J. Math. Biol.
DOI 10.1007/s00285-011-0443-3
Mathematical Biology
From Markovian to pairwise epidemic models
and the performance of moment closure approximations
Michael Taylor · Péter L. Simon ·
Darren M. Green · Thomas House · Istvan Z. Kiss
Received: 20 September 2010 / Revised: 11 May 2011
© Springer-Verlag 2011
Abstract Many if not all models of disease transmission on networks can be linked
to the exact state-based Markovian formulation. However the large number of equa-
tions for any system of realistic size limits their applicability to small populations. As
a result, most modelling work relies on simulation and pairwise models. In this paper,
for a simple SI S dynamics on an arbitrary network, we formalise the link between
a well known pairwise model and the exact Markovian formulation. This involves
the rigorous derivation of the exact ODE model at the level of pairs in terms of the
expected number of pairs and triples. The exact system is then closed using two dif-
ferent closures, one well established and one that has been recently proposed. A new
interpretation of both closures is presented, which explains several of their previously
observed properties. The closed dynamical systems are solved numerically and the
results are compared to output from individual-based stochastic simulations. This is
done for a range of networks with the same average degree and clustering coefficient
but generated using different algorithms. It is shown that the ability of the pairwise
system to accurately model an epidemic is fundamentally dependent on the underlying
large-scale network structure. We show that the existing pairwise models are a good fit
M. Taylor (
B
) · I. Z. Kiss
School of Mathematical and Physical Sciences, Department of Mathematics,
University of Sussex, Falmer, Brighton BN1 9QH, UK
e-mail: mt264@sussex.ac.uk
P. L. Simon
Institute of Mathematics, Eötvös Loránd University Budapest, Budapest, Hungary
D. M. Green
Institute of Aquaculture, University of Stirling, Stirling FK9 4LA, UK
T. House
Department of Biological Sciences, Mathematics Institute, University of Warwick,
Gibbet Hill Road, Coventry CV4 7AL, UK
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M. Taylor et al.
for certain types of network but have to be used with caution as higher-order network
structures may compromise their effectiveness.
Keywords Network · Epidemic · Markov chain · Moment closure
Mathematics Subject Classification (2000) 60J27
1 Introduction
The spread of diseases within a population depends not only on the nature of the
pathogen but also on the way in which infectious individuals come into contact with
susceptible individuals. The network of these contacts provides the supporting struc-
ture on which the disease transmission process takes place. There is a large body of
research examining network epidemic models with the aim of understanding how net-
work properties impact on disease invasion, spread and control (Keeling and Eames
2005). Many different modelling approaches have been proposed, which fall into three
broad classes: exact Markovian or state-based models (Simon et al. 2011), individual-
based stochastic simulation or micro models (Keeling and Eames 2005) and deter-
ministic ODE-based macro models (Sato et al. 1994; Rand 1999; Keeling 1999; van
Baalen 2000). This classification is not application specific and it simply refers to the
scale (e.g. individual level or population level) at which the modelling is being carried
out. The links between state-based, micro and macro models are explored in detail by
Gustafsson and Sternad (2010).
State-based systems, given by the master equation, or Kolmogorov equation, con-
tain information about all possible states of the system along with the associated rates
of transition from one state to another. Solving the resulting set of differential equa-
tions provides a full system description with no need for simulation. This approach
has typically been used for small networks (Keeling and Ross 2008) due to the number
of equations increasing exponentially with system size (e.g. SI S type dynamics on a
network with N individuals results in 2N − 1 equations). With significant increases
in computing power, this approach provides a realistic alternative to individual-based
stochastic simulation of small populations, although we are unable yet to solve a full
state-based set of ODEs for realistic network sizes. For special classes of graphs how-
ever, using the lumping technique discussed by Simon et al. (2011), large reductions
in the system size can be achieved and the state-based models become a viable alter-
native even for large networks. However, for problems involving large networks with
complex structure, individual-based simulation remains the most realistic approach.
The advantages offered by individual-based modelling come at the cost of little
or no analytical tractability. To overcome this problem, ‘moment-closure’ type ODE-
based models have been developed and formulated, offering faster computational time
and more analytical tractability. These differ from classic compartmental-based ODE
models in that the evolution equations for the expected number of individuals involves
the expected number of pairs and higher-order structures. Many such models have
been derived heuristically (Sato et al. 1994; Rand 1999; Keeling 1999; van Baalen
2000; House and Keeling 2010) but recently their direct link to Markovian models
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