key: cord-0198214-xtnoudku
authors: Azimi, Vahid; Sharifi, Mojtaba; Fakoorian, Seyed; Nguyen, Thang Tien
title: State Estimation-Based Robust Optimal Control of Influenza Epidemics in an Interactive Human Society
date: 2020-05-27
journal: nan
DOI: nan
sha: b10eedf76cbe1eff57478eac370615fcab49323d
doc_id: 198214
cord_uid: xtnoudku

This paper presents a state estimation-based robust optimal control strategy for influenza epidemics in an interactive human society in the presence of modeling uncertainties. Interactive society is influenced by the random entrance of individuals from other human societies whose effects can be modeled as a non-Gaussian noise. Since only the number of exposed and infected humans can be measured, states of the influenza epidemics are first estimated by an extended maximum correntropy Kalman filter (EMCKF) to provide a robust state estimation in the presence of the non-Gaussian noise. An online quadratic program (QP) optimization is then synthesized subject to a robust control Lyapunov function (RCLF) to minimize susceptible and infected humans, while minimizing and bounding the rates of vaccination and antiviral treatment. The joint QP-RCLF-EMCKF meets multiple design specifications such as state estimation, tracking, pointwise control optimality, and robustness to parameter uncertainty and state estimation errors that have not been achieved simultaneously in previous studies. The uniform ultimate boundedness (UUB)/convergence of error trajectories is guaranteed using a Lyapunov stability argument. The soundness of the proposed approach is validated on the influenza epidemics of an interactive human society with a population of 16000. Simulation results show that the QP-RCLF-EMCKF achieves appropriate tracking and state estimation performance. The robustness of the proposed controller is finally illustrated in the presence of modeling error and non-Gaussian noise.

Influenza viruses can cause epidemic human diseases that are currently a worldwide health concern. Proper control of influenza epidemics is a crucial task that can mitigate economic and epidemiological burdens. Recent years have witnessed numerous studies in analysis, modeling, and control of influenza epidemiological systems [1] - [6] . Mathematical model of influenza epidemics can provide an opportunity to design model-based control strategies and to analyze the stability of closed-loop systems. Several mathematical models have been proposed for influenza epidemic systems [1] , [2] , [7] . In [1] , compartmental models of the influenza were proposed while considering the vaccination and antiviral treatment as control inputs. In [7] , influenza dynamics were modeled by a set of nonlinear differential equations. In [2] , a nonlinear SEIAR model of the influenza with two control inputs and five states was described. In this model, the positive state variables S, E, I, A, and R are the Susceptible, Exposed, Infected, Asymptomatic, and Recovered individuals while rates of vaccination and antiviral treatment are considered positive control inputs.

To recover all individuals of a society, the best intervention strategy is desired to be designed for the influenza epidemics. Optimal control is one of the widely-used approach that has been employed to determine the treatment strategies [3] , [4] , [8] - [10] . In [3] , an optimal control problem was employed to minimize the number of infected individuals at minimal efforts of the vaccination. Different optimal control strategies were suggested in [4] to minimize the impact of influenza pandemics involving antiviral treatment and/or the isolation measures. In [8] , prevention of the pandemic influenza was enhanced towards evaluating time-dependent optimal prevention policies and considering its execution cost. In [9] , a dynamic model of an influenza pandemic model was formulated with the existence of vaccination and treatment, and then analyzed in terms of the vaccine intake variations. In [10] , a prioritization scheme for allocation of a sizeable quantity of influenza vaccine and antiviral drug was described for a stratified population.

Note that the above-mentioned optimal control strategies were formulated with the assumption of fully-known dynamic terms and parameters. However, mathematical models of the influenza epidemics may contain modeling uncertainties that should be taken into account in the control design structure. In [11] , a least squares method was employed to estimate unknown parameters of two influenza epidemic models. Although the estimation performance was validated, no any control strategy was designed to minimize the infected population. In [2] , a robust adaptive sliding mode controller was designed for a nonlinear SEIAR model of the influenza in the presence of parametric uncertainties. In that work, convergence of susceptible and infected humans to zero was provided by tracking some descending scenarios. Two robust terms were also incorporated in their devised controller whose gains were updated using adaptation laws to compensate for the parameter uncertainties. Stability of closed-loop influenza epidemic system was then proved using a Lyapunov framework and the Barbalat's lemma.

However, that recent paper [2] suffers from several drawbacks. (i) The main one is that the controller requires accurate measurement of state variables, while only the population of exposed and infected humans can be measured in practice.

