A PDE-ODE Coupled Spatio-Temporal Mathematical Model for Fire Blight During Bloom

TL;DR

This paper develops and analyzes a coupled PDE-ODE mathematical model to describe the spread of fire blight in orchards during bloom, providing insights into disease dynamics and potential management strategies.

Contribution

It introduces a novel PDE-ODE coupled model for fire blight spread and proves the existence of travelling wave solutions under certain conditions.

Findings

Existence of travelling wave solutions demonstrated

Numerical simulations support wave propagation hypothesis

Model offers potential for disease management insights

Abstract

Fire blight is a bacterial plant disease that affects apple and pear trees. We present a mathematical model for its spread in an orchard during bloom. This is a PDE-ODE coupled system, consisting of two semilinear PDEs for the pathogen, coupled to a system of three ODEs for the stationary hosts. Exploratory numerical simulations suggest the existence of travelling waves, which we subsequently prove, under some conditions on parameters, using the method of upper and lower bounds and Schauder's fixed point theorem. Our results are likely not optimal in the sense that our constraints on parameters, which can be interpreted biologically, are sufficient for the existence of travelling waves, but probably not necessary. Possible implications for fire blight biology and management are discussed.