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A continuous time Markov chain model for a plantation‐nursery system
Author(s) -
Gani J.,
Stals L.
Publication year - 2005
Publication title -
environmetrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.68
H-Index - 58
eISSN - 1099-095X
pISSN - 1180-4009
DOI - 10.1002/env.740
Subject(s) - markov chain , laplace transform , mathematics , expected value , value (mathematics) , statistics , combinatorics , mathematical analysis
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 $\hat p_j$ , j = 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 . Copyright © 2005 John Wiley & Sons, Ltd.