Finding the probability of infection in an SIR network is NP-Hard.

It is the purpose of this article to review results that have long been known to communications network engineers and have direct application to epidemiology on networks. A common approach in epidemiology is to study the transmission of a disease in a population where each individual is initially susceptible (S), may become infective (I) and then removed or recovered (R) and plays no further epidemiological role. Much of the recent work gives explicit consideration to the network of social interactions or disease-transmitting contacts and attendant probability of transmission for each interacting pair. The state of such a network is an assignment of the values {S,I,R} to its members. Given such a network, an initial state and a particular susceptible individual, we would like to compute their probability of becoming infected in the course of an epidemic. It turns out that this and related problems are NP-hard. In particular, it belongs in a class of problems for which no efficient algorithms for their solution are known. Moreover, finding an efficient algorithm for the solution of any problem in this class would entail a major breakthrough in computer science.