Physica A 389 (2010) 561–576 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Stochastic epidemics and rumours on finite random networks Valerie Isham a , Simon Harden a,∗ , Maziar Nekovee b,c a Department of Statistical Science, University College London, London, UK b Centre for Computational Science, University College London, Gordon Street, London WC1H 0AJ, UK c Mobility Research Centre, BT, Polaris 134, Adastral Park, Martlesham, Suffolk IP5 3RE, UK article info Article history: Received 19 May 2009 Received in revised form 24 September 2009 Available online 7 October 2009 Keywords: Epidemic models Rumour models Random networks Stochastic models abstract In this paper, we investigate the stochastic spread of epidemics and rumours on networks. We focus on the general stochastic (SIR) epidemic model and a recently proposed rumour model on networks in Nekovee et al. (2007) [3], and on networks with different random structures, taking into account the structure of the underlying network at the level of the degree–degree correlation function. Using embedded Markov chain techniques and ignoring density correlations between neighbouring nodes, we derive a set of equations for the final size of the epidemic/rumour on a homogeneous network that can be solved numerically, and compare the resulting distribution with the solution of the corresponding mean-field deterministic model. The final size distribution is found to switch from unimodal to bimodal form (indicating the possibility of substantial spread of the epidemic/rumour) at a threshold value that is higher than that for the deterministic model. However, the difference between the two thresholds decreases with the network size, n, following a n −1/3 behaviour. We then compare results (obtained by Monte Carlo simulation) for the full stochastic model on a homogeneous network, including density correlations at neighbouring nodes, with those for the approximating stochastic model and show that the latter reproduces the exact simulation results with great accuracy. Finally, further Monte Carlo simulations of the full stochastic model are used to explore the effects on the final size distribution of network size and structure (using homogeneous networks, simple random graphs and the Barabasi–Albert scale-free networks). © 2009 Elsevier B.V. All rights reserved. 1. Introduction In the 21st century, there are many infectious diseases that pose substantial threats to human and animal populations. There are concerns over the spread of HIV/AIDS, the emergence of new infections such as SARS, the threat of a new human influenza pandemic arising from avian influenza, possible outbreaks of smallpox resulting from terrorist action, and many others. Mathematical models have an important role to play in controlling the spread of such infections. In addition, the infectious agents and the infected hosts need not be biological systems. Individual personal computers and servers are regularly targeted by viruses spread across computer networks, while the effective transmission of information over the internet and the behaviour of social interaction networks are topical research interests. In many ways, the spread of information resembles that of infection, and the models that have been developed have many features in common. In each case, individuals are classified as being of one of three types: susceptible, infected or removed for infections; ignorant, spreader or stifler for information or rumours. The simplest models assume homogeneous mixing of hosts, so that susceptibles become infected at a rate proportional to the current numbers of susceptibles and infectives, and similarly for ∗ Corresponding author. Tel.: +44 2076791872; fax: +44 2073834703. E-mail address: simon@stats.ucl.ac.uk (S. Harden). 0378-4371/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2009.10.001