ \mathbf{1}_{(y \neq 0)} y^{-\rho} + y^{\rho} \ge \mathbf{1}_{(I \neq 0)} I^{-\rho} + I^{\rho} $$

Then we obtain the inequality

$$ \dot{I} \le -I^{1-\rho} - I^{1+\rho}, $$

leading to FxTS, according to Subsection 3.3, at least in the situations where the condition $I \le 1$ is fulfilled.

Let us now discuss the general case of $I$.

* If $\frac{\Lambda}{\mu} \le 1$, then we have $I \le 1$ (since in $\Omega$, we have $I \le \frac{\Lambda}{\mu}$). In this case, one can allow the value $\epsilon = 1$ in the expression of the steering control.

* In the case $\frac{\Lambda}{\mu} > 1$, we consider the control:

$$ u = \left( \frac{\beta \Lambda}{\mu} - \epsilon(\mu + \gamma) \right) + \mathbf{1}_{(y \neq 0)} y^{-\rho} + y^{\rho}, $$

with $0 < \rho < 1$ and $0 < \epsilon < 1$, so that

$$ \dot{I} \le -(1 - \epsilon)(\mu + \gamma)I := -hI, $$

which implies

$$ I(t) \le I_0 e^{-ht} \le \frac{\Lambda}{\mu} e^{-ht}. $$

Therefore, we can choose $t_1 > 0$, independent of the initial conditions, such that

$$ I(t) \le 1, \quad \forall t \ge t_1. $$

Thus, the settling time can be chosen as $T_* = \sup(t_1, \frac{\pi}{2\rho})$. □
**Corollary 9** The system (10) is FxTS with the control law:

$$u = \left( \frac{\beta\Lambda}{\mu} - \epsilon(\mu + \gamma) \right) + c(\mathbf{1}_{(y \neq 0)}y^{-\rho} + y^{\rho}), \quad 0 < \rho < 1/2, \ 0 < \epsilon < 1,$$

where $c > 0$ is again control that depends on the given prescribed time.

## 4.2 FTS and FxST when measuring a proportion of the incidence

Here we assume that the measurable output is a proportion of the new infected individuals per unit time (information usually provided by Public Health Services through epidemiological surveillance). That is $y(t) = \alpha(t) \beta S(t) I(t)$.

Therefore we consider the following system:

$$\left\{
\begin{aligned}
\dot{S} &= \Lambda - \mu S + \eta R - \beta SI + (1 - p) u, \\
\dot{I} &= \beta SI - \mu I - \gamma I - u I, \\
\dot{R} &= \gamma I - (\mu + \eta) R + p u I, \\
y &= \alpha(t) \beta S(t) I(t)
\end{aligned}
\right.
\quad (11)$$

where $0 < \alpha(t) < 1$ is continuous.

### 4.2.1 FTS with output control when measuring a proportion of the incidence

**Theorem 10** The system (11) is FTS with the output control law:

$$u = \left( \frac{\beta\Lambda}{\mu} - (\mu + \gamma) \right) + \mathbf{1}_{(y \neq 0)} y^{-\rho}, \quad 0 < \rho < 1.$$

*Proof* We take the following output-feedback control that uses only the output $y$:

$$u = \left( \frac{\beta\Lambda}{\mu} - (\mu + \gamma) \right) + \mathbf{1}_{(y \neq 0)} y^{-\rho}, \quad 0 < \rho < 1.$$

We then have

$$\dot{I} = \beta SI - \mu I - \gamma I - u I = (\beta S - \frac{\beta\Lambda}{\mu} - \mathbf{1}_{(y \neq 0)} y^{-\rho}) I$$

Taking into account that $S \le \frac{\Lambda}{\mu}$, we obtain $\dot{I} \le -\mathbf{1}_{(y \neq 0)}y^{-\rho} I = -(\alpha(t) \beta S(t))^{-\rho} I^{1-\rho}$.

Hence, we get again

$$\dot{I} \le - \left( \alpha(t) \frac{\beta\Lambda}{\mu} \right)^{-\rho} I^{1-\rho}$$

Using that $0 < \alpha(t) < 1$, we get

$$\dot{I} \le - \left( \frac{\beta\Lambda}{\mu} \right)^{-\rho} I^{1-\rho},$$

which leads to FTS. □
Remark 5 • Notice that one can also integrate the first inequality, which may improve the bound of the settling time, method that we avoid as $\alpha(t)$ is unknown.

