Stability analysis and observer design for discrete-time SEIR epidemic models

Abstract

This paper applies Micken's discretization method to obtain a discrete-time SEIR epidemic model. The positivity of the model along with the existence and stability of equilibrium points is discussed for the discrete-time case. Afterwards, the design of a state observer for this discrete-time SEIR epidemic model is tackled. The analysis of the model along with the observer design is faced in an implicit way instead of obtaining first an explicit formulation of the system which is the novelty of the presented approach. Moreover, some sufficient conditions to ensure the asymptotic stability of the observer are provided in terms of a matrix inequality that can be cast in the form of a LMI. The feasibility of the matrix inequality is proved, while some simulation examples show the operation and usefulness of the observer.

Ibeas et al. Advances in Dierence Equations (2015) 2015:122 DOI 10.1186/s13662-015-0459-x

R E S E A R C H Open Access

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Web End = Stability analysis and observer design for discrete-time SEIR epidemic models

Asier Ibeas1*, Manuel de la Sen2, Santiago Alonso-Quesada2 and Iman Zamani3

*Correspondence: [email protected]

1Departamento de Telecomunicaci i Enginyeria de Sistemes, School of Engineering, Universitat Autnoma de Barcelona, Bellaterra, Barcelona, 08193, Spain Full list of author information is available at the end of the article

Abstract

This paper applies Mickens discretization method to obtain a discrete-time SEIR epidemic model. The positivity of the model along with the existence and stability of equilibrium points is discussed for the discrete-time case. Afterwards, the design of a state observer for this discrete-time SEIR epidemic model is tackled. The analysis of the model along with the observer design is faced in an implicit way instead of obtaining rst an explicit formulation of the system which is the novelty of the presented approach. Moreover, some sucient conditions to ensure the asymptotic stability of the observer are provided in terms of a matrix inequality that can be cast in the form of a LMI. The feasibility of the matrix inequality is proved, while some simulation examples show the operation and usefulness of the observer.

Keywords: epidemics; SEIR; discrete models; stability; observer design

1 Introduction

Mathematical models have become an important tool in analyzing the causes, dynamics, and spread of epidemics []. Thus, its study is crucial in order to obtain valuable knowledge of the underlying aspects of the disease. Moreover, the analysis of mathematical models describing epidemics spreading allows authorities to make decisions regarding the vacci-nation policies, quarantine application and so on. These models can be classied into two main categories: continuous-time and discrete-time models.

Continuous-time models were the rst proposed to describe epidemics spreading since modeling has traditionally been focused from a dierential equations point of view. Therefore, many specic features regarding these models have been studied in many works such as the presence of bifurcations [, ], the construction of approximate solutions [, ], the fractional dynamics [], the existence of equilibrium points and oscillatory behavior [], and the presence of waves [], to cite but a few. However, model stability has been by far the most important property to be studied [].

On the other hand, discrete-time models have gained substantial importance during the last years as the increasing number of papers concerning this type of models reveals []. There are many reasons that explain this tendency. Among them we can nd the simplicity of its simulation, the fact that it is easier to adjust systems parameters from statistical data in discrete-time models than in continuous-time ones and the fact that discrete-time models may exhibit a richer dynamic behavior than its continuous-time counterparts []. Until now, as [] states, the study of discrete epidemic models was

2015 Ibeas et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Ibeas et al. Advances in Dierence Equations (2015) 2015:122 Page 2 of 21

focused on the computation of the reproduction number, the existence and stability (both local and global) of the equilibrium points, and the extinction, permanence, and persistence of the disease. In addition, these studies have been performed basically on SI, SIR, and SIRS models.

Moreover, there are two ways to face the problem of setting up a discrete-time epidemic model. The rst one consists in modeling the disease spreading directly in a discrete domain. The second one faces the problem by discretizing a previously existing continuous-time model. The latter approach will be used in this paper. In this way, several discretization methods have been used in the literature including the implicit and explicit Euler, mixed, and Runge-Kutta methods []. However, discretization procedures do not always preserve the structural properties exhibited by continuous-time models such as the conservation of the total population or the positivity of the solutions. Therefore, much research has been done in the direction of obtaining discretization methods able to preserve all these properties. One of the most important breakthroughs in this respect is the research made by Micken on the use of nonstandard nite dierence (NSFD) methods []. These methods combine implicit and explicit Euler approximations in the incidence rate function. Therefore, this paper will apply Mickens NSFD discretization method to obtain a SEIR discrete-time epidemic model while stating its main properties such as the well-posedness of the model, the positivity of solutions and the existence and stability of equilibrium points. Since most of the previous literature is conned to SI, SIR, and SIRS models, this study will contribute to the application of NSFD approaches to a wider class of models. In addition, the properties will be proved based on an implicit representation of the equations, in comparison with previous approaches [, ], where an explicit description of the model is obtained prior to analysis.

