# Dynamic Processes Spreading on Networks

#### Lecturer

Dr Joel C Miller, La Trobe University

#### Synopsis

Networks form the substrate along which many infectious diseases and ideas spread. Understanding how the structure of a network interacts with a spreading process is an important step in its control. The importance of infectious disease control has been clear for some time. More recently it has become clear that our social networks are highly susceptible to the spread of false information, which can be used by adversaries to damage countries.

We will develop differential equations models that perform well at predicting the spread of stochastic simulations in random network networks. We will also compare those models with simulations in networks generated from real observations (there are a number of sources of such measured networks).

Our goal will be to gain experience developing relevant models, and to understand how interactions between individuals can affect how a disease or idea spreads. We will use this to provide some insight into how to reduce (or enhance) a spreading process.

#### Course Overview

• Introduction to Random Networks (the “Configuration Model”)
• Introduction to Python
• Biological contagions
• Social contagions
• Interacting contagions

#### Prerequisites

• Differential Equations
• Need to be able to solve dI/dt = alpha I
• Understanding how to get from a model description to a differential equation
• Understanding stability of equilibria will be helpful.
• Basic knowledge of probability
• the sum of probabilities is 1,
• The probability a set of independent events all happen is the product of the probabilities of all the individual events.
• Experience with Markov Chains will put you ahead of the game
• Programming experience (especially with Python) would be helpful, but if you don’t have the experience, Python is one of the simplest languages to start learning and the introduction will be gentle. If you want a job in academia or in industry, programming experience will be a big benefit to you. (Project Euler is a great resource for programming for mathematical topics: https://projecteuler.net/)

#### Assessment

• Mid-School Assignment: 40%
• Final Examination: 60%

• Epidemic processes in complex networks (Pastor-Satorras et al)
• Dynamical Systems on Networks (Porter & Gleeson)
• Mathematics of epidemics on networks: from exact to approximate models (Kiss, Miller & Simon)
• Epidemics on Networks Python package https://epidemicsonnetworks.readthedocs.io/en/latest/

(all are probably available free of charge through your institution’s library. Do not spend money on them unless you really want them. The first two are also available on arxiv)

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Dr Joel Miller,
La Trobe University

Joel Miller is an applied mathematician who develops mathematical models to guide policy decisions for infectious disease spread. Much of his career has been in interdisciplinary institutions such as:

• Los Alamos National Laboratory
• The British Columbia Centre for Disease Control
• The Harvard School of Public Health
• The Institute for Disease Modeling

He is particularly interested in understanding how the small-scale structure of a population affects disease spread.