*5.2. Example 2: Ebola Outbreak in West Africa*

The 2014 West Africa Ebola outbreak [47] is considered using the following parameters to demonstrate the efficiency of the suggested technique having a period of one day as

$$\begin{array}{ll} a = 0.0099/day, & b = 0.00128/day, \ c = 0.1887/day, \\ d = 0.1/day, & \wp = 0.4/day. \end{array}$$

Using these parameters, we get matrix *A*, as follows:


The output measurement noise *v*(*k*) is uniformly distributed with given bounds − *V* ≤ *v*(*k*) ≤ *V*; *V* = 0.001.

We assumed that (*k*) = ℘*S*(*y* − *v*) is unknown but bounded with the following constraints:

$$
\overline{\varphi}(k) - \mathfrak{r} = \overline{\varphi}(k) \le \overline{\varphi}(k) \le \overline{\varphi}(k) = \overline{\varphi}(k) + \mathfrak{r}; \quad \mathfrak{r} = 0.0001.
$$

The observability matrix is obtained using *A* and *C* as


The disturbance matrix to obtain bounds on the uncertain input, and uncertain birth and death rate is computed as

*<sup>φ</sup>*(*k*) = <sup>Σ</sup>*<sup>A</sup>* <sup>−</sup> <sup>Δ</sup>*y*(*k*)Σ*CA* with <sup>Δ</sup>*y*(*k*) = *<sup>A</sup>*3!−<sup>1</sup> and


Similarly, the measurement noise matrix is calculated as *ϕv*(*k*) = −Δ*y*(*k*). The initial unknown bounded states are part of theinterval *x*(0) ≤ *x*(0) ≤ *x*(0) with *x*(0)=[0.88 0.06 0.049 0] and *x*(0)=[0.93 0.08 0.052 0.05].

The bounds on the uncertain input, uncertain birth-death rate and measurement noise are obtained using (10), (11) and (13), respectively. For the proposed SEIR model (4), we have *n* = 4; therefore, interval predictor (24) is used for *k* = 1, 2 whereas (9) is used for *k* > 2 to obtain guaranteed bounds on *x*(*k*) provided that *x*(0) ≤ *x*(0) ≤ *x*(0).

The simulation results of the proposed method and the one in [27] are depicted in Figure 6 to compare the observers' dynamics. As shown in Figure 6, the bounds generated by the developed method are tighter than those resulting from the work of Degue et al. [27]. In addition, the proposed method is easy to implement compared with [27] as it does not need observer gain and nonnegativity of the system dynamics to design the interval estimator. Moreover, Figures 7 and 8 show that the interval estimation errors *e*− *<sup>i</sup>* = *xr* − *xr*,

*e*+ *<sup>i</sup>* = *xr* − *xr* for *r* = 1, 2, 3, 4; *i* = *S*, *E*, *I*, *R*, respectively, provided by our work are much smaller compared with [27].

**Figure 6.** Interval estimations by proposed method vs. the method given in [27] for each state variable *x*1, *x*2, *x*3, *x*<sup>4</sup> corresponding to *S*, *E*, *I*, *R*.

**Figure 7.** Upper-bound error: (a) proposed method; (b) method given in [27].

**Figure 8.** Lower-bound error: (a) proposed method; (b) method given in [27].