Furthermore, control theory has recently emerged as one of the proposed approaches to deal with the design of vaccination campaigns []. Thus, dierent types of vaccination laws based on control theory have appeared in the literature during the last years, such as the state-feedback control [, ], robust sliding control [], or feedback linearization []. Nevertheless, most of these works calculate the vaccination eort based on the values of the subpopulations, namely, susceptible, exposed, infectious and immune of the SEIR model. From a practical point of view only the total population and the number of infectious may be obtained from clinical data at health centers. Therefore, the number of susceptible, exposed, and immune cases may not be available to on-line compute the control vaccination law. Under these circumstances, a state observer is necessary to estimate the value of these subpopulations.

State observers are largely used in control applications to cope with the inability to measure some state variables. Thus, we can nd many applications in chemical [], aeronautic [], and electrical engineering [], to cite just a few. Also, there are few previous works where a state observer is designed for epidemic models [, ]. In these works, an ob-server is designed for a continuous-time SEIR epidemic model whose values are then used to implement a vaccination strategy. In the present work, however, a state observer is designed for a discrete-time SEIR model obtained by a NSFD discretization method, which is for the rst time being considered in the literature. The design of such a state observer must face some particularities, which make it dierent from previous approaches to ob-server design:

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NSFD schemes lead to implicit systems rather than explicit. Thus, the design of the state observer must be done implicitly. This makes a big dierence from traditional observer design where the problem is usually formulated in an explicit way.

Epidemic models are singular systems so that there exists an algebraic relation between the dynamic variables of the system.

The number of infectious can be measured, implying that there is no need to estimate this state component, which leads to a reduced-order observer.

In this way, the combination of all the above items makes the design of state observers for discrete-time SEIR epidemic models rather dierent from standard approaches in the literature. In addition, the asymptotic stability of the observer will be discussed, while some simulation examples will show its operation. Some preliminary results of this paper were rst presented in [] and [].

The paper is organized as follows. Section describes the model description and problem formulation. Section studies the main properties of the discrete model. The design of the state observer is carried out in Section . Section is devoted to the study of the analytic properties of the observer. Finally, Section shows some numerical simulation examples while our Conclusions end the paper.

2 Model description

Consider the SEIR continuous-time epidemic model described by []

S(t) = S(t) + R(t)

S(t)I(t)

N(t) (

S(t)I(t)

N(t) +

N(t), ()

E(t) =

+ )E(t),

I(t) = ( + )I(t) + E(t),

R(t) = ( + )R(t) + I(t), ()

where S(t), E(t), I(t), and R(t) denote the subpopulations of susceptible, exposed, infectious and immune, respectively, dening the state vector xT(t) = [S E I R]. N(t) denotes the total population at time t (i.e. N(t) = S(t) + E(t) + I(t) + R(t)), is the rate of deaths from causes unrelated to the infection, denotes the birth rate and is the rate of losing immunity. The term S(t)I(t)N(t) is referred to as the disease incidence rate and denes the total number of new infections per unit of time at time t. and are, respectively, the average duration of the latent and infectious periods. All the above parameters are assumed to be positive so as to represent a real situation. This model is also referred to as the SEIRS model in the literature (see, for instance, []) due to the presence of the terms

R(t) in () and (). The total population dynamics can be calculated by summing up all the equations ()-(), leading to

N(t) = ( )N(t). ()

It can be deduced from () that the total population is constant when = , increases when > and decreases when < . This model is discretized by using Mickens NSFD

Ibeas et al. Advances in Dierence Equations (2015) 2015:122 Page 4 of 21

method [], to get

Sk+ Sk

h =

Sk+ + Rk+

Sk+Ik

Nk +

Nk+, ()

