*4.3. Finite-Time Convergence*

The finite-time convergence of the interval width [*x*(*k*)] is one of the primary issues concerning the tight initialization and stability of the interval estimator. Therefore, this section proves that [*x*(*k*)] converges to a known upper-bounded value in finite time provided by the uncertain quantities. Hence, we introduce the following Lemma to compute the upper bound on [*x*(*k*)].

**Lemma 2.** *The following inequality determines the upper bound on the width of the interval vector provided by (9) and (24):*

$$\mathcal{IG}\left[\mathbf{x}(k)\right] \le \left\|\boldsymbol{\phi}(k)E\_{n-1}\right\|\_{\infty} \mathcal{O}\left[\mathbb{G}\right] + \left\|\boldsymbol{\phi}(k)\right\|\_{\infty} \mathcal{O}\left[w\right] + \left\|\left|\boldsymbol{\varphi}\_{v}(k)\right\|\_{\infty} \mathcal{O}\left[v\_{n}\right], \quad \forall k \ge 3. \tag{28}$$

**Proof.** Firstly, the recursive system (24) is employed as an interval predictor during the initialization phase for *k* = 0, 1, 2 to provide tight bounds on the interval vector of the SEIR model (4). As a result, the upper limit on the width of interval vector provided by (24) is given by

$$\mathcal{U}\left[\mathbf{x}(k)\right] \le \left\| \prod\_{\ell=1}^{k} A(k-\ell) \right\|\_{\infty} \mathcal{U}\left[\mathbf{x}(0)\right] + \mathcal{U}\left[\eth(k-1)\right].\tag{29}$$

Secondly, the proposed interval state estimator (9) is used for *k* ≥ 3. Then, Equation (23) implies

$$\left|\mathbb{U}\left[\mathbf{x}(k)\right]\right| \le \left\|\left[\phi(k)E\_{n-1}\right]\right\|\_{\infty} \left\|\mathbb{U}\left[\mathbb{S}\right] + \left\|\left[\phi(k)\right]\right\|\_{\infty} \left\|\mathbb{U}\left[w\_{n-1}\right] + \left\|\left[\phi\_{v}(k)\right]\right\|\_{\infty} \left\|\mathbb{U}\left[v\_{n}\right] \right\|\right.\tag{30}$$

whereas

$$\begin{split} & \| \| \boldsymbol{\phi}(k) \boldsymbol{E}\_{n-1} \| \_{\infty} \boldsymbol{\eth} \left[ \boldsymbol{\eth} \right] + \| \boldsymbol{\phi}(k) \| \_{\infty} \boldsymbol{\eth} \left[ \boldsymbol{w}\_{n-1} \right] + \| \boldsymbol{\phi}\_{\boldsymbol{\nu}}(k) \| \_{\infty} \boldsymbol{\eth} \left[ \boldsymbol{v}\_{\boldsymbol{\nu}} \right] \\ & \leq \| \boldsymbol{\phi}(k) \boldsymbol{E}\_{n-1} \| \_{\infty} \boldsymbol{\eth} \left[ \boldsymbol{\eth} \right] + \| \boldsymbol{\phi}(k) \| \_{\infty} \boldsymbol{\eth} \left[ \boldsymbol{w} \right] + \| \boldsymbol{\phi}\_{\boldsymbol{\nu}}(k) \| \_{\infty} \boldsymbol{\eth} \left[ \boldsymbol{v}\_{\boldsymbol{\nu}} \right]. \end{split} \tag{31}$$

Based on (30) and (31), one can easily determine that

$$\mathbb{G}\left[\mathbf{x}(k)\right] \le \left\|\boldsymbol{\phi}(k)E\_{n-1}\right\|\_{\infty} \mathbb{G}\left[\mathbb{G}\right] + \left\|\boldsymbol{\phi}(k)\right\|\_{\infty} \mathbb{G}\left[w\right] + \left\|\boldsymbol{\varphi}\_{v}(k)\right\|\_{\infty} \mathbb{G}\left[v\_{n}\right].\tag{32}$$

This completes the proof of Lemma 2.

**Remark 1.** *It is worth noting that we do not need observer gain to design the interval estimator. Therefore, the impact of gain that leads to pessimistic state enclosures for the traditional-type interval observer design method [27,28] is avoided. However, it is more demanding in term of computation time. In addition, unlike Kalman filter-type state estimators, the exact values of exogenous signals are not necessarily known and hence represent an advantage while dealing with practical applications.*