(ii) In that approach, studied human society was assumed to be isolated from other societies. However, a random entrance of individuals from other societies into the main society of interest results in degrading the control performance. This kind of society is called "interactive society" and the effects from the other societies can be modelled as a non-Gaussian noise as shown in Fig. 1. (iii) In their method, although the convergence of system solutions was obtained and the robustness of closed-loop systems against parametric uncertainties was demonstrated, control optimality, as an important design specification, has not been taken into account. In other words, tracking, robustness, and minimizing the rates of vaccination and antiviral treatment should be achieved at the same time by devising an appropriate control strategy. (iv) In the normalized SEIAR model, the control signals should be always positive and less than 1. However, the approach in [2] was not able to bound the rates of vaccination and antiviral treatment in the controller implementation while facing with high parameter uncertainties and disturbances. It should also be noted that the rest of the above-mentioned papers suffer from the shortcomings mentioned in Items (i) and (iv). The Kalman filter [12] is still the most common method for state estimation of linear systems because of its optimality and simplicity. However since the mathematical model of the influenza comprises a set of nonlinear differential equations, the extension of Kalman filters, namely the extended Kalman filter (EKF) and the unscented Kalman filter (UKF) [13] - [21] , can be alternatively used for the state estimation purpose. A Kalman filter is derived based on the minimum mean square error (MMSE) criterion, which follows that it uses only second-order information of the signal and it is optimal until the gaussianity of noises is preserved. However, in this paper, the performance of the ordinary EKF may break down for the influenza epidemics of an interactive society that is disturbed by non-Gaussian noise (when the society is not isolated). To solve this issue, the maximum correntropy Kalman filter (MCKF) can be utilized to provide robustness for the Kalman filter in the presence of non-Gaussian noise or large outliers [22] - [24] . The MCKF uses the correntropy criterion instead of MMSE through which higher-order information of process and measurement noises is used [25] , [26] .

Motivated by the aforementioned shortcomings of the existing controllers, that have been already designed for the influenza epidemic systems, and the desire to develop a new multi-objective controller for such systems, this work is the first step towards designing a state estimation-based robust optimal controller for influenza epidemics in an interactive human society (demonstrated in Fig. 1 ) in the presence of modeling uncertainties and non-Gaussian noise. The main contributions of this paper are as follows: (i) the state estimation of the influenza epidemics in an interactive human society; (ii) the design of a robust optimal controller to minimize the population of susceptible and infected humans, while minimizing and bounding the rates of vaccination and antiviral treatment; (iii) the proof of the UUB/convergence of tracking errors; and (iv) the robustness of the proposed algorithm in the presence of parameter perturbation and random entrance of individuals from the other societies.

In this paper, we begin by formulating an extended MCKF (EMCKF) algorithm to estimate the states of an influenza dynamical system while using the number of exposed and infected humans as measurement. With the aim of achieving the boundedness/convergence of system's errors with a minimal control effort, an online quadratic program (QP) is synthesized subject to a robust control Lyapunov function (RCLF). The joint QP-RCLF finds the optimal balance between control effort and stability of closed-loop system. The robust term is incorporated in the QP-RCLF framework to compensate for state estimation error and modeling uncertainties. The unified state estimation-based controller QP-RCLF-EMCKF provides the convergence of susceptible and infected populations to a small neighborhood around the origin, while minimizing and bounding the control effort. The UUB/convergence of tracking errors is finally proven using a Lyapunov stability argument. To assess the performance of the proposed approach QP-RCLF-EMCKF, simulation results are carried out for the influenza epidemic model. Results show that the proposed controller successfully achieves the promised design specifications such as tracking and state estimation for this epidemiological system. Tests show that the QP-RCLF-EMCKF strategy provides appropriate robustness in the presence of parametric uncertainties and random entrance of humans from other societies to the society of interest.

The paper is organized as follows: Section II describes an influenza epidemic model and the problem statement. Section III presents the state estimation framework using EMCKF algorithm. Section IV presents our proposed control strategy QP-RCLF-EMCKF. Section V provides the simulation results. Section VI presents discussion, conclusion, and future work.

In this section, we begin by describing a dynamical model for the influenza epidemics and then present the problem statement.

