Observer design for a class of nonlinear piecewise systems. Application to an epidemic model with treatment

Mathematical Biosciences 271 (2016) 128–135

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Observer design for a class of nonlinear piecewise systems. Application to an epidemic model with treatment Abdessamad Abdelhedi a, Driss Boutat b, Lassaad Sbita a, Ramdane Tami b, Da-Yan Liu b,∗ a b

National Engineering School of Gabes, Research Unit of Photovoltaic, Wind and Geothermal Systems, Tunisia INSA Centre Val de Loire, Université d’Orléans, PRISME EA 4229, 18022 Bourges, France

a r t i c l e

i n f o

Article history: Received 27 January 2014 Revised 17 July 2015 Accepted 9 November 2015 Available online 19 November 2015 Keywords: Nonlinear system Extended observer Epidemic model Treatment function

a b s t r a c t Susceptible Exposed Infectious and Recovered epidemic model endowed with a treatment function (SEIR-T model) is a well-known model used to reproduce the behavior of an epidemic, where the susceptible population and the exposed population need to be estimated to predict and control the propagation of a contagious disease. This paper focuses on the nonlinear observer design for a class of nonlinear piecewise systems including SEIR-T models. For this purpose, two changes of coordinates are provided to transform the considered systems into an extended nonlinear observer normal form, on which a high gain observer can be applied. Then, the proposed method is applied to a SEIR-T model. Finally, simulation results are given to show its efficiency.

1. Introduction The spread of a contagious disease is a complex phenomenon, since it involves thousands of individuals and uncountable contamination factors, such as wind and merchandizes. A verbal description can be used to show its global aspect. However, a mathematical model is indispensable in order to give rigorous analysis [7]. Many suitable models have been developed to reproduce the behavior of an epidemic, such as Susceptible Infectious and Recovered (SIR) model and Susceptible Exposed Infectious and Recovered (SEIR) model [23,25]. The SEIR epidemic model subdivides the population into several groups: susceptible population, exposed population, infectious population and recovered population. This model has greatly enhanced our ability to understand and study the phenomenon so as to control the spread and optimize the disease fighting actions. In 1981, a model on the interaction between fox populations and rabies was considered [2], where a quadratic control function has been integrated in the total population dynamics in order to stop the spread of rabies. Afterwards, a treatment function proportional to the number of infectious population was proposed to study bifurcation of the SIR model [19]. Then, using a similar way to [2], a quadratic treatment function was proposed [9], which takes into account not only the limits of the available resources but also the fact that they are subject to decrease. Recently, the bi-stability and the bifurcation of

∗

Corresponding author. Tel.: +33 248484091. E-mail addresses: [email protected] (A. Abdelhedi), [email protected] (D. Boutat), [email protected] (L. Sbita), [email protected] (R. Tami), [email protected] (D.-Y. Liu). http://dx.doi.org/10.1016/j.mbs.2015.11.002 0025-5564/© 2015 Elsevier Inc. All rights reserved.

© 2015 Elsevier Inc. All rights reserved.

the SEIR model endowed with a quadratic function treatment (SEIR-T model) were studied [11]. When using a SEIR-T model, in order to predict and control the propagation of a contagious disease and the virus mutation, the future tendency of the disease needs to be reconstructed through a population and to analyze how the treatment function influences the behavior of an epidemic. For these purposes, the measurements of the susceptible population, the exposed population, the infectious population and the total population should be provided. Unlike the infectious population and the total population, the susceptible population and the exposed population are usually not measurable in a real situation. Consequently, their estimations are important. However, the existing works were basically interested in the study of the behaviors of epidemic models, such as the classification of singularities, the stability, the bifurcation, the existence of limit cycles, the chaotic behavior, etc. (see, e.g., [8,9,11,12,16,19,20]). In [1], an observer-based vaccination was designed for a SEIR epidemic model, where a control law was synthesized via an exact feedback input-output linearization approach. To the best of our knowledge, the problem of estimating the susceptible population and the exposed population in a SEIR-T model has not been studied, especially using the extended nonlinear observer normal forms concept. The observer design is useful to estimate the state of a dynamical system. Hence, it is important in control theory. However, it is usually difficult to design an observer for a nonlinear system. In order to solve this problem, [15] proposed to transform a nonlinear system into a simple observer normal form, on which an existing observer technique can be applied. Inspired by this idea, an output observer normal form was developed in [18,21,24]. Recently, an extended

