Contact network epidemiology can provide quantitative input even before pathogen is fully characterized.

Effectively controlling infectious diseases requires quantitative comparisons of quarantine, infection control precautions, case identification and isolation, and immunization interventions. We used contact network epidemiology to predict the effect of various control policies for a mildly contagious disease, such as severe acute respiratory syndrome, and a moderately contagious disease, such as smallpox. The success of an intervention depends on the transmissibility of the disease and the contact pattern between persons within a community. The model predicts that use of face masks and general vaccination will only moderately affect the spread of mildly contagious diseases. In contrast, quarantine and ring vaccination can prevent the spread of a wide spectrum of diseases. Contact network epidemiology can provide valuable quantitative input to public health decisionmaking, even before a pathogen is well characterized.

Public concern regarding emerging infectious diseases is on the rise. The 21st century began with the emergence or reemergence of zoonotic diseases like severe acute respiratory syndrome (SARS) (

In response to this problem, we have found that mathematical models of disease transmission can be used to evaluate and optimize control strategies. Such quantitative predictions can be empirically tested through randomized comparative trials, and mathematical models increasingly contribute to public health decisions regarding policy and intervention (

We use contact network epidemiology to compare intervention strategies for airborne ^{2}

Contact network models capture and estimate interpersonal contacts that lead to disease transmission within a community (

Contact network epidemiology allows us to assess the vulnerability of a population to an infectious disease on the basis of the structure of the network (its degree distribution) and on the average transmissibility (

We built an urban contact network model with 2,000 households with an average household size of 2.6 (5,154 persons) based on demographic information for the Greater Vancouver Regional District, British Columbia, Canada. We used publicly available data from sources such as Statistics Canada to estimate the distribution of ages, household sizes, school and classroom sizes, hospital occupancy, workplaces, and public spaces (

Most of the edges in the network are undirected, meaning that transmission may occur in either direction (black edges in

Schematic diagram of a directed network. Each black vertex represents a member of the general population; gray vertexes represent healthcare workers.

In an urban setting, not all encounters are equally likely to lead to disease transmission. We capture this heterogeneity in 2 ways. First, in the simulated urban network, the probability of a contact between 2 persons depends on the location and nature of their overlapping daily activities. For example, persons in the same household are connected to each other with probability 1, while persons who encounter each other in a public space are connected to each other with a probability from 0.003 to 0.300. Second, after these connections are determined, we assign a distinct transmissibility, _{ij}

In any given network exists a critical transmissibility value, _{c}_{c}_{c}_{c}_{c}

The epidemic potential of disease is commonly estimated by using the basic reproductive number _{0}, the number of secondary infections arising from a single infection in a relatively naïve population (_{0} = _{c}_{0} = 1. Public health interventions aim to reduce the number of new infected cases, ideally decreasing the effective reproductive number of the disease below the epidemic threshold, _{eff}

The difference between average transmissibility _{0} is important. While both have threshold values that distinguish epidemic from nonepidemic scenarios (_{0} = 1 and _{c}_{0} depends on all of these factors, particularly on the numbers of interactions within the community. For example, consider a single airborne pathogen spreading through a hospital, where abundant close contacts exist, and through a rural community, where close contacts are rare. The per contact probabilities of transmission (_{ij}_{0} will be substantially higher in the hospital than in the rural setting.

The heterogeneous spread of SARS worldwide suggested context-dependent patterns of transmission with relatively rapid spread through hospitals and relatively slow spread through communities (_{0}. We take this approach to evaluating disease control strategies in an urban setting (

A primary public health goal is to bring disease from a value above an epidemic threshold to a value below the threshold, thereby eliminating the threat of a large-scale epidemic. This goal can be achieved through interventions that directly affect the transmissibility of the pathogen (_{c}

