Stability analysis and observer design for discretetime SEIR epidemic models
 Asier Ibeas^{1}Email author,
 Manuel de la Sen^{2},
 Santiago AlonsoQuesada^{2} and
 Iman Zamani^{3}
https://doi.org/10.1186/s136620150459x
© Ibeas et al.; licensee Springer. 2015
Received: 7 January 2015
Accepted: 6 April 2015
Published: 17 April 2015
Abstract
This paper applies Micken’s discretization method to obtain a discretetime SEIR epidemic model. The positivity of the model along with the existence and stability of equilibrium points is discussed for the discretetime case. Afterwards, the design of a state observer for this discretetime 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.
Keywords
1 Introduction
Mathematical models have become an important tool in analyzing the causes, dynamics, and spread of epidemics [1]. 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 vaccination policies, quarantine application and so on. These models can be classified into two main categories: continuoustime and discretetime models.
Continuoustime models were the first proposed to describe epidemics spreading since modeling has traditionally been focused from a differential equations point of view. Therefore, many specific features regarding these models have been studied in many works such as the presence of bifurcations [2, 3], the construction of approximate solutions [4, 5], the fractional dynamics [6], the existence of equilibrium points and oscillatory behavior [7], and the presence of waves [8–10], to cite but a few. However, model stability has been by far the most important property to be studied [11–17].
On the other hand, discretetime models have gained substantial importance during the last years as the increasing number of papers concerning this type of models reveals [18–22]. There are many reasons that explain this tendency. Among them we can find the simplicity of its simulation, the fact that it is easier to adjust system’s parameters from statistical data in discretetime models than in continuoustime ones and the fact that discretetime models may exhibit a richer dynamic behavior than its continuoustime counterparts [22]. Until now, as [22] states, the study of discrete epidemic models was 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 discretetime epidemic model. The first one consists in modeling the disease spreading directly in a discrete domain. The second one faces the problem by discretizing a previously existing continuoustime 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 RungeKutta methods [18–22]. However, discretization procedures do not always preserve the structural properties exhibited by continuoustime 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 finite difference (NSFD) methods [21]. These methods combine implicit and explicit Euler approximations in the incidence rate function. Therefore, this paper will apply Micken’s NSFD discretization method to obtain a SEIR discretetime epidemic model while stating its main properties such as the wellposedness of the model, the positivity of solutions and the existence and stability of equilibrium points. Since most of the previous literature is confined 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 [18, 19], 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 [23–29]. Thus, different types of vaccination laws based on control theory have appeared in the literature during the last years, such as the statefeedback control [23, 25], robust sliding control [30], or feedback linearization [24]. Nevertheless, most of these works calculate the vaccination effort 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 online compute the control vaccination law. Under these circumstances, a state observer is necessary to estimate the value of these subpopulations.

NSFD schemes lead to implicit systems rather than explicit. Thus, the design of the state observer must be done implicitly. This makes a big difference 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 reducedorder observer.
The paper is organized as follows. Section 2 describes the model description and problem formulation. Section 3 studies the main properties of the discrete model. The design of the state observer is carried out in Section 4. Section 5 is devoted to the study of the analytic properties of the observer. Finally, Section 6 shows some numerical simulation examples while our Conclusions end the paper.
2 Model description
3 Main properties of the discretetime model
Initially, the wellposedness of (12) will be proved.
Lemma 1
Proof
Furthermore, we have the following.
Theorem 1
\(S_{k}\), \(E_{k}\), \(I_{k}\), and \(R_{k}\) remain nonnegative for all discrete time \(t_{k} = kh\), \(k \in\mathbb{N}\) provided that \(S_{0}\), \(E_{0}\), \(I_{0}\), \(R_{0} \geq0\), \(1+ (\mu \nu)h >0\), and all the system’s parameters are positive.
Proof
Remark 1
Notice that Theorem 1 states that the system is always well defined, since \(I_{k} \geq0\) and \(N_{k} \geq0\) for all \(k \geq0\) implies that \(\mathcal{M} (x_{k})\) is always invertible due to Lemma 1 and makes possible the calculation of \(x_{k+1}\) at each time step through (12).
Remark 2
Remark 3
3.1 Existence and stability of equilibrium points
This section studies the existence and stability of equilibrium points in the discretetime SEIR model given by (4)(5).
Lemma 2
 (1)
If \(\mu\ne\nu\) then the only equilibrium point is given by the trivial one \(S^{*} = E^{*} =I^{*} =R^{*} =N^{*}=0\).
 (2)If \(\mu= \nu\) the total population is constant, i.e. \(N(t) = N = N(0) = S(0) + E(0) + I(0) + R(0)\) and there are three potential equilibrium points given by:
 (a)
The trivial one, \(S^{*} = E^{*} = I^{*} = R^{*} = N^{*} = 0\).
 (b)
\(S^{*} = N\), \(E^{*} = I^{*} = R^{*} = 0\), i.e. the total population is susceptible. This point is usually referred to as the diseasefree equilibrium point.
 (c)
The socalled endemic equilibrium point:
 (a)
Proof
The proof of the first part is straightforward from (1)(2) and (4)(5) while the equilibrium points are calculated in [25]. □
Theorem 2
The discretetime epidemic SEIR model (4)(5) is locally asymptotically stable around the diseasefree equilibrium point when \(\mu= \nu\) provided that \(\mathcal {R}_{0} := \frac{\beta\sigma}{(\mu+ \gamma)(\mu+ \sigma)} < 1\).
On the other hand, the following theorem holds when \(\mathcal{R}_{0} >1\).
Theorem 3
Proof
 (1)
All the state variables converge to a constant finite value.
 (2)
Some of the state variables do not converge to a constant value, and therefore, oscillate.
Theorem 3 represents an alternativetype theorem to establish the behavior of the solutions when \(\mathcal{R}_{0} >1\). 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 (23)(24) makes it difficult to obtain analytical conditions based entirely on the parameters to distinguish in between. This result is also in line with those presented in [36] for the continuoustime model, where the conditions under which the endemic equilibrium point is globally stable are presented. The important feature regarding the case \(\mathcal{R}_{0} >1\) 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 effort. Nevertheless, the number of susceptible, exposed, and immune cases may be very difficult 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 discretetime model.
4 Design of the state observer
The purpose of the state observer is to obtain an estimation of the state, \(\hat{x}_{k}\), from the unique available data, the infectious subpopulation, \(I_{k}\). 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 1
The total population \(N_{k}\) is known at all \(k \geq0\).
Assumption 2
The number of infectious is available at all \(k \geq0\).
Remark 4
The total population may be obtained from (7) if birth and mortality rates are known or by direct measurement. This assumption is particularly feasible in the case of nonlethal diseases, for which the population remains almost constant during the spreading stage.
Remark 5
The number of infectious may be obtained from clinical data at health centers.
5 Stability analysis of the observer
Theorem 4
Proof
Remark 6
It will be proved below that the stability condition (46) is feasible so that it is always possible to design an asymptotically stable state observer.
Remark 7
Equation (46) can be expressed as a linear matrix inequality (LMI) by using Schur complements. Therefore, the observer design problem can be efficiently solved by available numerical suites.
Remark 8
Since \(e_{k} \rightarrow0\), we have \((E_{k}  \hat{E}_{k}) \rightarrow0\) and \((R_{k}  \hat{R}_{k}) \rightarrow0\) as \(k \rightarrow \infty\). As a consequence, (33) implies \((S_{k}  \hat{S}_{k}) \rightarrow0\) as \(k \rightarrow\infty\) and all immeasurable state variables are observed.
The following corollary appears from Theorem 4.
Corollary 1
If \(\mathcal{Q}_{k}\) satisfies the matrix equation (46), then it is invertible.
Proof
In this way, if the gain vector \(L_{k}\) meets Theorem 4’s requirements, the observer is well defined and asymptotically stable. Furthermore, we can prove now that inequality (46) is feasible.
Theorem 5
There exists at least one gain vector \(L_{k}^{T} = [l_{1k} \ l_{2k}]\) such that (46) holds.
Proof

