between persistence and extinction of disease. The idea of conditional dependence between variables of
*
Corresponding author.
E-mail addresses: isthri@bioss.ac.uk (I. Krishnarajah), glenn@bioss.ac.uk (G. Marion), gavin@ma.hw.ac.uk
(G. Gibson).
www.elsevier.com/locate/mbs
Mathematical Biosciences 208 (2007) 621–6430025-5564/$ - see front matter  2007 Elsevier Inc. All rights reserved.interest underlies these mixture approximations. In the first approximation, we assume that the distribu-
tion of infectives (I) conditional on population size (N) is governed by the beta-binomial and for the sec-
ond form, we assume that I is governed by zero-modified beta-binomial distribution where in either case
N follows a log-Normal distribution. We analyse the impact of coupling and inter-dependency between
population variables on the behaviour of the approximations developed. Thus, the approximations are
applied in two situations in the case of the SIS model where: (1) the death rate is independent of disease
status; and (2) the death rate is disease-dependent. Comparison with simulation shows that these mixture
approximations are able to predict disease extinction behaviour and describe transient aspects of the
process.
 2007 Elsevier Inc. All rights reserved.Novel bivariate moment-closure approximations
Isthrinayagy Krishnarajah
a,
*
, Glenn Marion
a
, Gavin Gibson
b
a
Biomathematics and Statistics Scotland, JCMB, The King’s Buildings, Edinburgh EH9 3JZ, UK
b
School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK
Received 19 December 2005; received in revised form 27 November 2006; accepted 4 December 2006
Available online 21 December 2006
Abstract
Nonlinear stochastic models are typically intractable to analytic solutions and hence, moment-closure
schemes are used to provide approximations to these models. Existing closure approximations are often
unable to describe transient aspects caused by extinction behaviour in a stochastic process. Recent work
has tackled this problem in the univariate case. In this study, we address this problem by introducing
novel bivariate moment-closure methods based on mixture distributions. Novel closure approximations
are developed, based on the beta-binomial, zero-modified distributions and the log-Normal, designed
to capture the behaviour of the stochastic SIS model with varying population size, around the thresholddoi:10.1016/j.mbs.2006.12.002

of the system at time t is described by the population size, N(t), and the number of infectives, I(t).
622 I. Krishnarajah et al. / Mathematical Biosciences 208 (2007) 621–643To capture both short-term transient aspects and represent quasi-equilibrium dynamics of the SIS
model we develop two novel bivariate closure approximations based on the univariate beta-bino-
mial and zero-modified beta-binomial distributions [21]. We aim to analyse the impact of couplingKeywords: Log-normal; Beta-binomial; Mixture distribution; Zero-modified distribution; Moment-closure; Bivariate
SIS; Markov process
1. Introduction
The application of stochastic models to study the evolution of disease or population dynamics
has gained considerable popularity in recent years [2–15]. A particularly well developed applica-
tion of stochastic processes is to epidemics. In common with many applications, disease-related
models often exhibit nonlinear transition probabilities, for example ([4,9,16]). It is straightforward
to construct forward equations for the transition probability describing these models. However, it
is currently extremely difficult to obtain analytic solutions to these equations. Simulating the
process to understand the underlying behaviour is common but can be time consuming in certain
circumstances. Hence, approximation methods are an important tool, useful in exploring the
underlying behaviour of such models.
Moment-closure approximation is one such method used to provide analytic insights into mod-
el behaviour, which can also help to validate the results of complex simulations. Closure methods
are typically based on differential equations describing the temporal evolution of moments or
cumulants of a stochastic process, for example the expected number of infectives or susceptibles
in the evolution of disease dynamics. The technique of moment-closure, introduced by [1],is
increasingly seen in the study of epidemiology [4,7,9,16–19,21] and ecology [6,15,22–24,35].
Closure methods can be applied to explore both transient and equilibrium dynamics in univariate
as well as multivariate contexts. However, existing closure approximations based on distributional
assumptions such as the Normal [4,1] and log-Normal [22] typically fail to describe transient as-
pects caused by extinction behaviour or in situations where the population distribution is highly
skewed [7,21]. This is also true for the widely used truncation procedures in which for a set of k
moment or cumulant equations, the (k + 1)th and higher order central moments or cumulants are
set to 0. These problems were tackled in the context of the univariate SIS model by introducing
moment-closure approximations based on the beta-binomial and zero-inflated (or zero-modified)
mixture distributions [21]. These approximations were able to capture transient aspects as well as
quasi-equilibrium dynamics of the univariate SIS process. Our main interest is to extend the
application of distributional based moment-closure method to approximate multivariate
nonlinear stochastic models. The use of mixture distributions in moment-closure may potentially
be extended to develop tailored approximations that also describe extinction behaviours in two-
dimensional models. Since closure based on mixture approximations worked well in the univariate
case [21], we aim to extend the application of moment-closure approximation to address similar
problems in a bivariate context by considering mixtures of distributions.
As a starting point here we consider a simple two-dimensional system, namely the stochastic
SIS (susceptible–infected–susceptible) model with varying population size [5,25,26] where the stateand inter-dependency between population variables on the behaviour of the approximations