# Solution for the Mathematical Modeling and Future Prediction of the COVID-19 Pandemic Dynamics
## 3 Types of Mathematical Models Used
### Transfer Function Model  
Transfer functions are often utilized in control theory to characterize the input-output dependencies in the case of linear time-invariant systems (these types of systems can be modelled in the time domain using ordinary differential equations with constant coefficients) [38]. The dependency between the input signal and the output signal written in the complex domain in the case of a SISO system (Single Input Single Output) is presented in (1):  
\[\mathrm{Y(s)}=\frac{\mathrm{num(s)}}{\mathrm{den(s)}}\mathrm{U(s)}+\mathrm{E(s )}, \tag{1}\]  
Figure 2: MISO process structure. where: Y(s), U(s), and E(s) represent the Laplace transforms of the output signal, input signal and noise signal, respectively; num(s) and den(s) represent the numerator and the denominator of the transfer function which mathematically connects the U(s) and Y(s) signals; and num(s) and den(s) represent polynomials of \"s\" complex variable associated to the Laplace transform. A linear time-invariant system [39] is considered, it is modelled by the following differential Equation (2):  
\[\mathrm{a_{0}y(t)+a_{1}\frac{d}{dt}y(t)+\ldots+a_{n-1}\frac{d^{n-1}}{dt^{n-1}}y(t )+a_{n}\frac{d^{n}}{dt^{n}}y(t)=b_{0}x(t)+b_{1}\frac{d}{dt}x(t)+\ldots+b_{m-1} \frac{d^{m-1}}{dt^{m-1}}x(t)+b_{m}\frac{d^{m}}{dt^{m}}x(t), \tag{2}\]  
where y is the output signal of the system and x is the input signal. In (2), the \(\mathrm{a_{i}}\) (\(\mathrm{i}\in\{1,2,\ldots,n\}\)) and \(\mathrm{b_{j}}\) (\(\mathrm{j}\in\{1,2,\ldots,m\}\)), coefficients are constant. Consequently, the differential equation presented in (1) can be used for describing the operation of a linear system. Due to the fact that in (1), only one input signal and only one output signal occur, this type of differential equation can be used for modelling a SISO (Single Input Single Output) system. By applying the Laplace transform for zero initial conditions to (1), it results in (3):  
\[\mathrm{H(s)=\frac{Y(s)}{X(s)}=\frac{b_{m}\cdot s^{m}+b_{m-1}\cdot s^{m-1} \cdot\ldots\ldots\cdot+b_{1}\cdot s+b_{0}}{a_{n}\cdot s^{n}+b_{n-1}\cdot s^{n-1 }\cdot\ldots\cdot+a_{1}\cdot s+a_{0}}, \tag{3}\]  
where \(Y(s)=L[y(t)]\) and \(X(s)=L[x(t)]\) represent the Laplace transforms of the output, with respect to the input signal, and \(H(s)\) represents the transfer function of the modelled system. A system is practically feasible (practically implementable) only if \(n\geq m\). If \(n>m\) the system is strictly proper (the case of the majority of practically feasible systems); if \(n=m\) the system is proper. The main advantage of applying the Laplace transform is the conversion of the differential Equation (2) (in the time domain) into an algebraic equation (in the complex domain), an equation which is further much simple to be mathematically processed. The considered biological system (process) of the SARS-CoV-2 transmission dynamics is a MIMO one (Multiple Input Multiple Output) due to its complexity and due to the multitude of factors which influence it. However, in order to obtain a higher accuracy for the proposed mathematical models, the SARS-CoV-2 transmission process is modelled as a family of MISO (Multiple Input Single Output) models. More exactly, the cumulative effect of all the input signals to each output signal is modelled separately. The general linear differential equation usable for modelling a MISO system with 5 input signals and one output signal is presented in (4):  
\[\sum_{i=0}^{n}a_{i}\cdot\frac{d^{i}(y(t))}{dt^{i}}= \sum_{i=0}^{m_{1}}b_{1i}\frac{d^{i}(x_{1}(t))}{dt^{i}}+\sum_{k=0}^{m_{2}}b_ {2k}\cdot\frac{d^{k}(x_{2}(t))}{dt^{k}}+\sum_{p=0}^{m_{3}}b_{3p}\cdot\frac{d^{ p}(x_{3}(t))}{dt^{p}}+\sum_{c=0}^{m_{4}}b_{4c}\cdot\frac{d^{c}(x_{4}(t))}{dt^{c}}+ \sum_{c=0}^{m_{5}}b_{3c}\cdot\frac{d^{c}(x_{5}(t))}{dt^{c}}, \tag{4}\]  
where \(y(t)\) is the output signal and \(x_{1}(t)\), \(x_{2}(t)\), \(x_{3}(t)\), \(x_{4}(t)\) and \(x_{5}(t)\) are the input signals.