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By substituting Eqs. (48), and (43) and utilizing ab≤a^2/2+b^2/2, one can write

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By substituting equations (48) and (43) and utilizing the inequality \( ab \leq \frac{a^2}{2} + \frac{b^2}{2} \), one can write:

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where D  is a constant which denotes the bound of Δd(t) and may not be available. Substituting (49) into (55), it can obtain that

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where \( D \) is a constant that denotes the bound of \( \Delta d(t) \) and may not be readily available. Substituting equation (49) into equation (55) yields:

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By directly integrating differential inequality (56), we have

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By directly integrating the differential inequality (56), we obtain:

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With the application of virtual controllers (25), (26), (33), parameters update laws (39), (40) 
the stability of the closed-loop system is guaranteed.

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With the application of the virtual controllers (25), (26), and (33), as well as the parameter update laws (39) and (40), the stability of the closed-loop system is guaranteed.

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please edit: In this section, we apply the proposed controller scheme to an uncertain bi-model SEIAR influenza epidemiological model with five saturated control inputs. The system dynamics account for input saturation, uncertainty, and external disturbances. The values of the influenza epidemic parameters used in the simulations are taken from [2,16] and are provided in Table I. We validate the effectiveness of the proposed system through simulation analysis using the MATLAB Simulink environment. Furthermore, we assume the following saturation levels for input control.

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In this section, we apply the proposed controller scheme to an uncertain bi-modal SEIAR influenza epidemiological model with five saturated control inputs. The system dynamics incorporate input saturation, uncertainty, and external disturbances. The values of the influenza epidemic parameters used in the simulations are sourced from references [2, 16] and are provided in Table I. We validate the effectiveness of the proposed system through simulation analysis in the MATLAB Simulink environment. Additionally, we assume the following saturation levels for the control inputs:

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The simulations are based on a final treatment time of t_f=60 days, representing the duration for eradicating the disease in the population, as determined by the designer. To control the influenza epidemic, the target reduction of the susceptible and infected compartments 〖(I〗_d,〖 E〗_d,S_d) over time is defined as follows

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The simulations are based on a final treatment time of \( t_f = 60 \) days, which represents the duration required to eradicate the disease from the population, as determined by the designer. To control the influenza epidemic, the target reductions for the susceptible and infected compartments \( (I_d, E_d, S_d) \) over time are defined as follows:

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The simulations are based on a final treatment time of t_f=60 days, which represents the duration required to eradicate the disease from the population, as determined by the designer. To control the disease epidemic, the target reduction of the susceptible and infected compartments 〖(I〗_d,〖 E〗_d,S_d) over time is defined as follows:

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The simulations are based on a final treatment time of \( t_f = 60 \) days, which represents the duration required to eradicate the disease from the population, as determined by the designer. To control the disease epidemic, the target reductions for the susceptible and infected compartments \( (I_d, E_d, S_d) \) over time are defined as follows:

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Where a is the adjustable population reduction rate and is chosen as a=0.2. I_f, E_fi, and S_fi are the desired steady-state numbers of infected, exposed, and susceptible individuals, respectively, they are assumed to be zero (I_f=0,E_f=0,〖 S〗_f=0 ). I_0,E_0, and S_0 are the initial numbers of infected, exposed, and susceptible persons. The initial values of the influenza variables in these simulations are also given in Table II. Parameters in controller laws are presented in Table III. Additionally, the model parameters are perturbed by 80% and
∆d(t)=〖[ 0.3sin(4t)   0.5sin(3t)   0.8cos(2t)  0.2sin(5t)  0.4cos(t)]〗^T represents unidentified time-varying external disturbances, unmodeled dynamics, and perturbations introduced into the system.

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Where \( a \) is the adjustable population reduction rate, chosen as \( a = 0.2 \). The desired steady-state numbers of infected, exposed, and susceptible individuals—denoted as \( I_f \), \( E_f \), and \( S_f \)—are assumed to be zero (i.e., \( I_f = 0 \), \( E_f = 0 \), \( S_f = 0 \)). The initial numbers of infected, exposed, and susceptible individuals are represented as \( I_0 \), \( E_0 \), and \( S_0 \), respectively. The initial values of the influenza variables in these simulations are provided in Table II, while the parameters in the controller laws are presented in Table III.

Additionally, the model parameters are perturbed by 80%, and 

\[
\Delta d(t) = \begin{bmatrix} 
0.3 \sin(4t) \\ 
0.5 \sin(3t) \\ 
0.8 \cos(2t) \\ 
0.2 \sin(5t) \\ 
0.4 \cos(t) 
\end{bmatrix} 
\]

represents unidentified time-varying external disturbances, unmodeled dynamics, and perturbations introduced into the system.

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The text presents two simulation scenarios to compare the population dynamics of the two-group SEIAR mathematical model under varying conditions. The scenarios include: (I) the absence of control measures, (II) applying a standard ABSMC approach to a system with restrictions on its control signal. By comparing these scenarios, we gain insights into the effects of control strategies and saturation constraints on the population dynamics of the two-group SEIAR model. This analysis underscores the importance of careful controller design and the consideration of practical limitations in achieving effective disease control

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The text presents two simulation scenarios to compare the population dynamics of the two-group SEIAR mathematical model under varying conditions. The scenarios include: (I) the absence of control measures, and (II) the application of a standard Adaptive Backstepping Sliding Mode Control (ABSMC) approach to a system with restrictions on its control signal. 

By comparing these scenarios, we gain insights into the effects of control strategies and saturation constraints on the population dynamics of the two-group SEIAR model. This analysis underscores the importance of careful controller design and the consideration of practical limitations in achieving effective disease control.

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please edit

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The text presents two simulation scenarios to compare the population dynamics of the two-group SEIAR mathematical model under different conditions. The scenarios include: (I) the absence of control measures, and (II) the implementation of a standard Adaptive Backstepping Sliding Mode Control (ABSMC) approach in a system with limitations on its control signal.

By comparing these scenarios, we gain insights into the effects of control strategies and saturation constraints on the population dynamics of the two-group SEIAR model. This analysis highlights the importance of careful controller design and the consideration of practical limitations in achieving effective disease control.

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