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Modelling the effects of media during an influenza epidemic
- Shannon Collinson^{1, 2} and
- Jane M Heffernan^{1, 2}Email author
https://doi.org/10.1186/1471-2458-14-376
© Collinson and Heffernan; licensee BioMed Central Ltd. 2014
Received: 4 September 2013
Accepted: 4 April 2014
Published: 17 April 2014
Abstract
Background
Mass media is used to inform individuals regarding diseases within a population. The effects of mass media during disease outbreaks have been studied in the mathematical modelling literature, by including ‘media functions’ that affect transmission rates in mathematical epidemiological models. The choice of function to employ, however, varies, and thus, epidemic outcomes that are important to inform public health may be affected.
Methods
We present a survey of the disease modelling literature with the effects of mass media. We present a comparison of the functions employed and compare epidemic results parameterized for an influenza outbreak. An agent-based Monte Carlo simulation is created to access variability around key epidemic measurements, and a sensitivity analysis is completed in order to gain insight into which model parameters have the largest influence on epidemic outcomes.
Results
Epidemic outcome depends on the media function chosen. Parameters that most influence key epidemic outcomes are different for each media function.
Conclusion
Different functions used to represent the effects of media during an epidemic will affect the outcomes of a disease model, including the variability in key epidemic measurements. Thus, media functions may not best represent the effects of media during an epidemic. A new method for modelling the effects of media needs to be considered.
Keywords
Background
Influenza causes annual epidemics and occasional pandemics, which have claimed millions of lives throughout history. In the past century four worldwide pandemic outbreaks of influenza have occurred: 1918, 1957, 1977 and 2009, [1, 2]. According to the Public Health Agency of Canada, inter-pandemic (or seasonal) influenza affects approximately 20,000 Canadians, with approximately 2,000 to 8,000 deaths annually [3]. In the USA, it has been reported that flu-associated deaths can range from 3,000 to 49,000 individuals per year [4].
Mass media can affect disease transmission during an influenza epidemic or pandemic. Attention to health news has been increasing in importance during the last few decades, and thus, media reports can play an important role in defining health issues, since they serve as a major source of information and are able to incite changes in behaviour in the public [5]. Individual response to a disease threat depends on risk perception that is gained largely through information reported by governments to mass media: number of infections, hospitalizations and deaths, as provided by the government [6, 7].
We have recently seen the use of mass media reports during two infectious disease outbreaks. The first novel infectious disease of the twenty-first century was SARS. It had distinct features such as rapid spatial spread and self-control, and vast media coverage [6, 7] that used to inform the public, provide a means of contract tracing, and advise isolation of exposed individuals.
Media coverage was substantial during the H1N1 epidemic in 2009 as well, which may have had an effect on the transmission of influenza by promoting social distancing and self-isolation [8]. The media coverage of this influenza pandemic was widespread, with an increased sense of urgency since this influenza strain was related to the 1918 pandemic strain that caused approximately 50 million deaths worldwide [1].
Mathematical modelling has been used to study the effect of mass media on epidemics by employing the well-known Susceptible-Exposed-Infectious-Recovered (SEIR) model and various extensions [6–14]. In these studies, mass media has been incorporated using different, but qualitatively similar, functions that directly affect disease transmission and susceptibility. In general, the chosen functions are decreasing functions with respect to the current number of infected individuals in the population. However, the choice of function seems to be arbitrary. It is possible that the choice of function can change study results. For example, public health officials could be interested in epidemic measurements such as the peak number of infection, peak time, total number of infections and end of epidemic, which are all directly related to important public health resources (i.e. number of hospital bed, antiviral stockpile, vaccination doses). These key measurement may vary depending on the chosen media function. Therefore, a sensitivity analysis on these functions is required.