• We can also use a gain control $c > 0$ in the output part of the control:

$$u = \left( \frac{\beta\Lambda}{\mu} - (\mu + \gamma) \right) + \mathbf{1}_{(y \neq 0)}(cy)^{-\rho},$$

so that

$$\dot{I} \leq - \left( c \alpha(t) \frac{\beta \Lambda}{\mu} \right)^{-\rho} I^{1-\rho}$$

which give

$$\dot{I} \leq - \left( c \frac{\beta \Lambda}{\mu} \right)^{-\rho} I^{1-\rho}$$

Thus, by taking $c$ large enough, we can improve the upper bound on the settling time.

### 4.2.2 FxTS and PrTS with output control when measuring a proportion of the incidence

**Theorem 11** The system (11) is FxTS with the control law:

$$u = \left( \frac{\beta\Lambda}{\mu} - \epsilon(\mu + \gamma) \right) + \mathbf{1}_{(y \neq 0)} y^{-\rho} + y^{\rho}, \quad 0 < \rho < 1/2, \ 0 < \epsilon < 1.$$

*Proof* First, by working in $\Omega$, we can assume that $y \leq \beta \frac{\Lambda}{\mu} I$.

Then, if $\beta \frac{\Lambda}{\mu} \leq 1$, we obtain the FxTS, as in the previous subsection, by using the following feedback law:

$$u = \left( \frac{\beta\Lambda}{\mu} - (\mu + \gamma) \right) + \mathbf{1}_{(y \neq 0)} y^{-\rho} + y^{\rho}, \quad 0 < \rho < 1.$$

Otherwise, as in the above subsection, we consider the control:

$$u = \left( \frac{\beta\Lambda}{\mu} - \epsilon(\mu + \gamma) \right) + \mathbf{1}_{(y \neq 0)} y^{-\rho} + y^{\rho}, \quad 0 < \rho < 1,$$

where $0 < \epsilon < 1$.

□

**Corollary 12** The system (11) is PxTS with the control law:

$$u = \left( \frac{\beta\Lambda}{\mu} - \epsilon(\mu + \gamma) \right) + c(\mathbf{1}_{(y \neq 0)}y^{-\rho} + y^{\rho}), \quad 0 < \rho < 1/2, \ 0 < \epsilon < 1, c > 0,$$

where the control gain is chosen according to the prescribed time.
# 5 Stabilization in finite-time of a generalized SIRS model

Consider the system:

$$ egin{cases} \dot{S} = \Lambda - \mu S + \eta R - \beta(t)f(S)I + upI \\ \dot{I} = \beta(t)f(S)I - \mu I - \gamma I - uI \\ \dot{R} = \gamma I - (\mu + \eta)R + uqI \end{cases} \tag{12} $$

where $p, q \ge 0$ and $p + q = 1, \mu, \eta, \gamma$, and $\Lambda$ are the system's parameters, and the time-varying parameter $\beta(t)$ is continuous, while $u$ is the control. The function $f$ is continuous on $[0, \Lambda]$ with $f(0) = 0$.

This system can be written in the following form:

$$ \dot{x} = X(t, x) + uY(x), \tag{13} $$

where

$$ x = \begin{pmatrix} S \\ I \\ R \end{pmatrix}, \quad X(t, x) = \begin{pmatrix} \Lambda - \mu S + \eta R - \beta(t)f(S)I \\ \beta(t)f(S)I - \mu I - \gamma I \\ \gamma I - (\mu + \eta)R \end{pmatrix} \quad \text{and} \quad Y(x) = \begin{pmatrix} pI \\ -I \\ qI \end{pmatrix}. $$

The system (13) possesses the Disease Free Equilibrium point (DFE) $E_0 = (\frac{\Lambda}{\mu}, 0, 0)$.

## 5.1 Finite-time stabilization

Let us begin with a necessary condition for finite-time stability (FTS). To this end, consider the uncontrolled version of system (13):

$$ \dot{\varphi} = X(t, \varphi), \quad \varphi(0) = x_0 \tag{14} $$

Similar to Subsection 3.1, we have the following necessary result for FTS.