Ek+ Ek

h =

Sk+Ik

Nk (

+ )Ek+,

Ik+ Ik

h = (

+ )Ik+ + Ek+,

Rk+ Rk

h = (

+ )Rk+ + Ik+, ()

where h > denotes the integration step while Sk, Ek, Ik, and Rk denote the value of the susceptible, exposed, infectious and immune at sampling time tk = kh which dene the discrete-time state vector xTk = [Sk Ek Ik Rk]. The dynamics of the total population in discrete-time can be obtained by summing up again all the equations ()-() to get

Nk+ Nk

h = (

)Nk+, ()

which corresponds to the NSFD discretization of the continuous-time equation (). This fact implies that the discretization method is consistent with the total population dynamics given by the continuous-time model, property not satised by all discretization methods []. Equation () can be re-written as

Nk+ =

+ ( )hNk ()

which provides the explicit value of Nk+ in terms of the systems parameters and Nk. For the sake of simplicity, let us rearrange ()-() in the following implicit matrix form:

M(xk)xk+ = xk + hNk+ = xk +

h + ( )hNk = Dxk, ()

where

T = [ ],

D = I +

h + ( )h

, ()

M(xk) =

A B H C

G D F J

()

with

A = + h +

hNk Ik, B =

h, C = + ( + )h,

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D = + ( + )h, J = + ( + )h, F = h,

G = h, H =

CDJ BFG BCF BCD DHJ ADJ BHF BDH GHJ AGJ ACJ BGH FGH AFG ACF ACD

N pro

vided that S, E, I, R , + ( )h > , and all the systems parameters are positive.

Proof The proof is performed by mathematical induction. Thus, let us assume that x is

non-negative. Now suppose that xk is non-negative and it will be proved that so is xk+.

h

Nk Ik. ()

Note that the matrix M(xk) depends explicitly on xk through entries A and H, since they

depend on Nk and Ik as () shows. Equation () allows one to isolate the value of state vector xk+, when possible, as

xk+ = M(xk)Dxk. ()

For the SEIR model, the closed-form explicit representation given by () is intricate. Therefore, all the properties of this model will be stated starting from the implicit form () rather than from () in comparison with some previous works [, ], where an explicit (or semi-explicit) formulation of the model is obtained prior to analysis. The following section is devoted to the study of the main properties of this scheme.

3 Main properties of the discrete-time model

Initially, the well-posedness of () will be proved.

Lemma M(xk) is invertible, its inverse is given by () and all the entries of the matrix

M(xk) are non-negative for any set of positive parameters and any Ik , Nk . We

have

M(xk) =

ACDJ HGBF

. ()

Proof Firstly, notice that, since all the parameters (including the time step h) are assumed to be positive and Ik, Nk , A, B, C, D, J, F, G > and H from (). Now, the determi

nant of M(xk) can be symbolically computed as

det M(xk) = ACDJ HGFB. ()

Hence, if Ik, Nk then ACDJ > , A > H, C > G, D > F, and J > B according to (). There

fore, ACDJ > HGFB, det M(xk) = ACDJ HGFB > , and M(xk) is invertible. Moreover, its

inverse can be symbolically computed as (). Finally, it can be directly seen from () that all its entries are non-negative since all the quantities involved are non-negative from (). Thus, the lemma is proved.

Furthermore, we have the following.

Theorem Sk, Ek, Ik, and Rk remain non-negative for all discrete time tk = kh, k

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In this way, since xk is non-negative so is Nk and, in particular, Ik. This fact implies that

M(xk) is invertible due to Lemma . In addition, its inverse is given by (), whose entries

are non-negative. Then the (k + )th state vector, xk+, is calculated through () as

xk+ = M(xk)xk + M(xk)

h + ( )hNk. ()

Since + ( )h > , all the elements in the right-hand side of () are non-negative, so is xk+ and the theorem is proved.

Remark Notice that Theorem states that the system is always well dened, since Ik and Nk for all k implies that M(xk) is always invertible due to Lemma and makes

possible the calculation of xk+ at each time step through ().

Remark Moreover, Theorem can also be used to establish the range of values of the time step h that guarantees the well-posedness of the discrete-time model. Thus, from + ( )h > we see that h must fulll:

< h < if > ,

h (, ) otherwise.

()

Since h must meet this constraint, the method is a NSFD-based one, rather a pure NSFD method.

