Modelling song popularity as a contagious process

Popular songs are often said to be ‘contagious’, ‘infectious’ or ‘viral’. We find that download count time series for many popular songs resemble infectious disease epidemic curves. This paper suggests infectious disease transmission models could help clarify mechanisms that contribute to the ‘spread’ of song preferences and how these mechanisms underlie song popularity. We analysed data from MixRadio, comprising song downloads through Nokia cell phones in Great Britain from 2007 to 2014. We compared the ability of the standard susceptible–infectious–recovered (SIR) epidemic model and a phenomenological (spline) model to fit download time series of popular songs. We fitted these same models to simulated epidemic time series generated by the SIR model. Song downloads are captured better by the SIR model, to the same extent that actual SIR simulations are fitted better by the SIR model than by splines. This suggests that the social processes underlying song popularity are similar to those that drive infectious disease transmission. We draw conclusions about song popularity within specific genres based on estimated SIR parameters. In particular, we argue that faster spread of preferences for Electronica songs may reflect stronger connectivity of the ‘susceptible community’, compared with the larger and broader community that listens to more common genres.


Introduction
Music is ubiquitous in society; everyone listens to it and most people prefer certain styles [1]. This ubiquity results in an enormous variety of music and a huge number of songs for listeners to choose from. In spite of this abundance, a remarkably small number of popular songs are almost immediately recognizable to most people at a given time. How does a song become popular and how is it that certain songs become so much more popular than others? What are the underlying social mechanisms that drive these processes?
There are many similarities between the release of a new hit song and the outbreak of an infectious disease. When an infectious disease first enters a population, it is transmitted from person to person via social interactions. Prevalence eventually reaches a peak and then declines as the susceptible pool is exhausted and/or infectious individuals recover. After a new hit song is released, it also 'spreads' rapidly through a population, from person to person and through various media, eventually reaching some peak popularity and then diminishing in appeal [2]. At the end of a disease epidemic, a large proportion of the population will have been infected with the disease, whereas at the end of a hit song's period of extreme popularity, a large proportion of the population will recognize that song.
Could the same social processes that facilitate spread of infectious disease in a population also drive song popularity? Popular songs are often described as 'viral' or 'catchy' as if they could 'infect' people; perhaps this description is more apt than has been previously recognized. In fact, the download time series for many popular songs that we examine in this study are similar in shape to time series for infectious diseases. This resemblance suggests that it is possible that there are social mechanisms underlying song popularity similar to the social mechanisms that drive the spread of an infectious disease, and has acted as our motivation to investigate standard epidemiological models as a tool to study song popularity.
Here, we consider how well a standard epidemiological model and a purely phenomenological spline can fit download time series for popular songs. For comparison, we also fit the same mechanistic and phenomenological models to stochastically simulated epidemic data. If song popularity is driven by a contagious process, then we would expect a mechanistic epidemic model to perform as well relative to a spline when the two models are applied to song download data as it does when they are applied to infectious disease data. If this is the case, then we can attach meaning to the epidemiological parameters estimated for popular songs based on disease transmission model fits and interpret these parameters to draw mechanistic conclusions about song popularity, which we cannot do with purely phenomenological models like splines.
We study data from a large and detailed database of song downloads from 2007 to 2014, a period when downloading (as opposed to streaming) was a primary method of music consumption (for further context on downloading versus streaming, see Aguiar [3]).

