Hannah Lea Callender

Transmission of Infectious Diseases: Data, Models, and Simulations

Abstract This chapter introduces students to various aspects of mathematical modeling in epidemiology, including data collection, development of models, and deriving and interpreting predictions of models. The agent-based models of this chapter are explored by using a teaching tool, IONTW (Infections On NeTWorks), which was developed by the authors using the NetLogo programming language. The chapter discusses in detail the process of building mathematical models of disease transmission. Students will frequently be challenged to critically examine the simplifying assumptions that are involved in this process. It also provides an intuitive introduction to thinking about simulation models in terms of pseudocode, intended to empower students who have no prior knowledge of computer programming to analyze simulation models at a semiformal level. The chapter includes numerous conceptual and simulation exercises aimed at developing deeper understanding of the concepts.

Disease Transmission Dynamics on Networks: Network Structure Versus Disease Dynamics

Abstract The first part of this chapter covers compartment-based models that are constructed under the uniform mixing assumption. The role of the basic reproductive ratio in these models is illustrated both by theoretical results and simulations with IONTW (Infections On NeTWorks), a teaching tool that was developed by the authors and uses the NetLogo language. The second part introduces students to network-based models of disease transmission. In these models it is assumed that the infection can be transmitted only during a direct contact between two hosts who are adjacent in the underlying contact network. Models of this kind allow for incorporation of more realistic details than do compartment-based models that assume uniform mixing. Students are guided through sequences of conceptual and simulation exercises to discover how the network structure influences the predictions of the model, including predictions about the effectiveness of possible control measures.

Modeling epidemics on a regular tree graph

We will first provide a brief introduction to models of disease transmission on so-called contact networks, which can be represented by various structures from the mathematical field of graph theory. These models allow for exploration of stochastic effects and incorporation of more biological detail than the classical compartment-based ordinary differential equation models, which usually assume both homogeneity in the population and uniform mixing. In particular, we use an agent-based modelling platform to compare theoretical predictions from mathematical epidemiology to results obtained from simulations of disease transmission on a regular tree graph. We also demonstrate how this graph reveals connections between network structure and the spread of infectious diseases. Specifically, we discuss results for how certain properties of the tree graph, such as network diameter and density, alter the duration of an outbreak.

Mathematical modelling and analysis of cellular signalling in macrophages

Cell signalling pathways play a crucial role in proper cell development and behaviour, with implications to survival, chemotaxis, proliferation, and even programmed cell death known as apoptosis. In this article, we outline a mathematical model of the G-protein signalling pathway in a particular cell line of macrophages, focusing on activation of a particular G-protein-coupled receptor, P2Y6. The model is based on the kinetics of P2Y6 surface receptors, inositol trisphosphate, cytosolic calcium, and differential dynamics of multiple species of diacylglycerol. Insight into the dynamics of the system is given through recently available experimental results and incorporated into the model. Mathematical analysis of the model, including establishment of global existence, uniqueness, positivity, and boundedness of solutions, and global stability of a unique steady-state solution is discussed.

Proportional Insulin Infusion in Closed-Loop Control of Blood Glucose

A differential equation model is formulated that describes the dynamics of glucose concentration in blood circulation. The model accounts for the intake of food, expenditure of calories and the control of glucose levels by insulin and glucagon. These and other hormones affect the blood glucose level in various ways. In this study only main effects are taken into consideration. Moreover, by making a quasi-steady state approximation the model is reduced to a single nonlinear differential equation of which parameters are fit to data from healthy subjects. Feedback provided by insulin plays a key role in the control of the blood glucose level. Reduced β-cell function and insulin resistance may hamper this process. With the present model it is shown how by closed-loop control these defects, in an organic way, can be compensated with continuous infusion of exogenous insulin.