Elsevier

Mathematical Biosciences

Volume 203, Issue 2, October 2006, Pages 301-318
Mathematical Biosciences

Semi-empirical power-law scaling of new infection rate to model epidemic dynamics with inhomogeneous mixing

https://doi.org/10.1016/j.mbs.2006.01.007Get rights and content

Abstract

The expected number of new infections per day per infectious person during an epidemic has been found to exhibit power-law scaling with respect to the susceptible fraction of the population. This is in contrast to the linear scaling assumed in traditional epidemiologic modeling. Based on simulated epidemic dynamics in synthetic populations representing Los Angeles, Chicago, and Portland, we find city-dependent scaling exponents in the range of 1.7–2.06. This scaling arises from variations in the strength, duration, and number of contacts per person. Implementation of power-law scaling of the new infection rate is quite simple for SIR, SEIR, and histogram-based epidemic models. Treatment of the effects of the social contact structure through this power-law formulation leads to significantly lower predictions of final epidemic size than the traditional linear formulation.

Introduction

In traditional modeling of disease epidemic dynamics, the expected number of new infections per day per infectious person is taken to be proportional to the fraction of the population that is susceptible. This linear formulation implements the homogeneous-mixing assumption, by which each susceptible person is equally likely to become the next victim. In real social structures, however, some people have a greater chance to receive and transmit disease than others. In an epidemic, highly connected people tend to be infected earlier than less-connected people. As an epidemic progresses, not only are there fewer susceptible people, but those that remain tend to have fewer social contacts. Recent literature suggests a power law formulation for the degree distribution of the social contact structure. We have found that epidemic models can practically incorporate inhomogeneous mixing by taking the number of new infections per day per infectious person to scale as a power (greater than one) of the fraction of the population that is susceptible.

A large-scale simulation, in which the social contact structure of a large urban population is implemented with unprecedented fidelity, enables computation of epidemic dynamics with a realistic social contact structure. Synthetic populations have been constructed for three major US metropolitan areas, and epidemics of avian-related influenza have been simulated on these populations. From these simulations, the number of new infections per day per infectious person is found to scale not linearly with the fraction of the population that is susceptible, but as a power law. High-fidelity simulation can be used to determine the scaling exponent semi-empirically.

The power-law mixing formulation enables traditional epidemic models to capture the effects of social contact structures on epidemic dynamics in an empirical way. This power-law formulation gives significantly different epidemic dynamics than the traditional linear scaling. In particular, for a given basic reproductive number, power-law scaling predicts much less severe epidemics than linear scaling model. This semi-empirical power-law formulation can easily be implemented into existing epidemic models to account for the social contact structure within populations.

From the outset of modern epidemic modeling, the number of new infections per unit time per infectious person has been modeled as proportional to the fraction of the population that remains susceptible [1], [2], [3]. This linear scaling is a consequence of the homogenous mixing assumption, wherein every susceptible individual in the population is assumed to have an equal chance of becoming the next victim. Under the homogeneous mixing assumption, the number of new infections per unit time at time t, q(t), is given byq=αIS/P,where S(t) is the number of susceptible persons, I(t) is the number of infectious persons, P is the number of people in the initial population, and the coefficient α is the average number of disease transmissions per day per infectious person that would be expected if the entire population were susceptible.

The homogeneous mixing formulation of the new infection rate is used in traditional few-component stock-flow models (e.g., susceptible-infectious (SI), susceptible-infectious-susceptible (SIS), susceptible-infectious-removed (SIR), and susceptible-incubating-infectious-removed (SEIR) formulations), as well as in stock-flow models employing many components to represent disease states, treatment status, and victim behaviors [4], [5]. It is also used in histogram-based time-binned disease progression models [6]. Traditional stochastic epidemic models also incorporate the homogeneous mixing assumption [7], making use of the probabilistic interpretation of Eq. (1), in which q dt is the probability that a new infection occurs in dt about t.