Transmission- vs. contact-reduction intervention. A) Transmission-reduction intervention: solid curves show the average size of an outbreak (left panel) and the probability of a large-scale epidemic (right panel). The horizontal axes cover the spectrum of disease transmissibility (from 0 to 1) such that a single disease is associated with a unique value on either the left curve (if T<Tc) or the right curve (if T>Tc). The epidemic threshold Tc separates the 2 zones. For better visualization, we chose 2 different scales for horizontal axes of the 2 panels. Consider a disease with T = 0.245 (top black circle). A transmission-reduction intervention causes the black circle to slide on a new position on the curve. A successful intervention is the one that lowers T to a value <Tc. B) Contact-reduction intervention: solid curves in the top panel show the epidemiologic vulnerability of the original network. Contact-reduction interventions alter the structure of the contact network and shift the epidemic curves to the right (solid curves in bottom panel). The 2 dashed vertical lines show the critical transmissibility threshold for the old (left) and new (right) networks. Consider the disease denoted by the black circle: the contact-reduction intervention raised the epidemic threshold above transmissibility of the disease and thereby eliminated the possibility of an epidemic.

In our simulated urban contact network, the critical transmissibility threshold is _{c}_{0} = 5 for this contact network and thus corresponds to a moderately infectious disease like smallpox (_{c}_{c}

The second strategy involves modifying the contact network itself. Interventions such as quarantine and closing schools and other public places effectively eliminate potential contacts (edges) between persons. Interventions such as immunization and the prophylactic use of antibacterial or antiviral drugs are tantamount to removing persons (vertexes) from the contact network and therefore also alter the network structure. We mathematically assess the effect of such strategies by deleting edges and vertexes from the contact network and predicting the new probability of an epidemic and expected distribution of cases within the community.

We evaluated a variety of commonly implemented public health interventions by changing the contact patterns within the network, transmissibility of the disease, or both. For each strategy, we calculated several epidemiologic quantities: 1) the epidemic threshold, _{c}_{c}_{c}_{prob}_{c}

Comparing the effect of face masks for the general public and healthcare workers (HCWs). Mask efficiency is the percent reduction in transmissibility to or from a person correctly using a mask. Compliance is the fraction of the population adopting the intervention. Results are for a mildly contagious disease with a transmissibility T = 0.075 and a moderately contagious disease with a transmissibility T = 0.245. The equivalent basic reproductive number for these diseases are R0 = 1.545 and R0 = 5.047, respectively. Without intervention, both of these diseases have T above the epidemic threshold for the community (Tc = 0.048) and thus may ignite a large-scale epidemic. The probabilities that such epidemics will occur (without intervention) are Sprob = 0.50 and Sprob = 0.97, respectively. Some interventions may not bring T below the epidemic threshold and thus only reduce the probability of an epidemic (gray boxes), while others succeed in containing transmission to a small outbreak (white boxes). Gray boxes give the probability of an epidemic, and white boxes give the expected size of an outbreak. Outbreak size may not be an integer since s is an average taken from all possible outbreaks in the community.

Comparing the effect of isolation and quarantine. Isolation alone reduces the infectious period by a specified percentage. Quarantine involves both isolation and sequestering a fraction of all case contacts. See the

Comparing general vaccination and ring vaccination strategies. General vaccination protects a percentage of persons chosen randomly from the population with an efficacy determined by the vaccine itself. Ring vaccination involves isolating the patient (and the associated reduction in the infectious period) followed by targeted vaccination of contacts. The degree to which contacts are successfully protected depends on the success of contact tracing and the efficacy of the vaccine. See the

Although general use of face masks may have a moderate effect, its success hinges on correct use and level of compliance. For instance, face masks that are 75% effective will only prevent a large-scale epidemic of a SARS-like disease if ≥60% of the general population complies perfectly (

One of the factors that influences the transmissibility

Contacts between infected and susceptible persons can be eliminated during an outbreak through measures such as quarantine, closing public venues, and ring vaccination, or they can be eliminated preventatively through general vaccination strategies.

In _{eff}_{eff}_{eff}_{0}) is not a universal constant but instead critically depends on structure of the host community.

Intervention projections in terms of Reff. This figure presents the results in the lower panel of

A general vaccination strategy is one in which a substantial proportion of the population is vaccinated at random. The success of this measure depends on proportion (coverage), vaccine efficacy, and disease transmissibility. The availability of a vaccine, therefore, does not guarantee prevention unless both delivery and vaccine-induced immunity are sufficient. For example,

Ring vaccination of close contacts, on the other hand, is a very effective approach overall. This intervention, like quarantine, involves both transmission and contact reduction. Identifying the index patient results in a reduced infectious period. Subsequent identification and protection of his or her contacts through vaccination further limits the potential spread of the pathogen.

