Elsevier

Mathematical Biosciences

Volume 303, September 2018, Pages 139-147
Mathematical Biosciences

The distribution of the time taken for an epidemic to spread between two communities

https://doi.org/10.1016/j.mbs.2018.07.004Get rights and content

Highlights

  • We use mathematical models to assess interventions to control epidemic spread between communities.

  • We evaluate the spreading probability and the distribution of the time taken.

  • Approximations are developed to efficiently evaluate these quantities.

  • Controlling infection at its source prevents/delays epidemic spread most effectively.

  • For certain parameter regions, model choice affects assessment of interventions.

Abstract

Assessing the risk of disease spread between communities is important in our highly connected modern world. However, the impact of disease- and population-specific factors on the time taken for an epidemic to spread between communities, as well as the impact of stochastic disease dynamics on this spreading time, are not well understood. In this study, we model the spread of an acute infection between two communities (‘patches’) using a susceptible-infectious-removed (SIR) metapopulation model. We develop approximations to efficiently evaluate the probability of a major outbreak in a second patch given disease introduction in a source patch, and the distribution of the time taken for this to occur. We use these approximations to assess how interventions, which either control disease spread within a patch or decrease the travel rate between patches, change the spreading probability and median spreading time.

We find that decreasing the basic reproduction number in the source patch is the most effective way of both decreasing the spreading probability, and delaying epidemic spread to the second patch should this occur. Moreover, we show that the qualitative effects of interventions are the same regardless of the approximations used to evaluate the spreading time distribution, but for some regions in parameter space, quantitative findings depend upon the approximations used. Importantly, if we neglect the possibility that an intervention prevents a large outbreak in the source patch altogether, then intervention effectiveness is not estimated accurately.

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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