# mj1.atMichael Jaros' Techblog

25Mar/120

## Infection modeling: The branching process model

When examining the spread of diseases inside a population, not only the contagiousness of the disease, but also the structure of the network connecting the population determine the progress of the infection, as Easley and Kleinberg describe in1. Because messages in a social network spread in a very similar way as diseases or ideas, we try to model the discovery phase of a tweetflow invocation using infection modelling.

In tweetflow terms, the contagiousness of a disease for a node corresponds to the payoff (reward - effort) of the tweetflow and the skill of the node. The length of the infectious period corresponds to the period of validity (time to live, ttl). The severity of the infection could match the priority of a tweetflow, if applicable.

Diseases spread in a population as members of the population infect other members (biological contagion). Ideas can spread in the same way inside a social network (social contagion). However, there is an important difference in the way these types of infections are usually analyzed: In biological contagion, there is no decision-making, but a random choice (infection or no infection). Sometimes, these randomized models are useful for social contagion too, if the decision processes are too complex to model or have too many unknown parameters.

The simplest model for infections is the branching process. The contact network is considered a regular tree with k children per node, and the distance from the root is measured in waves. Beginning from this root, the (infected) originator, each infected person passes on the infection to each of the k people in the subsequent wave with probability p. The basic reproductive number $R_0=pk$ is the expected number of new infections caused by a single infected node. If $R_0>1$, the disease persists in the network, if R_0 < 1[/math], it will die out after a finite number of waves. So both a high infection probability and high numbers of connected nodes are factors of persistence of the disease.

1. Easley and Kleinberg (2010): Networks, Crowds, and Markets []