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Modelling the effects of media during an influenza epidemic

Modelling the effects of media during an influenza epidemic 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: Mass media, Epidemic, Influenza, Agent-based Monte Carlo simulation Background source of information and are able to incite changes in Influenza causes annual epidemics and occasional pan- behaviour in the public [5]. Individual response to a dis- demics, which have claimed millions of lives throughout ease threat depends on risk perception that is gained history. In the past century four worldwide pandemic largely through information reported by governments to outbreaks of influenza have occurred: 1918, 1957, 1977 mass media: number of infections, hospitalizations and and 2009, [1,2]. According to the Public Health Agency deaths, as provided by the government [6,7]. of Canada, inter-pandemic (or seasonal) influenza affects We have recently seen the use of mass media reports approximately 20, 000 Canadians, with approximately during two infectious disease outbreaks. The first novel 2, 000 to 8, 000 deaths annually [3]. In the USA, it has been infectious disease of the twenty-first century was SARS. It reported that flu-associated deaths can range from 3, 000 haddistinctfeaturessuchasrapid spatialspreadand self- to 49, 000 individuals per year [4]. control, and vast media coverage [6,7] that used to inform Mass media can affect disease transmission during an the public, provide a means of contract tracing, and advise influenza epidemic or pandemic. Attention to health news isolation of exposed individuals. has been increasing in importance during the last few Media coverage was substantial during the H1N1 epi- decades, and thus, media reports can play an important demic in 2009 as well, which may have had an effect on role in defining health issues, since they serve as a major the transmission of influenza by promoting social dis- tancing and self-isolation [8]. The media coverage of this *Correspondence: jmheffer@yorku.ca influenza pandemic was widespread, with an increased Department of Mathematics & Statistics, York University, Toronto, Canada sense of urgency since this influenza strain was related to Modelling Infection and Immunity Lab, Centre for Disease Modelling, York Institute for Health Research, York University, Toronto, Canada © 2014 Collinson and Heffernan; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Collinson and Heffernan BMC Public Health 2014, 14:376 Page 2 of 10 http://www.biomedcentral.com/1471-2458/14/376 −p γ I the 1918 pandemic strain that caused approximately 50 f (I, p ) = e (1) million deaths worldwide [1]. f (I, p ) = (2) Mathematical modelling has been used to study the 1 + p I effect of mass media on epidemics by employing the well- known Susceptible-Exposed-Infectious-Recovered (SEIR) f (I, p ) = 1 − (3) p + I model and various extensions [6-14]. In these studies, mass media has been incorporated using different, but where I is the number of infectious individuals in a pop- qualitatively similar, functions that directly affect disease ulation, γ is the recovery rate, and p , i = 1, 2, 3 is a transmission and susceptibility. In general, the chosen parameter used to represent media effects in these func- functions are decreasing functions with respect to the tions. In general, these functions are decreasing functions current number of infected individuals in the population. of I, which represents the fact that, as the number of infec- However, the choice of function seems to be arbitrary. It tions increases in a population, and is reported by mass is possible that the choice of function can change study media, susceptible individuals will practice social distanc- results. For example, public health officials could be inter- ing or control measures, which decreases susceptibility. ested in epidemic measurements such as the peak number Comparing the three functions, we see that, for a given I of infection, peak time, total number of infections and and γ , p and p can be written in terms of p : 1 2 3 end of epidemic, which are all directly related to impor- tant public health resources (i.e. number of hospital bed, − ln p +I antiviral stockpile, vaccination doses). These key mea- 3 c p = (4) surement may vary depending on the chosen media func- γ I tion. Therefore, a sensitivity analysis on these functions is p =.(5) required. p I 3 c In addition to that mentioned above, there is a further drawback of the previous studies which include media These equations demonstrate that, if p is increased, then functions, in that, deterministic systems of ordinary dif- p and p must decrease to have the functions remain the 1 2 ferential equations are employed. Deterministic models same value at a chosen I . can describe the mean behaviour of an epidemic, but In Figure 1, we plot Functions (1)-(3) (dotted, dashed, information surrounding any variability in key epidemic solid lines respectively) for media parameters p , i = 1, 2, measurements cannot be made. A stochastic model is 3 (dotted, dashed, solid lines) with γ = 1/4and I = 300 well suited to this task. Various methods of introducing (top) and I = 1000 (bottom). Note that the functions are stochasticity into disease models exist: [15-19]. Agent- equal when I = 300 (top) and 1000 (bottom). Here, p = Based Monte Carlo (ABMC) simulations, provide a way 10 (left panel), 100 (middle panel), and 1000 (right panel), in which individuals with certain disease characteristics and p1and p2 are determined using Equations (4) and (5). can be tracked in a population. This method also pro- vides flexibility, as changes in biological assumptions can SEIR framework be easily incorporated, which are difficult to include in To compare the effects of different media Functions (1)-(3), other methods. we must choose a standardized model. For the purposes of our study, we have chosen the basic Susceptible-Exposed- In the sections that follow we give an overview of the