In addition to that mentioned above, there is a further drawback of the previous studies which include media functions, in that, deterministic systems of ordinary differential equations are employed. Deterministic models can describe the mean behaviour of an epidemic, but information surrounding any variability in key epidemic measurements cannot be made. A stochastic model is well suited to this task. Various methods of introducing stochasticity into disease models exist: [15–19]. Agent-Based Monte Carlo (ABMC) simulations, provide a way in which individuals with certain disease characteristics can be tracked in a population. This method also provides flexibility, as changes in biological assumptions can be easily incorporated, which are difficult to include in other methods.
In the sections that follow we give an overview of the functions used to describe media in the disease modelling literature. The functions are then incorporated into a standardized SEIR model, and model results are compared. A stochastic agent-based Monte Carlo (ABMC) simulation is then introduced, and is employed to study the variability within an epidemic depending on the media function chosen. A sensitivity analysis is also completed in order to determine the importance of certain model parameters on various epidemic outcomes for each media function.
Methods
Media functions
These equations demonstrate that, if p _{3} is increased, then p _{1} and p _{2} must decrease to have the functions remain the same value at a chosen I _{ c }.
SEIR framework
Initial conditions and parameter values for model (6), functions (1)-(3) and the ABMC
Description | Value | Range | Unit | Reference | |
---|---|---|---|---|---|
Population | |||||
S | Susceptible | 10,000 | |||
E | Exposed | 0 | |||
I | Infectious | 10 | |||
R | Recovered | 0 | |||
Parameter | |||||
R _{0} | Basic reproductive ratio | 1.5 | 1.3-1.7 | ||
β | Contact transmission rate | 3.71287e-5 | (person-day) ^{-1} | Eq. (7) | |
σ | Transition rate E to I | 1/2 | (day) ^{-1} | [20] | |
γ | Recovery rate | 1/4 | (day) ^{-1} | [20] | |
p _{ i } | Media parameter | varies | varies | [7] | |
ABMC | |||||
S/β | Mean time to transmission | S(t)/β | Eq. (7) | ||
1/σ | Mean exposed time | 2 | days | [20] | |
1/γ | Mean infectious time | 4 | days | [20] | |
1/p _{ i } | Media parameter | varies | varies | [7] |
Agent-based Monte Carlo simulation
Results and discussion
Basic reproductive ratio
where N = S _{0} is the total population size of susceptibles at time zero. Note that the calculation of R _{0} is independent of f(I,p).
Comparison of media functions
Model and fitted parameters β and p _{ i }
Model | β | p _{ i } | R _{0} |
---|---|---|---|
no media | 3.29 × 10^{-5} | 1.32 | |
(1) | 3.33 × 10^{-5} | 0.00365 | 1.33 |
(2) | 3.32 × 10^{-5} | 5.2 × 10^{-5} | 1.33 |
(3) | 3.33 × 10^{-5} | 1000 | 1.33 |
Key epidemic measurements for the SEIR model with (a) no media, (b) media function (1), (c) media function (2), and (d) media function (3)
Model | Peak Magnitude | Peak time | Epidemic end | Total |
---|---|---|---|---|
(E+I) | (days) | (days) | (I) | |
(a) | 2676.1 | 21.155 | 76.6065 | 9254.8 |
2623.9±204.09 | 21.005±19.9 | 77.1863±15.59 | 9542.4±255.37 | |
(b) | 1061.5 | 21.7841 | 162.3647 | 7537.5 |
1095.6±46.35 | 21.001±19.99 | 168.077±31.24 | 7578.13±460.95 | |
(c) | 858.176 | 18.544 | 176.8229 | 7557.9 |
902.15±78.85 | 17.7±15.99 | 177.07±23.6 | 7622.2±328.8 | |
(d) | 1144.7 | 23.5169 | 154.3623 | 7586.0 |
1175.9±175.09 | 22.01±20.99 | 157.07±28.005 | 7643.9±229.2 |
Sensitivity analysis
Within public health settings, a goal is to identify key characteristics of an epidemic that drive the infection with the hope of determining public health measures that can be implemented so that control or eradication of the pathogen can be achieved. By performing sensitivity analysis on System (6) with Equations (1)-(3), parameters that most affect epidemic outcomes for each media function can be identified, informing policy makers so that appropriate public health measures can be put into place. To conduct the sensitivity analysis, we use Latin Hypercube Sampling (LHS) and partial rank correlation coefficients (PRCC) [23].
