Epidemic Spread in Networks: Existing Methods and Current Challenges

¹ School of Mathematical Sciences, School of Biological Sciences, and Monash Academy for Cross & Interdisciplinary Mathematics, Monash University, VIC 3800, Australia
² School of Mathematical and Physical Sciences, Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, UK

^⋆ JCM dedicates this work to the memory of Bob Borrelli, who taught him how to use integrating factors and much more.
^⋆⋆ Corresponding author. E-mail: joel.miller@monash.edu

Abstract

We consider the spread of infectious disease through contact networks of Configuration Model type. We assume that the disease spreads through contacts and infected individuals recover into an immune state. We discuss a number of existing mathematical models used to investigate this system, and show relations between the underlying assumptions of the models. In the process we offer simplifications of some of the existing models. The distinctions between the underlying assumptions are subtle, and in many if not most cases this subtlety is irrelevant. Indeed, under appropriate conditions the models are equivalent. We compare the benefits and disadvantages of the different models, and discuss their application to other populations (e.g., clustered networks). Finally we discuss ongoing challenges for network-based epidemic modeling.