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  "body": "\n## Example\n\nLet $d \\ge 1$ and $\\lambda \u003e 0$.\nFor $x \\in \\Z^d$ and $\\eta \\in \\set{0, 1}^{\\Z^d}$, let\n$$c(x, \\eta) = \\begin{cases} 1 \u0026 \\text{if } \\eta(x) = 1, \\\\ \\lambda \\, \\#\\set{y \\in \\Z^d : \\abs{y - x} = 1, \\ \\eta(y) = 1} \u0026 \\text{if } \\eta(x) = 0. \\end{cases}$$\nThe *contact process* on $\\Z^d$ with infection rate $\\lambda$ is the spin system on $\\set{0, 1}^{\\Z^d}$ with these flip rates.\nHere $\\eta(x) = 1$ means that site $x$ is infected.\nAn infected site recovers at rate $1$, and a healthy site is infected at rate $\\lambda$ times its number of infected neighbours.\nThe rates are bounded by $\\max(1, 2d\\lambda)$ and each depends on $2d + 1$ sites, so it is an [[interacting particle system|GR49AM3PSZ]].\n\n## Remarks\n\nThe configuration with no infected site is absorbing.\nThere is a critical value $\\lambda_c \\in (0, \\infty)$: started from one infected site, the infection dies out almost surely for $\\lambda \u003c \\lambda_c$ and survives forever with positive probability for $\\lambda \u003e \\lambda_c$.\nA lower bound is $\\lambda_c \\ge 1/(2d)$.\nIndeed, the number of infected sites is dominated by a branching process in which each individual dies at rate $1$ and gives birth at rate $2d\\lambda$.\nThis branching process dies out almost surely when $2d\\lambda \\le 1$.\n\n## Explanations\n\n### L1 human\n\nIt is a model of an epidemic on a grid.\nSick people heal at a fixed rate and pass the illness to healthy neighbours at rate $\\lambda$ each.\nIf $\\lambda$ is small the epidemic dies out; if $\\lambda$ is large it may last forever.\n",
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