In this work, we apply the nonlinear filtering theory to the estimation of the partially observed dynamics of anthracnose which is a phytopathology. The signal here is the inhibition rate and the observations are the fruit volume and the rotted volume. We propose stochastic models based on deterministic models studied previously in the literature, in order to represent the noise introduced by uncontrolled variations on parameters and errors on the measurements. Under the assumption of Brownian noises, we prove the well-posedness of the models in either they take into account the space variable or not. The filtering problem is solved for the nonspatial model giving Zakai and Kushner–Stratonovich equations satisfied respectively by the unnormalized and the normalized conditional distribution of the signal with respect to the observations. A prevision problem and a discrete filtering problem are also studied for the realistic cases of discrete and possibly incomplete observations. We illustrate the filter behavior through figures displaying the average estimation relative error and a 95% confidence region obtained after a hundred of numerical simulations with initial conditions taken randomly with respect to uniform law.
Data assimilation algorithm based on an approximate solution of the optimal nonlinear filtering problem
