Modeling control strategies of respiratory pathogens

Abstract

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.

Figure 1

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

Figure 2

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 TTc). 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

Figure 3

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.

Figure 4

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 Figure 3 caption for further details.

Figure 5

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 Figure 3 caption for further details.

Figure 6

Intervention projections in terms of Reff. This figure presents the results in the lower panel of Figure 4 expressed in terms of effective reproductive number rather than the projected size of an outbreak. If Reff<1 outbreaks will die out, while if Reff>1, epidemics may ensue. Note that the shading indicates epidemic potential and coincides perfectly with the shading in Figure 4.

Figure 7

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.

Figure A1

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

Figure A2

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 http://www.statcan.ca/english/Pgdb/famil53e.htm

Figure A3

School size distribution for Vancouver. The data presented here are in the Vancouver School Board December 2002 Ready Reference, publicly available online from http://www.vsb.bc.ca/board/publications.htm

Figure A4

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.

Figure A5

Average size of small outbreaks (top) and the epidemic probability (bottom) for 5 different networks introduced in Figure A4.

Figure A6

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 (2,3). We assumed this distribution of per day transmission probabilities across the entire network in our analysis. The first peak corresponds to household contacts, while the second peak (with higher probability of transmission) corresponds to all contacts occurring in healthcare settings. Note that this distribution includes the per day probabilities of transmission for all possible contacts in the network, and thus has a smaller mean than the estimates reported for SARS in which the probability of transmission was estimated for particular settings (2,3).