Background (a) Song popularity research
Song popularity has been the subject of much research. Some authors have sought to predict the peak and duration of a song's popularity based solely on its previous popularity rankings [4,5]. Others have attempted to determine what musical features make a song popular [6][7][8][9]. While there may be specific musical characteristics that predict popularity, social processes also affect how a song gains popularity. Given that complex interactions undoubtedly exist between musical and social factors, disentangling the influence of each on a song's popularity is a difficult task.
Previous research has found both support for [6,7] and evidence against [8,9] the idea that musical features can predict a song's popularity. Nunes & Ordanini [7] used audio information to show that songs that were number 1 hits on the Billboard Hot 100 Charts in the past 55 years had distinctly different instrumentation than songs that never climbed above the 90th position royalsocietypublishing.org/journal/rspa Proc. R [6] found that audio and lyric information about a song could each be used to generate better-than-random predictions about whether a song would be a hit. However, in their study of harmonic and timbral trends in the Billboard Hot 100 Charts over the past 50 years, Mauch et al. [9] found that the frequency of specific timbral characteristics cycled in the Hot 100 Charts as musical styles with different types of instrumentation came in and out of fashion. In addition, Pachet & Roy [8] failed to predict songs' popularity based on audio characteristics, regardless of whether these characteristics were quantified from an audio signal or from human input.
Several studies [10][11][12][13] have found that information from social media sites, social music sites or peer-to-peer file-sharing networks can predict song popularity, which hints at the underlying social processes driving song popularity. Bischoff et al. [10] built a model that predicted song popularity based on various Last.fm tags relating to user listening habits and previous popularity of the artist in question; Koenigstein et al. [12] demonstrated that search queries from Gnutella could be used to predict a song's peak position in the Billboard Hot 100 Charts. Schedl et al. [13] used Last.fm play-count data to predict popularity of artists in specific countries. They compared this method with predictions based on (i) user posts from Twitter, (ii) information from shared folders in Gnutella, and (iii) the number of pages returned by search engines that were related to an artist in a specific country. Kim et al. [11] also used Twitter posts to predict song popularity, finding that hashtags related to music listening behaviour of users could help forecast rankings of songs on Billboard charts. Zangerle et al. [14] extended this work to look at data over a longer time period. They found that although Twitter data alone were not sufficient to generate good song popularity predictions, using these data in multivariate predictive models significantly increased the models' predictive ability. Support for the idea that social interactions have a high impact on song popularity was presented in a study by Salganik et al. [15]. They played the same set of new music for several distinct groups of participants; song popularity was much less predictable, and between-song differences in popularity more extreme, when others' opinions of songs were presented with the music.
Lastly, researchers have used neural imaging to examine the influence of a song's overall popularity on adolescents' rankings of that song [16]. The functional magnetic resonance imaging (fMRI) data collected in this study suggested that teenagers are more likely to change their evaluation of a song to more closely align with its overall popularity rating as a result of the anxiety created by a difference between their opinion and the opinion of others. Neural activity in specific regions of the brain while listening to songs significantly correlated with sales data for that song over the next 3 years, even though subjective ratings of the songs from participants did not [17].

(b) Epidemiological modelling
Infectious disease spread is commonly studied using a compartmental framework, in which individuals are classified according to disease state [18,19]. In typical situations where individuals acquire immunity after recovering, the simplest framework involves compartments containing 'susceptible', 'infectious' and 'recovered' individuals. This model is known as the SIR model (figure 1). The rate at which individuals move among the three compartments is given by a system of ordinary differential equations, where β is the transmission rate and γ is the recovery rate. The SIR model also yields three slightly more intuitive parameters: the mean infectious period, 1/γ , the initial epidemic growth rate, r = β − γ , and the basic reproduction number, R 0 = β/γ . R 0 is the expected number of individuals one infectious individual would infect in a wholly susceptible population. The expected final size Z of an epidemic can also be calculated based on R 0 [20,21] by solving for Z in the final size relation, Z is the expected proportion of the initially susceptible population that will have been infected over the entire course of the epidemic. If the SIR model is interpreted in the context of popular songs, individuals are classified as being 'susceptible to', 'infected with' or 'recovered from' a song. The mean infectious period 1/γ measures the average time period for which an individual will continue to enjoy listening to a song, during which they may tell others about this song, thus 'spreading' it through the population. The basic reproduction number R 0 measures the average number of people in a wholly susceptible population who will be influenced to download a new song by one individual who is actively listening to and talking about this song. An individual has 'recovered' from a song when they are no longer actively listening to the song and spreading it to others.
When applied to infectious diseases, the SIR model can be used to draw a number of useful conclusions about an epidemic, such as how long an epidemic will last, what the final size of an epidemic will be and how quickly a disease will spread in a population. If it were successfully applied to song spread, these might be translated into conclusions about an epidemic of song downloads. For instance, it might be possible to estimate the duration of a song's popularity, how many people in total will download it or how quickly it will become popular in a population. This is not the first time epidemiological models have been applied to the dynamics of song popularity. A similar idea was employed by Tweedle & Smith? [22], who studied the effects of positive and negative media attention on 'Bieber Fever'. However, while they were working with an epidemiological model, their study was entirely theoretical-they did not apply the ideas to any data. Rather than considering the general dynamics of song popularity, Tweedle & Smith? [22] focused on the excessive popularity of a particular artist within a specific demographic.

