Spatial SIR epidemic simulation

Is there a sharp threshold above which a local outbreak becomes a lattice-wide epidemic — and where is it?

▶ Run the simulationSee the measured result

Units: dimensionless (open-bond probability = transmissibility T; square-lattice bond-percolation threshold, exact — Kesten 1980). Secondary knowns: triangular-lattice bond threshold 2·sin(π/18)=0.3472963 exact; Bethe/tree branching threshold at coordination z=4 is 1/(z−1)=0.33333.

How the lab tests it

Run the simplest spatial SIR: each infected cell infects every susceptible 4-neighbour with probability T, then recovers (immune). From one seed, sweep T and measure the final immune fraction and the spanning probability (does the outbreak reach the boundary?); locate T_c at the Π = ½ crossing.

What it checks

a sharp epidemic threshold at T_c ≈ ½ — exactly the square-lattice bond-percolation threshold (Grassberger's SIR↔percolation map, exact by duality; the live single-seed crossing on the finite 151² lattice reads just under ½, ≈0.495, converging to ½ as the lattice grows) — and notably ABOVE the well-mixed mean-field prediction R₀ = 1 (T ≈ ⅓ on a 4-contact graph), because spatial clustering wastes infections on already-hit neighbours