The white entries in

Left panel: variation of outbreak sizes as a function of transmissibility. We generated 1,000 epidemics for each of 20 values of T from 0 to the epidemic threshold. The solid curve represents the mean of outbreak size (m), the dashed curve represents 1 standard deviation above the mean (m + s), and the dotted line at the bottom shows the minimum size of an outbreak, which is always equal to 1, meaning that after the introduction of the first infected case the disease did not spread further. Right panel: sensitivity of epidemic probability to network stochasticity. We generated 100 different networks, each with 2,000 households. Because of the stochastic nature of contact formation during network generation, these 100 networks contain different numbers and configurations of edges and therefore have different degree distributions. The solid curve shows the mean probability of an epidemic across the 100 networks for transmissibilities above the epidemic threshold, and the dashed curves are 95% confidence limits for the mean probability of an epidemic.

Our mathematical predictions are based on a single simulated urban network with 2,000 households with an average of 2.6 people per household. To address the sensitivity of the predictions to the particular pattern of contacts in the network, we stochastically generated 100 urban networks of equal size and predicted the probability of an epidemic for the range of _{c}

Using contact network epidemiology, we evaluated various airborne infection control policies for a simulated urban setting like Vancouver. This approach explicitly captures the heterogeneous patterns of interpersonal contacts that lead to disease transmission and allows rapid mathematical prediction of the probability and distribution of an epidemic. This analysis does not depend on computationally intensive simulations. Furthermore, the approach allows one to quantitatively compare strategies that directly reduce the transmissibility of a pathogen or limit opportunities for a pathogen to spread. Although each strategy has been considered on its own, these methods can easily predict the effect of combined interventions for an entire spectrum of airborne infectious diseases, including SARS, smallpox, influenza, and meningococcal meningitis, among others.

Although the qualitative results of this analysis are applied to urban settings, the work is meant to be a proof of concept rather than to provide specific quantitative recommendations for urban control of communicable diseases such as SARS and smallpox. Until we have developed contact network models for a wide range of communities and assessed their generality, contact network epidemiology will need to be applied on a case-by-case basis. For example, hospitals can use these methods to improve control of nosocomial airborne infections. To start, each facility should model its particular network of patient–healthcare worker interactions, then calculate the effect of measures such as respiratory droplet precautions, grouping patients in cohorts, modifications to healthcare worker assignments, and vaccination (

The success of contact network epidemiology depends not only on realistic models of contact patterns but also on reliable estimates of the average transmissibility of the pathogen, _{0}, based on the doubling time of case counts in the early phase of an outbreak or epidemic. The value of _{0}, however, may vary substantially, depending on the population in which it is measured. For example, recent estimates of _{0} for SARS ranged from 1.2 to 3.6 (_{0}. For each case, one must measure not just the number of secondary cases, but also the total number of contacts of the case-patient during the infectious period and then divide the first value by the second.

Just as enormous molecular and technological resources are often mobilized to develop vaccines and diagnostic tools for emerging infectious diseases of public health importance, we should also harness the powerful quantitative mathematical tools that help assess disease interventions. When an airborne pathogen strikes, public health officials should be able to make scientifically grounded decisions about the competing medical, economic, and social implications following deployment of control measures. We illustrate that contact network epidemiology can provide detailed and valuable insight into the fate and control of an outbreak. Integrating these tools into public health decision making should facilitate more rational strategies to manage emerging diseases, bioterrorist events, and pandemic influenza in situations in which empiric data are not yet available to guide decision making.

