Disease spreads only through existing contact links β not uniformly. The network topology is the primary driver of outbreak size and speed.
Highly-connected hub nodes (yellow rings) transmit at twice the rate and seed most secondary cases β the 20/80 rule of epidemiology.
Each node is Susceptible, Infected or Recovered. Rβ = Ξ²Β·<k>/Ξ³ tells you whether an epidemic takes off (Rβ > 1) or dies out.
This simulation models a classic SIR (Susceptible-Infected-Recovered) epidemic propagating across a contact network rather than through a well-mixed population. Each node represents an individual and each edge a social or physical contact through which disease can pass. When the network is built with preferential attachment (Barabasi-Albert model), it produces a scale-free degree distribution: most individuals have few contacts while a small number of highly-connected hubs β super-spreaders β accumulate dozens. These hubs are marked with yellow rings and transmit at twice the baseline rate, mirroring the empirical 20/80 rule observed in measles, SARS, and COVID-19 outbreaks.
Transmission probability Ξ² and recovery probability Ξ³ per simulated day combine with the average node degree <k> to yield the basic reproduction number Rβ = Ξ²Β·<k>/Ξ³, displayed live in the panel. When Rβ > 1 an epidemic takes off; when Rβ < 1 it fizzles. Compare the scale-free network (heterogeneous hubs) with the random Erdos-Renyi graph (homogeneous connectivity) to see why network structure profoundly alters outbreak speed, peak infected count, and the fraction eventually infected. Adjust Ξ² to mimic different pathogen transmissibility and Ξ³ to model recovery time.
What is the SIR model?
SIR divides a population into three compartments: Susceptible (never infected), Infected (currently contagious), and Recovered (immune). At each time step every infected individual can transmit to susceptible neighbours with probability Ξ² and recover with probability Ξ³. The simplicity makes it the foundation of mathematical epidemiology, used for influenza, measles, and COVID-19 modelling alike.
What is a super-spreader?
A super-spreader is an individual who causes significantly more secondary infections than the average. On a contact network super-spreaders typically correspond to high-degree hub nodes β people with many social contacts. They accelerate outbreak ignition, generate large clusters of secondary cases, and are priority targets for vaccination or quarantine interventions.
What is Rβ and why does it matter?
Rβ (the basic reproduction number) is the expected number of secondary infections produced by a single case in a fully susceptible population. When Rβ > 1 the epidemic grows exponentially; when Rβ < 1 it dies out. On a network Rβ approximates Ξ² times average degree divided by Ξ³. Reducing Ξ² (e.g., masks) or Ξ³ (faster recovery) directly lowers Rβ.
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 β new members join social networks by connecting to already-popular members β 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 Rβ = 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 β the epidemic peak β 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 Ξ² 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.