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

• Published in: Environmetrics (London, Ont.) Vol. 16; no. 8; pp. 849 - 861
• Main Authors: Gani, J., Stals, L.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 01.12.2005; Wiley
• Subjects: Animal, plant and microbial ecology; Biological and medical sciences; continuous time Markov chain; Earth sciences; Earth, ocean, space; Engineering and environment geology. Geothermics; Exact sciences and technology; Fundamental and applied biological sciences. Psychology; General aspects. Techniques; infection; Methods and techniques (sampling, tagging, trapping, modelling...); nursery; plantation; Pollution, environment geology; virus; Markov chain analysis; models; infection; Disease; probability; Plant juvenile growth stage; Nursery (animal); trees; virus; solution; nursery; continuous time Markov chain; plantation
• ISSN: 1180-4009; 1099-095X
• DOI: 10.1002/env.740

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 pj(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.