A. Abdelhedi et al. / Mathematical Biosciences 271 (2016) 128–135

nonlinear observer normal form was introduced in [14] and then developed in [3–5,17,22]. Remark that all these proposed observer normal forms can be considered as a powerful tool to design robust observers for nonlinear dynamical systems, where the states are globally estimated. Bearing these ideas in mind, the aim of this paper is to propose an observer design to transform a SEIR-T model into an extended nonlinear observer normal form, on which a high gain observer can be applied to estimate the susceptible population and the exposed population using the infectious population and the total population. This paper is organized as follows: Section 2 provides the background of the proposed observer design by recalling the extended nonlinear observer normal form and a class of high gain observers. Section 3 introduces the considered nonlinear piecewise system, which can be transformed into an extended nonlinear observer normal form by applying two changes of coordinates. Section 4 concerns the application of the proposed method to a SEIR-T epidemic model. Then, simulation results are given to show the efficiency . Finally, some conclusions and perspectives in Section 5. 2. Background of observer design Since nonlinear dynamical systems are usually characterized by complex structures, it is difficult to design a controller or an observer for them. In order to solve this problem, these dynamical systems are transformed into an observer normal form by applying a change of coordinates. Among the existing observer normal forms, the output depending extended nonlinear observer normal form is generally considered, which is given as follows:



z˙ = A(y, w )z + β (y, w ), w˙ = γ (y, w ), y = Cz,

(1)

where z ∈ Rn and y ∈ R are respectively the real state and the real output (the measurement) of a considered dynamical system, w ∈ R T is an extra output variable, β (y, w ) = [β1 (y, w ), . . . , βn (y, w )] , C = [0, . . . , 0, 1] is an 1 × n matrix and

⎛

0 ⎜α2 (y, w ) ⎜ 0 A(y, w ) = ⎜ ⎜ .. ⎝ . 0

0 ··· 0 ··· α3 (y, w ) · · · .. .. . . 0 ...

0 0 0 .. . αn (y, w )

⎞

0 0⎟ ⎟ 0⎟ .. ⎟ .⎠ 0

T zˆ˙ = A(y, w )zˆ + B(y, w ) −  −1 (y )R−1 ρ C (C zˆ − y¯ ), T T 0 = ρ Rρ + G Rρ + Rρ G − C C,

where (y ) = diag[

⎛

0 ··· ⎜1 · · · G = ⎜. . ⎝ .. . . 0 ···

n

i=2 αi (y ),

⎞

(2)

(3)



T e˙ = zˆ˙ − z˙ = A(y, w ) −  −1 (y )R−1 ρ C C e.

i=3 αi (y ), . . . , αn (y ), 1], and

0 0 0 0⎟ .. .. ⎟ ⎠. . . 1 0

Consequently, by taking an appropriate value of ρ , the exponential stability of the observation error can be guaranteed by the boundedness of the states and the outputs (y, w). 3. A class of extended observer systems In this paper, the following three states and single output dynamical system is considered:

⎧ ⎪ ⎨χ˙ 1 = ϕ1,1 (y )χ1 + ψ1,2 (y )χ2 + ψ1,3 (y ) + T1,σ (y ), χ˙ 2 = ϕ2,1 (y )χ1 + ψ2,2 (y )χ2 + ψ2,3 (y ) + T2,σ (y ), ⎪χ˙ 3 = ψ3,1 (y )χ2 + ψ3,3 (y ) + T3,σ (y ), ⎩ y = χ3 ,

