# A Generalized Framework for the Estimation of Edge Infection Probabilities
## 1 Introduction  
Infection models are frequently used in many real-life applications in sociology, economics and epidemics. A large variety of models are available for different applications, the most popular ones being compartmental models and their variants [(1; 2)]. They are regularly used to model infectious diseases [(2)], the spread of information and behavior on social networks [(3)], or the spread of economic events [(4; 5; 6; 7)]. Some of the tasks associated with these models include influence maximization, that is, finding the set of individuals yielding the largest expected infection [(8; 9; 10; 11)], or the prediction of a posteriori infection values, i.e., finding the probability of infection for all nodes of the network [(12)]. Many infection models are network-based; they assume a set of nodes and the connections between them to be given. Most also assume a specific value to be available on the links of the networks; these are called _edge infection values_, and they represent the probability that the infection spreads from one node to another. A common challenge in the application of infection models is the lack of information on the edge infection values on the links of networks. Recently, several authors have proposed models to estimate these values. Many of the models assume that the time stamps of the infection for each node are given [(13; 14; 15; 16; 17; 18; 19; 20)], although some approaches do not require this property [(21; 22; 23)]. Current estimation methods are based on a variety of infection models [(15)], but most often, the SIR model or one of its variants is used [(16; 17; 18; 21)]. Most of these models were developed independently of each other, and most of them take a different, often application-specific approach to the problem. The Generalized Inverse Infection Model (GIIM) was recently proposed to solve the problem of estimating edge infection probabilities in two case studies. The model was applied to estimate the risk of infection between geographical regions during (1) the 2009 H1N1 pandemic in Sweden [(24)] and (2) the 2016 Zika outbreak in the Americas [(25)]. In both cases, the model was able to formulate edge infection values as a function of knowndisease-relevant attributes, such as socioeconomic, climate and travel determinants, and was also shown to have good predictive capabilities. The properties of the GIIM have only been described within the strict boundaries of these two case studies, however. As such, there is no existing scientific work that examines the theoretic properties and general applicability of the model. In this paper, we show that:  
1. The GIIM can account for an arbitrary number of observations on the infection process. 2. The GIIM allows for flexibility in the type of observations provided as input. 3. The GIIM allows for flexibility in the underlying infection model. An infection-spreading process may be fully or partially observable. The GIIM is able to handle an _arbitrary number of observations_ on such processes, enabling it to adapt to the specific requirements of applications. In this paper, we experiment with the two extremes of such requirements. In fully observed scenarios, we know at exactly what time a node was infected. In two-observation cases, we can only observe who was infected at the beginning and at the end. Both fully observed and two-observation approaches and any in between them can be naturally formulated in the framework of the GIIM. Observations on the infection process might be available in various forms, so the model should be flexible enough to handle _different forms of input data_. Two types of observations are investigated in this paper. The first one is based on the common assumption that at a certain time step, we know which nodes were infected and which nodes were not. In the second type of observation, we have real values indicating how likely it is for the nodes to become infected at a given time step. We propose additional types of observations that we plan to investigate in future work.