Pick a scenario; every slider below stays editable.
Press the sweep: 20 outbreaks per point on the current network.
The science
Contact network. Each house is a person or household; each arc is a regular contact. Three structures with the same average number of contacts <k>: random (Poisson degrees), small-world (a ring of neighbours with a few long-range shortcuts, high clustering) and hub-dominated (preferential attachment: a few people have very many contacts).
Disease. Discrete-time SIR (step 0.1 days in the checks, 0.25 d on screen): an infectious person infects each susceptible contact with probability p = 1 - e-beta dt per step and recovers with r = 1 - e-gamma dt. The probability that an infectious person ever transmits along one given contact is T = p / (p + r - p r) (about beta/(beta+gamma) for small steps).
Network R0 and thresholds. A neighbour reached along an edge has on average <k(k-1)>/<k> further contacts, so R0 = T <k(k-1)> / <k> (Newman 2002). Big outbreaks need R0 > 1, i.e. T > Tc = <k> / <k(k-1)>. Hubs make <k²> large, so hub-dominated networks have a tiny threshold. Removing a random fraction phi of nodes by vaccination behaves like T -> T(1-phi) for the threshold, giving phic = 1 - Tc/T; removing the best-connected people first needs far less.
Final size. With degree distribution generating functions G0, G1, solve u = 1 - T + T G1(u); the outbreak that takes off infects a fraction 1 - G0(u). This is exact for large tree-like random graphs and is the independent check in the page (it uses only the degree list, not the epidemic code).
Honest simplifications. Fixed network (no changing contacts), everybody equally susceptible, perfect lifelong immunity, a vaccine that works 100 % at once, no latent period, no symptoms or behaviour change, 240 nodes (so stochastic fade-outs are common). The theory is approximate for the small-world network (clustering breaks the tree assumption) and slightly off for the hub network. Parameter values are illustrative, not a disease. Targeted vaccination is idealised: real programmes cannot see everyone's contacts.