**6. Example**

The appearance of territories with low economic status always causes the growth of immigration. The early 2000s were remarkable for the formation of several such territories in Northern and Central Africa, the Near East, Afghanistan, etc. As a result, tens of millions of migrants moved to the EU as the level of life in these territories dropped below the subsistence minimum. The EU countries have to allocate considerable financial resources for their filtering and accommodation, which are often unacceptable. An example below illustrates the use of soft randomization for estimating and forecasting of immigration flows from Syria (1) and Libya (2) (the system SL) to Germany (1), France (2), and Italy (3) (the system GFI).

*1. Randomized model, parameters, measurement errors, time intervals, and real data collections.* Choose the randomized mathematical model (Equation (25)) with the normalized variables

$$p\_n[sh] = \frac{K\_n[sh]}{K\_{max}}, \qquad n = \overline{1,3}.\tag{45}$$

This gives

$$p\_n[(s+1)h] = -(1 - b\_1 m\_n) p\_n[sh] + h b\_1 b\_2 \sum\_{i=1, i \neq n}^3 m\_i l\_{in} p\_i[sh] + h f\_n[sh],$$

$$f\_n[sh] = \sum\_{i=1}^M b\_{in} b\_3^{c\_{in}} \quad \qquad n = \overline{1,3}, \tag{46}$$

$$T[sh] = \sum\_{n=1}^3 \sum\_{i=1}^2 c\_{in} b\_{in} b\_3^{c\_{in}}.$$

The state variables of the system GFI and also the immigration flows from the system SL are normalized, i.e.,

$$0 \le p\_n[sh] \le 1, \quad 0 \le f\_n[sh] \le 1, \qquad n = \overline{1, N}. \tag{47}$$

The variable *z*<sup>∗</sup> characterizes the entropy operator of the immigration process and satisfies the last equation in Equation (46). The values of the parameters *mi*, *hin*, *bin*, and *cin* are combined in Table 1, where columns are different values of corresponding parameter. Recall that the two lowest rows of Table 1 indicate the values of the parameters *cin*. By assumption, the regions of both systems have the same specific cost.

**Table 1.** Values of relative parameters.


In accordance with this table, *mmax* = 0.5, *hmax* = 0.5, *bmin* = 0.3, *bmax* = 0.4, and *cmax* = *cmin* = *c* = 0.5. The measurement errors of population sizes ¯ *ξ*[*sh*] ∈ *R*<sup>3</sup> (in normalized units) belong to the intervals

$$\vec{\xi}[sh] \in \Xi = [\vec{\xi}^\dagger, \vec{\xi}^\dagger], \qquad \vec{\xi}^\pm = 0.01,\tag{48}$$

and by assumption they have the same limits for times *sh*.

The normalized observation (model output) has the form

$$\mathbf{v}[sh] = \mathbf{p}[sh] + \vec{\zeta}[sh]. \tag{49}$$

The random parameter model in Equation (46) was employed for estimating parameter characteristics and testing on corresponding time intervals with step *h* = 1*year*:

• T*est* = 2009–2013 as the estimation interval; and

• T*tst* = 2014–2018 as the testing interval.

*2. Entropy estimation of PDFs of model parameters and measurement noises (interval* T*est).* This problem was solved using available data on regional population distribution for Germany (*n* = 1), France (*n* = 2), and Italy (*n* = 3) and also on the shared cost of immigrants maintenance on the estimation interval (see Table 2 and UNdata service at https://data.un.org/).


**Table 2.** Input and output data collections.

In this model, the random parameters *b*1, *b*2, and *b*3 take values within the intervals

$$b\_1 \in \mathcal{B}\_1 = [1.0, 2.5]; \quad b\_2 \in \mathcal{B}\_2 = [0.5, 1.8], \quad b\_3 \in \mathcal{B}\_3 = [0.3, 1.5]. \tag{50}$$

In accordance with Equation (24),

$$\mathcal{U}\_1 = 0.5; \; \mathcal{U}\_2 = 0.5; \; \mathcal{U}\_3 = 1.2; \; \mathcal{U}\_4 = 0.986. \tag{51}$$

Then, the soft RML procedure yields the following optimal PDFs of the model parameters and measurement noises:

$$\begin{array}{rcl} \mathcal{W}^\*(\mathbf{b}) &=& \frac{\exp\left(-0.5b\_1 - 0.5b\_1b\_2 - 1.2b\_3^{0.5} - 0.986\right)}{\mathcal{W}}, \\\mathcal{Q}^\*(\boldsymbol{\xi}) &=& \frac{\exp\left(-\sum\_{n=1}^3 \xi\_n^2\right)}{\mathcal{Q}}, \end{array} \tag{52}$$

where

$$\mathcal{W}^{\flat} = \int\_{\mathcal{B}\_1} \int\_{\mathcal{B}\_2} \int\_{\mathcal{B}\_3} \exp\left(-0.5b\_1 - 0.5b\_1b\_2 - 1.2b\_3^{0.5} - 0.986\right) db\_1 db\_2 db\_3,$$

$$\mathcal{Q}^{\flat} = \prod\_{n=1}^{3} \int\_{-0.01}^{0.01} \exp(-\xi^2) d\xi. \tag{53}$$

The two-dimensional sections of the three-dimensional PDFs of the model parameters are shown in Figure 1a–c, while the graphs of the PDFs of the measurementnoises in Figure 2.



*3. Model testing.* The randomized model in Equation (49) with the optimal PDFs in Equations (52) and (53) was tested using the above data on regional population sizes from the UNdata service

(https://data.un.org/) (see Table 3). This table also presents the testing results in terms of the ensemble-average trajectories *p*¯1[*sh*], *p*¯2[*sh*], and *p*¯3[*sh*].


**Table 3.** Input and output data collections.

Testing was performed via sampling of the randomized interval parameters with the PDFs in Equations (52) and (53) and construction of the corresponding trajectories by Equation (49). Figure 3a–c shows ensembles of such trajectories *<sup>v</sup>*1[*sh*], *<sup>v</sup>*2[*sh*], *v*[*sh*] as well as the ensemble-average trajectories *v* ¯ <sup>1</sup>[*sh*], *<sup>v</sup>*¯2[*sh*], *v*¯3[*sh*] (Graph 1); the real trajectories *y*1[*sh*], *y*2[*sh*], *y*3[*sh*] of regional population sizes (Graph 2); and the limits of the variance pipes *p*¯∗1 [*sh*] ± *σ*1, *p*¯∗2 [*sh*] ± *σ*2, *p*¯∗3 [*sh*] ± *σ*3 (Graph 3).

**Figure 3.** (**a**) *v*¯1[4], (**b**) *v*¯2[4], (**c**) *v*¯3[4].

The testing accuracy was estimated in terms of the relative root-mean-square error

$$\delta\_n = \frac{\sqrt{\sum\_{s=0}^4 \left(p\_n[sh] - y\_n[sh]\right)^2}}{\sqrt{\sum\_{s=0}^4 (p\_n[sh])^2} + \sqrt{\sum\_{s=0}^4 (y\_n[sh])^2}} \tag{54}$$

In the example under study, it constituted 4.6% (Region 1), 3.5% (Region 2), and 2.6% (Region 3).