The solution Rρ of the algebraic observer equations given in (3) is explicitly given as follows:

j−1

(6)

where χ i for i = 1, 2, 3, are the state variables, y is the output (the measurement), σ is a switched rule, which determines the piecewise behavior of this dynamical system. It is assumed that functions ϕ i, j and ψ i, j are smooth enough. Moreover, in order to ensure the observability, it is also assumed that ϕ 1, 1 (y) = 0, ϕ 2, 1 (y) = 0 and ψ 3, 1 (y) = 0 in their domain. This means that the state variables can be written as a function of the output and its derivatives [13]. The main objective of this paper is to estimate the state variables χ i for i = 1, 2, 3, from the output (the measurement) using an observer of kind (3). In order to achieve this, the remainder of this article is devoted to address two changes of coordinates that transform the dynamical system (6) into the extended nonlinear observer normal form given by (1). Thus, an observer of kind (3) can be used in this last form. These transformations attempt to eliminate the terms containing χ 1 and χ 2 in the first dynamic equation of (6) and the term containing χ 2 in the second dynamic equation. For this purpose, we propose to successively apply two different changes of coordinates in the two following subsections.

In this subsection, we aim to annihilate the terms containing χ 2 in the two first dynamic equations given in (6). For this purpose, we have the following result. Lemma 1. The following change of coordinates:

ξ1 = χ1 − and

ξ3 =





y 0 y

0

ψ1,2 (s ) ds, ξ2 = χ2 − ψ3,1 (s ) 1

ψ3,1 (s )



y 0

ψ2,2 (s ) ds, ψ3,1 (s )

ds

(7)

transforms the dynamical system (6) into the following preliminary form:

where

 (−1 )i+ j i + j − 2 Rρ (n + 1 − i, n + 1 − j ) = i+ j−1 , j−1 ρ

i+ j−2

(5)

⎧ ⎨ξ˙1 = ϕ1,1 (y )ξ1 + ϕ1,3 (y ) + T¯1,σ (y ), ξ˙ = ϕ2,1 (y )ξ1 + ϕ2,3 (y ) + T¯2,σ (y ), ⎩ ˙2 ξ3 = ξ2 + ϕ3,3 (y ) + T¯3,σ (y ),

n

for i = 1, . . . , n and j = 1, . . . , n, where ficients.

It can be shown that the dynamics of the corresponding observation error is governed by the following equation:

3.1. First change of coordinates

is an n × n matrix. In order to guaranty the observability of the extended nonlinear observer normal form given by (1), it is assumed that the functions α i ( ·, ·) for i = 2, . . . , n, are nonvanishing. Then, according to [10], a high gain observer can be applied to this extended nonlinear observer normal form. Thus, it yields:



129

(4)

are the binomial coef-



ψ1,2 (s ) ds + ψ1,3 (y ) − 0 ψ3,1 (s ) ψ1,2 (y ) T ( y ), T¯1,σ (y ) = T1,σ (y ) − ψ3,1 (y ) 3,σ  y ψ1,2 (s ) ϕ2,3 (y ) = ϕ2,1 (y ) ds + ψ2,3 (y ) − 0 ψ3,1 (s ) ψ2,2 (y ) T ( y ), T¯2,σ (y ) = T2,σ (y ) − ψ3,1 (y ) 3,σ ϕ1,3 (y ) = ϕ1,1 (y )

y

(8)

ψ1,2 (y ) ψ ( y ), ψ3,1 (y ) 3,3 ψ2,2 (y ) ψ ( y ), ψ3,1 (y ) 3,3

130

A. Abdelhedi et al. / Mathematical Biosciences 271 (2016) 128–135

ϕ3,3 (y ) = T¯3,σ (y ) =



ψ2,2 (s ) ψ3,3 (y ) ds + , ψ3,1 (s ) ψ3,1 (y )

Then, using the first and the fourth dynamic equations given in (11), we get:

1

z˙ 1 = e−

y 0

ψ3,1 (y )

T3,σ (y ).

