Master equation solution of a plant disease model

Abstract

We develop the exact solution for a stochastic plant disease model with nonlinear density dependence and time-decaying susceptibility. In our biological application only the transient behaviour, rather than the stationary state, is relevant. To model the population noise we use the Master equation, leading to a linear ordinary differential equation system with finally time-independent coefficients. A numerically stable procedure is given by the Padé approximation of the solution in form of an exponential of a matrix. On the basis of this solution the parameter estimation is performed using experimental data from epidemiological microcosms. In the Master equation formalism, in order to generalize to more complicated models, for example, including a time-independent latent period as a first multivariate model, we finally suggest a numerical procedure to estimate the likelihood comparing data time series with simulated Master equation time series.