## 2 Model Definition
### The Population \(\mathbb{P}\)  
The model considers a constant population \(\mathbb{P}\) with a fixed number of individuals \(N\equiv|\mathbb{P}|\). Each individual belongs to one of three groups: susceptible \((S)\), exposed \((E)\), or infected \((I)\), such that \(N=S+E+I\). Individuals in the susceptible group have no immunity and are susceptible to infection. When an individual in the susceptible group \((S)\)  
Figure 1: A schematic view of the model’s dynamics. \(F_{out}\) and \(F_{in}\) are the force of air getting out and in (with the pathogen’s particles) during an individual’s breathing process. is exposed to the pathogen, the individual is transferred to the exposed group (\(E\)) at a rate \(\beta\). The individuals stay in the exposed group (\(E\)) for \(\xi\) time steps. This duration is mainly defined by the pathogen and its interaction with the immune system. After \(\xi\) time steps, the individuals are transferred to the infected group (\(I\)). The epidemiological state of the population follows the equations:  
\[\frac{dS(t)}{dt}=-\beta(t)I(t)S(t),\ \frac{dE(t)}{dt}=\beta(t)I(t)S(t)-\xi E(t),\ \frac{dI(t)}{dt}=\xi E(t). \tag{3}\]  
This \(SEI\) model is a private case of the \(SEIR\) model, such that \(\forall t:R(t)=0\). This case better suits short-duration pandemic spread at the beginning of the pandemic, where there is still a small (or even neglected) number of recovered or vaccinated individuals, such as the one of interest in this model. Formally, let \(\mathbb{P}\) be a non-empty set of individuals, such that each individual \(p\in\mathbb{P}\) is defined by a tuple \(p:=(l,b,\
u,c)\), where \(l=(l_{x},l_{y},l_{z},l_{\alpha},l_{\beta})\) is the location and orientation of the individual's face, such that \((l_{x},l_{y},l_{z})\in E\) is the three-dimensional location in the environment, and \(l_{\alpha}\in[0,2\pi],l_{\beta}\in[0,\pi]\) are the angles in the \(xy\) and \(xz\) plans, respectively; \(b=(\phi_{\delta},\phi_{r},\phi_{g})\) are the breathing-related properties, such that \(\phi_{\delta}\) is the minimal number of pathogen particles needed to be inside the individual's body to cause infection, exposing a susceptible individual to the pathogen; \(\phi_{r}\) is the natural reduction rate of the pathogen particle for susceptible individuals; \(\phi_{g}\) is the amount of pathogen particle the individual generates during breathing out when infected; \(\
u\in\{s,e,i\}\) is the individual's current epidemiological state; \(c\) is the number of pathogen particles currently found in the individual's body. At each point in time, each individual in the population is either breathing in, out, or neither. During the inhaling phase, a force \(F_{in}\) originated at \((l_{x},l_{y},l_{z})\) is manifested. The \(F_{in}\) force influences the air movement as well as the pathogen particles in its close proximity. In particular, the pathogen particles that are located at a distance \(\gamma\) from \((l_{x},l_{y},l_{z})\) during the inhaling are removed from the environment and inserted into the individual's body. In a similar manner, during an exhaling phase, a force \(F_{out}\) originated at \((l_{x},l_{y},l_{z})\) is manifested. If the individual is infected (i.e., \(\
u=i\)), then \(\phi_{g}\) pathogen particles are generated at each breath and added to the environment in an exponentially decaying distribution as a function of the distance from \((l_{x},l_{y},l_{z})\) with weights \(\omega_{x},\omega_{y}\), and \(\omega_{z}\) for the \(x,y\), and \(z\) axis. Individuals follow a breathing pattern as follows: inhaling, not breathing, exhaling, and not breathing. This pattern is repeated for each individual during the entire model's dynamics. A susceptible individual (\(\
u=s\)) becomes exposed (\(\
u=e\)) if \(c>\phi_{\delta}\). During the time an individual belongs to either the exposed or infected epidemiological states, the change in the number of pathogen particles in the body (\(c\)) does not have any effect on its epidemiological state.