On methods for studying stochastic disease dynamics

Abstract

Models that deal with the individual level of populations have shown the importance of stochasticity in ecology, epidemiology and evolution. An increasingly common approach to studying these models is through stochastic (event-driven) simulation. One striking disadvantage of this approach is the need for a large number of replicates to determine the range of expected behaviour. Here, for a class of stochastic models called Markov processes, we present results that overcome this difficulty and provide valuable insights, but which have been largely ignored by applied researchers. For these models, the so-called Kolmogorov forward equation (also called the ensemble or master equation) allows one to simultaneously consider the probability of each possible state occurring. Irrespective of the complexities and nonlinearities of population dynamics, this equation is linear and has a natural matrix formulation that provides many analytical insights into the behaviour of stochastic populations and allows rapid evaluation of process dynamics. Here, using epidemiological models as a template, these ensemble equations are explored and results are compared with traditional stochastic simulations. In addition, we describe further advantages of the matrix formulation of dynamics, providing simple exact methods for evaluating expected eradication (extinction) times of diseases, for comparing expected total costs of possible control programmes and for estimation of disease parameters.

Stochastic dynamics and logistic population growth

“…However, if the typical population size is not large, internal fluctuations can lead to the extinction of the population [14]. The effects of internal fluctuations have been studied in predator-prey models [15,16], epidemic models [17][18][19][20][21][22][23], cell biology [24], and ecological systems [13]. In particular, extinction of a stochastic population [11,25,26], which is a crucial concern for population biology [27] and epidemiology [28,29], has also attracted scrutiny in cell biochemistry [30] and in physics [31,32].…”

From Markovian to pairwise epidemic models and the performance of moment closure approximations

“…Solving the resulting set of differential equations provides a full system description with no need for simulation. This approach has typically been used for small networks [19] due to the number of equations increasing exponentially with system size (e.g. SIS type dynamics on a network with N individuals results in 2 N − 1 equations).…”

Outbreak properties of epidemic models: The roles of temporal forcing and stochasticity on pathogen invasion dynamics

“…Although the power of master equations for simulating and deriving analytical insight into the behaviour of stochastic epidemic models has been recognised previously (Chen & Bokka, 2005;Grabowski & Kosinski 2004;Keeling and Ross, 2008;Rozhnova & Nunes, 2009), we illustrate here how insight into the effects of temporal variability in transmission on stochastic infectious disease dynamics may also be incorporated within this framework. This is a key advance in the study of infectious disease dynamics, as much of the insight to date on the effects of seasonality on disease dynamics has resulted largely from analytical or numerical analysis of deterministic epidemic (or endemic) models (Bailey, 1975;Bolker and Grenfell, 1993;Dietz, 1976;Moneim, 2007;Stone et al, 2007).…”

“…The use of master equations, whereby the probability of occurrence of each possible disease state is simultaneously considered, to understand the behaviour of stochastic infectious disease models has been described elsewhere (Keeling & Ross, 2008), along with applications to epidemic processes in homogeneous models (Chen & Bokka 2005) and structured/hierarchical models (Grabowski & Kosinski 2004;Rozhnova and Nunes 2009), yet by comparison to their deterministic counterparts, relatively little infection modelling work has adopted such methods.…”