Proof. By taking the time derivatives of the transformations given in (7), we get:

⎧ ψ (y ) ⎪ ˙ y, ⎪ξ˙1 = χ˙ 1 − 1,2 ⎪ ψ3,1 (y ) ⎪ ⎪ ⎨ ψ (y ) ˙ ξ˙2 = χ˙ 2 − 2,2 y, ψ3,1 (y ) ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ξ˙3 = ˙ y. ψ3,1 (y )

(9)

(10) Finally, this proof can be completed using again (7). 

In this subsection, we aim to annihilate the term containing ξ 1 in the first dynamic equation given in (8). For this purpose, the following dynamical system is considered, where an auxiliary dynamics is added to the dynamical system given in (8):

(11)

where w ∈ R is an auxiliary variable considered as an output, and η (·, ·) is a function of the outputs y and w. In the following theorem, the function η( ·, ·) is given such that the augmented dynamical system given by (11) can be transformed into the extended nonlinear observer normal form given by (1). Theorem 1. Assume that η (y, w ) = κ (w )ϕ1,1 (y ) = 0, then the augmented dynamical system given by (11) can be transformed into the extended nonlinear observer normal form given by (1), by applying the following change of coordinates:

β1 (y, w ) = e−

w 0

(12)

ds

κ (s )

 ϕ1,3 (y ) + T¯1,σ (y ) ,

β2 (y, w ) = ϕ2,3 (y ) + T¯2,σ (y ), β3 (y, w ) = ϕ3,3 (y ) + T¯3,σ (y ), w

α2 (y, w ) = e 0 α3 (y, w ) = 1.

ds κ (s )

ϕ2,1 (y ),

Proof. Firstly, by taking the time derivative of the first dynamic equation given in (12), we obtain:

z˙ 1 = e

−

w 0



ds

κ (s )

ξ˙1 −

ξ1 w˙ . κ (w )



κ (s )

(13)

Secondly, substituting ξ 1 by z1 in the second dynamic equation given in (11) yields: w 0

ds

κ (s )

ϕ2,1 (y )z1 + ϕ2,3 (y ) + T¯2,σ (y ) = α2 (y, w )z1 + β2 (y, w ).

Finally, according to the third dynamic equation given in (11), we have:

z˙ 3 = ξ˙3 = ξ2 + ϕ3,3 (y ) + T¯3,σ (y ) = z2 + β3 (y, w ).

(15) 

Remark 1. The auxiliary dynamic w˙ = η (y, w ) is chosen in such a way that η (y, w ) = κ (w )ϕ1,1 (y ) guaranties the boundedness of w. 4. Application 4.1. SEIR-T epidemic model

3.2. Second change of coordinates: dynamics extension

where

0

(14)

⎧ ψ (y ) ⎪ ξ˙1 =ϕ1,1 (y )χ1 + ψ1,3 (y ) + T1,σ (y ) − 1,2 (ψ3,3 (y ) + T3,σ (y )), ⎪ ⎪ ψ3,1 (y ) ⎪ ⎪ ⎨ ˙ξ2 =ϕ2,1 (y )χ1 + ψ2,3 (y ) + T2,σ (y ) − ψ2,2 (y ) (ψ3,3 (y ) + T3,σ (y )), ψ3,1 (y ) ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ξ˙3 =χ2 + (ψ (y ) + T3,σ (y )). ψ3,1 (y ) 3,3

⎧ w ⎨z1 = e− 0 κds(s) ξ1 , z =ξ , ⎩z2 = ξ2 , 3 3

ds

ϕ1,1 (y )ξ1 + ϕ1,3 (y ) + T¯1,σ (y )  1 − κ (w )ϕ1,1 (y )ξ1 κ (w ) w

 − 0 κds (s ) ϕ ¯ =e 1,3 (y ) + T1,σ (y ) = β1 (y, w ).

