# Fractional-Order SIR Epidemic Model for Transmission Prediction of COVID-19 Disease
## 3 Fractional-Order Generalization of SIR Model  
It is well known that the integer-order derivative can be generalized by the fractional-order counterpart. In this case, the derivative is presented in the form of \(\mathcal{D}^{\alpha}\), where the order \(\alpha\in\mathbb{R}\). Note that the derivative is usually defined in one of three definitions, e.g., Caputo, Rieman-Liouville, and Grunwald-Letnikov ones. In the paper, Grunwald-Letnikov [19] is exploited hereinafter, where  
\[\mathcal{D}^{\alpha}x(t)=h^{-\alpha}\lim_{h\to 0+}\sum_{j=0}^{\lfloor\frac{t}{ \beta}\rfloor}(-1)^{j}\binom{\alpha}{j}x\left(t-\frac{j}{h}\right) \tag{3}\]  
with \(t\) is continuous-time, \(h\) is sampling interval and \((\cdot)\) denoting newton binomial, which (for \(j>0\)) can be calculated using the Pochhammer function as  
\[\binom{\alpha}{j}=\frac{\alpha_{j}}{j!}=\frac{\prod_{i=0}^{j-1}(\alpha-i)}{j!} =\frac{a_{j-1}}{(j-1)!}\frac{\alpha-j+1}{j} \tag{4}\]  
In application, we use fixed step method, where we use nabla difference defined in discrete-time \(\mathbb{T}=\{0,1,2,...\}\), such that \(t=\mathbb{T}h\). The nabla difference is defined as  
\[\
abla^{\alpha}x(t) = h^{-\alpha}\sum_{j=0}^{\lfloor\frac{t}{\beta}\rfloor}(-1)^{j} \binom{\alpha}{j}x\left(t-\frac{j}{h}\right) \tag{5}\] \[\
abla^{\alpha}x_{k} = h^{-\alpha}\sum_{j=0}^{k}(-1)^{j}\binom{\alpha}{j}x_{k-j} \tag{6}\]  
Note that, for satisfactory low \(h\), we have \(\mathcal{D}^{\alpha}x(t)\approx\
abla^{\alpha}x(t)\), so that we can effectively use the discrete-time difference to model continuous time processes. It is well known that the incorporation of fractional-order derivative into model equations can lead to improving model parameters in several physical processes. In particular, various diffusion processes can be better modeled by use of fractional-order differential equations [20, 21, 22]. Taking into account that virus transmission is a similar class of process to the above, the fractional-order generalization of the SIR model can significantly improve the model performances. In this case, the integer-order differences are substituted by the fractional-order counterparts \(\alpha_{i}\in\mathbb{R}\), \(i\in\{1,\ 2,\ 3\}\), and the fractional-order SIR model can be presented in form  
\[\mathcal{D}^{\alpha_{1}}S(t)= -\frac{\beta I(t)S(t)}{N}\] \[\mathcal{D}^{\alpha_{2}}I(t)= \frac{\beta I(t)S(t)}{N}-\gamma I(t) \tag{7}\] \[\mathcal{D}^{\alpha_{3}}R(t)= \gamma I(t)\]  
where \(\alpha_{i}\in(0,\ 2)\), \(i\in\{1,\ 2,\ 3\}\), \(\beta\) and \(\gamma\) are as in Equation (1). The fractional-order generalization of the SIR model dynamics is different from the 'classical' counterpart. The fractional-order model's several properties are different from in the integer-order one, e.g., the total population number \(N\) is time-various phenomena. Therefore, the SIR model's fractional-order generalization should be considered to be a heuristics model with no direct dependency on physical phenomena. Moreover, it is worth emphasizing that, in this case, we have three additional parameters for estimation \(\{\alpha_{1},\alpha_{2},\alpha_{3}\}\). Taking into account that, in most countries, COVID-19 statistics are presented once per day, to the prediction of the virus transmission, we can use discrete-time nabla difference as in Equation (5) where sampling period \(h=1\), so we have  
\[ \begin{split}\
abla^{\alpha_{1}}S_{k+1}=&-\frac{ \beta l_{k}S_{k}}{N}\\ \
abla^{\alpha_{2}}I_{k+1}=&\frac{\beta l_{k}S_{k}}{ N}-\gamma I_{k}\\ \
abla^{\alpha_{3}}R_{k+1}=&\gamma I_{k}\end{split}  \tag{8}\]  
Now, combining Equations (6) and (8), we can easily present model (8) in the form of  
\[ \begin{split} S_{k+1}=&\left(\frac{-\beta l_{k}}{N} +\alpha_{1}\right)S_{k}-\sum_{j=2}^{k+1}(-1)^{j}\binom{\alpha_{1}}{j}S_{k-j}\\ I_{k+1}=&\left(\frac{\beta S_{k}}{N}-\gamma+\alpha _{2}\right)I_{k}-\sum_{j=2}^{k+1}(-1)^{j}\binom{\alpha_{2}}{j}I_{k-j}\\ R_{k+1}=&\gamma I_{k}-\sum_{j=2}^{k+1}(-1)^{j} \binom{\alpha_{3}}{j}R_{k-j}\end{split}  \tag{9}\]  
which can be easily used in the simulation of the process.