MSc Math Defence: A Deterministic Discrete Stage-Structured SIR Model with Indirect Transmission

CANDIDATE:  ABBY ANDERSON

ABSTRACT:

In this thesis we propose a deterministic discrete stage-structured SIR model with indirect transmission. We show the model is well-posed using fundamental Ordinary Differential Equations (ODE) theory. We derive a sufficient but not necessary condition for stability of the disease-free equilibrium (DFE) using Gershgorin’s Theorem. We calculate the basic reproduction number, R₀, using the Next Generation Method. In our numerical simulations, we further explore the stability of the DFE via Gershgorin’s Theorem and the eigenvalues of the Jacobian matrix evaluated at the DFE. Additionally, we establish R₀ as a sharp criterion for disease persistence and establish a relationship between the model’s transience and R₀. We compare the stage-structured SIR model against its non-stage-structured counterpart, demonstrating that the latter gives a more refined description of disease dynamics. We conclude by proposing a model for the spread of Nosema ceranae in the Western honeybee which includes discrete stage-structure and indirect transmission.