Conceived and designed the experiments: DLC MEH IMLJ. Performed the experiments: DLC. Analyzed the data: DLC. Contributed reagents/materials/analysis tools: DLC VJO. Wrote the paper: DLC MEH VJO IMLJ.
The authors have declared that no competing interests exist.
Mathematical and computer models of epidemics have contributed to our understanding of the spread of infectious disease and the measures needed to contain or mitigate them. To help prepare for future influenza seasonal epidemics or pandemics, we developed a new stochastic model of the spread of influenza across a large population. Individuals in this model have realistic social contact networks, and transmission and infections are based on the current state of knowledge of the natural history of influenza. The model has been calibrated so that outcomes are consistent with the 1957/1958 Asian A(H2N2) and 2009 pandemic A(H1N1) influenza viruses. We present examples of how this model can be used to study the dynamics of influenza epidemics in the United States and simulate how to mitigate or delay them using pharmaceutical interventions and social distancing measures. Computer simulation models play an essential role in informing public policy and evaluating pandemic preparedness plans. We have made the source code of this model publicly available to encourage its use and further development.
Computer simulations can provide valuable information to communities preparing for epidemics. These simulations can be used to investigate the effectiveness of various intervention strategies in reducing or delaying the peak of an epidemic. We have made a detailed influenza epidemic simulator for the United States publicly available so that others may use the software to inform public policy or adapt it to suit their needs.
Mathematical and computer models of epidemics have contributed to our understanding of the spread of infectious disease and the measures needed to contain or mitigate them
We have released the source code for a new stochastic model of influenza epidemics, FluTE. FluTE is an individual-based model capable of simulating the spread of influenza across major metropolitan areas or the continental United States. The model's structure is based on previously published work
FluTE is an individual-based simulation model of influenza epidemics. In this section, we describe the model's community structure, natural history of influenza, and simulated interventions. Briefly, all individuals in the model are members of social mixing groups, within which influenza is transmitted by random mixing. The model can simulate several intervention strategies, and these can either change the transmission characteristics of influenza (e.g., vaccination) or change the contact probabilities between individuals (e.g., social distancing). Interventions can occur before the epidemic or in response to an ongoing epidemic.
The simulation creates synthetic populations based on typical American communities. The population is divided into census tracts, and each tract is subdivided into communities of 500–3000 individuals based on earlier models
33% | single adult |
34% | two people (two adults or a parent and child) |
13% | two adults, one child |
10% | two adults, two children |
7% | two adults, three children |
2% | two adults, four children |
1% | two adults, five children |
Data from
Most working-age adults (about 72% of 19–64 year-olds) are employed. Employment rates are determined on a tract-by-tract basis using data from the US Census 2000's Summary File 3, table PCT35. Employed individuals often work outside of their home communities. Each employed individual is assigned to work in a destination census tract based on commuting data taken from Part 3 of the Census Transportation Planning Package (
Individuals can engage in short-term, long-distance domestic travel to represent vacations and other trips. Travel in our model is based on the implementation in
New infected individuals are introduced to a simulation by infecting randomly selected people. This epidemic seeding process can occur once at the beginning of a simulation or daily. In addition, one can simulate an epidemic that is seeded from international travelers. In this scenario, randomly selected individuals in the counties with one of the United States' 15 busiest international airports are infected each day, proportional to the daily traffic of these airports (see
Airport | City | Passengers/year |
JFK | New York, NY | 21,842,544 |
LAX | Los Angeles, CA | 17,019,166 |
MIA | Miami, FL | 15,509,279 |
ORD | Chicago, IL | 11375367 |
EWR | Newark, NJ | 10,812,993 |
ATL | Atlanta, GA | 9,166,055 |
SFO | San Francisco, CA | 8,648,219 |
IAH | Houston, TX | 7,627,942 |
IAD | Washington, DC | 5,893,142 |
DFW | Dallas/Ft. Worth, TX | 4,872,207 |
DTW | Detroit, MI | 3,887,481 |
PHL | Philadelphia, PA | 3,734,127 |