Infectious-Recovered (SEIR) model with a constant popu- functions used to describe media in the disease modelling lation size N: literature. The functions are then incorporated into a stan- dardized SEIR model, and model results are compared. A S =−f (I, p)βSI stochastic agent-based Monte Carlo (ABMC) simulation E = f (I, p)βSI − σ E is then introduced, and is employed to study the variabil- I = σ E − γ I (6) ity within an epidemic depending on the media function R = γ I chosen. A sensitivity analysis is also completed in order to N = S + E + I + R . determine the importance of certain model parameters on various epidemic outcomes for each media function. Briefly, susceptible individuals (S) are infected by infec- tious individuals (I)withrate β and become exposed Methods (E). Transmission can also be reduced as determined by Media functions f (I, p). Exposed individuals become infectious with rate From the disease modelling literature [6-14] we have iden- σ , and infectious individuals recover with rate γ . The ini- tified three distinct functions employed to present the tial conditions and parameters values for this study can be effects of mass media: found in Table 1. Collinson and Heffernan BMC Public Health 2014, 14:376 Page 3 of 10 http://www.biomedcentral.com/1471-2458/14/376 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 0 500 0 500 0 500 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 0 1000 2000 0 1000 2000 0 1000 2000 Number of Infecteds (I) Figure 1 Media Functions f (I, p) for different values of p. Functions (1), (2) and (3) (dotted, dashed, solid lines respectively) are shown when p = 10 (left), 100 (middle), 1000 (right), γ = 1/4and I = 300 (top), 1000 (bottom). p and p are calculated using equations (4) and (5). 3 c 1 2 Agent-based Monte Carlo simulation by Heffernan and Wahl, [16,22]. Briefly, the ABMC sim- For further comparison of the SEIR model with different ulation moves forward in time following event times: the media functions, we utilize an Agent-Based Monte Carlo next time that an individual changes state within the sys- (ABMC) simulation to capture the variability in the epi- tem. Agents in each of the susceptible, exposed, infectious demic infection curve (that cannot be determined using a and recovered compartments are assigned event times system of deterministic equations like that of System (6)). corresponding to Table 1 when they are introduced into There are various ways of developing an ABMC simula- the simulation, for each event that allows such an individ- tion. We have employed a previous method as developed ual from that compartment to change state. For example, Table 1 Initial conditions and parameter values for model (6), functions (1)-(3) and the ABMC Description Value Range Unit Reference Population S Susceptible 10, 000 EExposed 0 IInfectious 10 R Recovered 0 Parameter R Basic reproductive ratio 1.5 1.3 − 1.7 [20,21] −1 β Contact transmission rate 3.71287e−5 (person-day) Eq. (7) −1 σ Transition rate E to I 1/2 (day) [20] −1 γ Recovery rate 1/4 (day) [20] p 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 Media parameter varies varies [7] f(I,p) f(I,p) Collinson and Heffernan BMC Public Health 2014, 14:376 Page 4 of 10 http://www.biomedcentral.com/1471-2458/14/376 Table 2 Model and fitted parameters β and p Infection Model β p R i 0 Exposed Susceptible Time to infected −5 no media 3.29 ×10 1.32 −5 (1) 3.33 ×10 0.00365 1.33 Become −5 −5 (2) 3.32 ×10 5.2 ×10 1.33 infectious −5 (3) 3.33 ×10 1000 1.33 Based on 100 days of data generated using System (6) with Media Function (3) Infectious Recovered with σ and γ values listed in Table 1 and p = 1000. Time to infect Recover from Time to recovered infection Figure 2 Schematic of the Agent Based Monte Carlo simulations the means of the lifetime distributions in the ABMC are corresponding to Table 1. simply the reciprocals of the rates used in System (6). However, so that infection event times always depend on the current size of the susceptible population S(t) (similar exposed individuals are assigned a time to become infec- to what is assumed in System (6)), the infection time dis- tious, and infectious individuals are assigned a time to tribution mean is continuously updated. Figure 2, shows recovery and a time to infect a susceptible. The mini- the progression of an individual through the epidemic. mum event time determines the next event to occur in the simulation. The event is then carried out, and the next event is determined. To compare to the SEIR system as Results and discussion described above, we assume exponential distributions for Basic reproductive ratio all parameters. Table 1 lists the parameter values of the The basic reproductive ratio R is defined as the number SEIR model and the mean value of the parameter distri- of newly infectious individuals produced by one infec- butions for the ABMC simulation. Note that, in general, tious individual in a totally susceptible population. For S E I R 10000 400 500 6000 8000 4000 6000 2000 4000 0 0 0 0 500 0 500 0 500 0 500 10000 400 500 6000 8000 4000 6000 2000 4000 0 0 0 0 500 0 500 0 500 0 500 10000 400 500 6000 8000 4000 6000 2000 4000 0 0 0 0 500 0 500 0 500 0 500 Time (days) Figure 3 Epidemic curves using the different values for each media function. Functions (1) (dotted line), (2) (dashed line), and (3) (solid line) are shown when p = 10 (top), 100 (middle), 1000 (bottom). p and p are determined by equations (4) and (5) with γ = 1/4and Ic = 300. For 3 1 2 comparison, the standard SEIR model 6 with no media effect is also shown (dash-dotted line). Population Size Collinson and Heffernan BMC Public Health 2014, 14:376 Page 5 of 10 http://www.biomedcentral.com/1471-2458/14/376 a description of different methods for calculating R see [22]. For System (6), βN R =.