## 1 Introduction
### The Susceptible-Infectious-Removed (SIR) Model  
Despite the development of various extensions of the SIR model, the standard SIR model remains the preferred first approach for analyzing the spreading of an infectious disease (especially in the beginning or first phase) and it is reasonably predictive [11]. The SIR model splits a given population \(N\) into three compartments (non-intersecting classes) at any given time \(t\) (measured in days) namely, (i) susceptible (not yet infectious and disease free individuals at \(t\)) denoted \(S_{t}\), (ii) infectious (confirmed or isolated individuals) denoted \(I_{t}\), and (iii) removed (no longer infectious or dead) denoted \(R_{t}\). The number of individuals in each compartment vary over time. In general, the dynamics is described with a large number of susceptible individuals at the beginning, since the entire population that is not infected is considered to be susceptible, while infectious individuals remain low at the beginning of a pandemic. At subsequent times, the number of infectious individuals will increase, the number of removed individuals will begin to gradually increase, and the number of susceptible individuals will decrease. Finally, towards the end of a pandemic, the number of removed individuals will increase, the number of infectious individuals will decrease gradually, while the number of susceptible individuals will remain the lowest. The disease dynamics according to SIR model can be visualized, as in Figure 1. The variation rate over time in each compartment is modeled using a system of non-linear ordinary differential equations (ODEs),  
\[\frac{dS}{dt}=-\frac{\beta S1}{N},\quad\frac{dI}{dt}=\frac{\beta S1}{N}-\gamma I,\quad\frac{dR}{dt}=\gamma I. \tag{1}\]  
The primary assumption of SIR model is that the population is closed and fixed, and it is the sum of individuals in all of t compartments i.e., \(N=S+I+R\) for all \(t\). The epidemiological parameters of interest are, i) the transmission rate \(\beta\) and ii) the recovery rate \(\gamma\). Accordingly, the average transmission (from an infectious individual to a susceptible individual) period is \(1/\beta\) days and the average infectious period is \(1/\gamma\) days. Estimation of the SIR parameters is a critical task, since there is no closed-form analytical solution to the SIR model. Numerical approximation methods, such the Runge-Kutta methods, are often employed in order to solve the SIR model with estimated parameters. Thus, the quality of the simulation heavily relies on the estimates of the epidemiological parameters. Apart from that, a good estimate of the parameters is crucial in assessing the transmissibility of an infectious diseases in real-time through the effective reproduction number \(\mathcal{R}_{t}\). One of the approaches for inferring \(\mathcal{R}_{t}\) is by using compartmental epidemiological models, in which \(\mathcal{R}_{t}\) is treated at the deflation of the basic reproduction number \(\mathcal{R}_{0}\)[20], which can be estimated using the relationship among the epidemiological parameters [21] that is given by:  
\[\mathcal{R}_{0}\approx\frac{\beta}{\gamma}. \tag{2}\]  
Figure 1: Infectious disease dynamics according to SIR model. _Adapted from_[11].