Description of the data
The database used for this study contains information on almost 1.4 billion individual song downloads. These data were obtained through a data-sharing agreement with MixRadio. Downloads occurred through Nokia cell phones in 33 countries over a 7 year period (2007-2014). 1 Each data entry includes information about the download such as track title, artist name, artist genre classification and time of download. Various metadata about users are also housed in the database, including user ID, total number of downloads and user country. Because of its size and the nature of the data it contains, the database is an excellent tool for cultural and social investigations relating to music [23][24][25][26].
We focused our investigation on popular songs in Great Britain (GB), a country with an active downloading history. A list of the top 1000 songs ranked by number of downloads in GB was royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 477:

Methods (a) Top 1000 songs in Great Britain (i) Data extraction and aggregation
The top 1000 songs in GB were defined as those with the most downloads in the database between 2007 and 2014. The database was queried using the open-source MySQL implementation of SQL [27]. The list of songs on which to test the SIR model was narrowed down by eliminating the 50 Christmas and holiday songs found in the top 1000 songs downloaded in GB. The time series for these songs display a pattern similar to seasonal epidemics, which the simple SIR model (equation (2.1)) cannot generate [18,19] (meaning that fitting the SIR model to these songs is not really a fair test). Minute-by-minute download counts for the remaining 950 songs were extracted, then aggregated at coarser time scales using the R statistical programming language [28]. The finest time scale used for most songs was daily since aggregating at time scales finer than this yielded noisy download time series and poor fits; however, some songs gained popularity so quickly that aggregating at the daily level obscured an initial increase in downloads. For these songs, the beginning of the time series was aggregated at a finer time scale than the rest to produce a time series that the SIR model could be fitted to; finer aggregation was conducted up to the point where the peak number of daily downloads occurred.

(ii) SIR fits
The SIR model was fitted to each resulting time series using the package fitsir in R; this package employs least-squares fitting to match solutions of the SIR model to a given time seriescode available at https://github.com/bbolker/fitsir. To account for the possibility of multiple least-squares solutions, Latin hypercube sampling of the parameter space [29] was used to generate 100 possible SIR fits for each song and the best fit was then selected. The parameter space consisted of the transmission rate β, the recovery rate γ and the initial condition (S 0 , I 0 , R 0 ), in which the initially recovered population is R 0 = 0, making the initially susceptible population S 0 and initially infectious population I 0 dependent on each other.

(iii) Spline fits
Cubic splines [30] were also fitted to each of the aggregated download time series in R. Three degrees of freedom (corresponding to one interior knot and two boundary knots) were used in all spline fits to match the three free parameters in our SIR model (the transmission rate β, the recovery rate γ and the initial condition (S 0 , I 0 , R 0 = 0)).

(iv) Goodness of fit
To compare the epidemiological model and the phenomenological model (cubic spline), we calculated the relative root mean squared error (RRMSE) to measure goodness of fit. This relative fit measure was calculated by finding the average relative distance between model trajectory and song download data point, where c represents a download data point (i.e. the number of downloads that occurred at a particular point in time) and s represents a point on the SIR or spline model trajectory (i.e. the predicted number of downloads at the same time). A lower relative fit measure implies a better fit.
(v) Minimum R 0 and Z A minimum possible basic reproduction number R 0 was also determined for songs released in GB. This was done using the final size formula (2.2) and the assumption that the final size Z for an epidemic of downloads for a particular song was the proportion of the initially susceptible population (S 0 ) that actually downloaded the song, The minimum possible final size Z for the country was calculated by taking the number of users in GB as the greatest possible susceptible population (S 0 ) and the smallest number of downloads for a song in the top 1000 most downloaded songs as the smallest possible number of 'total downloads'. Since the final size formula (2.2) implies that Z strictly increases with R 0 , it could then be used to determine the minimum possible R 0 (or R min ) for GB, based on this minimum possible Z.

(vi) Criteria for satisfactory fits
At this point, the set of songs being considered was further restricted to those that yielded reasonable fits, meaning that any songs with an estimated R 0 less than R min were excluded from further analysis. In addition, visual inspection determined that songs for which the relative fit measure (see equation (4.1)) was greater than 11 gave a poor fit, so these songs were also excluded. Epidemiological parameter estimates were obtained based on the SIR fits for the remaining subset of songs and analysed.