(7) where N = S is the total population size of susceptibles at 120 time zero. Note that the calculation of R is independent of f (I, p). Comparison of media functions Figure 3 plots System (6) using Media Functions (1)-(3) for p = 10, 100, and 1000 (solid line), with and p (dot- 3 1 ted line) and p (dashed line) determined by Equations (4) and (5), and γ = 1/4and I = 300. For comparison, the 0 50 100 150 200 250 300 350 400 c Time (days) standard SEIR model with no media effect is also shown Figure 4 Data and fitted models. The epidemic curves are shown (dash-dotted line). This figure demonstrates that mass resulting from fitting System (6) with Media Functions (1)-(3) (dotted media will affect the epidemic curve. It also demonstrates line, dashed line, solid line) and no media (dash-dotted line) to 100 that the epidemic curve varies depending on the media days of epidemic data (boxes) generated using Media Function (3) function used. with p = 1000 and γ and σ from Table 1. For a further comparison of the media functions, we determine the values of the basic reproductive ratio R and the media constant p of f (I, p) based on epidemic data S E I R 10000 2000 10000 5000 1000 5000 0 0 0 0 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 10000 2000 10000 5000 1000 5000 0 0 0 0 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 10000 2000 10000 5000 1000 5000 0 0 0 0 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 10000 2000 10000 5000 1000 5000 0 0 0 0 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 Figure 5 Epidemic curves for different media functions. The epidemic curves are shown for Model (6) (dashed line) and the ABMC simulation (gray lines). There are 100 simulations plotted for each. In each panel, we also show the mean of the simulations (sold line). The first row has no media effect and rows (2-4) correspond to Functions (1)-(3). Here, I = 300 and p = 1000, γ = . c 3 Infected Population (I) Population Size Collinson and Heffernan BMC Public Health 2014, 14:376 Page 6 of 10 http://www.biomedcentral.com/1471-2458/14/376 Table 3 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 The results of system 6 and the ABMC are listed in the top and bottom row of each section respectively with p = 1000, p and p determined using equations (4) and 3 1 2 (5), and γ = 1/4 and I = 300. 500 simulation runs are used for the ABMC calculations. generated from our models. For an example, Table 2 lists and no media (dash-dotted line). Table 2 and Figure 4 the values of β and p determined through a fitting rou- demonstrate that even though all of the models have a tine (in MATLAB - various routines were used and similar very similar shape and basic reproductive ratio R at the results were obtained) over the first 100 days of an epi- beginning of the epidemic, the resulting epidemic curves demic, with data generated using Media Function (3) with can still vary drastically depending on the media function σ and γ values listed in Table 1 and p = 1000. Figure 4 chosen. Similar observations were made using different plots the resulting epidemic curves of System (6) using numbersofdaysofdatausedtofit β and p, and differ- Media Functions (1)-(3) (dotted, dashed and solid lines) ent Media Functions to generate the data (not shown). Figure 6 LHS-PRCC results for I = 300 for peak magnitude. This sensitivity analysis is done with 1000 bins. The rows correspond to Function (1), Function (2) and Function (3), respectively. The columns of this PRCC figure correspond to p = 10, p = 100 and p = 1000. Here I = 300 and 3 3 3 c γ = . 4 Collinson and Heffernan BMC Public Health 2014, 14:376 Page 7 of 10 http://www.biomedcentral.com/1471-2458/14/376 Figure 7 LHS-PRCC results for I = 1000. This sensitivity analysis is done with 1000 bins. The rows correspond to Function (1), Function (2) and Function (3), respectively. The columns of this PRCC figure correspond to p = 10, p = 100 and p = 1000. 3 3 3 Therefore, it can be concluded that key epidemic mea- Sensitivity analysis surements – peak number of infections, time of peak, end Within public health settings, a goal is to identify key of epidemic, and total number of infections – will also vary characteristics of an epidemic that drive the infection depending on the Media Function chosen. with the hope of determining public health measures that System (6) is useful in describing the mean behaviour can be implemented so that control or eradication of the of an epidemic, but it is unable to provide estimates of pathogen can be achieved. By performing sensitivity anal- variation of important public health measures. A stochas- ysis on System (6) with Equations (1)-(3), parameters that tic simulation lends itself well to demonstrating variation most affect epidemic outcomes for each media function within an epidemic. Here, we employ an Agent-Based can be identified, informing policy makers so that appro- Monte Carlo (ABMC) simulation. Figure 5 shows 100 sim- priate public health measures can be put into place. To ulation runs of the ABMC simulation when no media is conduct the sensitivity analysis, we use Latin Hypercube involved (top), and when Media Functions (1)-(3) are used Sampling (LHS) and partial rank correlation coefficients (second to bottom rows). Each simulation run (gray line) (PRCC) [23]. is shown, as well the mean of the simulation runs (solid We first conduct the sensitivity analysis with I , p c 3 line) and solution of the System (6) with the correspond- and γ constant to directly compare the media functions. ing media functions (dashed line). Note that the mean Figures 6 and 7 show the PRCC values determined for (solid line) and solution to System (6) are in agreement. peak magnitude when I = 300 (Figure 6) and 1000 This figure demonstrates that variability in the key epi- (Figure 7) for Functions (1)-(3) (top to bottom) when p = demic measurement occurs. Table 3 lists the mean and 10, 100 and 1000 (left, middle and right columns respec- standard error of each epidemic measurement when no tively). These figures demonstrate that for each set of media is considered (section a), and Media Functions (1)- parameters the PRCC value is similar. This is true for each (3) are included in the simulation (Sections b-d). In