Reproduction Number And Asymptotic Stability For The Dynamics of a Honey Bee Colony with Continuous Age Structure

TL;DR

This paper develops a mathematical model using partial differential equations to analyze honey bee colony dynamics with age structure, including disease effects, and derives conditions for stability and disease spread.

Contribution

It introduces a novel analytical approach for stability analysis and a method to compute the basic reproduction number in age-structured disease models.

Findings

Colony recovers if losses are addressed early.

Derived explicit equilibrium age distribution.

Provided a new method to calculate R0 for age-structured models.

Abstract

A system of partial differential equations is derived as a model for the dynamics of a honey bee colony with a continuous age distribution, and the system is then extended to include the effects of a simplified infectious disease. In the disease-free case we analytically derive the equilibrium age distribution within the colony and propose a novel approach for determining the global asymptotic stability of a reduced model. Furthermore, we present a method for determining the basic reproduction number $R_0$ of the infection; the method can be applied to other age-structured disease models with interacting susceptible classes. The results of asymptotic stability indicate that a honey bee colony suffering losses will recover naturally so long as the cause of the losses is removed before the colony collapses. Our expression for $R_0$ has potential uses in the tracking and control of an infectious disease within a bee colony.