*4.2. Interval Prediction for k* < 3

It is worth mentioning that to use the developed interval state estimator (9) for the given SEIR model (4), the initial *n* − 1 = 3 values of input-output should be accessible. However, in many real life scenarios, these values are not always available for measurements, such as in the given case, where only the initial values are given with some known bounds. Therefore, we proposed the following recursive system as an interval predictor that provides a bound on the system's states for *k* = 0, 1, 2.

**Proposition 2.** *The following interval predictor generates* [*x*(*k*)] *such that for k* = 0, 1, 2*, we have x*(*k*) ∈ [*x*(*k*)] *as*

$$\begin{array}{l} [\mathbf{x}(k)] = \prod\_{\ell=1}^{k} A(k-\ell)[\mathbf{x}(0)] + [\partial(k-1)],\\ [\partial(k)] = A(k)[\partial(k-1)] + E[\mathbb{S}] + [w],\\ [\partial(0)] = E[\mathbb{S}] + [w].\end{array} \tag{24}$$

**Proof.** To prove Proposition 2, we use mathematical induction. As for the initial case *k* = 0, the input and output are available. Therefore, we consider the case *k* = 1. Hence, the first cycle of the SEIR model produces the following equations:

$$\begin{array}{l} x(1) = A(0)x(0) + E \odot (0) + w(0), \\ [x(1)] \in A(0)[x(0)] + E[\hearrow] + [w], \\ [x(1)] \in A(0)[x(0)] + [\partial(0)], \end{array} \tag{25}$$

which implies the correctness of (24) for *k* = 1. Next, we demonstrate that (24) is true for *k* = 2. Once again, considering (4) for *k* = 2, we get

$$\begin{aligned} \mathbf{x}(2) &= \prod\_{\ell=1}^{2} A(2-\ell)\mathbf{x}(0) + E\Diamond(1) + w(1) + \sum\_{m=0}^{1} \left\{ \prod\_{\ell=1}^{2} A(2-\ell) \right\} \{E\Diamond(m) + w(m)\}, \\ \mathbf{x}(2) &\in \prod\_{\ell=1}^{2} A(2-\ell)[\mathbf{x}(0)] + A(1)[\partial(0)] + E[\Diamond] + [w], \end{aligned} \tag{26}$$

$$\mathbb{E}\left[\mathbf{x}(2)\right] \in \prod\_{\ell=1}^{2} A(2-\ell)[\mathbf{x}(0)] + [\partial(1)].\tag{27}$$

Thus, by simple mathematical induction, one can easily show that (27) is true for *k* = 2 as well. This completes the proof of Proposition 2.
