A closed space with poor ventilation, a large crowd. People talk on top of the general noise at the wedding. They sing at the choir practice or church service. They breathe rapidly in a Zumba class. And one of them feels unwell – or doesn’t yet, but will in a few days. How many people will they infect?

Answering this question is very different from calculating how many people a typical virus carrier infects. On average, the carrier of the novel coronavirus (absent restrictive measures) infects three people. But each of the situations described above is a real case from the COVID-19 pandemic, where 40–80 people were infected directly by the carrier. This is called a superspreading event – when one person transmits a virus to a disproportionately large number of people.

Although the average indicator, the basic reproduction number (R₀), is widely used in research, the shape of the epidemic mainly depends on such atypical situations. Therefore, along with R₀, a dispersion parameter k is used to describe how the virus spreads. It reflects the difference in how many people are infected by the virus carriers.

If all the carriers of the virus are lined up according to how many people they have infected, then at the beginning of the row there will be superspreaders, and at the end – those who have not infected anyone. The parameter k will indicate where the group of most active spreaders ends and the group of less active ones begins. When almost all infections originated from a small group of superspreaders, and most carriers did not spread the virus, k is close to zero. When all the carriers have infected approximately the same number of people, so there are no superspreaders, k will rise to 10 and above.

For the Spanish flu pandemic, this parameter was about one, which corresponds to a more uniform spread of the virus. But for the coronavirus epidemics, k was close to zero, reflecting the notable role of superspreading (0.16 for SARS, 0.25 for MERS). This may be due to the airborne and aerosol transmission of coronaviruses, as these viruses are emitted even with loud speech and are suspended in the air long enough to be inhaled by many people.

A COVID-19 preprint gives an even lower estimate of k at 0.1, though earlier studies suggested that the role of superspreading in this pandemic is not higher than for other coronavirus epidemics. Moreover, the vast majority of carriers do not infect anyone.

Virus transmission depends on how carriers behave in different situations. If a person is self-isolating with pronounced disease symptoms, they will most likely not infect others – not only because they do not meet anyone, but also because the longer it is after the onset of symptoms, the less one emits the virus (Bullard et al. 2020; Cheng et al. 2020). But a few days before or after the onset of symptoms, carriers are the most infectious (He et al. 2020). Therefore, if during an asymptomatic period a person spent more than 10 minutes in crowded closed spaces and was not wearing a mask, it is very likely that they have already become a superspreader. Researchers believe that diminishing the likelihood of each virus carrier to appear in such situations, as well as tracking the close contacts of the carriers, can significantly reduce the number of virus transmission (Liu, Eggo, and Kucharski 2020).

To learn more about superspreading, we have performed a systematic search in the scientific literature database Scopus and have built a map of publications based on their reference lists (Figure 1). Proximity in this map and belonging to the same cluster mean that the papers cite the same publications, therefore the papers are likely to consider similar issues. The map is built using VOSviewer software.

Publications on superspreading are split into six clusters:

• light blue, top left: 2003 SARS epidemic,
• yellow, center left: 2015 MERS epidemic,
• navy, center: ecological and spatial factors,
• purple, top right: social networks,
• blue, bottom left: tuberculosis spreading,
• orange, bottom right: vector-borne diseases (not covered in the review).

The most important paper in the whole map shows that the number of people infected by a virus carrier is very unevenly distributed (Lloyd-Smith et al. 2005). The authors test this hypothesis on the data obtained by tracking contacts during the epidemics of eight different viruses, and it turns out that this is a common feature of all diseases transmitted between people. So, for SARS, in more than 80% of transmission cases, the virus was spread by 20% of carriers. Researchers note that targeted control measures (isolation of virus carriers and tracking their contacts) in such a situation are more effective than general quarantine. The authors also propose a mathematical definition of superspreading to predict the frequency of such cases with known R₀ and k.

Light blue cluster: 2003 SARS epidemic

The 2003 SARS epidemic has started in a Hong Kong hotel, where tourists got infected from a sick traveler and spread the virus to different countries. In the article by Shen et al. (2004), SARS superspreading is defined as the transmission of the virus to at least eight people, and researchers at that time discovered four such cases in China.

At the same time, according to statistics, on average, the virus was transmitted to 2.7 people in the first stages of the epidemic and a smaller number afterwards (Riley et al. 2003). Nosocomial transmission of the virus was quite common, and if not for this, the epidemic would have taken on a much smaller scale (Small et al. 2005, Yu et al. 2007).

By Li et al.’s estimations (2004), 71% of infections in Hong Kong and 75% in Singapore occurred during superspreading events. They happened partly because patients went to the doctor no earlier than four days after the onset of symptoms. In this regard, the authors emphasize the importance of early diagnosis and isolation of the infected.