A generalized infection rate of the formq=αI(S/P)νwith scaling power ν greater than one, provides a semi-empirical formulation in which the homogeneous mixing assumption is relaxed. The actual functional dependence of the new infection rate on the susceptible fraction will obviously be more complicated than Eq. (2). Nevertheless, the use of Eq. (2), with the exponent determined by high-fidelity simulation or historic epidemiology in actual cities, can give much better epidemic dynamic modeling than the homogeneous mixing model. Herd immunity can be incorporated in a straight-forward way:q=αI(S0/P)(S/S0)ν,where S0 is the initial number of susceptible individuals, but for simplicity, the following analysis will assume fully susceptible populations, i.e., S0 = P.

The power-law scaling of Eq. (2) has appeared in the literature [8], but the exponent ν was restricted to a value less than 1; the exponent was written as 1-b, and b was called the safety-in-numbers-power. The formulation was unrelated to social contact structure, nor was data used to place a value on the exponent. The power-law formulation of Eq. (2) has been used in an SIR model [9], but the only result reported was that with an exponent of ν = 2, an epidemic threshold occurs at ατI = 1, where τI is the average duration of the infectious period.

The coefficient α is related to the basic reproductive number, or can be obtained in alternative ways. The reproductive number, R0, is the mean number of disease transmissions per index case that would be expected if the population was entirely susceptible [3]. If the disease is uniformly infectious over an infectious period lasting τI days, the coefficient would beα=R0/τI.For many diseases, infectiousness depends on how long a person has been infectious. Defining i(τ) as the mean number of new cases per day transmitted by a person who has been infectious for τ days (assuming a completely susceptible population), the coefficient would be the average infectiousness of all infectious persons:α=0τIdτI(t,τ)i(τ)/I(t),where I(t) is the number of infectious people at time t; I(t, τ) dτ is the number of people that have been infectious for time in dτ about τ at time t, and τI is the maximum infectious period. α will depend weakly on how fast the epidemic is growing or decreasing, which in turn depends on time, or on the susceptible fraction. This dependence is eliminated here by examining a disease model in which infectiousness is constant over the infectious period so that Eq. (4) can be used. Alternatively, α could be formulated as a product of the number of contacts per person, times the transmission probability per day per contact [10].

The epidemic size, F, also known as the attack rate, is defined as the total number of people infected during the epidemic divided by the initial population:F=0dtq(t)/P.For homogeneous mixing, with the number of new cases per day given by Eq. (1), the epidemic size satisfies the well-known relation [2]FH=1-e-R0FH.This result for epidemic size is not affected by the presence or absence of an incubation stage, or by the distribution of infectious stage durations. For R0 > 1, F takes a non-zero value and an epidemic is said to occur.

For the number of new cases per day given by the power-law formulation of Eq. (2), a derivation along the lines of that leading to Eq. (7) obtains the new result that the epidemic size satisfies the relation:FI=1-[1+(ν-1)R0FI]-1/(ν-1).For ν = 2, the epidemic size is simply Fν=2 = 1  1/R0.

Eqs. (7), (8) are relations that define the epidemic size as an implicit function of R0, which can be evaluated with a few Newton–Raphson iterations. Fig. 1 shows the epidemic size as a function of the basic reproductive number R0, for homogeneous mixing and for the power-law formulation with scaling exponents of 1.8, 2.0, and 2.2. For a given R0, the power-law formulation leads to a smaller epidemic size than the traditional epidemic formulation would predict. For a scaling power of ν = 2, the inhomogeneous-mixing model predicts a 50% epidemic size for R0 = 2, while the homogeneous-mixing model predicts 80% epidemic size for the same reproductive number. The homogeneous-mixing model would obtain the same 50% epidemic size with a much less contagious disease (i.e., with R0 = 1.38).

For small epidemics, the epidemic size found by expansion of Eq. (7) or Eq. (8) is F  2(R0  1)/ν, where ν = 1 gives the linear homogeneous-mixing formulation. For small epidemics, infecting under a few percent of the population, the power-law formulation predicts an epidemic size of about half (i.e., 1/ν) as large as that predicted by the linear model, for the same R0.