Remark The non-negativity and well-posedness of the discrete-time system are crucial properties for the discrete-time model []. These properties are obtained easier by using the NSFD method than by using other discretization schemes. For instance, consider the forward Euler approximation of the derivative applied to the left-hand side of ()-() [], to obtain

Sk+ Sk

h =

Sk + Rk

SkIk

Nk +

Nk, ()

Ek+ Ek

h =

SkIk

Nk (

+ )Ek,

Ik+ Ik

h = (

+ )Ik + Ek,

Rk+ Rk

h = (

+ )Rk + Ik. ()

These equations can be cast in the matrix form:

xk+ = Mxk + h

SkIk

Nk , ()

where

M =

+ ( )h h h ( + )h

( + )h h ( + )h h ( + )h

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and xk = [Sk Ek Ik Rk]T, = [ ]T. If we want to apply now the same argument as in () to prove the non-negativity of the system (i.e. the non-negativity of all the elements of the right-hand side of ()) we nd two main drawbacks: (i) the rst one is that the vector possesses negative components, and, secondly, (ii) all the entries of the matrix M are

non-negative provided that the following constraints hold:

+ ( )h , () ( + )h , ( + )h , ( + )h . ()

Thus, the simple argument used in () is not directly applicable here, while it may lead to further conditions on the sampling period h arising from ()-():

< h min(+ ,+ ,+ , ) if < ,

< h min(

+ ) otherwise.

+ )( + )

N, ()

E = (

+ )( ( + )( + ))

(( + + )( + ) + ) N,

( ( + )( + ))

(( + + )( + ) + )N. ()

()

Therefore, the NSFD method reveals as a very useful one to treat the discretization problem of the SEIR model since its properties are proved easily and under mild mathematical conditions.

3.1 Existence and stability of equilibrium points

This section studies the existence and stability of equilibrium points in the discrete-time SEIR model given by ()-().

Lemma The discrete-time equation systems ()-() and continuous-time equation systems ()-() posses the same equilibrium points, given by

() If = then the only equilibrium point is given by the trivial one

S = E = I = R = N = .

() If = the total population is constant, i.e.

N(t) = N = N() = S() + E() + I() + R() and there are three potential equilibrium points given by:(a) The trivial one, S = E = I = R = N = .

(b) S = N, E = I = R = , i.e. the total population is susceptible. This point is usually referred to as the disease-free equilibrium point.

(c) The so-called endemic equilibrium point:

S = (

+ )( + )( ( + )( + ))

(( + + )( + ) + ) N,

I = (

R =

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Proof The proof of the rst part is straightforward from ()-() and ()-() while the equilibrium points are calculated in [].

Now, the stability of the equilibrium points is discussed. We will focus on the two nontrivial equilibrium points obtained when = : the disease-free and the endemic equilibrium points.a For this, the nonlinear system () is linearized around these equilibrium points and the rst Lyapunov theorem (linearization theorem) is used. Thus, the linearized equation readsb

(DM(xk)xk+)xk+ x vk+ = vk, ()

where x denotes the corresponding equilibrium point and vk = xk x is the incremental variable. In this way, the results are obtained starting from the implicit equation () rather than from the explicit one () as some previous works do [, ]. The advantage of this approach is that the calculus of the Jacobian matrix ((D

M(xk)xk+)

xk+ )x is easier implicitly than explicitly. In order to perform the calculation of the Jacobian, we must rstly write

M(xk) xk+ in terms of only xk+ since, otherwise, we would be omitting some terms in

the Taylor development. Therefore

M(xk) xk+ =

+ h + hN(GEk+ + DIk+) B hN(GEk+ + DIk+) C

G D F J

xk+. ()

Thus, the Jacobian is given by

(DM(xk)xk+)xk+ x

= D

+ h + hN(GE + DI) hNGS

N DS B hN(GE + DI) C + hNGS hNDS

G D F J

h

,

so that around the disease-free equilibrium point

(DM(xk)xk+)xk+ xdf

= D

+ h hG hD B

C + hG hD G D F J

and the linearized system is given by

D

+ h hG hD B

C + hG hD G D F J

vk+ = vk. ()

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Since the incremental variables satisfy v,k + v,k + v,k + v,k = due to the constraint Sk +

Ek + Ik + Rk = N we can omit one variable in our analysis, since it can be obtained from this algebraic equation. Therefore, if we omit the rst one we obtain the reduced system,

C + hG hD G D

F J

v,k+

v,k+

v,k+

=

v,k

v,k

v,k

. ()