BOS | Boston, MA | 3,673,748 |
FLL | Fort Lauderdale, FL | 3,062,384 |
SEA | Seattle, WA | 2,766,576 |
Data from
The current modeling of the natural history of influenza is as follows: An individual is infectious for six days starting the day after becoming infected. The individual's infectiousness is proportional to the log of the daily viral titers taken from a randomly chosen one of the six experimentally infected patients described in
When a susceptible individual is infected (at time
0 days | 1 day | 2 days | |
Preschool-age children | 0.304 | 0.575 | 0.324 |
School-age children | 0.203 | 0.498 | 0.375 |
Adults | 0.100 | 0.333 | 0.167 |
Data from
The simulation runs in discrete time, with two time steps per simulated day to represent daytime and nighttime social interactions. The contact probability of two individuals in the same mixing group is the probability that they will have sufficient contact for transmission during a time step. Contact probabilities of individuals within families were tuned so that the simulated household secondary attack rates match estimates from
Addy 1991 | simulated ( |
||||
child | adult | child | adult | ||
29.0% | 14.2% | 28.6% | 13.5% | ||
10.3% | 15.6% | 9.3% | 16.2% |
Age group | Asian A (H2N2) | Hong Kong A (H3N2) | Age group | simulated |
1957–8 | 1968–9 | ( |
||
Pre-school children | 35% | 34% | 0–4 years | 38% |
School-age children | 55% | 35% | 5–18 years | 53% |
Young adults | 25% | 35% | 19–29 years | 26% |
Middle adults | 20% | 32% | 30–64 years | 28% |
Old adults | 14% | 31% | 23% | |
Overall | 31% | 34% | 33% |
household | 32% | 31% | 29% |
schools/daycares | 30% | 24% | 21% |
workplace | 10% | 13% | 15% |
neighborhood/community | 18% | 21% | 23% |
0.8 | 0.8 | 0.35 | 0.35 | 0.35 | |
0.25 | 0.25 | 0.4 | 0.4 | 0.4 | |
0.08 | 0.08 | 0.035 | 0.035 | 0.035 | |
0.025 | 0.025 | 0.04 | 0.04 | 0.04 | |
0.0000435 | 0.0001305 | 0.000348 | 0.000348 | 0.000696 | |
0.0000109 | 0.0000326 | 0.000087 | 0.000087 | 0.000174 | |
0.05 | 0.05 | ||||
0.28 | |||||
0.12 | |||||
0.0348 | |||||
0.03 | |||||
0.0252 |
Transmission probabilities in the simulation are adjusted by multiplying all contact probabilities by a scalar,
(A) Observed secondary cases vs
The simulated case generation time, or the time between infection of an individual and the transmission to susceptibles, was 3.4 days for a wide range of
The primary pharmaceutical intervention is vaccination. Vaccinated individuals in the simulation have a reduced probability of becoming infected (VE
Vaccines do not reach full efficacy immediately – their protective effects may gradually increase over several weeks. The default behavior in the model is that the vaccine takes two weeks to reach maximum efficacy, with the efficacy increasing exponentially starting the day after the vaccination. Because of the delay in reaching maximum efficacy, it may be necessary to vaccinate the population early. In the simulation, vaccines can be administered at least four weeks before the epidemic (i.e., pre-vaccination), during the epidemic (reactive), or one dose can be administered at least three weeks before the epidemic and the boost can be administered reactively (prime-boost).
Antiviral agents (neuraminidase inhibitors) can be used for treatment of cases and for prophylaxis of susceptibles. A single course of antiviral agents is enough for 10 days of prophylaxis or 5 days of treatment. In the model, 5% of individuals taking antiviral agents prophylactically stop after 2 days and 5% taking them for treatment stop after 1 day
Several non-pharmaceutical interventions can be simulated in the model.
During an epidemic, individuals may be requested to stay at home if they become ill. When simulating
During an epidemic, those living with symptomatic individuals may be requested to stay home
FluTE is written in C/C++ and is released under the GNU General Public License (GPLv3, see
A configuration file is used to specify the population to use for the simulation, the parameters for starting the epidemic, the transmissibility of the infectious agent, and the desired intervention strategies. The configuration file is text-based and can be typed in by a user or generated with a script. The simulation outputs results to text files, which can be easily parsed for plotting or statistical analysis.
A parallelized version of the code supports simulations of large populations (up to the entire continental United States). This version of the program assigns the populations of different counties to different processors, and OpenMPI is used to update the status of individuals who travel between communities that are located on different processors and to update the global status of the epidemic and the interventions (e.g., the total number of vaccines used). The simulation uses approximately 80 megabytes of memory per million simulated individuals.
