= Spatial Bernoulli infection model
{title2=$P_{it}=1-e^{-\alpha A_{it}}$}
Given the currently infected set $I_t$ and susceptible set $S_t$, define $A_{it}(\eta)=\sum_{j\in I_t}K_\eta(d_{ij})$ and $P_{it}=1-e^{-\alpha A_{it}}$. Conditionally <independent> new-infection indicators $Y_{it}$ have <likelihood>
$$
L(\alpha,\eta)=\prod_t\prod_{i\in S_t}(1-e^{-\alpha A_{it}(\eta)})^{Y_{it}}e^{-\alpha A_{it}(\eta)(1-Y_{it})}.
$$
A nonnegative infection intensity requires $\alpha\geq0$; an unrestricted <normal distribution> <prior distribution> for $\alpha$ must therefore be restricted or replaced. With $K_\beta(d)=d^{-\beta}$ or $K_\gamma(d)=e^{-\gamma d}$ and positive distances, the conditional <posterior> kernels are not generally conjugate normal <probability densities>. <Metropolis–Hastings algorithm> updates, optionally on $\log\alpha$ with its <Jacobian determinant>, provide a direct sampling method.
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