We consider a discrete time, finite horizon, eco-epidemiological model for the management of an outbreak impacting fruit trees. The model is designed at the landscape scale, with contamination both within and between patches, according to a linear diffusion model. No treatment is available, and management consists in detecting and removing infected trees. Agents have to decide whether they are willing to perform detection at a given cost. Within this framework, we evaluate the tradeoff between keeping infected trees with less valuable production and removing them in order to avoid new infections. We compare the decentralized solution in which each agent maximizes his own utility function, with the central planner solution which maximizes the global production. Within this general framework, we analyze a two-periods-two-patches model. Using a feedback information structure, we show the existence of Nash equilibria. Exploration of the initial condition space shows that there is often a single equilibrium. However for some particular parameters and initial conditions we find multiple Nash equilibria, meaning that we are in presence of coordination problems. Mixing theoretical results and numerical simulations, we build a mapping between initial conditions (infected trees for each patch) and equilibria. Those results are compared to the central planner solution and we characterize situations in which private management leads to inefficiency. This work has been inspired by the management of the Sharka outbreak in France. Sharka is a viral disease responsible for important damages on fruit trees from the Prunus genus. A possible extension of our work would be to build a case study using our theoretical model together with parameters and data corresponding to this particular outbreak