Several factors contribute to inhomogeneous mixing whereby some susceptible individuals are more likely than others to become the next victim. The susceptibility (i.e., the likelihood of becoming infected, after receiving a specified dose) can vary from individual to individual. Highly susceptible individuals tend to become infected earlier, leaving the remaining population to be composed of less and less susceptible persons as the epidemic progresses. Although variation in susceptibility is usually modeled with a log-normal distribution, an analysis of epidemic dynamics with an assumed power-law distribution of susceptibility has been conducted [11], but global results for new case rate are not given.

Another potential contributor is the spatial distribution of the infectious source. An analysis of epidemic dynamics with a fractal spatial distribution of the infectious source [12] did not report the scaling of new infections per day per infectious person with respect to the susceptible fraction of the population.

A third mechanism contributing to inhomogeneous mixing is that each individual belongs to a finite number of mixing groups, each of finite size. People in infected mixing groups are much more likely to become the next victim than those not belonging to infected groups. One approach to implementing mixing-group structure into an epidemic dynamics model is to compute disease transmission among hundreds or thousands of individuals, each having been assigned to several pre-constructed mixing groups (representing household, school, work, hospitals, casual contacts, etc.) [13], [14]. Each mixing group allows a specified number of persons to interact, and is characterized by a transmission probability per day per fellow-mixing-group-member ranging from high for households, to moderate for schools, to low for neighborhoods. While such an approach could generate the number of new cases per infectious person per day as a function of the remaining susceptible fraction of the population, that result has not been reported. A result that was reported [13] was that for an R0 value of 3.2 (characterizing smallpox in a fully susceptible population), the expected value of the epidemic size computed for populations of 2000 with a household-school-neighborhood social connection structure is 63%. This epidemic size is much smaller than the 95.2% that would be predicted with the homogeneous-mixing model, but agrees with the corresponding result for a power-law formulation with ν = 2.285.

In a cellular automata approach to modeling local mixing within a spatial framework, hundreds or thousands of individuals are placed on a 2D lattice, and transmission can occur only between neighboring individuals [15]. In the absence of long-range connections, outbreaks are highly localized on the 2D grid. Each individual belongs to a mixing group consisting of themselves and eight neighbors. The number of new cases per day per infectious person has not been reported from 2D cellular automata models.

A fourth mechanism contributing to inhomogeneous mixing is that different individuals have differing numbers of social contacts. Recent literature suggests a power law degree distribution, in which the number of people having k contacts is proportional to kγ, with γ in the range of 2–3 [16], [17], [18], [19]. A minimum number of connections, and an exponential roll-off factor is generally imposed to avoid divergences associated with high and low values of k. Systems of coupled epidemic dynamic equations have been formulated wherein groups consisting of all individuals with k contacts are described by separate dynamic equations [20], [21], [22], [23]. These models assume that disease is transmitted and received in proportion to the number of contacts that a person has. Results have been obtained to relate the epidemic size to the contact distribution parameters and the transmission probability per connection. Such analyses have not derived a formulation like Eq. (2), but make such a formulation plausible. A large scale simulation with 7 290 000 individuals connected with a power-law distribution of contacts per person has been used to simulate the dynamics of SARS-like epidemics [24], but the susceptible fraction in the reported runs never drops significantly below unity.

Bond-percolation analysis (in which individuals are connected by a contact graph, each link has a probability of being active, and statistics are evaluated on the sizes of the resulting subgraphs) have been conducted with a power-law distribution of number of contacts [25], [26]. The percolation approach does not give epidemic dynamics, but it does obtain epidemic thresholds and the epidemic size as a function of the average transmission probability on a contact. Although the reported results were not directly comparable with the results of this study, further bond-percolation analysis may corroborate the scaling of epidemic size with the power-law exponent.

