**1. Introduction**

There were around 30.2–45.1 million people living with HIV with 680,000 casualties in 2020, whereas an epidemic like seasonal influenza causes 3–5 million serious illness cases with 250,000–500,000 casualties each year worldwide according to the WHO [1,2]. The surveillance of infectious diseases plays a vital role in analyzing these epidemics, for instance, origin, spread and dynamics. PHS relies upon surveillance statistics collected by agencies such as the Chinese Center for Disease Control and Prevention (China CDC) for infected people to estimate the activity level of such diseases, intervention strategy preparation, and recommendations of design policies.

Mathematical modeling of epidemics plays a major role in organizing public health responses and developing early outbreak detection systems [3–7]. The first modern mathematical epidemic model, i.e., susceptible-infectious-recovered (SIR), was proposed by

**Citation:** Khan, A.; Bai, X.; Ilyas, M.; Rauf, A.; Xie, W.; Yan, P.; Zhang, B. Design and Application of an Interval Estimator for Nonlinear Discrete-Time SEIR Epidemic Models. *Fractal Fract.* **2022**, *6*, 213. https:// doi.org/10.3390/fractalfract6040213

Academic Editors: Thach Ngoc Dinh, Shyam Kamal, Rajesh Kumar Pandey and Norbert Herencsar

Received: 24 January 2022 Accepted: 7 April 2022 Published: 9 April 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Kermack et al. in 1927 for cholera (London 1865) and plague epidemics (Bombay 1906, London 1665–1666) [8]. According to the SIR model, a fixed number of the population can be divided into three compartments at any time: susceptible individuals (not yet infected but can be infected in future), infectious individuals (those who have an infection and can infect others), and recovered individuals (who are recovered from the infection and are immune now). The number of people for each compartment represents the states of a SIR epidemic model. The total number of individuals who are assumed to be mixed homogenously remains the same, which means the probability of each individual coming in contact with others is equal [3].

However, the generic SIR epidemic model must be expanded to include a fourth compartment in case of many infectious diseases, for instance, influenza-like illness, tuberculosis, and HIV/AIDS [3,9,10]. The state of the fourth compartment corresponds to the latency period of disease, i.e., someone who is infected but still unable to infect others. This modified model is called the SEIR epidemic model [11]. Several estimation techniques have been developed to track and estimate the states of these models [3,12,13]. To design these estimators to converge to actual states, one needs to know the exact values of the uncertain quantities. However, designing such estimators for SEIR models in a real scenario is challenging, especially when the uncertain parameters are not exactly known but are defined by an interval or polytope. Interval estimator techniques can solve such issues [14–22]. Based on the monotone system theory (MST), interval estimators are designed to estimate the real states at any time instant and generate a set of acceptable values known as the interval in [20,23–28]. The ability to deal with large and unknown uncertainties in the system is one of the key advantages of interval state estimator design [29–31]. However, getting cooperative/nonnegative systems are not always possible, and solving this issue for interval estimator design is still an open area of research.

This article proposes an interval estimation-based method to track and estimate the four states of the SEIR epidemic model subject to uncertain parameters. Diaby et al., 2015 [32] proposed the first interval estimator for the continuous-time epidemic model. The results obtained by Diaby et al. were adequate but not ideal because the observer gain was manually set. Instead, we consider the discrete-time SEIR model and use an efficient method based on the observability matrix to design the interval estimator without observer gain. Finite-time convergence for the interval vectors' width is derived to verify the boundedness of the estimation error that significantly improves the accuracy of the designed method.

It is worth mentioning that the observer gain used in the conventional interval observers' design determines the magnitude of the upper bound of the interval estimation errors (for instance, see [27,28]). As a result, interval observers that converge faster may result in a state enclosure that is too conservative at steady state. The noted issue is solved in this study since we do not require an observer gain to run the proposed state estimator. In contrast, set-membership state estimators address the optimization problem at each iteration, and the problem of finite convergence time is ignored. Therefore, the proposed result on finite time convergence is intriguing. However, it is a little more demanding in terms of computation time. Furthermore, compared with the Kalman filter-type estimators, the proposed interval state estimator requires less information on state disturbances and measurement noise to generate guaranteed enclosures of the real state vector. This knowledge is advantageous when dealing with real-world situations when state disturbances and measurement noise are poorly known. More specifically, compared with the existing results in the literature, the contributions are fourfold:


state matrix *A*. The bounds on the uncertain input are constructed before designing the interval estimator;


The remainder of this work continues with notations and interval analysis in Section 2. The problem statement is described in Section 3, and the key findings are shown in Section 4. Two numerical examples are given in Section 5, where a comparison with previous results in [27,28] is demonstrated. Finally, concluding remarks are presented in Section 6.