A state space representation of the influenza epidemics can be described by the following nonlinear SEIAR model [2] :

where z = [z 1 , z 2 , z 3 , z 4 , z 5 ] T = [S, E, I, A, R] T ∈ 5 denotes the state variables of the system with positive values; z 1 represents the population that is susceptible to get infected with influenza; z 2 is the number of people who are infected with influenza but not yet infectious (exposed); z 3 stands for population that is infected and also infectious with influenza symptoms; z 4 represents the number of individuals who are influenza carriers but without any symptoms (asymptomatic); z 5 denotes the number of recovered humans; u = [u 1 , u 2 ] T ∈ 2 is the vector of normalized control inputs such that 0 ≤ u i ≤ 1 for i = 1, 2; u 1 is the rate of vaccination of the susceptible population z 1 ; and u 2 is the rate of antiviral treatment of the infected population z 3 . More details about this epidemiological model can be found in [2] , [27] , [28] .

This paper aims to design a robust optimal controller to decrease the number of susceptible z 1 and infected z 3 populations while using the minimum possible rates of vaccination u 1 and antiviral treatment u 2 . More importantly, the normalized control inputs must be bounded between 0 and 1, which requires a set of control constraints to be incorporated in the controller design. For this purpose, an online QP control strategy is formulated by considering the RCLF and the abovementioned input constraints to generate a pointwise optimal control effort, while achieving the convergence of system's errors.

Since only z 2 and z 3 are measurable in practice, the proposed controller uses the estimate of system's states (populations) as feedback in closed-loop system. To achieve a robust state estimation of the influenza epidemics in an interactive human society in the presence of non-Gaussian noise, an EMCKF algorithm is employed and specifically developed for this dynamical system. A robust term is also designed to robutify the system against state estimation error and parametric uncertainties. The resulting state estimationbased control strategy QP-RCLF-EMCKF meets multiple design objectives such as tracking, control optimality, state estimation, and robustness. The UUB/convergence of all system solutions is proven using a Lyapunov framework and the proposed controller is finally validated by comprehensive simulation studies.

In this section, an EMCKF algorithm is described and presented to estimate the system states. This filter only uses the number of exposed and infected humans (z 2 and z 3 ) as possible measurements. Consider the following general form of a nonlinear stochastic continuous-time system for the influenza epidemic model (1)

where h (z, t) = [z 2 , z 3 ] T ∈ 2 is the vector of measurable variables (populations) in the influenza epidemics; w(t) ∈ 5 is the continuous-time process noise vector of the system with covariance matrix Q ∈ 5×5 ; v(t) ∈ 2 is the continuoustime measurement noise with covariance R ∈ 2×2 ; and Θ is the vector of actual system parameters as

Assumption 1: The noises w(t) and v(t) are both uncorrelated, Gaussian, and zero-mean. However, a shot noise is enforced to the measurement noise v(t) to model the effects from the other societies on the main interactive society, which results in a non-Gaussian noise as

(4)

The nonlinear functions f (.) ∈ 5 and h(.) ∈ 2 are sufficiently smooth in z, such that they can be linearized using the Taylor series expansions.

The EMCKF is similar to the EKF as they are based on linearization using first-order Taylor series expansion. Therefore, the following Jacobian matrices are used to linearize the system:

whereẑ andΘ are the estimations of z and Θ, respectively. The initialization of the filter is given as:

where E(.) stands for the expected value operation; P (0) is the covariance of the initial estimate; and z(0) andẑ(0) show the initial value of the states and its estimates, respectively. The state estimate and the EMCKF gain for the continuoustime nonlinear system (1) are formulated as follows [29] , [30] :

in which the time-varying gain ν(t) and the estimation error covariance matrix P (t) are defined as

with the kernel function G σ ( · ) defined as

where . stands for the Euclidean norm of a vector;

. R(t) −1 denotes a weighted Euclidean norm of a vector (i.e., 1 as a positive definite matrix); and σ is the user-specified bandwidth (kernel size).

The EMCKF algorithm is robust against large outliers or non-Gaussian noises, because when the system is perturbed by such noises, then ν(t) → 0 which prevents the divergence of the filter. It can be seen that by picking a large value of σ, ν(t) → 1 and the EMCKF reduces to the ordinary EKF.

Assumption 3: We assume that under the EMCKF algorithm, the state estimation error e e = z −ẑ ∈ 5 is bounded.

Remark 1: The EMCKF algorithm uses the estimate of system parametersΘ, the measurements (z 2 , z 3 ), and the control signal u.

The next section will formulate a state-estimation robust optimal control while utilizing the estimate of system's states provided by the EMCKF algorithm.