One more mechanism contributing to inhomogeneous mixing is the variation in the proximity and duration of contacts, and the variation in the ventilation in the room where the contact occurs. These mechanisms have not been analyzed in the literature.

Section snippets

EpiSims

EpiSims is an epidemic simulation tool that explicitly represents every person in a city, and every place within the city where people interact [19], [27]. Each person in the simulation is created according to actual demographic distributions drawn from census and other data, so that the synthetic population has the correct demographics, e.g., age distribution, household statistics, residential population density, etc.

A city is represented physically by a set of locations. The city’s road

Inhomogeneous mixing in SIR epidemic models

The semi-empirical treatment of inhomogeneous mixing can be implemented into the traditional SIR epidemic model:dS/dt=-q;dI/dt=q-I/τ;q=(R0/τ)I(S/P)ν.The epidemic curve obtained by numerical integration of the SIR equations is shown in Fig. 5 for parameters matching the EpiSims simulation of the baseline influenza epidemic in Los Angeles (P = 16 106 525, I(0) = 202, τ = 4.1 days, R0 = 1.34). The SIR model epidemic curves are given for homogeneous mixing with ν = 1, and power-law scaling with v = 2.06. The

Severe epidemics

The epidemics described so far are moderate, in the sense that if an index case infects an average of 1.34 persons total, his contact groups will not all become sick within a generation or two. For severe epidemics, in which each index case might infect an average of four or more persons, some of the contact groups (such as his household or workplace) may become completely infected, or nearly so. The secondary cases will then have fewer susceptible contacts, and the average new cases per day

Conclusions

The high-fidelity simulation EpiSims has been used to observe epidemic dynamics on social contact structures that statistically match actual populations of large metropolitan areas, in terms of where and with whom people live, work, play, drive, shop, and attend school. These contact structures emerge from millions of synthetic individuals engaging in realistic daily schedules, sometimes occupying locations simultaneously with other people, thus creating potential for disease transmission. For

References (32)

  • S. Del Valle et al.

    Effects of behavioral changes in a smallpox attack model

    Math. Biosci.

    (2005)
  • N.C. Severo

    Generalizations of some stochastic epidemic models

    Math. Biosci.

    (1969)
  • M.I. Meltzer

    The potential use of fractals in epidemiology

    Prev. Vet. Med.

    (1991)
  • W.O. Kermack et al.

    A contribution to the mathematical theory of epidemics

    Proc. Royal Soc. London, Series A

    (1927)
  • N.T.J. Bailey

    The Mathematical Theory of Infectious Diseases and its Applications

    (1975)
  • R. Anderson et al.

    Infectious Diseases of Humans: Dynamics and Control

    (1991)
  • E.H. Kaplan et al.

    Emergency response to a smallpox attack: the case for mass vaccination

    Proc. Natl. Acad. Sci.

    (2002)
  • M.I. Meltzer et al.

    Modeling potential responses to smallpox as a bioterrorist weapon

    Emerg. Infect. Dis.

    (2001)
  • S.A. Bozzette et al.

    A model for a smallpox-vaccination policy

    New Engl. J. Med.

    (2003)
  • V.J. Haas et al.

    Temporal duration and event size distribution at the epidemic threshold

    J. Bio. Phys.

    (1999)
  • M. Kretzschmar et al.

    Ring vaccination and smallpox control

    Emerg. Infect. Dis.

    (2004)
  • P. Sabatier et al.

    Fractals and epidemic process

    Int. J. Comput. Anticipat. Syst.

    (1998)
  • M.E. Halloran et al.

    Containing bioterrorist smallpox

    Science

    (2002)
  • J. Epstein, D. Cummings, S. Chakravarti, R. Singa, D. Burke, Toward a containment strategy for smallpox bioterror: an...
  • B. Eidelson et al.

    VIR-POX: An agent-based analysis of smallpox preparedness and response policy

    J. Artif. Soc. Social Simulat.

    (2004)
  • S.H. Strogatz

    Exploring complex networks

    Nature

    (2001)
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