each key epidemic measurement - the PRCC values are similar section we list the simulation mean and standard error between different values of p and I . However the PRCC 3 c (top row), and the solution of System (6). values between key measurements are different i.e. the Collinson and Heffernan BMC Public Health 2014, 14:376 Page 8 of 10 http://www.biomedcentral.com/1471-2458/14/376 Figure 8 PRCC for system (6). Results are shown for peak magnitude, peak time, epidemic end time, and total number of infections (left to right) when no mass media function is considered, and functions (1)-(3) are used (top to bottom). Here, p = 100 and p and p are determined by 3 1 2 Equations(4) and(5). PRCC values are different if comparing peak magnitude to This also explains the very different epidemic curves pro- epidemic end time (not shown). duced by System (6) when different media functions are For further sensitivity analysis, all parameters are employed, notwithstanding the similarities in the media allowed to vary. Figure 8 shows the result of the sensi- functions when plotted at a specific level of media. tivity analysis for all four outcomes key to public health: peak number of infections, peak time, epidemic end time, Conclusion and total number of infections. The results demonstrate Technology and media play an increasing role in daily life. that the SEIR model is more (or less) sensitive to certain Mass media that is transmitted via technological media parameters depending on what Mass Media Function (1)- can therefore be used to inform the public during pan- (3) is used. For example, changes in β have a large effect on demics and epidemics. An understanding of the effects peak magnitude when Function (1) is used, but it has lit- of media during an epidemic or pandemic threat can aid tletonoeffectwhenFunctions (2-3)represent theeffects in the development of public health policy. Of particu- of media. Variable p also has a large effect on peak mag- lar interest to public health are the effects of media on nitude when media is represented by Function (1). It has key epidemic measurements - peak magnitude of infec- a similar effect when Function (2) is used, but it has no tion, time of peak, end of epidemic, and total number of effect on the system utilizing Function (3). infections. The sensitivity analysis indicates that models that Mathematical modelling has been used to study the include mass media influence will greatly depend on effect of media on epidemics by employing functions in different parameters, depending on the media function the transmission terms of mathematical models [6-14]. chosen. This makes it very difficult for policy makers to A survey of the literature identified three functions that determine an effective public health intervention strategy. have been utilized to represent media in disease modelling Collinson and Heffernan BMC Public Health 2014, 14:376 Page 9 of 10 http://www.biomedcentral.com/1471-2458/14/376 [6-14]. We have conducted a comparison of these func- Inclusion of mass media reports and media waning in tions to determine the effects of media function on key a model of multiple seasons with seasonal forcing is an epidemic measurement and variability within these mea- interesting direction for study. surements. We first demonstrated that by including mass Competing interests media in System (6) the peak magnitude of the epidemic The authors declare that they have no competing interests. and the total number of infections would decrease. We also determined that the time to peak and the end of Authors’ contributions the epidemic would also occur earlier. However, we also MSC performed the literature review. JMH and MSC developed the models, conducted the analysis and wrote the paper. Both authors read and approved demonstrated that, although the functions are similar in the final manuscript. shape and magnitude, the resulting epidemic curve can be quite different (Figure 5). Therefore, the key epidemic Acknowledgements measurements that are used to inform public health policy The authors would like to thank Nicholas Geard, Amy Greer, James McCaw, will be different. Furthermore, we demonstrated that vari- Seyed Moghadas, Jianhong Wu and Huaiping Zhu for many helpful discussions. The authors would also like to thank the reviewers for their ability in the key epidemic measurements also depends on comments and suggestions that have added value to this study The work was the media function chosen (Figure 5 and Table 3). funded by an Ontario Early Researcher Award, Mathematics of Information A sensitivity analysis on System (6) with the differ- Technology and Complex Systems (MITACS), and the Natural Science and Engineering Research Council of Canada (NSERC). Simulations were ent media functions was also conducted. Obtained from conducted on Sharcnet. this analysis was the insight that some parameters are important for some outcomes and not for others. We can Received: 4 September 2013 Accepted: 4 April 2014 conclude that due to the different fixed functions result- Published: 17 April 2014 ing in very different epidemic behaviours, there is no clear References control strategy present. Also, as a reult of the differ- 1. 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Physica A: Stat Mech Appl 2013, 392(23):5824–5835. doi:10.1186/1471-2458-14-376 Cite this article as: Collinson and Heffernan: Modelling the effects of media during an influenza epidemic. BMC Public Health 2014 14:376. Submit your next manuscript to BioMed Central and take full advantage of: • Convenient online submission • Thorough peer review • No space constraints or color figure charges • Immediate publication on acceptance • Inclusion in PubMed, CAS, Scopus and Google Scholar • Research which is freely available for redistribution Submit your manuscript at www.biomedcentral.com/submit http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png BMC Public Health Springer Journals

Modelling the effects of media during an influenza epidemic