With the estimate of the system's states from the previous section in hand, this section is devoted to formulating the proposed controller in order to minimize the susceptible and infected populations. Defining z e = [z 1 , z 3 ] T ∈ 2 , the tracking objective reduces to the convergence of z e to its desired minimum value z d e . To achieve this objective, the first and third equations of Eq. (1) are taken into account and can be written as followsż

with

where the basis functions Φ 1 (z) and Φ 2 (z), the parameter vectors θ 1 and θ 2 , and the control map Z e including positive diagonal elements are defined as

Let us define e =ẑ e − z d e as the tracking error vector. Definingẑ e = [ẑ 1 ,ẑ 3 ] T and e e1,3 = [e e1 , e e3 ] T = z e −ẑ e , the tracking error can be redefined as

Assumption 4: Assume that the desired value z d e is bounded and of class C 1 (i.e., z d e is continuously differentiable) 2 .

Using Eqs. (10) and (13), the error dynamics are obtained asė

Using the notion of the feedback linearization, assuming thatė e1,3 = 0, and picking the following feedback control law

the error dynamics (14) are transferred to the linear systeṁ e = µ with µ as the virtual input vector. However, it should be pointed out that (i) the vectorė e1,3 is nonzero, (ii) the actual system parameters θ are not perfectly known, and (iii) the accurate measurement of state variables z is not available to the controller. To include the estimated stateẑ and parameters (θ 1 ,θ 2 ) (Items ii and iii), the feedback law (15) is modified as

Substituting the control law (16) into the error dynamics (14) in the presence of a nonzeroė e1,3 (Item i), one hasė

By rewriting the control map as

can be stated as

where e e1 = z 1 −ẑ 1 and e e3 = z 3 −ẑ 3 . Then, by defining

in which ∆ 1 ∈ 2 is defined as

Now, substituting Eqs. (20) and (21) into Eq. (18) yieldṡ

for which the uncertainty term ∆ ∈ 2 is described as

where ∆ 2 ∈ 2 is

In the next section, to provide the context for the uncertainty term ∆, its properties will be studied in detail.

A. Properties of the uncertainty term ∆ Throughout this section, we rely on the following property. Property 1: Let us define the whole population of the society as N = 5 i=1 z i whose variation can be obtained by the summation of all compartmental dynamics presented in (1)Ṅ

where α > 0 denotes the recovery rate for the symptomatic infected people and 0 < ζ << 1 is the fatality rate of the influenza. In view of (26) , it follows that the whole population N is a decaying upper bounded time-varying function such that N (t) ≤ N 0 , where N 0 > 0 is its initial magnitude. Hence, all compartmental variables z i for i = 1, . . . , 5 remain bounded during the treatment time such that z i ≤ N (t) ≤ N 0 . Whereby, according to Assumption 3, the estimates of all system variables z i are also bounded. In the following, we begin by expanding each of the components in (24) and then describe the uncertainty term ∆ as a linear function of e plus a bounded term.

1) Term ∆ 1 : Utilizing the definitions of the tracking and estimation errors from Assumption 3 and Eq. (13), the vectors ∆Φ 1 , ∆Φ 2 , andΦ 1 can be written as

with

In view of (27) and (28), the term ∆ 1 has the alternative form

where

According to Assumptions 3 and 4, and Property 1, all terms in the matrices A 1 , A 2 , and W 1 and the vectors W 2 , ∆Φ 2 , andΦ 2 are bounded. This coupled with the boundedness of the vectors θ i ,θ i , and ∆θ i for i = 1, 2 concludes that the term ∆ 1 is bounded by a linear function of e plus a bounded term W 3 such that

whereĀ 3 andW 3 are positive scalars such that A 3 ≤Ā 3 and W 3 ≤W 3 .

2) Term ∆ 2 : In view of (16), one obtains µ =Ŷ −ż d e −Ẑ e u using which the term ∆ 2 reduces to

with

In Section IV-C, we will synthesize a QP optimization problem through which the control input u i for i = 1, 2 is enforced to always stay between 0 and 1, i.e., u ≤ u 0 with a positive scalar u 0 . This bounding of the control signal along with the boundedness of e e1 and e e3 implies that

where W 4 ≤W 4 withW 4 > 0.

3) Termė e1,3 : In view of Eq. (7), the derivative of the estimation error for the number of susceptible and infected populations iṡ

where K 1,3 ∈ 2×2 is a matrix whose rows represent the first and third rows of the Kalman gain. Utilizing the definitions e ei = z i −ẑ i for i = 1, . . . , 5, and

with W 5 = −K 1,3 [e e2 , e e3 ] T . A careful inspection of Eq. (36) reveals that the first term is equal to the term ∆ 1 and therefore, one can writė

in which since A 3 , W 3 , W 4 , W 4 , K 1,3 , and u are all bounded, the bound for ė e1,3 is obtained as

whereW 5 is a positive scalar such that W 5 ≤W 5 .