The linearized reduced system () will be asymptotically stable provided that all the eigenvalues of the left-hand side matrix are strictly greater than unity in absolute value. In this way, the eigenvalues are given by

= J = + ( + )h,

= C + D +

hG + (C D) + hG(C + D) + Gh

,

= C + D +

hG (C D) + hG(C + D) + Gh

. ()

It can be readily seen that > , while

= + (

+ + )h + hG + (C D) + hG(C + D) + Gh

> . ()

The only eigenvalue that might threaten stability is , which may be re-written as

= C + D +

hG (C + D + hG) CD

. ()

Notice that the argument of the square root is positive from (). Therefore, () implies that must also be positive since CD > . The stability condition then reads > . Some algebra on () shows that this condition can be met, provided that

R :=

( + )( + ) < , ()

which corresponds to the same stability condition found in [] for the continuous-time system. Thus, not only does the discretization method preserve positivity and total population dynamics, but also the equilibrium points and their stability. In this way, the reproduction number of the discrete-time model is given by (). When () is met, the linearized system is globally stable around the disease-free equilibrium point, v,k, v,k, v,k, v,k and Sk N as k , while the nonlinear system is locally stable

around the equilibrium point. Thus, we have already proved the following theorem.

Theorem The discrete-time epidemic SEIR model ()-() is locally asymptotically stable around the disease-free equilibrium point when = provided that R :=

(+ )(+) < .

On the other hand, the following theorem holds when R > .

Theorem If = and R > then one of the following claims remains true:

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() The solution of the discrete-time SEIR model ()-() tends to the endemic equilibrium point.

() Some variables of the discrete-time model ()-() oscillate.

Proof Firstly note that the positivity of the model along with the constraint Sk + Ek +

Ik + Rk = N imply that Sk, Ek, Ik, Rk N. Therefore, all variables are globally bounded

through time. Now, since the state variables cannot diverge as k , there are only two

possibilities:

() All the state variables converge to a constant nite value.() Some of the state variables do not converge to a constant value, and therefore, oscillate.

If all the variables converge to a constant value, it must be an equilibrium point. The trivial point is not reachable due to the constraint Sk + Ek + Ik + Rk = N while the disease-free

equilibrium point is unstable when R > . In consequence, the nite constant limit must

be the endemic equilibrium point. On the other hand, if any of the variables does not

converge to a nite limit it must oscillate. Thus the theorem is proved.

Theorem represents an alternative-type theorem to establish the behavior of the solutions when R > . The particular case corresponding to one situation or another depends

on the particular values of the parameters. However, the complexity of the endemic point expressions ()-() makes it dicult to obtain analytical conditions based entirely on the parameters to distinguish in between. This result is also in line with those presented in [] for the continuous-time model, where the conditions under which the endemic equilibrium point is globally stable are presented. The important feature regarding the case R > is that the disease persists in the population (in a constant or oscillatory way),

and this is the reason why the equilibrium point is called endemic. In this case, the application of vaccination contributes to removing the disease. Many vaccination laws make use of the values of the subpopulations to generate the vaccination eort. Nevertheless, the number of susceptible, exposed, and immune cases may be very dicult to obtain in practice since only the infectious can be registered at health centers. Therefore, a state observer is designed in the next section for the discrete-time model.

4 Design of the state observer

The purpose of the state observer is to obtain an estimation of the state, xk, from the

unique available data, the infectious subpopulation, Ik. This issue is the main feature of the presented design process. In this way, the following feasible assumptions will be used in the sequel:

Assumption The total population Nk is known at all k .

Assumption The number of infectious is available at all k .

Remark The total population may be obtained from () if birth and mortality rates are known or by direct measurement. This assumption is particularly feasible in the case of non-lethal diseases, for which the population remains almost constant during the spreading stage.

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Remark The number of infectious may be obtained from clinical data at health centers.

