# Solution for the Mathematical Modeling and Future Prediction of the COVID-19 Pandemic Dynamics
## 3 Types of Mathematical Models Used
### ARX Model and ARMAX Model  
The ARX (autoregressive-exogenous) model is the most basic model that includes the input signal. As it is the outcome of solving linear regression equations in analytic form, the estimation of the ARX model is the most efficient of the polynomial estimation methods. As a result, the ARX model is preferable, especially when the model order is large. The ARX (n, m) model is defined in (17):  
\[y(k)=a_{1}y(k-1)+a_{2}y(k-2)+\ldots+a_{n}y(k-n)+b_{1}u(k-1)+b_{2}u(k-2)+b_{m}u (k-m)+v(k) \tag{17}\]  
where \(y\) is the model output signal; \(u\) is the model input signal; \(a_{i}\) is the \(i^{th}\) autoregressive (AR) parameter; \(b_{i}\) is the \(i^{th}\) exogenous (X) parameter; and \(v\) is the noise. When the system's disturbance \(v(k)\) is not white noise, the connection between the deterministic and stochastic dynamics might decrease the quality of the ARX model estimation. To reduce the error, the model order has to be set higher than the system order,especially when the signal-to-noise ratio is low. However, raising the model order might alter several dynamic properties of the model, such as its stability. ARMAX (autoregressive-moving-average) models are frequently utilized in system identification because they accurately capture a wide range of real-world processes with a minimal level of complexity. ARMAX models have the same stochastic context as Kalman filtering and are a common tool in control engineering for both, system model and control design. ARMAX models are part of the equation-error model family. Their result is given as the sum of three regression terms including historical inputs, outputs, and white noise samples. The parallel connection of a deterministic component is driven by the observed input signal \(u(t)\) and of a stochastic part driven by a remote white noise \(v(t)\). The deterministic part is represented by the discrete transfer function \(B(z^{-1})/A(z^{-1})\) and its output signal \(y(t)\) is not available. The stochastic part is denoted by the discrete transfer function \(C(z^{-1})/A(z^{-1})\) and its output signal is a white noise \(v(t)\) that can be the effect of non-accessible disturbances on the state of the deterministic part. Before estimating a model, it is necessary to select an appropriate model structure. The model structure is chosen based on an understanding of the physical systems. In system identification, three types of models are commonly used: black-box models, grey-box models, and user-defined models. The black-box model states that systems are unknown and that all model parameters may be adjusted without regard, for the physical environment. To change randomly all the settings is not allowed. The grey-box model states that some of the underlying dynamical or physical characteristics are known and that the model parameters may be constrained in some way. The user-defined model is based on the assumption that the frequently used parametric models cannot accurately reflect the model that has to be estimated. To aid in the modelling of an unknown system, several parametric model structures are available. Differential equations and transfer functions are used to explain systems in metric models. These models give insight into the mechanics of the system as well as compact model architectures. To test a variety of architectures to choose the optimal one is frequently advantageous. A parametric model structure is a black-box model that specifies either a continuous-time or discrete-time system. In particular, the technique described is based on the extension of the dynamic Frisch scheme. Once a high-order ARX model has been identified, the polynomials \(A(z^{-1})\), \(B(z^{-1})\) are estimated by using the properties of polynomials with common factors. Finally, part \(C(z^{-1})\) is estimated by considering the relation with its high-order autoregressive approximation.