**Theorem 13** If the system (13) is FTS toward an equilibrium $E_0$, then for all $x_0 \in \mathbb{R}^3$, we have:

$$ Y(\varphi_{x_0}(t)) = 0, \forall t \ge 0 \Rightarrow x_0 = E_0 $$

where $\varphi_{x_0}$ is the solution of (14) corresponding to the initial state $x_0$.

The following result provides sufficient conditions for finite-time stability.

**Theorem 14** Assume that $\beta(t) \ge \frac{\mu + \gamma}{M_{\Lambda}}$, where $M_{\Lambda} = \max_{S \in [0, \Lambda]} f(S)$. Then the component $I$ of the system (12) can be stabilized in finite time using the following positive feedback law, involving only the variable $I$.

$$ u = u(t, I) = (\beta(t)M_{\Lambda} - (\mu + \gamma)) + \mathbf{1}_{(I \neq 0)} I^{-\rho}, \quad 0 < \rho < 1, $$
Furthermore, this control guarantees the exponential convergence of the components S and R towards $\frac{\Lambda}{\mu}$ and 0 respectively.

*Proof* First of all, it can be easily shown that the following compact set is positively invariant:

$$ \Omega = \{(S, I, R)/ S, I, R \ge 0, S + I + R \le \frac{\Lambda}{\mu}\} $$

Let us set $z = (S, I, R)$. Since the component I of the state is positive, we can take the following candidate Lyapunov function: $V(z) = I \ge 0$. Then, using the second equation of (12), we can see that

$$ \dot{I} \le (\beta(t)M_{\Lambda} - (\mu + \gamma))I - uI. $$

First, observe that the control

$$ u(t) = \beta(t)M_{\Lambda} - (\mu + \gamma) + \mathbf{1}_{\{I \neq 0\}} I^{-\rho}, \quad 0 < \rho < 1, $$

is nonnegative whenever

$$ \beta(t) \ge \frac{\mu + \gamma}{M_{\Lambda}}. $$

Moreover, the invariance of $\Omega$ guarantees that the solution $(S, I, R)$ lies in $\Omega$ for any initial condition in $\Omega$.

Let $(S_0, I_0, R_0) \in \Omega$. Then, with the above choice of $u$, we obtain the following Lyapunov differential inequality:

$$ \dot{I} \le -I^{1-\rho}. $$

Let $[0, \tau)$ be the maximal time interval in which $I(t)$ does not vanish. Then for all $t \in [0, \tau)$, we have

$$ \dot{I}I^{\rho-1} \le -1, \forall t \in [0, \tau), $$

which by integrating gives

$$ I^{\rho}(t) \le I^{\rho}(0) - \rho t, \forall t \in [0, \tau) $$

from which, we derive $\tau \le \frac{I^{\rho}(0)}{\rho}$ and $I(t) = 0$ for all $t \ge T_* := \frac{I^{\rho}(0)}{\rho}$. In other words, the settling time for $I$ admits the upper bound

$$ T^* := \inf\{t \ge 0 : I(t) = 0\} \le \frac{I^{\rho}(0)}{\rho}. $$

Moreover, substituting back into the full system (12), we deduce from the third equation that:

$$ R(t) = R(T_*)e^{-(\mu+\eta)(t-T_*)}, \quad \forall t \ge T_* $$

Now, summing up the first and the last equation, we get

$$ \frac{d}{dt}(S + R) = \Lambda - \mu(S + R), \forall t \ge T_*, $$

which, together with the above estimate on $R$, gives:

$$ S(t) = \frac{\Lambda}{\mu} + \left(S(T_*) + R(T_*) - \frac{\Lambda}{\mu}\right) e^{-\mu(t-T_*)} - R(T_*)e^{-(\mu+\eta)(t-T_*)}, \forall t \ge T_* $$

We then deduce the exponential stabilization of the components $S(t)$ and $R(t)$.

□
## 5.2 Fixed and prescribed time stabilization

We have the following FxTS result which follows from similar arguments as in the Subsection 3.3.

**Theorem 15** Assume that $\beta(t) \geq \frac{\mu + \gamma}{M_{\Lambda}}$, where $M_{\Lambda} = \max_{S \in [0, \Lambda]} f(S)$.

