## 2 Materials and Methods
### Mathematical Models
#### 2.4.2 ODE and its Parameterization  
Dynamic systems (DS) are conventionally modeled by differential equations. In order to use differential equations as a solution, one needs to find a differential equation from the observational data. This is an inverse problem. Inverse problems are conventionally solved by analytical or numerical methods. Artificial neural networks are on the rise as a method for finding differential equations from data. The methods are convenient to use in parametrizing differential equations; another advantage is that they handle data noise pretty well--this research utilized machine learning to build differential equations. Consider the system in question as a dynamic system S, the variables (\(x(t)\), \(p(t)\), \(z(t)\)) being its response to some input \(u(t)\). Several events (bots, media) can be considered inputs. Let \(P\) be the modeled object; \(U\) be the managerial decisions, \(y=(x,p,z)\) be the system's response to a control input. With these definitions, state the key problems pertaining to the dynamic system (DS):  
1. (_U,P,y_)--DS behavior analysis;
2. (_U,P,y_)--DS control synthesis;
3. (_U,P,y_)--DS identification. This paper investigates the third problem. The other two have been solved in part. DS behavior was described by a system of autonomous differential equations:  
\[\left\{ \begin{array}{l}\frac{dx}{dt}=f_{1}(u,x,p,z)\\ \frac{dz}{dt}=f_{2}(u,x,p,z)\\ \frac{dy}{dt}=f_{3}(u,x,p,z)\end{array} \right. \tag{2}\]  
The right-side function in (2) was found by two methods, resulting in the ODE 1 and ODE 2 models. ODE 1 models were built in the local domain. ODE 1 models were linear systems of ordinary differential equations; ODE 2 models were nonlinear systems of ordinary differential equations. The occurrence rate of bots or posts in the media can serve as the variable \(\mathsf{u}(t)\) in (2). The variable was not used in modeling for this research; however, the variable can be introduced if data is available. Let us state the mathematical problem for problems written in (1) as forward and inverse ODE solution problems. Statement of the forward problem. \[\left\{ \begin{array}{l}\dot{y}=Fy\\ y(t_{0})=y_{0}\end{array} \right. \tag{3}\]  
where \(F=(f_{1},f_{2},f_{3})\), \(y(t)=(x(t),p(t),z(t))^{T}\), \(y_{0}\) are the initial conditions. For this forward problem, from the given initial conditions find \(y(t_{1}),\ y(t_{2}),\ y(t_{3}),\ \ldots\). The forward problem is stated to solve Problems 1, 2. Statement of the inverse problem. Readings \(y(t_{1}),\ y(t_{2}),\ y(t\,3),\ \ldots\) are known for some points. From these readings, find \(F(\cdot)\) in (3). To solve the inverse problem means to solve the problem of identifying the system, i.e., Problem 3. The ODE 1 model was built by cluster analysis using the Nelder-Mead method. Figure 1 shows the non-normalized time series in dynamics. It makes clear that the initial data do not have identical time series. To describe DS behavior in terms of differential equations, the observation interval [0; \(n\)] was split into several subintervals by cluster analysis. A model was built for each cluster. This approach is similar to modeling in the low-mod domain, where the behavior of a dynamic system can be described with only a few variables [41]. As a result, we were able to choose the simplest writing for the rightside in (3) to describe the processes and to track the change in the model parameters in order to analyze the processes in the system. DS behavior in each cluster was described by a system of autonomous differential equations:  
\[\left\{ \begin{array}{l}\frac{dx}{dt}=a_{11}x+a_{12}p+a_{13}z\\ \frac{dx}{dt}=a_{21}x+a_{22}p+a_{23}z\\ \frac{dp}{dt}=a_{31}x+a_{32}p+a_{33}z\end{array} \right. \tag{4}\]  
The parameters of the model (4) were estimated by the Nelder-Mead method [42].