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Appendix A. Technical details on parameter estimation.

1. The parameters d² and are positive, and we assumed a prior which is lognormal, the underlying normal distribution having mean log 0.1 and standard deviation 2. We used the Metropolis-Hastings algorithm with a jumping distribution which was lognormal with the current value as mean and standard deviation adjusted to give roughly a 30% accept ratio.
2. The parameters h and g are restricted to the range [0,1], and we assumed a flat prior distribution on this range. As the beta distribution is conjugate to the binomial distribution, we expressed the prior as Beta(1,1), in which case the full-conditional posterior distribution for h is Beta(H + 1, P – H + 1), where H = _i H_i and P = _i P_i are the total numbers of hosts and plants. We used the Gibbs sampler to update the value of h and the Metropolis-Hastings algorithm with logit-normal proposal distribution to update g.
3. Missing data I_i,t. We assumed a prior distribution that is discrete uniform in [I_i,t1, I_i,t2], where t₁ < t < t₂ are the closest time steps for which data is available. If there were several missing time steps between t₁ and t₂, the priors for each I_i,t were taken to be independent for each t. The likelihood of the missing data is zero if I_i,t > I_i,t+1, which restricts the set of feasible prior distributions. We used the Metropolis-Hastings algorithm with a jumping distribution which jumps with probability f to a value drawn from the discrete uniform distribution in [I_i,t-1, I_i,t+1] (and is thus independent of the previous value of I_i,t) and stays at I_i,t with probability 1 – f. The value of f was fixed to f = 0.05 to give a roughly 30% accept ratio. The joint distribution for missing data I_i,t was updated simultaneously for all i, but for a single t at a time.
4. Number of hosts H_i. As the prior distribution we used the discrete uniform distribution in [I_i,tmax, P_i] (note that the likelihood of the data is zero if P_i < I_i,tmax). The jumping distribution was based on the discrete uniform distribution in [I_i,tmax, P_i]. The number of hosts was updated separately for each quadrat i.

Figure A1 illustrates the performance of the algorithm against simulated data assuming biologically reasonable parameter values. While the fraction of hosts becomes very accurately estimated in this data set (4% relative error in the 95% highest posterior density interval; HPDI), the estimation of the other parameters proved to be slightly more difficult (10%–20% relative error in the 95% HPDIs).

FIG. A1. The performance of the estimation scheme against simulated data. The dots represent the assumed parameter values ( = 0.06, h = 0.7, g = 0.5, d = 0.277) and the lines the estimated posterior distributions. The number of plants and the initial number of infected plants was set to correspond to the data from population 1 in the year 2001.

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