z˙ 2 = e

Then, according to (6), we have:

⎧˙ ξ = ϕ1,1 (y )ξ1 + ϕ1,3 (y ) + T¯1,σ (y ), ⎪ ⎨ ˙1 ξ2 = ϕ2,1 (y )ξ1 + ϕ2,3 (y ) + T¯2,σ (y ), ⎪ ⎩ξ˙3 = ξ2 + ϕ3,3 (y ) + T¯3,σ (y ), w˙ = η (y, w ),

w

The Susceptible Exposed Infectious and Recovered (SEIR) epidemic model is a useful model to describe the spread of diseases through a host population. Let us denote S as the susceptible population, E as the exposed population without symptoms, I as the infectious population, R as the recovered population and N as the total population, which are functions of time. Then, the SEIR model with a treatment function (SEIR-T model) is defined as follows:

⎧ dS SI ⎪ ⎪ = bN − μS − β − pbE − qbI, ⎪ ⎪ dt N ⎪ ⎪ ⎪ ⎪ dE SI ⎪ ⎪ = β + pbE + qbI − (μ + ε )E, ⎪ ⎪ dt N ⎪ ⎨ dI = ε E − (a + δ + μ )I − Tσ , ⎪ dt ⎪ ⎪ ⎪ dR ⎪ ⎪ ⎪ ⎪ dt = aI − μR + Tσ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dN = (b − μ )N − δ I

(16)

dt

where b is the rate of the natural birth, μ is the rate of mortality, β is the transmission rate, δ is the death rate related to diseases, ε is the rate, at which the exposed population becomes infective, p is the rate of the offspring from an exposed population, q is the rate of the offspring from an infectious population, and a is the rate, at which the infectious individuals are recovered. Moreover, Tσ is a quadratic treatment function given by ( see [9] for more details):

Tσ = max(T (I ), 0 )

(17)

with

T (I ) = rI − gI2 ,

(18)

where r > 0 is the societal effort to fight the infection, g is a small positive value on the order of N (r0 ) such that if g = N (r0 ) then T( · )

vanishes at I = N (0 ) and reaches the maximal value Tmax = r N (40 ) at

I = N (20 ) . Hence, Tσ is a piecewise function (see for example Figs. 1 and 2). Let us consider the following domain  = {(S, I, E, R ) | S ≥ 0, E ≥ 0, I ≥ 0, R ≥ 0}. Then, according to whether Tσ = max(T (I ), 0 ) = 0

A. Abdelhedi et al. / Mathematical Biosciences 271 (2016) 128–135

131

Fig. 1. Behavior of the quadratic function T (I ) = rI − gI2 .

or T(I), the dynamical system defined in (16) can be subdivided into two subsystems. Hence, two space-works D1 = {(S, I, E ) | S, E, I ≥ 0 with T (I ) = rI − gI2 ≥ 0} and D2 = {(S, I, E ) | S, E, I ≥ 0 with T (I ) = rI − gI2 < 0} are defined. Thus, the system defined in (16) can be considered as a piecewise system. Since the total population N is given as follows:

N = S + E + I + R,

(19)

using the algebraical equations of N and R, the dynamical system defined in (16) becomes:

⎧ ⎪ ⎪ dS = bN − μS − β 1 SI − pbE − qbI, ⎪ ⎪ dt N ⎪ ⎪ ⎪ ⎪ dE 1 ⎪ ⎪ ⎪ ⎨ dt = β N SI + ( pb − μ − ε )E + qbI, dI ⎪ = ε E − (a + δ + μ )I − Tσ , ⎪ ⎪ dt ⎪ ⎪ ⎪ dN ⎪ ⎪ ⎪ ⎪ dt = (b − μ )N − δ I, ⎪ ⎩