BMC Public Health , Volume 14 (1): 10 – Dec 1, 2014

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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: Mass media, Epidemic, Influenza, Agent-based Monte Carlo simulation Background source of information and are able to incite changes in Influenza causes annual epidemics and occasional pan- behaviour in the public [5]. Individual response to a dis- demics, which have claimed millions of lives throughout ease threat depends on risk perception that is gained history. In the past century four worldwide pandemic largely through information reported by governments to outbreaks of influenza have occurred: 1918, 1957, 1977 mass media: number of infections, hospitalizations and and 2009, [1,2]. According to the Public Health Agency deaths, as provided by the government [6,7]. of Canada, inter-pandemic (or seasonal) influenza affects We have recently seen the use of mass media reports approximately 20, 000 Canadians, with approximately during two infectious disease outbreaks. The first novel 2, 000 to 8, 000 deaths annually [3]. In the USA, it has been infectious disease of the twenty-first century was SARS. It reported that flu-associated deaths can range from 3, 000 haddistinctfeaturessuchasrapid spatialspreadand self- to 49, 000 individuals per year [4]. control, and vast media coverage [6,7] that used to inform Mass media can affect disease transmission during an the public, provide a means of contract tracing, and advise influenza epidemic or pandemic. Attention to health news isolation of exposed individuals. has been increasing in importance during the last few Media coverage was substantial during the H1N1 epi- decades, and thus, media reports can play an important demic in 2009 as well, which may have had an effect on role in defining health issues, since they serve as a major the transmission of influenza by promoting social dis- tancing and self-isolation [8]. The media coverage of this *Correspondence: jmheffer@yorku.ca influenza pandemic was widespread, with an increased Department of Mathematics & Statistics, York University, Toronto, Canada sense of urgency since this influenza strain was related to Modelling Infection and Immunity Lab, Centre for Disease Modelling, York Institute for Health Research, York University, Toronto, Canada © 2014 Collinson and Heffernan; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Collinson and Heffernan BMC Public Health 2014, 14:376 Page 2 of 10 http://www.biomedcentral.com/1471-2458/14/376 −p γ I the 1918 pandemic strain that caused approximately 50 f (I, p ) = e (1) million deaths worldwide [1]. f (I, p ) = (2) Mathematical modelling has been used to study the 1 + p I effect of mass media on epidemics by employing the well- known Susceptible-Exposed-Infectious-Recovered (SEIR) f (I, p ) = 1 − (3) p + I model and various extensions [6-14]. In these studies, mass media has been incorporated using different, but where I is the number of infectious individuals in a pop- qualitatively similar, functions that directly affect disease ulation, γ is the recovery rate, and p , i = 1, 2, 3 is a transmission and susceptibility. In general, the chosen parameter used to represent media effects in these func- functions are decreasing functions with respect to the tions. In general, these functions are decreasing functions current number of infected individuals in the population. of I, which represents the fact that, as the number of infec- However, the choice of function seems to be arbitrary. It tions increases in a population, and is reported by mass is possible that the choice of function can change study media, susceptible individuals will practice social distanc- results. For example, public health officials could be inter- ing or control measures, which decreases susceptibility. ested in epidemic measurements such as the peak number Comparing the three functions, we see that, for a given I of infection, peak time, total number of infections and and γ , p and p can be written in terms of p : 1 2 3 end of epidemic, which are all directly related to impor- tant public health resources (i.e. number of hospital bed, − ln p +I antiviral stockpile, vaccination doses). These key mea- 3 c p = (4) surement may vary depending on the chosen media func- γ I tion. Therefore, a sensitivity analysis on these functions is p =.(5) required. p I 3 c In addition to that mentioned above, there is a further drawback of the previous studies which include media These equations demonstrate that, if p is increased, then functions, in that, deterministic systems of ordinary dif- p and p must decrease to have the functions remain the 1 2 ferential equations are employed. Deterministic models same value at a chosen I . can describe the mean behaviour of an epidemic, but In Figure 1, we plot Functions (1)-(3) (dotted, dashed, information surrounding any variability in key epidemic solid lines respectively) for media parameters p , i = 1, 2, measurements cannot be made. A stochastic model is 3 (dotted, dashed, solid lines) with γ = 1/4and I = 300 well suited to this task. Various methods of introducing (top) and I = 1000 (bottom). Note that the functions are stochasticity into disease models exist: [15-19]. Agent- equal when I = 300 (top) and 1000 (bottom). Here, p = Based Monte Carlo (ABMC) simulations, provide a way 10 (left panel), 100 (middle panel), and 1000 (right panel), in which individuals with certain disease characteristics and p1and p2 are determined using Equations (4) and (5). can be tracked in a population. This method also pro- vides flexibility, as changes in biological assumptions can SEIR framework be easily incorporated, which are difficult to include in To compare the effects of different media Functions (1)-(3), other methods. we must choose a standardized model. For the purposes of our study, we have chosen the basic Susceptible-Exposed- In the sections that follow we give an overview of the Infectious-Recovered (SEIR) model with a constant popu- functions used to