Using the previously computed bounds, the uncertainty term ∆ can be stated as a linear function of e plus a bounded term

whereĀ andW are two positive scalars. Employing the proposed feedback control law (16), the error dynamics (14) are partially linearized as presented in Eq. (23) . Then, the problem reduces to designing the virtual input µ to guarantee the UUB/convergence of error trajectory e while compensating for the uncertainty ∆. For this purpose, the next subsection will present a RCLF to ensure boundedness/convergence of the tracking error in a pointwise optimal fashion.

In this section, we begin by considering the special case of ∆ = 0 based on which the system (23) reduces tȯ

A function V (e) is an exponentially stabilizing control Lyanpunov function (ES-CLF) for the system (40), if the following conditions are met [31] :

where a 1 , a 2 , λ > 0. A candidate ES-CLF for the system (40) is then suggested as For this purpose, substituting the virtual input µ from Eq. (16) 

based on which φ 0 ∈ and φ 1 ∈ 2 are defined as

Then, substituting Eq. (45) into Eq. (42) , and using the definitions of φ 0 and φ 1 from Eq. (47), the inequality constraint (42) can be expressed as

which is called the CLF constraint. Now, a family of controllers that can minimize the control input u w.r.t. the inequality constraint (48) can be defined using the following pointwise min-norm control (PWMC) law [32] :

However, this control law can only guarantee the exponential convergence of e to zero in the absence of the quantity ∆. We now consider the general case in which ∆ = 0 for the error dynamics (23) . 

Substitute the calculated bound for ∆ from Section IV-A to haveV

By defining Λ = λ−2Ā > 0 with λ > 2Ā,V (e) < 0 outside the set

This implies that the tracking error norm remains less than B r at all time when φ 0 (e) > 0. In case that φ 0 (e) ≤ 0, we have u = 0 for which Eq. (50) becomeṡ

On the other hand, φ 0 (e) ≤ 0 implies that

Using Eq. (55) and Eq. (56) and following the same steps as in Eq. (53), we conclude that e is bounded in the same ball B r as in Eq. (66). The analysis can be further extended to show the exponential convergence of the tracking error vector to the set S 0 . For this purpose, apply the Youngs inequality for (53) on the termW e to obtaiṅ

whereΛ = Λ − 1 > 0 with Λ > 1. Applying the Comparison lemma [33] (Lemma 3.4), one obtains

This implies that V exponentially converges to a ball of sizeW 2 /(2Λ) with exponential converge rateΛ. Hence, since e ≤ 2V (e), the tracking error e will exponentially converge to the small compact set S 0 .

Remark 2: The size of the convergence ball B r is determined by the parameter Λ and the boundW , where the former can be tuned by users and the latter depends on the parameter uncertainties and the state estimation error.

Remark 3: The error trajectory e converges to a smaller ball for smaller state estimation error and parameter estimation error (smallerW ). The effect of the uncertainty ∆ can be also mitigated by choosing a sufficiently large value of λ. However, this may cause higher control effort and unpleasant system solutions.

It is seen that the PWMN control law (49) with defined φ 0 provides the boundedness of e in a compact ball with size B r . With the aim of compensating the uncertainty term ∆ and reducing the size of the ultimate ball without manipulating the convergence rate, the robust term

Employing Eq. (60), the inequality constraint (48) can be rewritten as

which is called the RCLF constraint. So now, the modified control law based upon φ 0 rob is suggested as

Theorem 2: Consider the Lyapunov function (43), the robust component (59), and the control law (62). Under the Assumptions 1, 2, 3, and 4 and Property 1, if K r <W , then e remains less than B r rob = 2(W − K r )/Λ at all time for all Θ ∈ 9 , any e(0) ∈ 2 , and any bounded e e (0) ∈ 5 . The convergence of e to the compact ball B r rob is globally exponential. However if K r ≥W , then e asymptotically converges to zero as t → ∞.