Therefore, Ik does not have to be estimated, since it is directly obtained from the actual data. This fact allows us to consider the design of a reduced-order observer. On the other hand, the constraint Nk = Sk + Ek + Ik + Rk for all k also permits the algebraic direct

calculation of one state variable in terms of the others. In this way, if xTk = [kkk Rk]

denotes the observed state, there is no need to estimatek, whilek will be calculated from

k = Nk k Rk Ik. ()

Thus, we just need to observe the values ofk and Rk. The rst step is to re-organize the state vector so as to put together the variables that are to be truly estimated. To this end dene the re-organized state vector zk = [Ek Rk Sk Ik]T = Txk with

T =

, ()

so that only the rst two components of zk have to be observed. With this notation, the implicit epidemic model () can be re-written as

TDM(xk)Tzk+ = Kkzk+ = zk, ()

where Kk = TDM(xk)T. Equation () will be the starting point to design the reduced-

order state observer. Now, () is expanded as

Kk Kk Kk

Kk Kk Kk

Kk Kk Kk

z,k+

z,k+

z,k+

=

z,k

z,k

z,k

()

with zT,k = [Ek Rk], z,k = Sk, and z,k = Ik. The reason to proceed with the observer design implicitly is that matrix Kk can be computed symbolically in an easy way. Hence, we have

Kk = diag(C, J), KTk = [H ], ()

KTk = [ F],

Kk =

h + h(G C) (B +

hJ) ,

Kk = A + H

h

h + ,

Kk =

h(F D)

h + ,

Kk = [G ], Kk = ,

Kk = D, ()

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with =

+()h . The last equation of () reads

Kkz,k+ + Kkz,k+ + Kkz,k+ = z,k ()

and will be used to reduce the order of the observer. Since z,k+ = Sk+ = Nk+ Ek+

Rk+ Ik+ = Nk+ [ ]z,k+ z,k+, () is transformed to

Kkz,k+ + Kk Nk+ [ ]z,k+ z,k+ + Kkz,k+ = z,k,

which nally becomes

Kk Kk[ ] z,k+

= z,k Kkz,k+ + Kkz,k+ KkNk+

k

R is the so-called observer gain. Note that () is an implicit observer since the observed variable z,k+ appears at both sides of equation. Therefore,

the observer is designed implicitly instead of explicitly. An explicit representation of the observer, more appropriate for implementation, can be obtained by re-arranging now () to

Qk z,k+ = z,k Lkk KkNk+ + (Kk Kk)z,k+,

Qk = Kk Kk[ ] LkKk + LkKk[ ], ()

so that

z,k+ = Qk z,k Lkk KkNk+ + (Kk Kk)z,k+ ()

whenever Qk exists. It will be proved in the sequel that this inverse matrix exists if the

observer is designed to be asymptotically stable. In this way, () and () provide together

. ()

Let us analyze the meaning of (). The right-hand side of (), k, is known since the infectious and the total population are known by Assumptions and . On the other hand, the left-hand side is unknown since it just involves the actual state, which is not known. This equation will be used as error signal to drive the state observer. The observer is then designed from the rst equation of ():

Kkz,k+ + Kk Nk+ [ ]z,k+ z,k+ + Kkz,k+ = z,k

by substituting the actual state by the observed one and by adding the observation error, leading to

Kk z,k+ + Kk Nk+ [ ]z,k+ z,k+ + Kkz,k+

= z,k + Lk Kk Kk[ ] z,k+ k , ()

which is the reduced-order observer to estimate the only two components that remain to be observed. LTk = [lk lk]

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an estimation of all the state variables. The derivation of the observer in an implicit way allows the symbolic calculation of the matrix Qk above as

Qk = C + lkG +

h IkNk h IkNk Glk J

vector LTk = [lk lk] such that the matrix inequality

QTkPkQk Pk+ Sk ()

holds for all k .

Proof Consider the positive-denite Lyapunov candidate sequence Vk = eTkPkek. Its time

variation is given by

Vk = Vk+ Vk = eTk+Pk+ek+ eTkPkek

= eTk+Pk+ek+ eTk+QTkPkQkek+

= eTk+ Pk+ QTkPkQk ek+

eTk+Skek+ . ()

Since Vk is positive denite and Vk is negative denite, the second Lyapunov theorem guarantees the convergence to zero of the observation error and the observed variables converge to the actual ones.

Remark It will be proved below that the stability condition () is feasible so that it is always possible to design an asymptotically stable state observer.

Remark Equation () can be expressed as a linear matrix inequality (LMI) by using Schur complements. Therefore, the observer design problem can be eciently solved by available numerical suites.

()

with the parameters dened by (). The explicit calculation of this matrix will allow us to analyze the stability properties of the observer.

