ReviewMathematical models to characterize early epidemic growth: A review
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
References (115)
- et al.
A generalized-growth model to characterize the early ascending phase of infectious disease outbreaks
Epidemics
(2016) - et al.
Dynamics and control of Ebola virus transmission in Montserrado, Liberia: a mathematical modelling analysis
Lancet Infect Dis
(2014) - et al.
The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda
J Theor Biol
(2004) - et al.
Spatiotemporal spread of the 2014 outbreak of Ebola virus disease in Liberia and the effectiveness of non-pharmaceutical interventions: a computational modelling analysis
Lancet Infect Dis
(2015) - et al.
The role of spatial mixing in the spread of foot-and-mouth disease
Prev Vet Med
(2006) - et al.
Quarantine in a multi-species epidemic model with spatial dynamics
Math Biosci
(2007) - et al.
A structured epidemic model incorporating geographic mobility among regions
Math Biosci
(1995) - et al.
Seven challenges for metapopulation models of epidemics, including households models
Epidemics
(2015) - et al.
Monogamous networks and the spread of sexually transmitted diseases
Math Biosci
(2004) - et al.
Network theory and SARS: predicting outbreak diversity
J Theor Biol
(2005)
The implications of network structure for epidemic dynamics
Theor Popul Biol
Modelling disease spread through random and regular contacts in clustered populations
Theor Popul Biol
Measures of concurrency in networks and the spread of infectious disease
Math Biosci
Nine challenges in incorporating the dynamics of behaviour in infectious diseases models
Epidemics
Infectious diseases of humans
Modeling infectious disease dynamics in the complex landscape of global health
Science
The prevention of malaria
Contributions to the mathematical theory of epidemics, IV: analysis of experimental epidemics of the virus disease mouse ectromelia
J Hyg (Lond)
The Western Africa Ebola virus disease epidemic exhibits both global exponential and local polynomial growth rates
PLoS Curr
Polynomial epidemics and clustering in contact networks
Proc R Soc Lond B, Biol Sci
Spatiotemporal evolution of Ebola virus disease at sub-national level during the 2014 West Africa epidemic: model scrutiny and data meagreness
PLoS ONE
Transmission dynamics of HIV infection
Nature
Risk behavior-based model of the cubic growth of acquired immunodeficiency syndrome in the United States
Proc Natl Acad Sci USA
Can growth be faster than exponential, and just how slow is the logarithm?
Math Gaz
Using phenomenological models to characterize transmissibility and forecast patterns and final burden of Zika epidemics
PLOS Curr
Comparative estimation of the reproduction number for pandemic influenza from daily case notification data
J R Soc Interface
How generation intervals shape the relationship between growth rates and reproductive numbers
Proc Biol Sci
Estimating the future number of cases in the Ebola epidemic—Liberia and Sierra Leone, 2014–2015
MMWR, Surveill Summ
Ebola virus disease in West Africa—the first 9 months of the epidemic and forward projections
N Engl J Med
Assessing the international spreading risk associated with the 2014 West African Ebola outbreak
Estimating the reproduction number of Zaire Ebola virus (EBOV) during the 2014 outbreak in West Africa
PLOS Curr
Ebola cases and health system demand in Liberia
PLoS Biol
Transmission dynamics and final epidemic size of Ebola virus disease outbreaks with varying interventions
PLoS ONE
Time series modelling of childhood diseases: a dynamical systems approach
Appl Stat
Travelling waves and spatial hierarchies in measles epidemics
Nature
Modeling household and community transmission of Ebola virus disease: epidemic growth, spatial dynamics and insights for epidemic control
Virulence
Three-scale network model for the early growth dynamics of 2014 West Africa Ebola epidemic
PLOS Curr
Modelling vaccination strategies against foot-and-mouth disease
Nature
The foot-and-mouth epidemic in Great Britain: pattern of spread and impact of interventions
Science
Smallpox transmission and control: spatial dynamics in Great Britain
Proc Natl Acad Sci USA
Containing bioterrorist smallpox
Science
Modelling disease outbreaks in realistic urban social networks
Nature
Characterizing the reproduction number of epidemics with early sub-exponential growth dynamics
The abundance threshold for plague as a critical percolation phenomenon
Nature
Plague outbreaks in prairie dog populations explained by percolation thresholds of alternate host abundance
Proc Natl Acad Sci USA
Utility of R0 as a predictor of disease invasion in structured populations
J R Soc Interface
The mathematical theory of infectious disease and its applications
The hidden geometry of complex, network-driven contagion phenomena
Science
Large-scale spatial-transmission models of infectious disease
Science
Cited by (328)
Exploring the impact of social stress on the adaptive dynamics of COVID-19: Typing the behavior of naïve populations faced with epidemics
2024, Communications in Nonlinear Science and Numerical SimulationTransmission matrices used in epidemiologic modelling
2024, Infectious Disease ModellingAn agent-based model with antibody dynamics information in COVID-19 epidemic simulation
2023, Infectious Disease ModellingMathematical modeling for Delta and Omicron variant of SARS-CoV-2 transmission dynamics in Greece
2023, Infectious Disease ModellingThe use of networks in spatial and temporal computational models for outbreak spread in epidemiology: A systematic review
2023, Journal of Biomedical Informatics