# A Generalized Framework for the Estimation of Edge Infection Probabilities
## 2 Definitions
### Infection Models  
Discrete infection models work by assigning states to the vertices of the graph. Each vertex may only be in one state at a given time, and all vertices must be in a state at all times. The states of the above models are susceptible (S), exposed (E), infected (I) and removed (R). The exposed and infected states have discrete time periods attached to them, and we denote these by \(\tau_{e}\) and \(\tau_{i}\). Infected nodes attempt to infect their susceptible neighbor nodes according to the corresponding edge weight probability \(w_{e}\). If successful, the newly infected nodes transition into an exposed state for \(\tau_{e}\) iterations, after which they attempt to infect susceptible neighbors for \(\tau_{i}\) iterations, after which point they become removed (dead or immune) and can no longer infect other nodes. The models differ in complexity, with the most complex being the SEIR model, which has all of the above states; in the SIR model, \(\tau_{e}=0\); in the IC model, \(\tau_{e}=0\) and \(\tau_{i}=1\); and in the SI model, \(\tau_{e}=0\) and \(\tau_{i}\rightarrow\infty\). Correspondingly, in the IC, SI and SIR models, the nodes are never in an exposed state, and in the SI model, the nodes are never removed; i.e., they may attempt to infect neighboring nodes indefinitely.  
We illustrate the mechanics of the infection process on the most general infection model, SEIR. Apart from the network itself, the infection process of the SEIR model has two inputs. The first is the assignment of edge infection probabilities \(W_{G}:E(G)\mapsto[0,1]\) for all edges. The second is the set of initially infected nodes \(A_{0}\subset V\). These nodes are considered to be infected at the start of the process. Let \(A_{i}\subseteq V\) be the set of nodes in the infected state in iteration \(i\). Each node \(u\in A_{i}\) tries to infect its susceptible neighbors \(v\) according to \(w_{u,v}\); if the attempt is successful, \(v\) will begin its exposed period. If more than one node tries to infect \(v\) in the same iteration, the attempts are made independently of each other in an arbitrary order within the same iteration. Still, in iteration \(i\), the status of infected nodes changes to removed if the difference between \(i\) and the time of their infection becomes greater than \(\tau_{e}+\tau_{i}\); exposed nodes become infected when the difference becomes greater than \(\tau_{e}\). The process terminates in iteration \(t\) if the set of infected and exposed nodes is empty. In a finite network, the process always terminates in a finite number of steps.  
The other infection models behave similarly, with minor differences. The SEIR infection processes take longer to complete because the nodes have to spend time in an exposed state before becoming infectious. Similarly, higher \(\tau_{i}\) corresponds to more infected nodes because there are more opportunities to infect neighboring nodes. Finally, in the SI model, given enough time, all vertices become infected with a probability of one if there is an initially infected node in every connected component.