🦠 Epidemic on a Network

A scale-free network has a degree distribution that follows a power law: the probability that a node has k connections is proportional to k raised to a negative exponent. A few hubs have very many links while most nodes have few. This topology emerges naturally from preferential attachment — new members join social networks by connecting to already-popular members — and is found in the Internet, citation networks, and real-world contact patterns.

In a well-mixed model all individuals are assumed equally likely to meet. On a heterogeneous network the hubs act as bridges that shortcut the population, allowing disease to travel across distant clusters quickly. Scale-free networks have no epidemic threshold for many disease parameters, meaning that even a very low transmission probability can eventually invade the whole network if hubs are not removed or immunised.

For a random network the epidemic threshold is R₀ = 1. For a scale-free network with the exponent typical of social contacts the threshold approaches zero as network size grows, because the ratio of variance to mean degree diverges. In practice this means some pathogens with low transmissibility that would die out in a homogeneous population can still cause large outbreaks in scale-free contact networks.

Removing or vaccinating the top few percent of highest-degree nodes destroys the short paths between clusters and raises the effective epidemic threshold. This targeted immunisation is far more efficient than random vaccination: protecting 5% of hubs can reduce peak infections by more than 50%, whereas random vaccination requires 60-70% coverage to achieve herd immunity on the same network.

The chart plots the fractions of the population in each SIR compartment over simulated days: blue for susceptible, red for infected, and green for recovered. The infected curve typically shows a characteristic bell shape — the epidemic peak — followed by decline as susceptibles are depleted and recovered individuals accumulate. The area under the infected curve represents the total disease burden.

The simulation begins with a small seed clique of nodes. Each new node joins and forms m edges, selecting existing targets with probability proportional to their current degree (preferential attachment). This rich-get-richer mechanism naturally generates the power-law degree distribution characteristic of real social, biological, and technological networks.

Lower β simulates physical distancing or mask use. Lower the network size and watch how small isolated communities slow spread. Switch to a random network to see how homogeneous connectivity changes outbreak dynamics compared with scale-free topology. Because super-spreaders are seeded first, restarting the simulation shows how early hub infection determines final epidemic size.