The distribution of the time taken for an epidemic to spread between two communities
Introduction
Infectious disease spread across regions has become more rapid due to increased global mobility [9]. Understanding how disease- and population-specific factors influence the timing of disease importation is important for effective interventions to prevent or delay disease spread.
As reviewed by Arino [1], metapopulation models are useful for modelling the spread of disease between regions, or ‘patches’, where within-patch disease transmission occurs more frequently than between-patch transmission [31]. In this study, we focus on the early stages of the epidemic, when it first begins to spread from its source patch. Previous studies of metapopulation models have used branching process approximations to calculate outbreak probabilities [4], [22]. However, these studies do not examine the timing of disease spread. Studies which address the timing of travel of infectious individuals, and thus of disease spread, have assumed deterministic within-patch disease dynamics [7], [12], [33]. The studies also do not explicitly include the effect of interventions. Importantly, all of these studies assume that travel results in permanent migration, but short-term travel may be a greater driving force in the spread of infection [17]. Threshold conditions for outbreaks assuming short-term travel have been previously determined [2], but the temporal dynamics of such models were not analysed.
In this study, using a two-patch susceptible-infectious-removed (SIR) model with stochastic disease dynamics and short-term travel between patches, we analyse the temporal aspect of disease spread, addressing limitations of previous studies. We evaluate the distribution of the time taken for the epidemic to spread to the second patch (the spreading time distribution), and the probability that such spread occurs (the spreading probability). In Section 2, we outline the methods used to perform these calculations. We proceed to explore how interventions change the spreading probability and median spreading time (Section 3.1), and how approximations can be made to calculate the spreading time distribution accurately and efficiently (Section 3.2).
Section snippets
Methods
In Section 2.1, we specify the ‘ground truth’ model for this study. However, exact calculation of the spreading time distribution for this model is computationally infeasible. Therefore, in Section 2.2.1 we make approximations to simplify the model, before deriving an expression for the spreading time distribution in Sections 2.2.2 and 2.2.3. Three methods for evaluating the spreading time distribution for the reduced model, with varying degrees of approximation, are presented in Section 2.3.
Results
This section focuses on the effect of interventions — reducing R0 (Eq. (1)) and the travel rate l in each patch — on the spreading time distribution (Section 3.1). We will additionally investigate the effect of the approximations used (Section 3.2). All results are obtained using Octave 4.0.0 [11], with plots made using MATLAB [24].
Conclusion
In this study, we have modelled disease spread between two regions. We constructed a branching process in Patch 2 and combined it with epidemic trajectories in Patch 1 to derive equations for the spreading probability and spreading time distribution. This approach decouples the contributions to the spreading time distribution by travel parameters and disease parameters in Patch 2, and by disease parameters in Patch 1. Previous studies (such as by Lahodny and Allen [22]) have used the branching
Acknowledgements
A. W. C. Y. was supported by an ACS Foundation Scholarship and an Australian Postgraduate Award (now Australian Government Research Training Program Scholarship). A. J. B. was supported by an Australian Research Council Discovery Early Career Researcher Award (DE160100690). J. V. R. is supported by an Australian Research Council Future Fellowship (FT130100254). A. W. C. Y., J. M. M. and J. V. R. are supported by the National Health and Medical Research Centre of Australia Centre for Research
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