Finding an individual's probability of infection in an SIR network is NP-hard

TL;DR

Determining an individual's probability of infection in an SIR network is computationally intractable (NP-hard), indicating no efficient algorithm exists for this problem in general, which impacts epidemic modeling and prediction.

Contribution

The paper proves that calculating infection probability for a specific individual in an SIR network is NP-hard, establishing its computational complexity.

Findings

The problem is NP-hard, requiring exponential time for exact solutions.

Implication that efficient algorithms for this problem are unlikely to exist.

Highlights the computational challenge in detailed epidemic modeling.

Abstract

The celebrated Kermack-McKendric model of epidemics studies 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. This ODE model arises as the limiting case of a network model where each individual has an equal chance of infecting every other. More recent work gives explicit consideration to the network of social interaction 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 problem is NP-hard. In particular, it belongs in a class of problems all of whose known solutions require an exponential amount of computation and for which it is unlikely that there will be more efficient solutions.