For any given $T > 0$, there is a gain control $c > 0$ for which, the component $I$ of the system (12) is stabilizable in fixed time with the feedback law:

$$u = (\beta(t)M_{\Lambda} - (\mu + \gamma)) + c(\mathbf{1}_{(I \neq 0)} I^{-\rho} + I^{\rho}), \quad 0 < \rho < 1.$$

Furthermore, this control guarantees the uniform exponential stability of the components $S$ and $R$.

# 6 Simulation of the SIRS model

## Closed-loop systems and parameters

The initial conditions are chosen such that $S(0), I(0), R(0) \geq 0$ and $S(0) + I(0) + R(0) \leq \frac{\Lambda}{\mu}$.

The system in closed-loop corresponding to finite time stability is as follows:

$$\left\{
\begin{aligned}
\dot{S} &= \Lambda - \mu S + \eta R - \beta SI + p \left( \frac{\beta \Lambda}{\mu} - (\mu + \gamma) \right) I + p I^{1-\rho} \\
\dot{I} &= \beta SI - \mu I - \gamma I - \left( \frac{\beta \Lambda}{\mu} - (\mu + \gamma) \right) I - I^{1-\rho} \\
\dot{R} &= \gamma I - (\mu + \eta) R + q \left( \frac{\beta \Lambda}{\mu} - (\mu + \gamma) \right) I + q I^{1-\rho}
\end{aligned}
\right.
\quad (15)$$

while the system with fixed-time stabilizing nonlinearities:

$$\left\{
\begin{aligned}
\dot{S} &= \Lambda - \mu S + \eta R - \beta SI + p \left( \frac{\beta \Lambda}{\mu} - (\mu + \gamma) \right) I + p(I^{1-\rho} + I^{1+\rho}), \\
\dot{I} &= \beta SI - \mu I - \gamma I - \left( \frac{\beta \Lambda}{\mu} - (\mu + \gamma) \right) I - (I^{1-\rho} + I^{1+\rho}), \\
\dot{R} &= \gamma I - (\mu + \eta) R + q \left( \frac{\beta \Lambda}{\mu} - (\mu + \gamma) \right) I + q(I^{1-\rho} + I^{1+\rho}).
\end{aligned}
\right.
\quad (16)$$

We will simulate these two systems with the following parameter values:

* $\Lambda = 1, \mu = 0.01, \eta = 0.02$

* $\beta = 0.005, \gamma = 0.1, \rho = 0.5$

* $p = 0, q = 1$

* With two different initial conditions: $S(0) = 20, I(0) = 50, R(0) = 2$ and $S(0) = 20, I(0) = 2, R(0) = 1$.

Figures 1, 2 and 3 illustrate the evolution in time of the states of the first system, while Figure 4 concerns the second system.
Obtained figures

<figure><img src="image_2.png" /><figcaption>Fig. 1 Simulation of (15) with $S(0) = 20, I(0) = 50, R(0) = 2. I(t) = 0$ for $t > 6$.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2 Simulation of (15) with $S(0) = 20, I(0) = 50, R(0) = 2$. $S(t)$ and $R(t)$ are asymptotically converging to their equilibria.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig. 3 Simulation of (15) with two different initial conditions (20, 50, 2) in red and (20, 2, 1) in blue.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4 Fixed-time SIR model simulation of (16) with two different initial conditions (20, 50, 2) in red and (20, 2, 1) in blue.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 5 Time evolution of $I(t)$ without control and with control (5), for the initial condition (70, 10, 2).</figcaption></figure>

## Comments and interpretation of the numerical results

The numerical simulations provide insightful information about the qualitative behavior of the two considered epidemic models.

In the first two figures, which correspond to the system (15), we observe that the infected population $I(t)$ vanishes in finite time. Importantly, the extinction time of $I(t)$ depends on the initial conditions. Specifically, in the first scenario (with $I(0) = 5$), the extinction occurs in less than 20 time units, while in the second scenario (with $I(0) = 7$), the extinction occurs later. This confirms that although finite-time stability holds, the convergence time is not uniform and varies with the initial state.