Proof: Utilizing the control law (62) in case that φ 0 (e) > 0,V (e) of Eq. (51) can be written aṡ

Once again, using the calculated bound of ∆ from Section IV-A, one haṡ

Here, two cases can be considered on selecting the robust gain K r : Case 1 (K r <W : uniform ultimate boundedness): In this case,V (e) < 0 outside the set

which follows that the size of the new convergence ball is 2(W − K r )/Λ. This implies that employing the robust term C rob with a positive gain that satisfies K r <W reduces the size of the ultimate bound on the tracking error e. In this case, the size of B r rob is determined by the parameter Λ and the discrepancy between the gain K r and the boundW . Once again, to ensure that the convergence of e to the set S 1 is exponential, we apply the Youngs inequality for (65) on the term (W − K r ) e to havė

for which applying the Comparison lemma yields

This concludes exponential convergence of V to a small neighborhood around the origin for which the size of the neighborhood is (W − K r ) 2 /(2Λ) and the exponential convergence rate isΛ. This coupled with the the radial unboundedness of the Lyapunov function V follows that the convergence of e to the set S 1 is globally exponential.

Case 2 (K r ≥W : asymptotic convergence): In this case, picking a sufficiently large robust gain in such a way that K r =W + κ r with κ r > 0 results iṅ

This concludes thatV becomes negative definite, which implies that e asymptotically converges to zero as t → ∞.

Although the larger robust gain K r provides better tracking performance, it results in a higher control signal (K r directly contributes to the control law u). On the other hand, the smaller K r provides a better control optimality, while the tracking error possesses a larger ultimate bound. Thus, a trade off should be made between control optimality and tracking performance when choosing the robust gain K r .

Remark 5: The proposed control strategy with the RCLF structure renders stronger conclusion for the stability of closed-loop system in the presence of uncertainty ∆.

With the formulation of the RCLF in hand, the next subsection will unify the EMCKF and the RCLF through synthesizing a QP optimization framework.

The quadratic program-based CLF (QP-CLF) technique is a contemporary control approach that guarantees stability of closed-loop systems while minimizing and bounding the control inputs [31] , [34] . However, modeling uncertainties and state estimation errors, i.e., ∆ = 0, degrade the performance of such controllers [35] , [36] . To mitigate this issue, in this section, we aim to design a robust optimal controller by the unification of the EMCKF algorithm (Section III) and the RCLF (Section IV-B) while utilizing the estimate of the system states. For this purpose, a QP optimization problem is employed to generate the same PWMC signal u(φ 0 rob , φ 1 ), which enables the incorporation of the RCLF constraint (61) as well as the required control bounds while using the estimates of the states and the system parameters.

We begin by recovering the virtual input µ from the main control signal (16) as

To formulate the QP-RCLF-EMCKF controller while minimizing the virtual input µ, the following cost function should be minimized:

(71) The control input u has to be also restricted between its prescribed minimum and maximum values such that u ≤ u i ≤ū, for i = 1, 2 with u = 0 andū = 1. Therefore, a QP optimization problem with the aforementioned tracking and control objectives can be formulated as:

where c is a relaxation coefficient for the RCLF constraint (61) when the control bound is enforced. Formally defining a QP problem, the above optimization can be presented in the following form

with

and

Equations (73), (74), and (75) show that the proposed controller uses the estimated states (by using EMCKF algorithm in Section III), the estimated parameters (best Parameter Description Values κ

Transition rate for the exposed 0.526 α Recovery rate for the infected 0.244 η Recovery rate for the asymptomatic 0.244 p Fraction of developing symptoms 0.667 ζ Fatality rate 0.98 Infectivity reduction factor for the exposed 0 δ

Infectivity reduction factor for the asymptomatic 1 q

Contact reduction by isolation 0.5 guess), tracking error, and the first derivative of the desired trajectory as a four-tuple (ẑ,θ, e,ż d e ). The general structure of the proposed QP-RCLF-EMCKF for the influenza epidemics in an interactive human society is illustrated in Fig. 1 .

In this section, the proposed control methodology QP-RCLF-EMCKF is implemented on the influenza epidemic model (1) whose parameters are shown in Table I . We aim to minimize the susceptible and infected individuals in an interactive human society with population of 16000. The Note that the summation of initial state variables is equal to the population of the main society. The simulation runs for 40 days. Table II provides the design parameters of the proposed approach for the state estimation algorithm explained in Section III and the controller formulated in Section IV. The design parameters are tuned to provide a (a) ∆Θ = +50% The effects from the other human societies on the main interactive society is modeled by a shot noise. Thus, the measurement noise is regarded as a non-Gaussian noise, which is a Gaussian noise that is affected by a shot noise as described in Eq. (4). In the simulation, the shot noise is seen as 20 impulses with magnitude of 200, which is randomly enforced to the measurement noise. This shot noise models the random entrance of 200 exposed and infected individuals from the other human societies into the human society of population 16000. Thus, the measurements (z 2 ,z 3 ) are affected by these 200 individuals during the simulation as shown in Fig. 2 .