(vii) Final size and initially susceptible population
A final size was determined for each individual song based on the estimated R 0 (using equation (2.2)). This final size Z is a proportion of the initially susceptible population S 0 , i.e. the number of individuals initially susceptible to downloading a song. Equation (4.2) could therefore be used to calculate S 0 for each song based on knowledge of total downloads and estimated final size.

(b) Simulated epidemic data
One thousand idealized infection curves were also generated using the Gillespie algorithm [31] to compute a stochastic solution for a given set of parameters. While it assumes that the intrinsic uncertainty comes entirely from an independent Poisson process (i.e. successive downloads are uncorrelated with each other-within the overall framework of the epidemic process, there is no additional heterogeneity such as 'super-spreading' events), this approach does generate reasonable levels of stochastic variation. The SIR model and cubic splines were fitted to each of the stochastic time series with the same methods used for song data. Goodness of fit was calculated using equation (4.1) as described above.

(a) SIR model fits and spline fits
Of the 950 songs in our sample set, 828 (87.2%) met our fitting criteria, i.e. they were well captured by the basic SIR model. Figure 3 compares the SIR model fit with the cubic spline fit for six of the songs that were well fitted by the SIR model. A cubic spline yielded a better fit for 484 of the 950 songs in our sample (50.9%); of the 828 songs that were well captured by the SIR model, 422 were better fitted by the SIR model than by the spline (51%; see figure 4). For the song download curves, the median relative fit measure for the SIR model was 1.50. This number was slightly lower (i.e. better) than the median relative fit measure for cubic splines, which was 1.85; however, for SIR fits, the distribution of the relative fit measure had a long tail ( figure 5). The SIR model often yielded a visually better fit for songs (i.e. a fit that better followed the main rise and fall of the download trajectory) even in cases where the spline yielded a better relative fit measure; this was particularly true for songs that had a very good SIR fit. Similar trends were seen in the performance of the SIR model against cubic splines when applied to simulated epidemic data, Of the 978 epidemic curves in our sample, 967 (98.9%) were well captured by the SIR model, based on our fitting criteria (they had a relative fit measure of less than 11). Of the entire set of 978 simulations, a cubic spline yielded a better fit for 462 of them (47.2%); of the 967 well captured curves, the SIR model yielded a better fit than a spline for 516 of them (53.4%; see figure 4). As with song download curves, the median relative fit measure for the SIR model applied to epidemic curves (1.53) was slightly lower than the median relative fit measure for cubic splines (1.62), with the SIR model fits again displaying a long tail in their relative fit measure distribution (figure 5).
These results show that the SIR model performed very similarly against cubic splines when fitted to songs as it did when fitted to epidemic curves ( figure 5). The SIR model is a good representation of the mechanisms that drive infectious disease spread; since it appears to fit song download data as well as it fits simulated SIR epidemic data, it is reasonable to propose that it captures some underlying social drivers of song popularity.

