A Mathematical Model for the Control and Eradication of a Wood Boring Beetle Infestation

Abstract

Abstract. We propose a mathematical model for an infestation of a wooded area by a beetle species in which the larva develop deep in the wood of living trees. Due to the difficulties of detection, we presume that only a certain proportion of infested trees will be detected and that detection, if it happens, will occur only after some delay, which could be long. An infested tree once detected is immediately cut down and burned. The model is stage structured and contains a second time delay, which is the development time of the beetle from egg to adult. There is a delicate interplay between the two time delays due to the possibility in one case for a larva to mature even in a tree destined for destruction. We present conditions sufficient for infestation eradication and discuss the significance of the conditions, particularly in terms of the proportion of infested trees that need to be detected and removed. If the infestation is successfully eradicated, there are always a number of trees that completely escape infestation, and we compute lower bounds and an approximation for this number. Finally, we present the results of some numerical simulations.

On a Diffusive Susceptible-Infected-Susceptible Epidemic Model with Mass Action Mechanism and Birth-Death Effect: Analysis, Simulations, and Comparison with Other Mechanisms

“…Nowadays it is widely recognized that spatial spread of an infection is closely related to the heterogeneity of the environment and the spatial-temporal movement of the hosts. This is well supported by numerous research on diseases including malaria [38,39], rabies [27,28,45], dengue fever [52], West Nile virus [34,53], hantavirus [1,2], Asian longhorned beetle [22,23], etc; see [51] and references therein. A popular way to incorporate spatial movement of hosts into epidemic models is to assume host random movements, leading to coupled reaction-diffusion equations.…”

An Optimal Control Framework for Resources Management in Agriculture

“…Delays should also be incorporated in modeling, and, thus, have a significant influence in the definition of optimal control strategies. An interesting problem of controlling and eradicating the infestation of wood boring beetle is considered in [31]. A key feature of the model consists in incorporating two delays: detection of infested trees and the development time of the beetle from egg to adult.…”

Global $L^\infty$-bounds and long-time behavior of a diffusive epidemic system in heterogeneous environment

“…On the other hand, it has been recognized that environmental heterogeneity and individual motility are significant factors that should be taken into consideration when studying the spread and control of infectious diseases; one may refer to, for instance, [15,48,60] for relevant discussions. Many reaction-diffusion epidemic models have been developed to investigate the impact of them on the dynamics of disease transmissions, such as malaria [45,46], rabies [27,28,53], dengue fever [61], West Nile virus [35,62], hantavirus [1,2], Asian longhorned beetle [20,21], etc. These models are derived from the ordinary differential equation (ODE) compartmental epidemic models by introducing random diffusion terms to describe the movement of individuals and the spatiotemporally dependent coefficients to describe the environmental heterogeneity.…”