describe media in the disease modelling lation size N: literature. The functions are then incorporated into a stan- dardized SEIR model, and model results are compared. A S =−f (I, p)βSI stochastic agent-based Monte Carlo (ABMC) simulation E = f (I, p)βSI − σ E is then introduced, and is employed to study the variabil- I = σ E − γ I (6) ity within an epidemic depending on the media function R = γ I chosen. A sensitivity analysis is also completed in order to N = S + E + I + R . determine the importance of certain model parameters on various epidemic outcomes for each media function. Briefly, susceptible individuals (S) are infected by infec- tious individuals (I)withrate β and become exposed Methods (E). Transmission can also be reduced as determined by Media functions f (I, p). Exposed individuals become infectious with rate From the disease modelling literature [6-14] we have iden- σ , and infectious individuals recover with rate γ . The ini- tified three distinct functions employed to present the tial conditions and parameters values for this study can be effects of mass media: found in Table 1. Collinson and Heffernan BMC Public Health 2014, 14:376 Page 3 of 10 http://www.biomedcentral.com/1471-2458/14/376 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 0 500 0 500 0 500 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 0 1000 2000 0 1000 2000 0 1000 2000 Number of Infecteds (I) Figure 1 Media Functions f (I, p) for different values of p. Functions (1), (2) and (3) (dotted, dashed, solid lines respectively) are shown when p = 10 (left), 100 (middle), 1000 (right), γ = 1/4and I = 300 (top), 1000 (bottom). p and p are calculated using equations (4) and (5). 3 c 1 2 Agent-based Monte Carlo simulation by Heffernan and Wahl, [16,22]. Briefly, the ABMC sim- For further comparison of the SEIR model with different ulation moves forward in time following event times: the media functions, we utilize an Agent-Based Monte Carlo next time that an individual changes state within the sys- (ABMC) simulation to capture the variability in the epi- tem. Agents in each of the susceptible, exposed, infectious demic infection curve (that cannot be determined using a and recovered compartments are assigned event times system of deterministic equations like that of System (6)). corresponding to Table 1 when they are introduced into There are various ways of developing an ABMC simula- the simulation, for each event that allows such an individ- tion. We have employed a previous method as developed ual from that compartment to change state. For example, Table 1 Initial conditions and parameter values for model (6), functions (1)-(3) and the ABMC Description Value Range Unit Reference Population S Susceptible 10, 000 EExposed 0 IInfectious 10 R Recovered 0 Parameter R Basic reproductive ratio 1.5 1.3 − 1.7 [20,21] −1 β Contact transmission rate 3.71287e−5 (person-day) Eq. (7) −1 σ Transition rate E to I 1/2 (day) [20] −1 γ Recovery rate 1/4 (day) [20] p 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 Media parameter varies varies [7] f(I,p) f(I,p) Collinson and Heffernan BMC Public Health 2014, 14:376 Page 4 of 10 http://www.biomedcentral.com/1471-2458/14/376 Table 2 Model and fitted parameters β and p Infection Model β p R i 0 Exposed Susceptible Time to infected −5 no media 3.29 ×10 1.32 −5 (1) 3.33 ×10 0.00365 1.33 Become −5 −5 (2) 3.32 ×10 5.2 ×10 1.33 infectious −5 (3) 3.33 ×10 1000 1.33 Based on 100 days of data generated using System (6) with Media Function (3) Infectious Recovered with σ and γ values listed in Table 1 and p = 1000. Time to infect Recover from Time to recovered infection Figure 2 Schematic of the Agent Based Monte Carlo simulations the means of the lifetime distributions in the ABMC are corresponding to Table 1. simply the reciprocals of the rates used in System (6). However, so that infection event times always depend on the current size of the susceptible population S(t) (similar exposed individuals are assigned a time to become infec- to what is assumed in System (6)), the infection time dis- tious, and infectious individuals are assigned a time to tribution mean is continuously updated. Figure 2, shows recovery and a time to infect a susceptible. The mini- the progression of an individual through the epidemic. mum event time determines the next event to occur in the simulation. The event is then carried out, and the next event is determined. To compare to the SEIR system as Results and discussion described above, we assume exponential distributions for Basic reproductive ratio all parameters. Table 1 lists the parameter values of the The basic reproductive ratio R is defined as the number SEIR model and the mean value of the parameter distri- of newly infectious individuals produced by one infec- butions for the ABMC simulation. Note that, in general, tious individual in a totally susceptible population. For S E I R 10000 400 500 6000 8000 4000 6000 2000 4000 0 0 0 0 500 0 500 0 500 0 500 10000 400 500 6000 8000 4000 6000 2000 4000 0 0 0 0 500 0 500 0 500 0 500 10000 400 500 6000 8000 4000 6000 2000 4000 0 0 0 0 500 0 500 0 500 0 500 Time (days) Figure 3 Epidemic curves using the different values for each media function. Functions (1) (dotted line), (2) (dashed line), and (3) (solid line) are shown when p = 10 (top), 100 (middle), 1000 (bottom). p and p are determined by equations (4) and (5) with γ = 1/4and Ic = 300. For 3 1 2 comparison, the standard SEIR model 6 with no media effect is also shown (dash-dotted line). Population Size Collinson and Heffernan BMC Public Health 2014, 14:376 Page 5 of 10 http://www.biomedcentral.com/1471-2458/14/376 a description of different methods for calculating R see [22]. For System (6), βN R =.