A. State estimation, tracking performance, and control effort Figure 3 shows the state estimation performance for the influenza epidemics along with the convergence of populations z 1 and z 3 . It is seen that the proposed EMCKF algorithm is able to accurately estimate the state variables while only measuring the populations z 2 and z 3 . This accurate estimation is achieved when the shot noise is enforced to the measurement noise, which represents an impulsive random entrance of the exposed and infected populations to the main human society of 16000. This implies that the proposed (a) ∆Θ = +50% (b) ∆Θ = −50% Fig. 6 : Vaccination rate u1 and antiviral treatment rate u2 under ±50% parameter uncertainty estimation algorithm has a strong robustness when the system is perturbed by non-Gaussian noises. Figure 3 also shows that the susceptible z 1 and infected z 3 individuals of the interactive human society are minimized in 14 days under the proposed control strategy. The convergence of variables z 1 and z 3 results in the convergence of populations z 2 and z 4 , and in turn the entire population z 5 is recovered. This implies that the proposed controller is able to recover all individuals of the human society with the population of 16000, even when the external infected individuals from other societies randomly invade the main society during a treatment time of 40 days. These results are in agreement with our main results presented in Section IV-B and Theorem 2 based on which UUB/convergence of system's errors is guaranteed. Figure 4 illustrates the rate of vaccination for susceptible individuals u 1 and the rate of antiviral treatment for the infected individuals u 2 . It is seen that the control signals generated by the proposed control technique fairly decreases to zero at the end of the treatment time. It can be also noted that none of the control signals hit the maximum control boundū as the peak controls are u 1max = 0.49 and u 2max = 0.64. Figure 4 also demonstrates the RCLF constraint violation during the simulation. It is seen that the RCLF violation is bounded by 0.05 when the relaxation coefficient is tuned as c = 10. A smaller value of c relaxes the RCLF constraint and decreases the possibility of its conflict with the control bound constraint; however, smaller c increases h and in turn deteriorates the tracking performance. For higher relaxation coefficient c, h is relatively zero and the RCLF constraint is never violated, but the QP may be infeasible due to the conflict of the RCLF constraint with the control bounds. Thus, the penalty coefficient c should be carefully selected to make a trade off between the tracking performance and the control constraints.

Different societies and populations can result in the influenza model (1) with different values of the system parameters Θ. To evaluate the robustness of the proposed control scheme against the parameter perturbation, the system parameters are deviated by ±50% from their nominal values. Figure 5 illustrates the state estimation and tracking perfor-mance of the influenza epidemics when the system parameters are perturbed by ±50%. It is seen that the proposed EMCKF algorithm can still provide an accurate state estimation under either case. Under +50% parameter perturbation, the number of susceptible z 1 and infected z 3 populations converges to a small ultimate ball around zero in 14 days using the proposed controller. In case that ∆Θ = −50%, although the estimated states z 2 and z 4 have a sluggish convergence to the actual states, the EMCKF algorithm can render a general convenient estimation performance. In this case, the convergence of z 1 and z 3 is also achieved in the same days as of ∆Θ = 0% and ∆Θ = +50%. This demonstrates that the proposed approach achieves good robustness against the parameter perturbation. These findings support the claim of our main results presented in Theorem 2 in which UUB/convergence of the tracking errors is ensured even in the presence of parameter uncertainties and state estimation error. Figure 6 shows the control signals under ±50% parameter uncertainty. It is observed that the rate of vaccination for z 1 (u 1 ) under both cases ∆Θ = +50% and ∆Θ = −50% has relatively similar magnitude and behavior compared to u 1 in no perturbation case. Under both ∆Θ = +50% and ∆Θ = −50%, the maximum value of u 1 is u 1max = 0.49. However, the rate of antiviral treatment for z 3 (u 2 ) under ∆Θ = +50% meets a higher magnitude in the first 10 days (u 2max = 0.75), which is 17% higher than u 2max in the case of no perturbation. Under ∆Θ = −50%, although u 2 hits the control boundū = 1 during t ∈ [15, 22] , convergence of z 3 is maintained. This implies that there is no conflict between the control bounds and the RCLF constraint such that they can be achieved at the same time. This demonstrates that the proposed approach is able to achieve convergence of system solutions and to satisfy the constraints in the presence of parameter perturbation and state estimation errors.