By contrast, the last two figures correspond to the system (16), which incorporates an additional nonlinear damping term $I^{1+\rho}$. In these cases, the infected population $I(t)$ still reaches zero in finite time, but notably, the extinction occurs at the same time regardless of the initial condition. This stronger property is known as fixed-time stability. That is, there exists a uniform settling time $T = \frac{\pi}{2\rho}$, with $\rho = 0.5$ in the first parameter set and $\rho = 0.6$ in the second, such that $I(t) = 0$ for all $t \ge T$, independently of the initial values $S(0)$, $I(0)$, $R(0)$.

## 7 Conclusion

This work has presented a novel control strategy for achieving finite-time and fixed-time eradication of infectious diseases within a bilinear SIRS model framework. In contrast to classical asymptotic approaches, which may lead to delayed intervention, the proposed method guarantees rapid suppression of the epidemic, a feature of crucial importance in urgent containment situations.

By introducing a nonlinear, state-dependent feedback law that modulates the infection
rate in a multiplicative manner, we have shown that the infected population can be
driven to zero either in finite time or in fixed time, depending on the choice of control
parameters. Prescribed finite-time stability is further investigated by incorporating
a control gain into the state-dependent terms of the feedback law, and it is proved
that such stability can be achieved through the use of high-gain control. Moreover, we
have established that the susceptible and recovered populations converge exponentially
toward their respective equilibrium values, thereby ensuring the overall stability of the
system. Numerical simulations corroborate the theoretical analysis and confirm the
effectiveness, robustness, and responsiveness of the proposed control strategy.
These results provide meaningful insights for the development of dynamic public health
intervention policies and open new perspectives for future research on finite-time and
fixed-time control of more complex epidemic models.

Declarations

* Funding: Not applicable

* Conflict of interest/Competing interests: The authors declare that they have no conflict of interest.

* Ethics approval and consent to participate: Not applicable

• Consent for publication: Not applicable

• Data availability: Not applicable

* Materials availability: Not applicable

* Code availability: Not applicable

* Author contribution: Not applicable

References

[1] Amaliki, Y., & Ouzahra, M. (2025). "Finite-time stabilization of a class of unbounded parabolic bilinear systems in Hilbert space." *Journal of Mathematical Analysis and Applications*, 129720.

[2] S. P. Bhat and D. S. Bernstein, "Finite-time stability of continuous autonomous systems," *SIAM Journal on Control and Optimization*, vol. 38, no. 3, pp. 751–766, 2000.

[3] P. Dorato, "An overview of finite-time stability," in *Current Trends in Nonlinear Systems and Control*, pp. 185–194. Birkhäuser, Boston, MA, 2006.

[4] W. M. Haddad and A. L'Afflitto, "Finite-time partial stability and stabilization, and optimal feedback control", *Journal of the Franklin Institute*, vol. 352, no. 6, pp. 2329–2357, 2015.

[5] C. Jammazi, "Continuous and discontinuous homogeneous feedbacks finite-time partially stabilizing controllable multichained systems," *SIAM Journal on Control and Optimization*, vol. 52, no. 1, pp. 520–544, 2014.
[6] E. Moulay and W. Perruquetti, "Finite time stability and stabilization of a class of continuous systems," *Journal of Mathematical Analysis and Applications*, vol. 323, no. 2, pp. 1430-1443, 2006.

[7] H. Najib and M. Ouzahra, "Output finite-time stabilisation of a class of linear and bilinear control systems," *International Journal of Control*, vol. 97, no. 4, pp. 690-700, 2023.

[8] S. Parsegov, A. Polyakov, P. Shcherbakov (2012). "Nonlinear fixed-time control protocol for uniform allocation of agents on a segment". *In 2012 IEEE 51st IEEE conference on Decision and Control (CDC)* (pp. 7732-7737). IEEE.

[9] Polyakov, A., Coron, J. M., & Rosier, L. (2018). On homogeneous finite-time control for linear evolution equation in hilbert space. *IEEE Transactions on Automatic Control*, 63(9), 3143-3150.

[10] E. Roxin, "On finite stability in control systems," *Rendiconti del Circolo Matematico di Palermo*, vol. 15, no. 3, pp. 273-282, 1966.

[11] K. Zimenko, D. Efimov, A. Polyakov, and A. Kremlev, "On necessary and sufficient conditions for output finite-time stability," *Automatica*, vol. 125, p. 109427, 2021.