**4. Interval Estimator Design for SEIR Model**

We will design the interval state estimator in this section for the SEIR model (4). In the presence of uncertain parameters, the primary goal of this research is to construct an interval estimator for the SEIR model such that the unknown state signals always satisfy the following inequality:

$$
\overline{\mathfrak{x}}(k) \le \mathfrak{x}(k) \le \overline{\mathfrak{x}}(k), \ \forall k \ge 0,\tag{5}
$$

where *x*(*k*), *x*(*k*) represent the highest and lowest values for the interval state bounds provided that *x*(0) ∈ [*x*(0), *x*(0)]. The proposed interval estimator can help to make a deciding rule for pandemic detection. The following definition and assumption are required to design the proposed interval state estimator for the given SEIR model.

**Proposition 1.** *As the state x*(*k*0) = *x*(0) *is determined uniquely for all u*(*τ*) *and y*(*τ*)*, τ* ∈ [*k*0, *k*1]*, the SEIR epidemic model described by (4) is observable over* [*k*0, *k*1]*.*

**Assumption 1.** *There are known bounds <sup>w</sup>*, *<sup>w</sup>* <sup>∈</sup> <sup>R</sup>4, *<sup>v</sup>*, *<sup>v</sup>* <sup>∈</sup> <sup>R</sup> *such thatw*(*k*) <sup>∈</sup> [*w*, *<sup>w</sup>*], *<sup>v</sup>*(*k*) <sup>∈</sup> [*v*, *<sup>v</sup>*]*.*

The given proposition and assumption are necessary for designing the proposed interval estimator. The bounds given by Assumption 1 determine the uncertainty of initial values, input disturbance and noise.