(b) Estimated and derived parameters
For the song download time series, the distributions of all estimated (i.e. fitted) or derived (calculated from the estimated parameters) parameters were explored. Once we concluded that the SIR model was a reasonable approximation of the scenario being studied, we could      draw conclusions based on the interpretation of these parameters in the context of song popularity. Interpretable parameters are one major advantage of a mechanistic model over a phenomenological model such as a spline. Table 1 gives the median estimated values for some key epidemiological parameters across the genres in our sample. The basic reproduction number R 0 has a natural interpretation in the context of songs; it tells us-in a fully 'susceptible' population-how many people would be influenced to download a song by an 'infectious' individual actively spreading that song (e.g. by talking about the song, playing it, sharing it on social media or requesting it on the radio). Figure 6    distribution of R 0 for the well-fitted songs from each genre in our sample set. The median R 0 varies substantially among genres, with Dance having the lowest median R 0 of 2.84 and Electronica having the highest median R 0 of 3430. One might expect Pop to have the highest R 0 , since this genre of music is the most widely played on the radio, at public events, on TV and in movies; however, its median R 0 of 35 is far outstripped by genres like Electronica, Rock (129) and Rap/Hip Hop (311). When looking at individual songs, the estimated initially susceptible population S 0 for a song (based on the song's fitted epidemic curve) was almost always less than the total download count for that song. Since the total download count should not exceed the initially susceptible population, we instead examined the S 0 values calculated using equation (4.2) (based on applying the final size relation equation (2.2) to obtain Z from the estimated R 0 and using the observed download count, as described in §4). Table 2 gives the median values by genre for download count, estimated final size and calculated initially susceptible population. The final size Z was usually close to 1, with 727/828 (87.8%) songs having Z > 0.9 and 640/828 (77.3%) having Z > 0.99, meaning that the calculated S 0 was generally very close to the download count. Since S 0 = 1/Z × (total downloads) (equation (4.2)), the songs for which Z is close to 1 form a straight line with slope slightly greater than 1 on the graph of calculated S 0 versus download count ( figure 7). However, Z being nearly constant with values close to 1 for most of the songs in our sample set does not mean that R 0 was nearly constant ( figure 6 shows the wide distribution of estimated R 0 values). This is because, according to equation (2.2), Z → 1 quite quickly as R 0 increases (Z = 0.98 already for R 0 = 4). The high estimates of R 0 for most songs (655/828 songs had an estimated R 0 > 4) explain why most estimates of final size Z were quite close to 1.

Discussion
The SIR model describes song download trends for popular songs well; it may be a good representation of the processes driving song popularity. Our results support the idea that the model could capture an underlying 'song transmission' mechanism or a contagious process that drives song popularity. Since the SIR model is mechanistic, the parameters that were estimated from SIR fits to song download data (table 1) and derived from those estimates (table 2) can provide insight into song popularity based on the mechanistic interpretation of these parameters. In particular, we can make inferences about the distinguishing characteristics of fans (susceptible populations) of different genres.  Although the median values for many of the estimated parameters differ significantly between genres (tables 1 and 2), median final size Z is quite consistent. Most songs infect their entire susceptible population, meaning that Z (and therefore median Z) is almost always nearly 1.
The median R 0 , however, varies substantially between genres. For Pop and Electronica, this difference between median R 0 tells an interesting story. Although we might expect Pop to have the highest median R 0 of all genres because of its name, it is in fact Electronica, a more niche genre, that holds this record by two orders of magnitude (table 1). However, this high median R 0 does not seem to correspond to many more downloads of Electronica songs than Pop songs; in fact, the median download count is relatively consistent across genres-a result we might expect so long as the songs for each genre were evenly distributed through the list of top 1000 most popular songs based on download count. Instead, we see Electronica's high R 0 values manifest in shorter, faster epidemics; the download time series for Electronica songs show the majority of downloads happening in a shorter time period than Pop songs, meaning that these songs appear to gain popularity faster than those in other genres, and to burn through their susceptible populations more quickly. Indeed, the median initial growth rate r is substantially higher for Electronica songs than for Pop songs, and the median doubling time is substantially shorter (table 1). These observations support previous work which showed that the pattern of downloads differs for songs depending on their genre [24,25].
What does this pattern tell us about fans of the different genres and how songs from each genre are transmitted? Perhaps fans of Electronica transmit these songs more actively or more effectively. The social network of Electronica fans might be more strongly connected than fan communities of other music genres such as Pop. Electronica fans may be more passionate about their favourite songs and bands than Pop fans, and therefore talk about and promote their favourite songs more. Perhaps Pop, being a more mainstream genre, is spread chiefly through more passive means like the radio. Such mechanistic hypotheses could not be derived from the analysis of a phenomenological model like our cubic spline fits.
This comparison between Pop and Electronica can be taken further if we consider susceptible populations. In our context, the susceptible population is defined as the group of individuals who may download a song if exposed to it. Since most individual songs had a final size Z very close to 1, their download count tended to be very close to their calculated susceptible population. Interestingly, the median download count and median susceptible population do not vary much between genres (table 2). This result again defies expectations about the Pop genre, namely that Pop songs would have the largest susceptible populations since it is literally 'popular music'. It is more likely instead that more people are exposed to Pop (and possibly that a higher number of people will tolerate listening to Pop), whereas only the susceptible population is exposed to a more niche genre like Electronica.
(a) Future research Although many people now consume music via streaming services, this study uses data from a large and detailed database of song downloads. Due to the nature of the data available, we have focused our analysis on song download behaviour. With a different dataset, the same type of analysis could be applied to streaming data, which might offer different information.
The SIR model is one of the most basic epidemiological models; as such, it neglects many aspects of disease transmission. Although our results show that the model describes song download dynamics well, it has too little structure to represent all the nuances of a song spreading through a population. Fitting other disease models, with more biological structure, to song download data might illuminate the most important processes driving 'song transmission'. Details worth considering include vital dynamics ('births' and 'deaths', i.e. users who join or leave a downloading service like MixRadio), the role that the conditions under which a song is released play in its success or failure (potentially analogous to heterogeneity in transmission, which has been included in simple disease models by modifying the transmission term to be nonlinear in S and/or I [32,33]), or the effect that social structure and human behaviour have on disease dynamics [34]. A reservoir model, such as those used to model spread of waterborne disease, might capture the influence of the radio and streaming services on the spread of a song [35,36]. A disease model that incorporated an initial pulse of infection might better represent the effect of mass media and marketing promotion that some songs receive. Imperfect vaccination models could be used to model the changing musical preferences of individuals within a population [37]. With access to live stream data, rather than just download data, one could apply a model that accounts for decay in immunity to the dataset, such as the SIS (susceptible-infectious-susceptible) or SIRS (susceptible-infectious-recovered-susceptible) model [38], or models with both decay of immunity and nonlinear incidence [32,33].
It might also be fruitful to pursue the concept of super-spreading in the context of song download epidemics, i.e. the idea that the basic reproduction number R 0 can vary substantially within a population when certain individuals have a higher degree of infectiousness than others [39]. In the context of songs, super-spreaders might be people who express their opinions of a song much more often and readily, more strongly/passionately and/or through a widely accessed social media platform. A potential avenue of future research would be to identify characteristics of song super-spreaders that would be detectable in the data.
The usual implementations of cubic splines result in smooth curves that maximize some measure of fit to the data, but they are not constrained to produce only unimodal curves (i.e. curves with an intermediate maximum, the typical shape of an epidemic curve). In our study, this means that a cubic spline that yielded a very good fit (according to our criterion equation (4.1)) might not follow the shape of the data it was fitting (e.g. 'Bad Romance', 'Breathe Slow' and 'Heartless' in figure 3). It would be interesting to investigate the performance of phenomenological models that have a built-in assumption of unimodality (such as those in [40,41]).