(7) where N = S is the total population size of susceptibles at 120 time zero. Note that the calculation of R is independent of f (I, p). Comparison of media functions Figure 3 plots System (6) using Media Functions (1)-(3) for p = 10, 100, and 1000 (solid line), with and p (dot- 3 1 ted line) and p (dashed line) determined by Equations (4) and (5), and γ = 1/4and I = 300. For comparison, the 0 50 100 150 200 250 300 350 400 c Time (days) standard SEIR model with no media effect is also shown Figure 4 Data and fitted models. The epidemic curves are shown (dash-dotted line). This figure demonstrates that mass resulting from fitting System (6) with Media Functions (1)-(3) (dotted media will affect the epidemic curve. It also demonstrates line, dashed line, solid line) and no media (dash-dotted line) to 100 that the epidemic curve varies depending on the media days of epidemic data (boxes) generated using Media Function (3) function used. with p = 1000 and γ and σ from Table 1. For a further comparison of the media functions, we determine the values of the basic reproductive ratio R and the media constant p of f (I, p) based on epidemic data S E I R 10000 2000 10000 5000 1000 5000 0 0 0 0 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 10000 2000 10000 5000 1000 5000 0 0 0 0 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 10000 2000 10000 5000 1000 5000 0 0 0 0 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 10000 2000 10000 5000 1000 5000 0 0 0 0 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 Figure 5 Epidemic curves for different media functions. The epidemic curves are shown for Model (6) (dashed line) and the ABMC simulation (gray lines). There are 100 simulations plotted for each. In each panel, we also show the mean of the simulations (sold line). The first row has no media effect and rows (2-4) correspond to Functions (1)-(3). Here, I = 300 and p = 1000, γ = . c 3 Infected Population (I) Population Size Collinson and Heffernan BMC Public Health 2014, 14:376 Page 6 of 10 http://www.biomedcentral.com/1471-2458/14/376 Table 3 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 The results of system 6 and the ABMC are listed in the top and bottom row of each section respectively with p = 1000, p and p determined using equations (4) and 3 1 2 (5), and γ = 1/4 and I = 300. 500 simulation runs are used for the ABMC calculations. generated from our models. For an example, Table 2 lists and no media (dash-dotted line). Table 2 and Figure 4 the values of β and p determined through a fitting rou- demonstrate that even though all of the models have a tine (in MATLAB - various routines were used and similar very similar shape and basic reproductive ratio R at the results were obtained) over the first 100 days of an epi- beginning of the epidemic, the resulting epidemic curves demic, with data generated using Media Function (3) with can still vary drastically depending on the media function σ and γ values listed in Table 1 and p = 1000. Figure 4 chosen. Similar observations were made using different plots the resulting epidemic curves of System (6) using numbersofdaysofdatausedtofit β and p, and differ- Media Functions (1)-(3) (dotted, dashed and solid lines) ent Media Functions to generate the data (not shown). Figure 6 LHS-PRCC results for I = 300 for peak magnitude. This sensitivity analysis is done with 1000 bins. The rows correspond to Function (1), Function (2) and Function (3), respectively. The columns of this PRCC figure correspond to p = 10, p = 100 and p = 1000. Here I = 300 and 3 3 3 c γ = . 4 Collinson and Heffernan BMC Public Health 2014, 14:376 Page 7 of 10 http://www.biomedcentral.com/1471-2458/14/376 Figure 7 LHS-PRCC results for I = 1000. This sensitivity analysis is done with 1000 bins. The rows correspond to Function (1), Function (2) and Function (3), respectively. The columns of this PRCC figure correspond to p = 10, p = 100 and p = 1000. 3 3 3 Therefore, it can be concluded that key epidemic mea- Sensitivity analysis surements – peak number of infections, time of peak, end Within public health settings, a goal is to identify key of epidemic, and total number of infections – will also vary characteristics of an epidemic that drive the infection depending on the Media Function chosen. with the hope of determining public health measures that System (6) is useful in describing the mean behaviour can be implemented so that control or eradication of the of an epidemic, but it is unable to provide estimates of pathogen can be achieved. By performing sensitivity anal- variation of important public health measures. A stochas- ysis on System (6) with Equations (1)-(3), parameters that tic simulation lends itself well to demonstrating variation most affect epidemic outcomes for each media function within an epidemic. Here, we employ an Agent-Based can be identified, informing policy makers so that appro- Monte Carlo (ABMC) simulation. Figure 5 shows 100 sim- priate public health measures can be put into place. To ulation runs of the ABMC simulation when no media is conduct the sensitivity analysis, we use Latin Hypercube involved (top), and when Media Functions (1)-(3) are used Sampling (LHS) and partial rank correlation coefficients (second to bottom rows). Each simulation run (gray line) (PRCC) [23]. is shown, as well the mean of the simulation runs (solid We first conduct the sensitivity analysis with I , p c 3 line) and solution of the System (6) with the correspond- and γ constant to directly compare the media functions. ing media functions (dashed line). Note that the mean Figures 6 and 7 show the PRCC values determined for (solid line) and solution to System (6) are in agreement. peak magnitude when I = 300 (Figure 6) and 1000 This figure demonstrates that variability in the key epi- (Figure 7) for Functions (1)-(3) (top to bottom) when p = demic measurement occurs. Table 3 lists the mean and 10, 100 and 1000 (left, middle and right columns respec- standard error of each epidemic measurement when no tively). These figures demonstrate that for each set of media is considered (section a), and Media Functions (1)- parameters the PRCC value is similar. This is true for each (3) are included in the simulation (Sections b-d). In each key epidemic measurement - the PRCC values are similar section we list