The simulation was written with several competing goals: to explicitly represent each individual in the population, to conserve memory, to run quickly, and to be (relatively) easy to read and modify. Each simulated individual is represented by a C structure that includes unique identifiers for the person and for each of the social mixing groups to which that person belongs, the age of the individual, the person's infection and vaccination status and dates, and other attributes. For each infected individual, the simulation identifies all susceptible individuals in that person's community who share a common mixing group, the infectiousness of the infected individual, the susceptibility of the susceptible, and the probability that transmission takes place for every time step. Although comparing each individual with every other within a community results in the number of comparisons increasing with the square of the number of individuals, community sizes are always smaller than 3,000 residents. Therefore, the number of comparisons made between individuals scales approximately linearly with the number of individuals in the simulation. More sophisticated algorithms could improve the simulation's performance, but may do so at the expense of the code's flexibility and readability.
The running time depends on the number of individuals infected during the course of a simulation. Simulating an epidemic in a population of 10 million people can take up to two hours (on a single processor on an Intel Core2 Duo T9400), but it may take only seconds if the virus is not highly transmissible (low
We illustrate the use of the model by simulating epidemics in metropolitan Seattle, a major metropolitan area with a population of approximately 560,000 according to the US 2000 Census. We ran simulations with different values of
(A) Daily prevalence of symptomatic influenza in simulations of metropolitan Seattle for various
The illness attack rates in the simulation are lower than those in a SIR model with random mixing (where
Simulated epidemics struck school-age children earlier than adults, which had been observed in earlier studies
Results plotted are from one simulation of metropolitan Seattle for each value of
One can simulate the population of the entire continental US using the parallel version of FluTE (mpiflute). The continental US had 280 million people in 64735 census tracts in 2000, based on the US 2000 Census. In our simulations, we found that the final illness attack rates for the US to be nearly identical to those of metropolitan Seattle, but the epidemic peak for a given
The color of each dot corresponds to the illness prevalence in a census tract. Image created using ArcGIS (Environmental Systems Research Institute, Inc.)
We have described a new publicly available influenza epidemic simulator, FluTE. It explicitly represents every individual in the simulation, so simulated epidemics can be studied in detail, even tracing individual transmission events. We illustrated the use of FluTE with examples in which we explored the effect of various intervention strategies on influenza epidemics in the United States and showed how transmissibility can be over-estimated early in an epidemic.
The simulation was written so that one can easily set the transmissibility, vaccination policies (e.g., fraction of the population to vaccinate), and other reactive strategies (e.g., school closures). These settings can be used to investigate questions such as: 1) What fraction of the population will become infected or ill? 2) How much vaccine coverage is required to mitigate an epidemic with a given
The model was calibrated to simulate epidemics of a virus similar to 1957/1958 Asian A(H2N2) and 2009 pandemic A(H1N1). We attempted to model realistic pharmaceutical and non-pharmaceutical interventions, but their effects on an epidemic have not been well quantified. The model's results are plausible and likely to be qualitatively correct, but there is insufficient data to calibrate it to produce quantitatively accurate results for the various possible disease parameters and mitigation strategies. Although the model generates realistic population-level results, the spatial dynamics of the epidemics it produces should be used for illustrative purposes only. When using the model to evaluate mitigation strategies, it is important to consider one's goals. For example, using antiviral agents to treat cases does not greatly reduce the final illness attack rate in the simulation, but it could greatly reduce mortality. The model does not directly evaluate the cost of interventions, but the numbers of cases in a simulated epidemic can be linked to cost and healthcare utilization data
Differential equation models are the most popular approach to disease modeling. The simplest of these (such as the SIR model
The current software supports a limited set of configuration options and is intended for batch runs using a scripting language. Using the model for scenarios not supported by the existing code, such as testing a novel intervention strategy or altering the contact parameters for a different attack rate pattern, would require modification of the source code, which we have released so that others can make such changes if needed. We decided to adopt the GNU General Public License (GPL), so that the source code of derivative works must be released. We believe this will facilitate the sharing of improvements. The availability of source code allows others to adapt the model to simulate outbreaks of other airborne infectious diseases such as smallpox
In the future, we would like to make our model more accessible to non-programmers. This may involve developing a user interface or adding new parameters to the configuration file. We would also like to include intervention strategies that best reflect government pandemic mitigation plans. Achieving these goals would depend upon close collaboration with public health officials to better understand their needs and to carefully simulate existing pandemic mitigation plans and capacities. Although we have calibrated our model to the best available data, more detailed and reliable information on the natural history of influenza, influenza transmission, human behavior in response to infection, and vaccine efficacy is needed. Sensitivity analyses of similar epidemic models have shown that results are robust to uncertainty in many parameters
We thank Brandon Dean for helpful discussions and Jon Sugimoto for producing the image in