Conclusion
This study has explored the utility of a mechanistic epidemic model (the SIR model) for describing song popularity by comparing its ability to capture song download patterns against that of a phenomenological model (cubic spline), and comparing this with each model's ability to capture infectious disease spread patterns. The SIR model performed similarly to the cubic spline both when fitted to song download data and when fitted to simulated epidemic curves, which is what we would expect if popular songs are indeed 'infectious'. Thus, our results indicate that song popularity may be driven by an underlying contagious process. Since the SIR model is mechanistic, we were able to make mechanistic inferences about song popularity based on parameters estimated from fits. Specifically, we drew conclusions about how the downloading and music-sharing behaviours of music fans may differ by genre.
To our knowledge, our analysis is the first comparison of the ability of splines with mechanistic epidemiological models to fit epidemic curves. When presented with infectious disease data, it is natural to consider mechanistic transmission models rather than phenomenological models. By contrast, if one is merely considering the possibility that some observed data were generated by a contagious process, then it is natural to consider a variety of models. Although the focus of this study is on song popularity, this work has an important methodological theme that has broader significance: to infer that 'mechanism X' was likely involved in generating a given dataset, the key question to ask is not 'Does model X do better than model Y?' but 'Does model X perform better than model Y to the same extent that it does for data known to have been generated by model X? ' We have shown that epidemic models offer a powerful tool for analysing music downloading trends and studying the mechanisms that drive song popularity. Applying some of the many possible extensions to the simple SIR model could help us to learn more about how songs become popular and how the mechanisms that drive song popularity relate to those that drive disease epidemics.
Data accessibility. The original database containing individual downloads from individual users is proprietary.
The aggregated data that were used for analysis and to make the figures in this paper (including fit measures and estimated/derived parameters) have been provided, along with a sample of song download proportion data. The data and scripts used for analysis are hosted in the github repository https://github.com/drosati/ SongDownloadEpidemics [42].