*4.1. Interval State Estimator Design*

The observability matrix ! ∈ <sup>R</sup>4×<sup>4</sup> for (4) is given by

$$
\bigcirc \begin{bmatrix} \bigcirc & & & \\ & \bigcirc A(k) & & \\ & \bigcirc A(k+1)A(k) & \\ \bigcirc A(k+2)A(k+1)A(k) & \end{bmatrix}
$$

Then, the SEIR model (4) can be written in the input/output form in the absence of uncertain quantities and exogenous signals as follows:

$$\mathbf{x}(k+4) = A(k+3)A(k+2)A(k+1)A(k)\mathbf{x}(k),$$

$$\begin{bmatrix} y(k) \\ y(k+1) \\ y(k+2) \\ y(k+3) \end{bmatrix} = \mathbf{\varPsi}(k:k+3) = \underset{\bigcirc}{\bigcirc} \mathbf{x}(k). \tag{6}$$

As a result, using available input-output values, (6) can be written as follows:

$$\mathbf{x}(k) = \bigcirc^{-1} \mathbf{\varPsi}(k:k+\mathfrak{z}) \tag{7}$$

Hence, the states of the SEIR model (4) can be obtained using the available input/output values for *k* − 3 ≥ 0 by

$$
\hat{\mathfrak{x}}(k) = \Delta\_{\mathfrak{Y}}(k)\Psi(k-\mathfrak{z}:k),
\tag{8}
$$

with

$$
\Delta\_y(k) = A(k-1)A(k-2)A(k-3)\bigcirc^{-1}.
$$

Then, the equation of our interval state estimator for the SEIR model (4) that generates certain bounds on the real states for *k* − 3 ≥ 0 subject to exogenous signals takes the following form:

$$\begin{aligned} \overline{\mathfrak{X}}(k) &= \mathfrak{X}(k) + \overline{D} + \overline{\Lambda}(k) + \overline{\mathcal{V}}, \\ \underline{\mathfrak{x}}(k) &= \mathfrak{x}(k) + \underline{\mathcal{D}} + \underline{\Delta}(k) + \underline{\mathcal{V}} \end{aligned} \tag{9}$$

where *<sup>D</sup>*, *<sup>D</sup>* <sup>∈</sup> <sup>R</sup>4×1, <sup>Λ</sup>(*k*), <sup>Λ</sup>(*k*) <sup>∈</sup> <sup>R</sup>4×<sup>1</sup> and *<sup>V</sup>*, *<sup>V</sup>* <sup>∈</sup> <sup>R</sup>4×<sup>1</sup> denote the upper and lower limits on uncertain birth and death rate, uncertain input, and measurement noise, respectively. We will define these terms one by one using Lemma 1 as follows.
