*Article* **Design and Application of an Interval Estimator for Nonlinear Discrete-Time SEIR Epidemic Models**

**Awais Khan 1,2,3, Xiaoshan Bai 1,3, Muhammad Ilyas 4, Arshad Rauf 5, Wei Xie 6, Peiguang Yan <sup>2</sup> and Bo Zhang 1,3,\***


**Abstract:** This paper designs an interval estimator for a fourth-order nonlinear susceptible-exposedinfected-recovered (SEIR) model with disturbances using noisy counts of susceptible people provided by Public Health Services (PHS). Infectious diseases are considered the main cause of deaths among the top ten worldwide, as per the World Health Organization (WHO). Therefore, tracking and estimating the evolution of these diseases are important to make intervention strategies. We study a real case in which some uncertain variables such as model disturbances, uncertain input and output measurement noise are not exactly available but belong to an interval. Moreover, the uncertain transmission bound rate from the susceptible towards the exposed stage is not available for measurement. We designed an interval estimator using an observability matrix that generates a tight interval vector for the actual states of the SEIR model in a guaranteed way without computing the observer gain. As the developed approach is not dependent on observer gain, our method provides more freedom. The convergence of the width to a known value in finite time is investigated for the estimated state vector to prove the stability of the estimation error, significantly improving the accuracy for the proposed approach. Finally, simulation results demonstrate the satisfying performance of the proposed algorithm.

**Keywords:** interval analysis; interval estimator; finite-time convergence; bounded uncertainties; infectious diseases; SEIR epidemic model
