Time Evolution of Disease Spread on Networks with Degree Heterogeneity
📝 Abstract
Two crucial elements facilitate the understanding and control of communicable disease spread within a social setting. These components are, the underlying contact structure among individuals that determines the pattern of disease transmission; and the evolution of this pattern over time. Mathematical models of infectious diseases, which are in principle analytically tractable, use two general approaches to incorporate these elements. The first approach, generally known as compartmental modeling, addresses the time evolution of disease spread at the expense of simplifying the pattern of transmission. On the other hand, the second approach uses network theory to incorporate detailed information pertaining to the underlying contact structure among individuals. However, while providing accurate estimates on the final size of outbreaks/epidemics, this approach, in its current formalism, disregards the progression of time during outbreaks. So far, the only alternative that enables the integration of both aspects of disease spread simultaneously has been to abandon the analytical approach and rely on computer simulations. We offer a new analytical framework based on percolation theory, which incorporates both the complexity of contact network structure and the time progression of disease spread. Furthermore, we demonstrate that this framework is equally effective on finite- and “infinite”-size networks. Application of this formalism is not limited to disease spread; it can be equally applied to similar percolation phenomena on networks in other areas in science and technology.
💡 Analysis
Two crucial elements facilitate the understanding and control of communicable disease spread within a social setting. These components are, the underlying contact structure among individuals that determines the pattern of disease transmission; and the evolution of this pattern over time. Mathematical models of infectious diseases, which are in principle analytically tractable, use two general approaches to incorporate these elements. The first approach, generally known as compartmental modeling, addresses the time evolution of disease spread at the expense of simplifying the pattern of transmission. On the other hand, the second approach uses network theory to incorporate detailed information pertaining to the underlying contact structure among individuals. However, while providing accurate estimates on the final size of outbreaks/epidemics, this approach, in its current formalism, disregards the progression of time during outbreaks. So far, the only alternative that enables the integration of both aspects of disease spread simultaneously has been to abandon the analytical approach and rely on computer simulations. We offer a new analytical framework based on percolation theory, which incorporates both the complexity of contact network structure and the time progression of disease spread. Furthermore, we demonstrate that this framework is equally effective on finite- and “infinite”-size networks. Application of this formalism is not limited to disease spread; it can be equally applied to similar percolation phenomena on networks in other areas in science and technology.
📄 Content
1 Time Evolution of Disease Spread on Networks with Degree Heterogeneity
Pierre-André Noël,1, 2 Bahman Davoudi,1 Louis J. Dubé,2, 3 Robert C. Brunham1 Babak Pourbohloul1, 2, 4,
1University of British Columbia Centre for Disease Control, Vancouver, British Columbia, Canada V5Z 4R4 2Département de Physique, de Génie Physique et d’Optique, Université Laval, Québec, Québec, Canada G1K 7P4 3Laboratoire de Chimie Physique-Matière et Rayonnement, Université Pierre et Marie Curie, 75231 Paris Cedex 05, France 4Department of Health Care & Epidemiology, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4
(Date: Oct 17, 2007)
*Corresponding Author:
Babak Pourbohoul, PhD Division of Mathematical Modeling UBC Centre for Disease Control 655 West 12th Avenue, Vancouver, BC Canada V5Z 4R4
2
Abstract
Two crucial elements facilitate the understanding and control of communicable disease spread within a social setting. These components are, the underlying contact structure among individuals that determines the pattern of disease transmission; and the evolution of this pattern over time. Mathematical models of infectious diseases, which are in principle analytically tractable, use two general approaches to incorporate these elements. The first approach, generally known as compartmental modeling, addresses the time evolution of disease spread at the expense of simplifying the pattern of transmission. On the other hand, the second approach uses network theory to incorporate detailed information pertaining to the underlying contact structure among individuals. However, while providing accurate estimates on the final size of outbreaks/epidemics, this approach, in its current formalism, disregards the progression of time during outbreaks. So far, the only alternative that enables the integration of both aspects of disease spread simultaneously has been to abandon the analytical approach and rely on computer simulations. Powerful modern computers can perform an enormous number of simulations at an incredibly rapid pace; however, the complex structure of “realistic” contact networks, along with the stochastic nature of disease spread, pose serious challenges to the computational techniques used to produce robust, real time analysis of disease spread in large populations. An analytical alternative to this approach is lacking. We offer a new analytical framework based on percolation theory, which incorporates both the complexity of contact network structure and the time progression of disease spread. Furthermore, we demonstrate that this framework is equally effective on finite- and “infinite”-size networks. Application of this formalism is not limited to disease spread; it can be equally applied to similar percolation phenomena on networks in other areas in science and technology.
3 The spread of communicable diseases is a dynamical process and as such, understanding and controlling infectious disease outbreaks and epidemics is pertinent to the temporal evolution of disease propagation. Historically, this aspect of disease transmission has been studied with the use of “coarse-grained” dynamical representation of populations, known as compartmental models.1, 2 In these models, a population is divided into a number of epidemiological “states” (or classes) and the time evolution of each is described by a differential equation. Figure 1a shows a schematic diagram of a simple Susceptible-Exposed-Infected-Removed (SEIR) model, in which every individual can be in the susceptible, exposed, infected or removed class at any given time. Although this approach, and its more complex variants, has been instrumental in understanding several features of infectious diseases over the past 3 decades, it comes with a major simplification. The simplifying assumption states that the population is “well mixed”, i.e., every individual has an equal opportunity to infect others. This assumption may be valid in the broader context of population biology. Human populations, however, tend to contact each other in a heterogeneous manner based on their age, profession, socio- economic status or behavior, and thus, the well-mixed approximation cannot portray an accurate image of disease spread among humans specifically in finite-size populations.3 Recent advances in network- and percolation- theories, have paved the way for physicists to bring a new perspective to understanding disease spread. Over the past decade, seminal work by Watts and Strogatz on small-world networks4, Barabasi et al. on scale-free networks5 and Dorogovtsev, Mendes6, Pastor-Satorras and Vespignani7 among others on the dynamics of networks has shed light on a number of intriguing aspects of epidemiological processes. In particular, groundbreaking work by Newman et al.8, 9, 10 has provided a strong foundation for the formulation of epidemiologi
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