In this section, we highlight superiority of the EMCKF algorithm over the ordinary EKF when the system is affected by the shot noise introducing the random entrance of exposed and infected individuals from other societies to the society of interest. Figure 7 illustrates the state estimation and tracking performance of the influenza epidemic system under the proposed controller but when an ordinary EKF is employed. It is seen that the estimated states z 2 , z 3 , and z 4 contain unpleasant impulses stemming from the shot noise and in turn do not converge to their actual states. This results in a steady state estimation error for the stateẑ 5 . Thus, it is seen that the estimation performance deteriorates when the system is disturbed by the shot noise and the EKF is employed.

Since the proposed controller uses the estimated states, inconvenient state estimation of the EKF negatively impacts the generated control signals as shown in Fig. 8 . Both the rate of vaccination for z 1 and the rate of antiviral treatment for z 3 intensively chatter after day 20 and even u 2 hits the control boundū. This shows that improper estimation performance of the EKF in the presence of shot noise causes the control Figure 8 also shows that the RCLF constraint violation is not smooth and chatters after day 20. This demonstrates that the proposed controller can not preserve its robustness for an interactive human society (when the main human society is not isolated from the other societies i.e., existing of non-Gaussian noise) when the ordinary EKF is employed instead of the proposed EMCKF algorithm.

Control of influenza epidemics in a human society is an important global health concern that imposes economic and epidemiological burdens. The optimal control strategy is one of the most popular design approaches that has been employed to control the influenza epidemics. However, previous optimal control approaches have been designed with the assumptions of fully-known dynamics and fullymeasurable states in addition to considering an isolated human society. The adaptive control strategy is an efficient design method for controlling the influenza epidemics in the presence of dynamic uncertainties. To cope with the modeling inaccuracies, an adaptive control method has been recently designed in [2] while still assuming that the system's states are measurable and the human society is isolated. In addition, that controller did not take the optimality of the vaccination and antiviral treatment rates into account.

Since the influenza dynamic models are a set of nonlinear differential equations, the EKF is a convenient algorithm for the state estimation of such systems. However, since the human society of interest is not isolated from the other societies (it is an interactive society that is impacted by non-Gaussian noise), performance of the ordinary EKF deteriorates in the presence of other societies' interactions.

Motivated by the aforementioned shortcomings of the existing works applied for the influenza epidemics and the aim of devising a new multi-objective controller for such systems, this paper presented a state estimation-based robust optimal control strategy for the influenza epidemics in an interactive human society in the presence of modeling uncertainties. An EMCKF algorithm was presented for state estimation purpose and a QP optimization problem was formulated w.r.t. a RCLF to recover the entire population of an interactive human society while compensating the state estimation error and the modeling error in an optimal fashion. The proposed QP-RCLF-EMCKF controller achieved multiple design specifications such as state estimation, tracking, control optimality, and robustness against the modeling error and the non-Gaussian noise stemming from the other societies' effects. A Lyapunov stability argument was used to prove the boundedness of the susceptible and infected populations to a small neighborhood around the origin. The convergence of the error solutions was also discussed under a proper selection of the robust gain. This boundedness/convergence was achieved at minimal rates of the vaccination and antiviral treatment. Simulation results illustrated that the proposed approach is able to provide accurate state estimation, tracking performance, and robustness to the modeling inaccuracies and the non-Gaussian noise associated with the nature of the interactive human societies. This was achieved in an optimal control fashion.

The control strategy developed in this study can be modified to be employed for a wide range of epidemiological diseases such as tuberculosis [37] , malaria [38] , Hepatitis C virus (HCV) [39] , HIV/AIDS [40] , [41] , and COVID-19 [42] , [43] . In terms of future studies, the following items will be considered: 1) In this paper, the system parameters Θ have to be guessed for use in the controller. However, to relieve the engineer of the need for such guess, future work is planned to design an adaptation mechanism to estimate these unknown parameters. 2) As illustrated in Figs. 3, 5, and 7, the exposed population z 2 peaks at the beginning of the simulation. It implies that the number of people who are infected with influenza but not yet infectious initially increases and then vanishes as time goes on. Future work is planned to design a controller such that the exposed population is maintained below a number during the treatment period. These items naturally encourage us to extend the presented approach by estimating the system parameters and creating a safe control structure in which the exposed population is kept below a specified level.

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