5 Stability analysis of the observer

This section is devoted to the stability analysis of the observer. For this purpose, denote the observation error as ek = z,k z,k. Thus, the error equation reads

Qkek+ = ek. ()

The stability of the observer is discussed in the following theorem, which provides sucient conditions to guarantee the observation of the states.

Theorem Consider the discrete-time model () satisfying Theorem constraints. Then the observation error ek of the observer dened by () under Assumptions and converges asymptotically to zero if there exist symmetric positive-denite matrices Pk, Sk

R and

Ibeas et al. Advances in Dierence Equations (2015) 2015:122 Page 14 of 21

Remark Since ek , we have (Ek k) and (Rk Rk) as k . As a con

sequence, () implies (Sk k) as k and all immeasurable state variables are

observed.

The following corollary appears from Theorem .

Corollary If Qk satises the matrix equation (), then it is invertible.

Proof Equation () can be re-written as

QTkPkQk Sk + Pk+. ()

Since Sk and Pk+ are symmetric positive-denite matrices, vT(Sk + Pk+)v > for any real vector v = and all k . Thus, from () we have

vTQTkPkQkv vT(Sk + Pk+)v > ()

for any v = implying that QTkPkQk is a positive-denite matrix. Therefore, det(QTkPkQk) >

. Since det(ABC) = det(A) det(B) det(C) we obtain from () det(Qk) det(Pk) > , imply

ing det(Qk) > and det(Qk) = .

In this way, if the gain vector Lk meets Theorem s requirements, the observer is well dened and asymptotically stable. Furthermore, we can prove now that inequality () is feasible.

Theorem There exists at least one gain vector LTk = [lk lk] such that () holds.

Proof It will be proved that the equality in () holds when Lk = and Sk = S is symmetric and positive denite. In this case, the matrix Qk reads

Qk = C +

, ()

whose eigenvalues are given by = C + h IkNk C = + ( + )h > and = J = +

( + )h > for all k . Since both eigenvalues are larger than unity at all time, we see

that Qk always exists and it is a convergent matrix. The proof is based on the explicit

calculation of a sequence of symmetric positive-denite matrices Pk satisfying (). The following notation will be used:

R

n

i=

L Ai = An . . . A A.

The rationale behind the above denitions is that R multiplies the matrices by the right

while L multiplies them by the left. Moreover, the following relation between the two

h IkNk h IkNk J

n

i=

Ai = A A . . . An,

Ibeas et al. Advances in Dierence Equations (2015) 2015:122 Page 15 of 21

denitions appears:

R

n

i=

Ai T = L

n

i=

ATi. ()

With this notation, we can now explicitly construct a solution to () as

Pk =

q=k

. ()

Now we can check that () is a valid solution to (): Convergence. Since Qk is a convergent matrix for all k the series () is

convergent and well dened. Positive deniteness. Equation () can be re-written as

Pk =

R

q

p=k QTp

S L

q

p=k Qp

L

q

p=k Qp

T

S L

q

p=k Qp

()

by using (). Thus,

xTPkx = xT

L

q

p=k Qp

T

S L

q

p=k Qp

x

xT L

q

p=k Qp

T

S L

q

p=k Qp

x

=

q=k

L

q

= p=k Qp

x TS L

q

p=k Qp

x

since S = ST > . Solution to (). A direct calculation yields

QTkPkQk Pk+

= QTk QTkSQk + QTkQTk+SQk+Qk + Qk Pk+

= S + QTk+SQk+ +

Pk+

Pk+

= S. ()

As a consequence, there exists at least one solution to () and the observer design problem is feasible.

Remark A further interpretation of Theorem is that the observer is asymptotically stable in the open-loop case since L = implies the vanishing of the error driving term in ().

Ibeas et al. Advances in Dierence Equations (2015) 2015:122 Page 16 of 21

6 Numerical examples

This section contains some simulation examples regarding the results introduced in the previous sections. The parameters of the system are assumed to be known and given by

= days, = . days, = days, = , = days, = . days.

()

Models parameters are taken from an outbreak of inuenza in a British boarding school in the late s, [], where the and parameters have been modied in such a way that a short-term simulation is enough to show the dynamic behavior of the system. In this way, in a period of some days the eect of natural growth can be noticed. The initial values of the populations are S = , E = , I = , R = for a total initial population of N = ,. Initially, Figures , , , and depict the dynamics of the continuous and discrete-time systems for time steps of h = hours and h = hours. These time steps are considered since they imply that data are gathered twice or once per day, respectively, which ts in a typical sampling procedure.

