## 2 Material and Methods
### Initialisation  
The initialisation was carried out in a two-dimensional environment \(R^{2}\), where agents \(A_{i}\) are distributed based on the demographic specifications of Joana Bezerra Island. We considered the population density, age distribution, and other demographic parameters to initialise these agents realistically. Positioning of Agents (A1, A2, A3):  
Agents A1: The initial position of agents A1 was determined by the residence location (agents A2) to which they are associated. This reflects a demographic distribution based on the following:  
\[\text{A1}_{\text{pos}}=\text{A2}_{\text{pos}}+\mathcal{N}(0,\sigma^{2})\]  
where (\(\text{A2}_{\text{pos}}\)) is the position of the associated agent A2 (residence) and \((\mathcal{N}(0,\sigma^{2}))\) represents a normally distributed variation around the residence position, indicating that the individual is within or near their home. Agents A2 (residences): The distribution of agents A2 in the environment is based on the demographic density of the studied area. Each agent A2 represents a residence in the neighbourhood, and its position is randomly determined within the environment:  
\[\text{A2}_{\text{pos}}=(x_{\text{rand}},y_{\text{rand}})\]  
where \((x_{\text{rand}})\) and \((y_{\text{rand}})\) are coordinates randomly selected within the bounds of the two-dimensional environment. Agents A3 (commercial and service establishments): These are positioned randomly in the environment, reflecting the location of commercial and service establishments on Joana Bezerra Island:  
\[\text{A3}_{\text{pos}}=(x_{\text{rand}},y_{\text{rand}})\]  
with \((x_{\text{rand}})\) and \((y_{\text{rand}})\) being coordinates selected to reflect the distribution of commercial and service establishments. Health State Initialisation (SEPAI3R3O)  
Each agent A1 was initialised in one of the states of the SEPAI3R3O model, with most starting in the Susceptible (S) state. The initial proportions of agents in other states (P, A, _I1_, _I2_, _I3_, _R1_, _R2_, _R3_, O) were defined based on prevalence and incidence data for COVID-19 on Joana Bezerra Island, indicated as \[\text{Initial\_State}_{\text{A1}}\sim\text{Distribution}(P_{S},P_{P},P_{A},\ldots,P_{O})\]  
where \((P_{S}),(P_{P}),(P_{A}),\ldots,(P_{O})\) represent the initial percentages of agents in each state of the SEPAI3R3O model. This table outlines the initial distribution of the population across different health states in the SEPAI3R3O model for Joana Bezerra Island. The values were derived from the data and references of the stated articles, ensuring that the simulation accurately reflected the region's epidemiological context. Through this careful initialisation, we addressed various crucial aspects for the realistic modelling of COVID-19 transmission, from the spatial distribution of agents to the initial configuration of their health state according to the specific disease dynamics in the study area. For better understanding, when a person of type A1 is created, they are assigned to a randomly chosen house of type A2. Parameter \(\delta_{7}\) defines the probability of A1 being economically active/employed. If a person is economically active but not included in group \(\delta_{8}\), an employer is randomly chosen among the available A3. We only created a single instance for agents of type A4 and A5. The age distribution of agent A1 is given by parameter \(\gamma_{4}\), considering the contact rate parameters \(\beta_{i}\) and immunity probability \(c_{i}\). The social stratum of A1, A2, and A3 is represented by the income distribution listed in Table 7; that is, the share of wealth represented by the GDP parameter \(\delta_{2}\). To clarify the wealth distribution process, it is essential to note that the total wealth of the simulation (\(\delta_{2}\)) is not distributed equally among all agents. Instead, it follows a stratified approach that reflects the economic inequality in real-world scenarios, particularly in areas of high social vulnerability such as Joana Bezerra Island.