## 4 Beyond the SIR Model
### Effect of Social Distancing on the Spread of the Virus  
The SIR model (Equation (2)) was modified in order to include the awareness term, i.e., social distancing, as a function of affected individuals (i.e., infected and recovered cases), as shown in Equation (10). In this study, two types of social distancing, namely, long- and short-term, were considered to investigate their leverage on the spread of COVID-19, according to previously published work [30].  
\[\begin{array}{l}\frac{dS}{dt}=-\beta\frac{S\dot{I}}{N}\psi(I,R),\\ \frac{dt}{dt}=\beta\frac{S\dot{I}}{N}\psi(I,R)-\gamma I,\\ \frac{dS}{dt}=\gamma I\end{array} \tag{10}\]  
Long-term social distancing can be simulated by reducing the interaction between individuals proportional to the rate of affected individuals (i.e., infected and recovered) according to Equation (11).  
\[\psi=\left(1-\frac{I+R}{N}\right)^{\xi} \tag{11}\]  
In short-term awareness models, the reduction of the interaction between individuals is proportional to the rate of infected individuals only (see Equation (12)).  
\[\psi=\left(1-\frac{I}{N}\right)^{\xi} \tag{12}\]  
The term \(\xi\) in both awareness models represents the reduction parameter. When \(\xi\) = 0, the original version of the SIR model was simulated with no social-distancing impact. On the other hand, the linear response of social distancing on the spread of the virus was simulated with \(\xi\) = 1, whilst nonlinear implications were seen when \(\xi\) > 1. In addition, the modified version of the SIR model (see Equation (10)) was fitted with the cumulative cases during the effective lockdown period (from 1 May 2020 to 14 June 2020) using short- and long-term types. Thereafter, we were able to define the optimized values for \(\beta\), \(\gamma\), \(S_{0}\), and \(\xi\). The optimal values of \(\xi\) when using short- and long-term types were 0.1 and 0.17, respectively.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **RMSE** & **R\({}^{2}\) (\%)** & **R\({}^{2}_{\rm adj}\) (\%)** & **AIC** \\ \hline Richards & 3819.01 & 99.78 & 99.78 & 1866.95 \\ \hline Generalized logistic & 2962.78 & 99.87 & 99.86 & 1809.57 \\ \hline Gompertz & 3201.23 & 99.85 & 99.84 & 1827.07 \\ \hline SIR & 2790.69 & 99.88 & 99.88 & 1796.05 \\ \hline \hline \end{tabular} RMSE: root-mean-square error; R\({}^{2}\) : R squared; R\({}^{2}_{\rm adj}\): adjusted R\({}^{2}\); AIC: Akaike’s information criterion.  
\end{table}
Table 2: Statistical measures of the epidemiological models.  
The simulation results shown in Figure 6 illustrated that the cumulative number of cases of infection reduced when the optimal exponent values (\(\xi\)), the linear (\(\xi=1\)) and non-linear (\(\xi=2\)) social-distancing models were applied compared to the nonmodified SIR model. The linear and nonlinear short-term social-distancing models showed a decrease of 4.63% and 9.1%, respectively, in comparison to the final size of the pandemic predicted by the nonmodified SIR model. Additionally, we applied the optimal exponent values of the short- and long-term lockdown types (0.1 and 0.17, respectively) acquired by fitting the modified SIR model (Equation (10)) with the data during the lockdown period. As consequence, short- (\(\xi=0.1\)) and long-term (\(\xi=0.17\)) lockdowns produced drops of 0.5% and 4.86%, respectively. The long-term awareness model showed a higher impact compared to the short-term awareness model (see Figure 6B).