4.2. Changes of coordinates and observer design Let us apply the first change of coordinates given in (7). Using (21), we obtain:

ξ1 = S +

pb

ε

y,

ξ2 = E −

1 ( pb − μ − ε ) y and ξ3 = y. ε ε

Then, taking the first order derivative with respect to time yields:

(20)

y = I,

where it is assumed that the output y = I and the total population N can be measured (are known). From now on, the objective is to estimate the susceptible population S and the exposed population E from the measurements of the infectious population I and the total population N. To do so, we will transform the dynamical system (20) into extended observer normal form. Remark 2. Although our model is endowed with two outputs, using y = I is sufficient to construct the change of coordinates and the auxiliary dynamics that transform the three first equations of (20) into the desired normal form. Moreover, the presence of N in the dynamical system (20) does not have any influence on the observer design. Indeed, the proposed observer works for any dynamical system of the form z˙ = A(π (t ))z + β (π (t )), where π (t) can be any known or measurable time-varying bounded function. This form is more general than (1) [10]. Now, the system obtained in (20) satisfies the form given in (6) with

⎧ ⎨ϕ1,1 = −(μ + βN y ), ψ1,2 (y ) = −pb, ψ1,3 (y ) = bN − qby. ϕ = β y, ψ (y ) = pb − μ − ε , ψ (y ) = qby. ⎩ψ2,1 y N= ε , 2,2ψ (y ) = −(a + δ + μ2,3)y. 3,1 ( ) 3,3

Consequently, the two changes of coordinates proposed previously will be applied in the next subsections.

⎧   ⎪ ˙1 = bN − μ + β y ξ1 − 1 (ε qb + pb(a + δ ))y ⎪ ξ ⎪ ⎪ N ε ⎪ ⎪ ⎪ ⎪ pb pb ⎪ ⎪ y2 − +β T , ⎪ ⎪ εN ε σ ⎪ ⎪ ⎪ 1 β pb ⎪ ⎨ξ˙2 = ξ1 y − β y2 + (ε qb + ( pb − μ − ε )(a + δ + μ ))y N εN ε pb − μ − ε ( ) ⎪ ⎪ + Tσ , ⎪ ⎪ ε ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ξ˙3 = ξ2 + ( pb − 2μ − ε − a − δ )y − Tσ , ⎪ ε ε ⎪ ⎪ ⎪ ⎪ N˙ = (b − μ )N − δ y ⎪ ⎩ y = εξ3 . (22) Hence, it can be seen that the system obtained in (22) via this change of coordinates has the form given in (8). Now, the system obtained in (22) is extended by adding a new dynamics w˙ = η (y, w ) = κ (w )ϕ1,1 (y ), where ϕ1,1 = −(μ + βN y ) is given in (21). This new dynamics and the dynamical system (22) give the form (11) with

  y β ϕ1,1 (y ) = − μ + β , ϕ2,1 (y ) = y, N

T¯1,σ (y ) = −

pb

ε

N

1 ( pb − μ − ε ) Tσ , T¯2,σ (y ) = Tσ , T¯3,σ (y ) = − Tσ .

ε

ε

Then, according to Theorem 1, the change of coordinates (21)

z1 = e−

w 0

ds

κ (s )

ξ1 , z2 = ξ2 and z3 = ξ3 =

y

ε

132

A. Abdelhedi et al. / Mathematical Biosciences 271 (2016) 128–135

Fig. 2. Behavior of the treatment function Tσ = max(rI − gI2 , 0 ).

Fig. 3. Estimation of the susceptible population S.

Fig. 4. Estimation of the exposed population E.

A. Abdelhedi et al. / Mathematical Biosciences 271 (2016) 128–135

133

Fig. 5. Comparison between the estimations of the infected population I with and without treatment.