The sensitivity analysis indicates that models that include mass media influence will greatly depend on different parameters, depending on the media function chosen. This makes it very difficult for policy makers to determine an effective public health intervention strategy. This also explains the very different epidemic curves produced by System (6) when different media functions are employed, notwithstanding the similarities in the media functions when plotted at a specific level of media.
Conclusion
Technology and media play an increasing role in daily life. Mass media that is transmitted via technological media can therefore be used to inform the public during pandemics and epidemics. An understanding of the effects of media during an epidemic or pandemic threat can aid in the development of public health policy. Of particular interest to public health are the effects of media on key epidemic measurements - peak magnitude of infection, time of peak, end of epidemic, and total number of infections.
Mathematical modelling has been used to study the effect of media on epidemics by employing functions in the transmission terms of mathematical models [6–14]. A survey of the literature identified three functions that have been utilized to represent media in disease modelling [6–14]. We have conducted a comparison of these functions to determine the effects of media function on key epidemic measurement and variability within these measurements. We first demonstrated that by including mass media in System (6) the peak magnitude of the epidemic and the total number of infections would decrease. We also determined that the time to peak and the end of the epidemic would also occur earlier. However, we also demonstrated that, although the functions are similar in shape and magnitude, the resulting epidemic curve can be quite different (Figure 5). Therefore, the key epidemic measurements that are used to inform public health policy will be different. Furthermore, we demonstrated that variability in the key epidemic measurements also depends on the media function chosen (Figure 5 and Table 3).
A sensitivity analysis on System (6) with the different media functions was also conducted. Obtained from this analysis was the insight that some parameters are important for some outcomes and not for others. We can conclude that due to the different fixed functions resulting in very different epidemic behaviours, there is no clear control strategy present. Also, as a reult of the different behaviours from the different media functions, we are unsure as to which is the best function to use to model mass media. This suggests that a function representing media may not be the best course for modelling the effects of media during an epidemic. We suggest that perhaps, a separate model compartment representing media reports, such as those incorporated into surveillance data [24] could be used. Development of such a model is a course for future work.
Mass media reports can affect social behaviour, that ultimately, affects transmission of disease. However, an individual’s response to a media stimulus will wane over time [13, 25–29]. Models that employ a media function such as those studied here are difficult to modify to involve a waning response to a media stimulus over the age of an epidemic. This can be incorporated easily into a model whereby media is represented as a model compartment. A study of the effects of ‘waning media’ is a course for future work.
The current study has employed a simple SEIR model that includes three parameters. This model implicitly assumes that individuals mix at random in the population, the age and sex of individuals is unimportant, and that the population size remains constant over the epidemic. These assumptions are not true reflections of reality. An interesting direction for future work is to consider the effects of mass media reports on individual decision making strategies, and study how mass media can impact the contact structure of a population network [30].
The simple SEIR model employed in this study was used to study one season of influenza. Typically, seasonal forcing is used to model multiple seasons of influenza. Inclusion of mass media reports and media waning in a model of multiple seasons with seasonal forcing is an interesting direction for study.
Declarations
Acknowledgements
The authors would like to thank Nicholas Geard, Amy Greer, James McCaw, Seyed Moghadas, Jianhong Wu and Huaiping Zhu for many helpful discussions. The authors would also like to thank the reviewers for their comments and suggestions that have added value to this study The work was funded by an Ontario Early Researcher Award, Mathematics of Information Technology and Complex Systems (MITACS), and the Natural Science and Engineering Research Council of Canada (NSERC). Simulations were conducted on Sharcnet.
Authors’ Affiliations
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