## 2 Material and Methods
### Epidemiological Model: SEPAI3R3O
The three groups of Infectious individuals were created to represent three distinct phases of infection: \(I1\), infected individuals with mild symptoms; \(I2\), infected individuals tested due to moderate symptoms (for which they may even be hospitalised); and \(I3\), infected individuals tested with severe symptoms. Following the same logic of the SEIR model, for a person to be hospitalised in severe condition (\(I3\)), they must have necessarily passed through the P stage, the \(I1\) stage (infectious but without symptoms and without being tested), and then through the \(I2\) stage (with sufficient symptoms to lead to testing). They only enter the \(I3\) stage if their condition becomes more serious, requiring intensive treatment in a hospital unit. Using this premise, only individuals in the \(I3\) group can die, moving to the O group. This means that in this model, the path for an individual to die is the following: \(\mathrm{S}\to\mathrm{P}\to I2\to I3\to\mathrm{O}\), \(\mathrm{S}\to\mathrm{P}\to I1\to I2\to I3\to\mathrm{O}\). For an individual to move to the recovered (R) group, they must first be in one of the recovering groups (\(R1\), \(R2\), and \(R3\)), which represent individuals recovering from the disease at various stages of intensity (i.e., \(I1\), \(I2\), and \(I3\) groups, respectively). The motivation behind creating such groups was the need to consider recovery time based on the severity of the infection, which allows us to model the demand for hospital beds satisfactorily. Therefore, the \(R1\), \(R2\), and \(R3\) groups do not communicate, meaning there are three distinct paths to recovery (R) for an infected person:  
1. \(I1\to R1\to R\) (i.e., faster recovery);
2. \(I1\to I2\to R2\to R\) (i.e., intermediate recovery time);
3. \(I1\to I2\to I3\to R3\to R\) (i.e., longer recovery time). In addition to the subgroups shown above, this model was conceived to consider the under-reporting rate (Psub) of the number of cases as, due to the division of subgroups, there is the possibility of controlling the proportion of people who migrate from \(I1\) to \(I2\) (i.e., cases that show enough symptoms for testing) or who migrate from \(I1\) to \(R1\). Thus, these individuals represent under-reported cases, as they contracted the disease but were never tested. The SEPAI3R3O model is represented in the Figure 1.  
Below are the ordinary differential equations of the model:  
\[\frac{dS}{dt}=-P_{sub}\times(P+A+I_{1}+I_{2}+I_{3}+O)S \tag{1}\] \[\frac{dE}{dt}=(\beta_{e}P+\beta_{0}A+\beta_{1}I_{1}+\beta_{2}I_{2}+ \beta_{3}I_{3})\times S-a_{0}E\] (2) \[\frac{dP}{dt}=a_{0}\times E-a_{1}P\] (3) \[\frac{dA}{dt}=f_{a1}\times P-c_{0}A\] (4) \[\frac{dI_{1}}{dt}=p_{0}\times((1-f)\times a_{1}\times P)-(c_{1}+p _{1})\times I_{1}\] (5) \[\frac{dI_{2}}{dt}=p_{1}\times I_{1}-(c_{2}+p_{2})\times I_{2}\] (6) \[\frac{dI_{1}}{dt}=p_{2}\times I_{2}-(c_{3}+\mu)\times I_{3}\] (7) \[\frac{dR_{1}}{dt}=(d_{0}+c_{0})\times I_{1}\] (8) \[\frac{dR_{2}}{dt}=(d_{1}+c_{1})\times I_{2}\] (9) \[\frac{dS}{dt}=-P_{sub}\times(P+A+I_{1}+I_{2}+I_{3}+O)S \tag{1}\]0 (10) \[\frac{dS}{dt}=-P_{sub}\times(P+A+I_{1}+I_{2}+I_{3}+O)S \tag{1}\]1 (11) \[\frac{dS}{dt}=-P_{sub}\times(P+A+I_{1}+I_{2}+I_{3}+O)S \tag{1}\]2  
where  
\(\beta_{i}\) is the rate at which infected individuals in class \(I_{i}\) come into contact with susceptible individuals and infect them;  
\(a_{i}\) is the rate of progression from the exposed class to the infected class;  
\(f\) is the fraction of all asymptomatic infections. Parameters \(\beta_{i}\)\(\forall_{i}=e,0,1,2\) represent the transmission rates for different phases of the disease; parameters \(c_{i}\)\(\forall_{i}=0,1,2\) represent the different recovery rates; \(u\) is the mortality rate; and parameters \(a_{i}\)\(\forall_{i}=0,1\) indicate the exit rate from classes E and P, respectively.