# Epidemic Spreading in Urban Areas Using Agent-Based Transportation Models
## 2 Background  
### Classical Epidemic Spread Models  
Traditional epidemiology models represent epidemics of communicable disease using a population-based, non-spatial approach. A major assumption of these mathematical models of epidemics is that the population can be divided into a set of distinct compartments. These so-called _compartmental models_ split the population into compartments that represent the different stages of a disease. The most general approach is the SIR model, which was described by [PERSON] and [PERSON] [3], considering three compartments: susceptible (S), infected (I), and recovered (R). Individuals that are susceptible have, in the case of the basic SIR model, never been infected, and they can catch the disease. Once they have it, they move into the Infected compartment. Infected individuals can spread the disease to susceptible individuals. The time they spend in the infected compartment is the infectious period, after which they enter the recovered compartment. The recovered compartment represents individuals that have recovered from the disease and no longer take an active part in the disease spread. The SIR model represents the virus transmission by a set of nonlinear ordinary differential equations (ODEs) that associate a transition rate to the mobility of agents between compartments.  
\[\begin{split}\frac{dS}{dt}&=-\beta S1\\ \frac{dI}{dt}&=\beta S1-\gamma I\\ \frac{dR}{dt}&=\gamma I\end{split} \tag{1}\]  
These equations describe the time evolution of a population between disease states susceptible, infected, and recovered. The rate of new infections can be defined as \(\beta S1\), where \(\beta>0\) is a parameter for infectivity. Infected individuals are assumed to recover with a constant probability at any time, which translates into a constant per capita recovery rate \(\gamma>0\), and thus an overall rate of recovery \(\gamma I\). Figure 1 illustrates the scheme of the basic SIR model.  
The basic SIR model assumes that the total population size remains constant over time \(N=S+I+R=const\). i.e., demographic turnovers (birth or death) are not considered. Another restriction of the original compartment model is that the model assumes that populations are completely mixed and, therefore, ignore spatial effects of spread epidemics; also, the interaction between individuals are neglected since they model populations as continuous entities [18]. Additionally, the SIR model is deterministic which implies that no randomness is involved. Hence, the same starting conditions give the same output.  
Based on this general model many variations of the disease-spreading model can be derived, including the SI Susceptible-Infected, SIS Susceptible-Infected-Susceptible, or SEIR Susceptible-Exposed-Infected-Recovered. The last model adds an intermediate stage \(E\) which represents a latent state in which individuals have been exposed to the disease but are not yet infective, i.e., the individuals in this stage have the virus but cannot infect others [9]. Furthermore, the models can be extended to account for vaccination strategies [19; 20].  
The advantage of these models is that they have only a small number of parameters, are easily extendable, can be solved numerically or even have analytical solutions [21; 22]. However, by considering an aggregated population, individual behavior cannot be taken into account, which has, in the presence of an infectious disease, a great influence on the epidemic dynamics [23].