Elsevier

Physics of Life Reviews

Volume 18, September 2016, Pages 66-97
Physics of Life Reviews

Review
Mathematical models to characterize early epidemic growth: A review

https://doi.org/10.1016/j.plrev.2016.07.005Get rights and content

Highlights

  • We review recent progress on characterizing early epidemic growth patterns.

  • We survey mathematical approaches for modeling early epidemic growth.

  • The standard SIR model can incorporate flexible early epidemic growth profiles.

Abstract

There is a long tradition of using mathematical models to generate insights into the transmission dynamics of infectious diseases and assess the potential impact of different intervention strategies. The increasing use of mathematical models for epidemic forecasting has highlighted the importance of designing reliable models that capture the baseline transmission characteristics of specific pathogens and social contexts. More refined models are needed however, in particular to account for variation in the early growth dynamics of real epidemics and to gain a better understanding of the mechanisms at play. Here, we review recent progress on modeling and characterizing early epidemic growth patterns from infectious disease outbreak data, and survey the types of mathematical formulations that are most useful for capturing a diversity of early epidemic growth profiles, ranging from sub-exponential to exponential growth dynamics. Specifically, we review mathematical models that incorporate spatial details or realistic population mixing structures, including meta-population models, individual-based network models, and simple SIR-type models that incorporate the effects of reactive behavior changes or inhomogeneous mixing. In this process, we also analyze simulation data stemming from detailed large-scale agent-based models previously designed and calibrated to study how realistic social networks and disease transmission characteristics shape early epidemic growth patterns, general transmission dynamics, and control of international disease emergencies such as the 2009 A/H1N1 influenza pandemic and the 2014–2015 Ebola epidemic in West Africa.

Introduction

Over the last few decades, mathematical models of disease transmission have been helpful to gain insights into the transmission dynamics of infectious diseases and the potential role of different intervention strategies [1], [2], [3], [4]. The use of disease transmission models to generate short-term and long-term epidemic forecasts has increased with the rising number of emerging and re-emerging infectious disease outbreaks over the last decades. This has highlighted the need to examine the underlying assumptions behind models of disease spread and control as well as understand how these assumptions affect estimates of key epidemiological parameters and associated epidemic predictions. In order to generate epidemic forecasts that are useful for public health decision-making, there is a need to design reliable models that capture the baseline transmission characteristics for specific pathogens and social contexts.

Recent research has renewed interest in identifying signature features of epidemic growth patterns, especially in the first few disease generations, which could help improve our understanding of the transmission dynamics of infectious diseases and inform the design of models of disease spread [5]. Important model ingredients include realistic population structures and their associated contact networks, appropriate heterogeneity configurations in susceptibility and infectivity, as well as the possibility of early reactive behavior changes that blunt the transmission rate. In this article we review how different mathematical modeling approaches incorporating realistic spatial structures [5], [6], [7], reactive behavior changes or inhomogeneous mixing parameters can yield different epidemic growth profiles ranging from sub-exponential to exponential growth dynamics.

The goals of this review are twofold. First, we describe recent progress using primarily phenomenological models to quantify the early epidemic growth patterns from infectious disease outbreak data. Second, we provide a review of the major mathematical modeling approaches that are useful to capture early epidemic growth profiles. Because mixing within and among populations affect early patterns of epidemic spread in a major way, a focus of this review is on how modelers can incorporate realistic population mixing structures in models ranging from metapopulation models to individual-based network models. In this process, we also examine how realistic social networks and disease transmission characteristics can shape early epidemic growth patterns. To do so, we analyze simulation data derived from detailed large-scale spatial models previously used to study transmission dynamics and control of international disease emergencies such as the 2009 A/H1N1 influenza pandemic and the 2014–2015 Ebola epidemic in West Africa.

Section snippets

Description of early epidemic growth profiles using phenomenological models

In this section, we describe recent progress using primarily phenomenological models to characterize the early epidemic growth profile from infectious disease outbreak data. We also discuss how the presence of a diversity of early epidemic growth profiles has implications for epidemic forecasting and understanding the transmission potential of infectious diseases.

Mechanistic models representing epidemic growth profiles

In this section, we reflect on several mechanisms that have been put forward to explain the sub-exponential epidemic growth patterns evidenced from infectious disease outbreak data [5], [6], [7]. These include spatially constrained contact structures shaped by the epidemiological characteristics of the disease (i.e., airborne vs. close contact transmission model), the rapid onset of population behavior changes, and the potential role of individual heterogeneity in susceptibility and infectivity.

Discussion

Early epidemic forecasts consisting of the likely short-term trajectory of an unfolding outbreak can help guide the type and intensity of interventions including healthcare infrastructure needs for diagnosis, isolation of infectious individuals, and contact tracing activities [114]. However, our ability to generate disease forecasts using epidemic models during the initial epidemic phase is not only hindered by a lack of reliable epidemiological information and case incidence data, but also by

Funding

GC acknowledges financial support from the NSF grant 1414374 as part of the joint NSF–NIH–USDA Ecology and Evolution of Infectious Diseases program; UK Biotechnology and Biological Sciences Research Council grant BB/M008894/1, NSF–IIS RAPID award #1518939, and NSF grant 1318788 III: Small: Data Management for Real-Time Data Driven Epidemic simulation, and the Division of International Epidemiology and Population Studies, The Fogarty International Center, US National Institutes of Health. SB and

Conflicts of interest

Authors declare no conflict of interest related to this article.

Acknowledgements

We are thankful to Stefano Merler and Alex Vespignani for facilitating simulation data of the early epidemic growth dynamics generated by their agent-based model of Ebola transmission dynamics in Liberia [28] and Maria Kiskowski for providing the best model fit curve to the Ebola situation in Liberia as derived from the household-community Ebola transmission model described in [27]. We also gratefully acknowledge high-performance computing resources (Orion) provided by Research Solutions at

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