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EXACT ANALYTICAL EXPRESSIONS FOR THE FINAL EPIDEMIC SIZE OF AN SIR MODEL ON SMALL NETWORKS

Part of: Markov processes Equations and systems with randomness General theory in ordinary differential equations

Published online by Cambridge University Press:  23 May 2016

K. MCCULLOCH Open the ORCID record for K. MCCULLOCH [Opens in a new window] ,
M. G. ROBERTS Open the ORCID record for M. G. ROBERTS [Opens in a new window]  and
C. R. LAING Open the ORCID record for C. R. LAING [Opens in a new window]
K. MCCULLOCH*
Affiliation:
Institute of Natural and Mathematical Sciences, Massey University, Albany, 0745, New Zealand email K.McCulloch@latrobe.edu.au
M. G. ROBERTS
Affiliation:
Institute of Natural and Mathematical Sciences, Massey University, Albany, 0745, New Zealand email K.McCulloch@latrobe.edu.au
C. R. LAING
Affiliation:
Institute of Natural and Mathematical Sciences, Massey University, Albany, 0745, New Zealand email K.McCulloch@latrobe.edu.au
*
✉K.McCulloch@latrobe.edu.au
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Abstract

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We investigate the dynamics of a susceptible infected recovered (SIR) epidemic model on small networks with different topologies, as a stepping stone to determining how the structure of a contact network impacts the transmission of infection through a population. For an SIR model on a network of $N$ nodes, there are $3^{N}$ configurations that the network can be in. To simplify the analysis, we group the states together based on the number of nodes in each infection state and the symmetries of the network. We derive analytical expressions for the final epidemic size of an SIR model on small networks composed of three or four nodes with different topological structures. Differential equations which describe the transition of the network between states are also derived and solved numerically to confirm our analysis. A stochastic SIR model is numerically simulated on each of the small networks with the same initial conditions and infection parameters to confirm our results independently. We show that the structure of the network, degree of the initial infectious node, number of initial infectious nodes and the transmission rate all significantly impact the final epidemic size of an SIR model on small networks.

Keywords

spread of infectioncontact networkprobability mass functionsnetwork topologyclustering coefficient

MSC classification

Primary: 60J28: Applications of continuous-time Markov processes on discrete state spaces
Secondary: 34A30: Linear equations and systems, general 34F05: Equations and systems with randomness
Type
Research Article
Information
The ANZIAM Journal , Volume 57 , Issue 4 , April 2016 , pp. 429 - 444
Copyright
© 2016 Australian Mathematical Society 

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