Fig. 6. Behavior of total population N(t) under and without treatment.

leads to the following desired extended nonlinear observer normal form:

  ⎧ w ds pb pb pb y2 ⎪ ˙ − = bN − qby − ( a + δ ) y + T e− 0 κ ( s ) , z β ⎪ σ 1 ⎪ ε ε N ε ⎪ ⎪ ⎪ ⎪ ⎪ y  w ds pb y2 ( pb − μ − ε ) ⎪ ⎪ z˙ 2 = β e 0 κ (s) z1 + qby + ( a + δ + μ )y−β ⎪ ⎪ N ε ε N ⎨ ( pb − μ − ε ) Tσ , + ⎪ ⎪ ε ⎪ ⎪ ⎪ 1 1 ⎪ ⎪z˙ 3 = z2 + ( pb − 2μ − ε − a − δ )y − Tσ , ⎪ ⎪ ε ε ⎪ ⎪ ˙ ⎪ ⎩N = (b − μ )N − δβy w˙ = −κ (w )(μ + N y ).

(23)

4.2.1. Observer design Remark that the system obtained in (23) can be written into the following observer normal form:

⎧ ⎪ ⎨z˙˙ = A(y, w )z + B(y, w ), N = ( b − μ )N − δ y ⎪ ⎩w˙ = η (y, w ),

(24)

y = Cz,

where

⎛

0 y 0w ⎝ A(y, w ) = β e N 0

ds κ (s )

⎞

0

0

0

0⎠,

1 0

134

A. Abdelhedi et al. / Mathematical Biosciences 271 (2016) 128–135

Fig. 7. Behavior of the auxiliary variable w(t) .

and

B(y, w ) =

⎛ 

lization in a quite high level. However, the right curve shows that the function of treatment allowed us to significantly reduce the number of infected individuals. Similar result in the case with treatment is shown in Fig. 6, where the total population increases.

  B1 B2 B3

bN − qby −

pb

( a + δ )y +

pb



w pb y2 β − Tσ e− 0 N ε

ds

⎞

κ (s )

⎜ ⎟ ε ε ⎜ ⎟ 2 ⎜ ⎟ = ⎜qby+ ( pb − μ − ε ) (a+ δ + μ )y−β pb y + ( pb − μ − ε ) T ⎟. σ⎟ ⎜ ε ε N ε ⎝ ⎠ 1 y ( pb − 2μ − ε − a − δ ) − Tσ , ε ε Consequently, according to the previous study in Section 2, a high gain observer can be applied to globally estimate the susceptible population and the exposed population [6]. 4.3. Simulation results In this subsection, the following parameters are taken in the model given in (20): b = 1/100, δ = 0.02, μ = 0.01, a = 0.0312, β = 1.25, p = 0.8, q = 0.95, ε = 0.5. The parameters for the treatment function Tσ are taken by r = 0.018 and g = 1.2766e−4 . The initial conditions for the model are S(0 ) = 140, E (0 ) = 0 , I (0 ) = 1, R(0 ) = 0, N (0 ) = 141, and the initial conditions for the observer are z1 (0 ) = 130.005, z2 (0 ) = 0.0227, z3 (0 ) = 0.4 and w(0 ) = 0. Moreover, in order to ensure the boundedness of extended dynamics w˙ (see Fig. 7 θ w = 0 is taken, the behavior of the auxiliary variable w) κ (w ) = sin ( θ w )2 for |w| < πθ , where θ is a parameter chosen to enlarge the interval of the variable w. In this simulation, θ = 0.0005, and ρ = 0.34 for the observer given in (3). The obtained quadratic function T and the corresponding treatment function Tσ are shown in Figs. 1 and 2, respectively. In order to show the the efficiency of the proposed method, the susceptible population, the exposed population and their estimations are shown in Figs. 3 and4, respectively. Hence, it can be seen that the obtained estimations give small estimation errors and fast convergence. In Fig. 5, the left curve corresponds to the case without treatment. It shows that the number of infected individuals increases until stabi2

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