Influenza epidemic spread simulation for Poland — a large scale, individual based model study

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Abstract

In this work a construction of an agent based model for studying the effects of influenza epidemic in large scale (38 million individuals) stochastic simulations, together with the resulting various scenarios of disease spread in Poland are reported. Simple transportation rules were employed to mimic individuals’ travels in dynamic route-changing schemes, allowing for the infection spread during a journey. Parameter space was checked for stable behaviour, especially towards the effective infection transmission rate variability. Although the model reported here is based on quite simple assumptions, it allowed to observe two different types of epidemic scenarios: characteristic for urban and rural areas. This differentiates it from the results obtained in the analogous studies for the UK or US, where settlement and daily commuting patterns are both substantially different and more diverse. The resulting epidemic scenarios from these ABM simulations were compared with simple, differential equations based, SIR models — both types of the results displaying strong similarities. The pDYN software platform developed here is currently used in the next stage of the project employed to study various epidemic mitigation strategies.

Introduction

Agent based models (ABMs, called also individual based models — IBMs) are well known tools and have been used for over three decades [1], [2] in the field of simulating the spread of an infectious disease. However, mostly due to the lack of both data and computer resources, their potential was not widely realized [3], until quite recently [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16].

In the ABMs methodology it is straightforward to represent complex social and physical systems. Although it might require vast amounts of data, especially when one wants to reproduce the realistic social structure of a given country or territory, an ABM description can easily reflect the heterogeneity of social contacts, and introduce a possibility of implementing dynamic social network. Both processes: the dynamical network changes and virus agent-to-agent transmission can be modelled taking into account their stochastic nature.

It does mean, however, that the possible advantages of applying ABMs in such simulations are strongly data dependent, while the required data are not always readily available. The threats posed by highly pathogenic respiratory diseases, like SARS, or the eventual spread of pandemic strain of influenza, require health authorities to plan and prepare for the possibility of a human pandemic. Especially, some control strategies need to be developed, and should they be deemed effective, implemented. School closure, targeted antiviral prophylaxis (TAP) and vaccination are among the mitigation means most often considered. Recently, Ferguson et al. [16] estimated the effect of school closure on influenza spread by using Markov chain Monte Carlo sampling and statistical modelling methods. Their previous models [7], [12], [15] were dealing with simulations of influenza spread in Thailand [12] and in follow-up studies concentrating on the UK and US [15] in each case by parametrisation of an individual, agent based simulation model of pandemic influenza transmission [7] or modelling a spread of the e.g., foot and mouth disease in the UK [17]. A similar approach by Germann et al. [14] also studied influenza spread in the US, implementing a very large scale simulation involving over 200 million geo-referenced software agents, using statistical models. In both independent US studies [14], [15] their findings were in good qualitative agreement, enhancing each other, but most importantly demonstrating the practical usefulness of multiple agent based, very large scale simulations. Such large scale epidemic simulations revealed the conditions of initial containment of an influenza outbreak in relationship to various preventive strategies.

Entirely different to the ABM simulations, is the approach to model the spread of disease by the use of differential equations based on uniform mixing assumptions [18]. Including the new effects, e.g. a spatio-temporal variability of coefficients or a coexistence of many virus strains, into differential equations model requires a reformulation of the entire system of equations. This results in new properties (e.g. attractor basins) and requires further stability studies. The interpretation of such results is usually a challenging issue, owing to the ambiguity of the calibration of the coefficients.

From such a perspective, the ABM approach seems to be a good alternative for epidemic modelling. Although it requires much more effort in preparing the system, it can be beneficial for straightforward implementation of the modules responsible for, e.g. an immunological history of individuals, variations in climate between regions, or a specific social structure, which are all together usually possible to calibrate at a lower scale. The agent based approach can also capture the epidemic course when a small number of individuals is involved.

In yet another approach, Eubank et al. propose the use of dynamic bipartite graphs to model the physical contact patterns that result from movements of individuals between specific locations. They have found that the contact network among individuals forms a strongly connected, small-world type of graph, with a well-defined scale for the degree distribution. Moreover, the locations’ graph is scale-free, which allows for highly efficient outbreak detection by placing sensors in the hubs of the network. Their results suggest that the outbreaks can be suppressed by a strategy of targeted vaccination combined with an early prophylaxis [11].

The construction of ABM models for epidemiological simulations requires: first, the generation of a virtual society with an appropriate demographic structure corresponding to the correct territorial population densities. Second, it requires a geo-referenced network of public objects (schools, workplaces, etc.), on which agents would operate and interact with each other. The issues involved in constructing such systems stem mostly from a need to recreate information contained or usually even hidden in the aggregated data as delivered by national census bureaus, and/or other such services. Therefore, there is a need to devise techniques allowing for the decomposition of various sources of the aggregated data into either their constituting components, or failing to allocate all necessary de-aggregated data (which is usually the case, except in the most lucky circumstances), to re-construct the underlying original data by building viable and geo-referenced software constructs which can serve as a reasonable approximation of the missing data, with detailed enough granulation. The virtual society of a type suitable for the purpose of the current study was constructed and tested earlier — the results are given in Ref. [19], some pertinent details are given below in the Section 2.

To date, only a few studies have reported very large scale, individual based, modelling efforts, covering country-wide epidemiological simulations. Ferguson et al. published results for Thailand, the US and UK [7], [12], [15], and Germann et al. [14] also for the US — all of them were concerned with the spread and suppression of influenza epidemics. The work presented here was directed to developing methods to construct an analogous, agent based, very large scale, country-wide influenza epidemiological model for Poland (38 million inhabitants).

Section snippets

Purpose

The aim was to acquire a possible scenario of infectious air-borne disease spread on the territory and society of Poland; both spatial and temporal epidemic profile might be achieved for various initial parameter values.

State variables and scales

The fundamental state variable is the Health Status of each agent. Each agent is treated separately and is described by its age, gender, main localization (primary context — a household), second most important context type, and their localization.

An agent is assigned to a

Parameter space scan

Following the assumptions employed in this work, a two-dimensional scan of the α and f parameters was performed in order to determine feasible scenarios of epidemic evolution. It should be noted, that the same R0 values may be obtained with different sets of α and f parameters, see Table 7. In our research the R0 value varies from 1.44 to 2.42, which is a range comparable to other ranges studied in this field [12], [14], [15], [16].

In principle, R0 values are linearly related to α for each of

Concluding remarks

Most of the past studies conducted in the field of computational epidemiology were focused on mitigation strategies. This is also the final goal of our ongoing effort, however, in this work we present only the preliminary stage necessary to perform, among others, studies of mitigation strategies and their means. Previously we have reported [19] the methods and the results of constructing a large scale, individual based virtual society — trying to mimic local population densities, the

Acknowledgements

We are greatly thankful for anonymous reviewers of this article for very many invaluable suggestions and discussions, we would also like to thank Andrew Churchard for looking over the English. The authors were partially supported by the EU project SSPE-CT-2006-44405, also FR and MG were partially supported from the 352/6.PR-UE/ 2007/7 grant.

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