Is there a sharp threshold above which a local outbreak becomes a lattice-wide epidemic — and where is it?
▶ Launch the interactive simulationRun 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.
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