the simulation mean and standard error between different values of p and I . However the PRCC 3 c (top row), and the solution of System (6). values between key measurements are different i.e. the Collinson and Heffernan BMC Public Health 2014, 14:376 Page 8 of 10 http://www.biomedcentral.com/1471-2458/14/376 Figure 8 PRCC for system (6). Results are shown for peak magnitude, peak time, epidemic end time, and total number of infections (left to right) when no mass media function is considered, and functions (1)-(3) are used (top to bottom). Here, p = 100 and p and p are determined by 3 1 2 Equations(4) and(5). PRCC values are different if comparing peak magnitude to This also explains the very different epidemic curves pro- epidemic end time (not shown). duced by System (6) when different media functions are For further sensitivity analysis, all parameters are employed, notwithstanding the similarities in the media allowed to vary. Figure 8 shows the result of the sensi- functions when plotted at a specific level of media. tivity analysis for all four outcomes key to public health: peak number of infections, peak time, epidemic end time, Conclusion and total number of infections. The results demonstrate Technology and media play an increasing role in daily life. that the SEIR model is more (or less) sensitive to certain Mass media that is transmitted via technological media parameters depending on what Mass Media Function (1)- can therefore be used to inform the public during pan- (3) is used. For example, changes in β have a large effect on demics and epidemics. An understanding of the effects peak magnitude when Function (1) is used, but it has lit- of media during an epidemic or pandemic threat can aid tletonoeffectwhenFunctions (2-3)represent theeffects in the development of public health policy. Of particu- of media. Variable p also has a large effect on peak mag- lar interest to public health are the effects of media on nitude when media is represented by Function (1). It has key epidemic measurements - peak magnitude of infec- a similar effect when Function (2) is used, but it has no tion, time of peak, end of epidemic, and total number of effect on the system utilizing Function (3). infections. The sensitivity analysis indicates that models that Mathematical modelling has been used to study the include mass media influence will greatly depend on effect of media on epidemics by employing functions in different parameters, depending on the media function the transmission terms of mathematical models [6-14]. chosen. This makes it very difficult for policy makers to A survey of the literature identified three functions that determine an effective public health intervention strategy. have been utilized to represent media in disease modelling Collinson and Heffernan BMC Public Health 2014, 14:376 Page 9 of 10 http://www.biomedcentral.com/1471-2458/14/376 [6-14]. We have conducted a comparison of these func- Inclusion of mass media reports and media waning in tions to determine the effects of media function on key a model of multiple seasons with seasonal forcing is an epidemic measurement and variability within these mea- interesting direction for study. surements. We first demonstrated that by including mass Competing interests media in System (6) the peak magnitude of the epidemic The authors declare that they have no competing interests. and the total number of infections would decrease. We also determined that the time to peak and the end of Authors’ contributions the epidemic would also occur earlier. However, we also MSC performed the literature review. JMH and MSC developed the models, conducted the analysis and wrote the paper. Both authors read and approved demonstrated that, although the functions are similar in the final manuscript. shape and magnitude, the resulting epidemic curve can be quite different (Figure 5). Therefore, the key epidemic Acknowledgements measurements that are used to inform public health policy The authors would like to thank Nicholas Geard, Amy Greer, James McCaw, will be different. Furthermore, we demonstrated that vari- Seyed Moghadas, Jianhong Wu and Huaiping Zhu for many helpful discussions. The authors would also like to thank the reviewers for their ability in the key epidemic measurements also depends on comments and suggestions that have added value to this study The work was the media function chosen (Figure 5 and Table 3). funded by an Ontario Early Researcher Award, Mathematics of Information A sensitivity analysis on System (6) with the differ- Technology and Complex Systems (MITACS), and the Natural Science and Engineering Research Council of Canada (NSERC). Simulations were ent media functions was also conducted. Obtained from conducted on Sharcnet. this analysis was the insight that some parameters are important for some outcomes and not for others. We can Received: 4 September 2013 Accepted: 4 April 2014 conclude that due to the different fixed functions result- Published: 17 April 2014 ing in very different epidemic behaviours, there is no clear References control strategy present. Also, as a reult of the differ- 1. 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Physica A: Stat Mech Appl 2013, 392(23):5824–5835. doi:10.1186/1471-2458-14-376 Cite this article as: Collinson and Heffernan: Modelling the effects of media during an influenza epidemic. BMC Public Health 2014 14:376. Submit your next manuscript to BioMed Central and take full advantage of: • Convenient online submission • Thorough peer review • No space constraints or color figure charges • Immediate publication on acceptance • Inclusion in PubMed, CAS, Scopus and Google Scholar • Research which is freely available for redistribution Submit your manuscript at www.biomedcentral.com/submit

Journal

BMC Public HealthSpringer Journals

Published: Dec 1, 2014

Keywords: public health; medicine/public health, general; epidemiology; environmental health; biostatistics; vaccine

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