The simulated urban center network that we used in our analysis is based on the demographic data for the Greater Vancouver Regional District (GVRD), British Columbia, Canada, with a population of 2 million people. The data that we use in our simulations are publicly available at Web sites for Statistics Canada (

The number and sizes of public places (schools, hospitals, shopping malls, workplaces, and generic public spaces) are largely based on publicly available statistics for Vancouver. For example, we assign students to schools on the basis of the distribution of elementary and secondary school sizes in Vancouver, which range from 100 to 2,100 students (

To make contacts between persons within these spaces, we typically assume a specified probability that 2 persons in the same place are in contact with each other. This probability ranges from 1 in the case of persons within the same household to very low values for persons who visit the same shopping center. This results in Poisson distributions of contact rates within specific settings.

Each household is a completely connected small network; that is, every person in a household is connected to every other person in that household. Since infection transmission occurs with probability

For the purposes of this analysis, we simulated an urban network with 2,000 households (≈5,200 persons). We used this relatively small network because it permitted extensive epidemic simulation as a means to verify our analytical predictions. Indeed, we found that the simulations agree well with the analysis (

The transmissibility value assumed for a mildly contagious disease (_{ij}_{ij}^{τ}_{i}_{ij}_{ij}_{ij}_{0} ≈ 5. For this case, we assumed that all contacts in the graph shared this average transmissibility.

For both diseases, we model intervention strategies by decreasing the duration of infectiousness, the probability of transmission for specific contacts in the network, or both. We then calculate the average transmissibility

In a mixed undirected-directed network (henceforth semidirected network), each vertex (person) has an undirected degree representing the number of undirected edges joining the vertex to other vertexes as well as both an in-degree and an out-degree representing the number of directed edges coming from other persons and going to other persons, respectively. The undirected degree and in-degree indicate how many contacts can spread disease to the person and thus are related to the likelihood that a person will become infected during an epidemic; the undirected degree and out-degree indicate how many contacts may be infected by that person should he become infected; thus, they are related to the likelihood that a person will ignite an epidemic.

Given the degree distribution of the contact network, one can analytically predict the fate of an outbreak. Let _{jkm}

In the main text, we report the following quantities: the basic reproductive number _{0}, the epidemic threshold _{c}_{prob}

The basic reproductive number can be written as

The expression for the transmissibility threshold value is

where

and therefore

The average size of an outbreak is given by

and the probability of an epidemic is given by the following expression:

where

and

We use numerical root finding methods to solve for

For a completely undirected network equation (1) can be simplified as

where _{k}^{2} = _{0} = _{0} =

These authors contributed equally to this work.

For the purposes of this manuscript, "airborne" refers to respiratory pathogens that are spread through respiratory secretions and can be either airborne, such as tuberculosis, or dropletborne, such as SARS.

We thank Mark Newman for technical guidance.

This work was supported in part by grants from the Canadian Institutes of Health Research (CIHR) (FRN: 67803) to the British Columbia SARS Investigators Collaborative that includes B.P., L.A.M., D.M.S., M.K., D.M.P. and R.C.B. and from the National Science Foundation (DEB-0303636) to L.A.M. The Santa Fe Institute and CIHR supported the working visits of B.P. and L.A.M.

Schematic diagram of an urban network. We used the demographic data for the Greater Vancouver Regional District to build the contact network.

Household size distribution in Vancouver, Toronto, and other Canadian metropolitan areas. The data presented here are publicly available online at the Statistics Canada Web site from

School size distribution for Vancouver. The data presented here are in the Vancouver School Board December 2002 Ready Reference, publicly available online from

The cumulative undirected-degree distributions for urban networks with 1,000, 2,000, 5,000, 10,000, and 20,000 households corresponding to population sizes 2,595, 5,337, 13,080, 25,722, and 51,590 persons.

Average size of small outbreaks (top) and the epidemic probability (bottom) for 5 different networks introduced in

Transmission probability distribution. The probability of transmission of respiratory pathogens depends on the amount of shedding, distance, duration of contact, and environmental factors such as temperature and humidity. Reports on SARS epidemiology suggest a bimodal distribution of transmission probabilities: close contacts in hospitals may have had high probabilities of transmission while typical contacts in schools, workplaces, and shopping malls may have had low probabilities of transmission (

Dr. Pourbohloul is the director of the Division of Mathematical Modeling at the University of British Columbia Centre for Disease Control. He is leading a research group, funded by the CIHR, in the application of network theory to the prediction and control of SARS and other respiratory infections.