**3. Problem Statement**

The SEIR discrete-time model demonstrated by Figure 1 and obtained using Euler discretization of the classical continuous-time model is given as follows [42,46]:

**Figure 1.** Block diagram for SEIR model.

$$\begin{aligned} S(k+1) &= (1 - a(k))S(k) + b(k)R(k) - \wp(k)S(k)I(k) + a(k), \\ E(k+1) &= (1 - a(k) - c(k))E(k) + \wp(k)S(k)I(k), \\ I(k+1) &= c(k)E(k) + (1 - a(k) - d(k))I(k), \\ R(k+1) &= d(k)I(k) + (1 - a(k) - b(k))R(k), \end{aligned} \tag{1}$$

where *S*(*k*), *E*(*k*), *I*(*k*), *andR*(*k*) represent state variables corresponding to the portion of population in each compartment of the model. The time-varying non-negative parameters *a* stands for the natural birth-death rate, whereas *b*, *c*, *d* denote the uncertain transition rates from one disease state to the other. The exact values of non-negative parameters *a*, *b*, *c*, *d* are unknown. We only know the lower bound and upper bound values, i.e., *<sup>a</sup>* <sup>∈</sup> [*a*, *<sup>a</sup>*], *<sup>b</sup>* <sup>∈</sup> [*b*, *<sup>b</sup>*], *<sup>c</sup>* <sup>∈</sup> [*c*, *<sup>c</sup>*] and *<sup>d</sup>* <sup>∈</sup> [*d*, *<sup>d</sup>*] with given *<sup>a</sup>*, *<sup>a</sup>*, *<sup>b</sup>*, *<sup>b</sup>*, *<sup>c</sup>*, *<sup>c</sup>*, *<sup>d</sup>*, *<sup>d</sup>* <sup>∈</sup> <sup>R</sup>+. The time-varying parameter ℘(*k*) is extremely uncertain, and no bounds on ℘(*k*) are available for measurements. The initial values for *<sup>x</sup>*(*k*) <sup>∈</sup> <sup>R</sup><sup>4</sup> are unknown but bounded with known bounds *<sup>x</sup>*(0), *<sup>x</sup>*(0) <sup>∈</sup> <sup>R</sup><sup>4</sup> such that *<sup>x</sup>*(0) <sup>≤</sup> *<sup>x</sup>*(0) <sup>≤</sup> *<sup>x</sup>*(0). At at any given time instant *<sup>k</sup>*, the death rate is exactly equal to birth rate *a*(*k*) in all the compartments. In fact, by summing up (1), one gets directly that the total population is constant, thus satisfying

$$N(k+1) = N(k) = N\_0 \quad \forall k \in \mathbb{Z}\_{0+\prime}$$

for

$$N(k) = S(k) + E(k) + I(k) + R(k), \quad \forall k \in \mathbb{Z}\_{0+}...$$

This results in

$$S(k+1) + E(k+1) + I(k+1) + R(k+1) = (1 - a(k))N\_0 + a(k), \quad \forall k \in \mathbb{Z}\_{0+\dots}$$

Hence, if the total population is initially in unity, then (1) remains as a normalized model for all samples with the total population remaining in unity through time; therefore

$$S(k+1) + E(k+1) + I(k+1) + R(k+1) = 1 - a(k) + a(k) = 1, \quad \forall k \in \mathbb{Z}\_{0+}.\tag{2}$$

The transmission of disease arises because of the interactions among susceptible and infectious individuals as described by (1). The disease is transferred to ℘(*k*) individuals through infectious individuals at each time instant. However, a new case only arises with probability *S*(*k*) when contact is directly made with the susceptible individual. Therefore, in compartment *S*, a fraction ℘(*k*)*I*(*k*) of people shift to exposed but non-infectious compartment *E* at time *k*. Similarly, a fraction *c* and *d* of individuals in compartments *E* and *I* migrate to the infectious *I* and recovered *R* compartments, respectively. It should be noted that the recovered compartment is composed of people not yet immune.

The output measured data consists of noisy counts of susceptible individuals obtained from different government sources such as census bureaus by PHS and are represented by the following output system model [3]:

$$y(k) = S(k) + v(k)\tag{3}$$

where *v*(*k*) ∈ *L*<sup>∞</sup> stands for unknown measurement noise with known bounds *v*(*k*), *v*(*k*) ∈ *L*<sup>∞</sup> such that *v*(*k*) ≤ *v*(*k*) ≤ *v*(*k*), ∀*k* ≥ 0. The unknown measurement noise consists of the uncertain number of susceptible people who did not visit the health care unit for diagnosis. Therefore, Equations (1) and (3) are rewritten as follows:

$$\begin{cases} x(k+1) = A(k)x(k) + E\mathfrak{J}(k) + w(k), \\ y(k) = Cx(k) + v(k), \end{cases} \tag{4}$$

where *<sup>x</sup>*(*k*)=[*S*(*k*) *<sup>E</sup>*(*k*) *<sup>I</sup>*(*k*) *<sup>R</sup>*(*k*)]*<sup>T</sup>* and (*k*) = <sup>℘</sup>(*k*)*S*(*k*)*I*(*k*) represent the unknown state vector to be determined and uncertain input, i.e., the newly confirmed infected people from the susceptible individuals at each time instant in the known population, respectively. The time-varying unknown matrix *A*(*k*) and constant matrices *E* and *C* in (4) are given by

$$A(k) = \begin{bmatrix} 1-a & 0 & 0 & b \\ 0 & 1-a-c & 0 & 0 \\ 0 & c & 1-a-d & 0 \\ 0 & 0 & d & 1-a-b \end{bmatrix},$$

$$E = \begin{bmatrix} -1 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \ w(k) = \begin{bmatrix} a(k) \\ 0 \\ 0 \\ 0 \end{bmatrix}, \ \mathbf{C} = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}^T.$$

The uncertain unknown bounded matrix *w*(*k*) for *w*, *w* ∈ *L*<sup>∞</sup> is defined as

$$
\underline{w} = \begin{bmatrix} \underline{a}(k) \\ 0 \\ 0 \\ 0 \end{bmatrix}, \overline{w} = \begin{bmatrix} \overline{a}(k) \\ 0 \\ 0 \\ 0 \end{bmatrix}'
$$

such that *w* ≤ *w* ≤ *w*.
