A continuous time Markov chain model for a plantation-nursery system

This article examines a continuous time Markov chain model for a plantation-nursery system in which diseased plantation trees are replaced at a daily rate λ by nursery seedlings. There is a random infection rate α caused by insects, and the disease is also spread directly between the N plantation trees at the rate β, starting with a diseased trees at time t = 0; in addition, some replacement seedlings prove to be infected with probability 0 < p < 1. We find a formal solution to the system in terms of the Laplace transforms p̂_j, y = 0, . . . , N, of the probabilities p_j(t) of j infected plantation trees at time t. A very simple example for N = 2, a = 1 is used to illustrate the method. We then consider numerically the effect of the parameters λ, α, and β on the system, and for small t study the influence of the initial number a of infected trees on the expected number of such trees at time t ≤ 365. As t → ∞, stationarity is achieved, irrespective of the initial value a.