It can be seen that the transient response depends slightly on the value of the time step, while maintaining the shape of the response and tending to be the same value in the steady-state. Moreover, since R > in this example, the disease is persistent and there are always

a number of infectious and exposed individuals among the population.

Now, the operation of the reduced state observer will be illustrated through some numerical examples. A time step of h = hours has been used. The observer gain has been selected as LT = l[ ] for the sake of simplicity so that there is only one free parameter.

Figure 1 S variable: susceptible population.

Figure 2 E variable: exposed population.

Ibeas et al. Advances in Dierence Equations (2015) 2015:122 Page 17 of 21

Figure 3 I variable: infectious population.

Figure 4 R variable: immune population.

Figure 5 Evolution of the actual and observed susceptible.

The initial values of the populations are S = , E = , I = , R = for a total initial population of N = . Since the observed populations are initially unknown we may choose the values of,, and R as desired while satisfying + + R = N I =

= . However, as a systematic way to initialize the observer we suggest to use the initial values = N I = and = R = . This is always a valid selection. Figures - show the evolution of the actual and observed populations when l = . while

Figure depicts the convergence of the observation errors to zero.

It can be noticed in Figures - that the actual states are observed and the estimation errors vanish asymptotically. Now, we may analyze the eect of varying the observer gain l into the dynamics of the observation. Thus, l will be changed from l = . to l = . Figure depicts the behavior of the observer for the susceptible when the gain is varied

Ibeas et al. Advances in Dierence Equations (2015) 2015:122 Page 18 of 21

Figure 6 Evolution of the actual and observed exposed cases.

Figure 7 Evolution of the actual and observed immune cases.

Figure 8 Dynamics of the observation errors.

from l = . to l = , while Figure displays the observation error. Also, Figures and show the observation error for the remaining variables. As can be deduced, the larger the observer gain is the faster the estimation of the state variables is. Hence, the most advisable tuning for the observer gain is the largest constant l satisfying the stability constraint ().

7 Conclusions

In this paper a discrete-time SEIR model has been obtained from a continuous-time one by using Mickens discretization method. Afterwards, the well-posedness of the model along with the existence and stability of the equilibrium points have been studied. Moreover, an implicit method has been used to analyze these properties instead of the explicit one used in previous works. It has been shown that this method preserves all the structural prop-

Ibeas et al. Advances in Dierence Equations (2015) 2015:122 Page 19 of 21

Figure 9 Behavior of the observer in S.

Figure 10 Observation error in S.

Figure 11 Observation error in E.

erties exhibited by the continuous-time model such as the expression of the reproduction number and the stability properties of the equilibrium points. In addition, the design of a state observer has been introduced for a discrete-time SEIR epidemic model. The ob-server was designed in an implicit way since the derivation of its stability properties as well as the design process itself are easier than when proceeding with an explicit model of the system. Moreover, some sucient conditions to ensure the asymptotic stability of the ob-server have been provided in terms of a matrix inequality that can be cast in the form of a LMI. The feasibility of these conditions has been proved while some simulation examples showed the operation and usefulness of the observer. The observer is shown to be capable of estimating the actual state variables, while all the observation errors are converging to zero.

Ibeas et al. Advances in Dierence Equations (2015) 2015:122 Page 20 of 21

Figure 12 Observation error in R.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally to the writing of this paper. All authors read and approved the nal manuscript.

Author details

1Departamento de Telecomunicaci i Enginyeria de Sistemes, School of Engineering, Universitat Autnoma de Barcelona, Bellaterra, Barcelona, 08193, Spain. 2Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country, P.O. Box 644, Bilbao, 48080, Spain. 3Young Researchers and Elite Club, Yasooj Branch, Islamic Azad University, Yasooj, Iran.

Acknowledgements

This work was partially supported by the Spanish Ministry of Economy and Competitiveness through grants DPI2012-30651 and DPI2013-47825-C3-1-R, by the Basque Government (Gobierno Vasco) through grant IE378-10 and by the University of the Basque Country through grant UFI 11/07.

Endnotes

a Note that this implies that the total population is constant, Nk = N = N0.

b The inverse of matrix D exists due to (9) and the constraint 1 + ( )h > 0.

Received: 7 January 2015 Accepted: 6 April 2015

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