Analysis of the Mathematical Model for the Spread of Pine Wilt Disease

Xiangyun Shi,Guohua Song

doi:10.1155/2013/184054

Abstract

This paper formulates and analyzes a pine wilt disease model. Mathematical analyses of the model with regard to invariance of nonnegativity, boundedness of the solutions, existence of nonnegative equilibria, permanence, and global stability are presented. It is proved that the global dynamics are determined by the basic reproduction numberℛ0and the other valueℛcwhich is larger thanℛ0. Ifℛ0andℛcare both less than one, the disease-free equilibrium is asymptotically stable and the pine wilt disease always dies out. If one is between the two values, though the pine wilt disease could occur, the outbreak will stop. If the basic reproduction number is greater than one, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region, and the disease persists at the endemic equilibrium state if it initially exists. Numerical simulations are carried out to illustrate the theoretical results, and some disease control measures are especially presented by these theoretical results.

Highlights

Pine wilt is a dramatic disease of pine caused by the pinewood nematode (Bursaphelenchus xylophilus), which constitutes a major threat to forest ecosystems worldwide, from both the economical point of view and the environmental perspective [1]
Rc > 1 indicates that pine wilt disease could occur, while R0 < 1 shows that the outbreak will stop since the infected pine decreases to the disease-free equilibrium
The following lemma is the main tool to prove the global stability of the endemic equilibrium with disease

Summary

Introduction

Pine wilt is a dramatic disease of pine caused by the pinewood nematode (Bursaphelenchus xylophilus), which constitutes a major threat to forest ecosystems worldwide, from both the economical point of view and the environmental (landscape) perspective [1]. Experience from control actions in Japan included aerial spraying of insecticides to control the insect vector (the cerambycid beetle Monochamus alternatus), injection of nematicides to the trunk of infected trees, slashing and burning of large areas out of control, beetle traps, biological control, and tree breeding programs [1]. These actions allowed some positive results, and unsuccessful cases due to the pinewood nematode spread and virulence.

Positivity and Boundedness of Solutions

Stability of the Disease-Free Equilibrium

Global Stability of the Endemic Equilibrium

Numerical Simulations and Disease Control