Convergence. Since \(\mathcal{Q}_{k}^{1}\) is a convergent matrix for all \(k \geq0\) the series (52) is convergent and well defined.

Positive definiteness. Equation (52) can be rewritten asby using (51). Thus,$$ P_{k} = \sum_{q=k}^{\infty} \Biggl( L \prod_{p=k}^{q} \mathcal{Q}_{p}^{1} \Biggr)^{T} S \Biggl( L\prod_{p=k}^{q} \mathcal{Q}_{p}^{1} \Biggr) $$(53)since \(S=S^{T} >0\).$$\begin{aligned} x^{T} P_{k} x =& x^{T} \sum_{q=k}^{\infty} \Biggl( L\prod_{p=k}^{q} \mathcal{Q}_{p}^{1} \Biggr)^{T} S \Biggl( L\prod_{p=k}^{q} \mathcal{Q}_{p}^{1} \Biggr) x \\ =&\sum_{q=k}^{\infty} x^{T} \Biggl( L \prod_{p=k}^{q} \mathcal{Q}_{p}^{1} \Biggr)^{T} S \Biggl( L\prod_{p=k}^{q} \mathcal{Q}_{p}^{1} x \Biggr) \\ =&\sum_{q=k}^{\infty} \Biggl( L\prod _{p=k}^{q} \mathcal{Q}_{p}^{1} x \Biggr)^{T} S \Biggl( L\prod_{p=k}^{q} \mathcal{Q}_{p}^{1} x \Biggr) \geq0 \end{aligned}$$

Solution to (46). A direct calculation yields$$\begin{aligned} &\mathcal{Q}_{k}^{T} P_{k} \mathcal{Q}_{k} P_{k+1} \\ &\quad= \mathcal{Q}_{k}^{T} \bigl( \mathcal{Q}_{k}^{T} S \mathcal{Q}_{k}^{1} + \mathcal{Q}_{k}^{T} \mathcal{Q}_{k+1}^{T} S \mathcal{Q}_{k+1}^{1} \mathcal{Q}_{k}^{1} + \cdots \bigr) \mathcal{Q}_{k} P_{k+1} \\ &\quad= S + \underbrace{ \bigl( \mathcal{Q}_{k+1}^{T} S \mathcal {Q}_{k+1}^{1}+ \cdots \bigr)}_{P_{k+1}}  P_{k+1} \\ &\quad= S . \end{aligned}$$(54)
6 Numerical examples
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 steadystate. Moreover, since \(\mathcal{R}_{0} > 1\) in this example, the disease is persistent and there are always a number of infectious and exposed individuals among the population.
7 Conclusions
In this paper a discretetime SEIR model has been obtained from a continuoustime one by using Micken’s discretization method. Afterwards, the wellposedness 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 properties exhibited by the continuoustime 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 discretetime SEIR epidemic model. The observer 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 sufficient conditions to ensure the asymptotic stability of the observer 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.
Declarations
Acknowledgements
This work was partially supported by the Spanish Ministry of Economy and Competitiveness through grants DPI201230651 and DPI201347825C31R, by the Basque Government (Gobierno Vasco) through grant IE37810 and by the University of the Basque Country through grant UFI 11/07.
Open Access 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.
Authors’ Affiliations
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