Yellow cluster: 2015 MERS epidemic 

Partly, the superspreading of the MERS coronavirus is also associated with nosocomial infections, since carriers of the virus went to different hospitals or were transferred from one hospital to another. For example, Kang et al. (2017) found no other clinical differences between superspreaders and other carriers of the virus than that superspreaders spent more time non-isolated in a hospital. Besides this, behavioral factors are also mentioned: many people visited the patients, and virus carriers entered large crowds and did not avoid physical contact during greetings (Al-Tawfiq and Memish 2016).

The study by Chowell et al. (2015) compares virus transmission in the case of MERS and SARS. Most cases of MERS transmission occurred because people were in the same hospital as the first infected person. In the case of SARS, more cases were concentrated among healthcare providers. These differences indicate the importance of special patient management to decrease the transmission of the virus.

Navy cluster: ecological and spatial factors

Paull et al. (2012) note that the difference in how many people the carrier of the virus infects is closely related to the differences in the environment, such as population density, temperature and humidity, ecosystem fragmentation, as well as the interaction of the environment and the genetic characteristics of the population. Hawley and Altizer (2011) urge studying the immune system and life history of superspreaders.

In the cluster, virus superspreading during human mobility (Bossak and Welford 2010) and animal migration (Craft et al. 2011) is also discussed.

Purple cluster: social networks

When epidemiologists model society as a network of contacts, they can use three main options for its structure. It can be a random network where everyone has an equal chance of communicating with everyone – this is a simple but not very realistic configuration. Or a scale-free network, where most nodes have few connections and only a small number of so-called network hubs have many contacts. Such a structure, for example, is used in modeling the spread of sexually transmitted diseases, or computer viruses. The third option is a “small-world” network. In this one, people are mainly connected at the local level, inside groups such as school, family, store visitors, metro passengers – but some connections are long-distance ones if a person communicates with several groups.

Depending on which structure fits reality best, proposed measures to identify superspreaders may vary. For example, Liu et al. (2015) discuss various centrality measures in the network, in particular, such a centrality analysis method as k-shell decomposition. In scale-free networks, the most central positions are occupied by hubs, people with a large number of social contacts – it is they who should be proactively tested for the virus and isolated at the first suspicion of illness.

But Masuda, Konno, and Aihara (2004) point out that in the case of SARS, the contact structure was resembling the “small world” model. Superspreaders were not necessarily too socially active. Rather, they led a normal life, but at the time when they were especially contagious, they were connected to groups susceptible to the virus.

Small, Tse, and Walker (2006) share this approach and model containment measures in these settings. As the main factor in the spread of the virus, they consider the period of more than six days, when the carriers of the virus spread the infection (for example, inside the hospital). Researchers suggest controlling the epidemic either by restricting far-reaching connections through partial quarantine or by quickly hospitalizing and isolating people who show symptoms of the disease.

Blue cluster: tuberculosis spreading

The publications of this cluster are devoted to the transmission of bacterial infection, Mycobacterium tuberculosis. It is more difficult to become infected with tuberculosis than coronaviruses, as it requires prolonged or regular contact.

Walker et al. (2013) suggest a new method to reconstruct the transmission pattern of this infection through Mycobacterium genome analysis. Previously, strains of tuberculosis were detected through MIRU-VNTR genotyping, and if the patients had different strains, this excluded the possibility that one of them transmitted the infection to another. However, if the strain was the same, it was difficult to understand how the infection was transmitted because patients often could not report such data. Molecular typing allowed only to indicate the presence or absence of superspreaders, based on the extent to which groups of patients with the same type of bacterium varied in size (Ypma et al. 2013).

The alternative method, whole-genome sequencing, identifies mutations in the genes and thus allows one to establish the pathways of infection. Since mutations most often accumulate, the dynamics of transmission can be traced between patients with different sets of mutations. In this way, it is possible to identify superspreaders that occupy central positions in the structure of the phylogenetic tree. Moreover, transmission from one spreader to many is characterized by a low diversity of Mycobacterium genomes (Comin et al. 2020).

Please proceed to page 2 to see general reviews on superspreading and the description of our data.

Imagine that an infection outbreak occurs in a completely vulnerable society. At once, five people become sick, and when communicating with healthy people, they spread the virus. If healthy and sick people can freely communicate, soon everyone becomes sick.

We could make it 100 people instead of five, or imagine that half of the sick do not contact anyone. All this would affect how much longer, or less, does it take for the disease to spread within the population.

The idea that the population is split into two groups, the healthy and the sick, is our model, that is, a simplified vision of the society. Here, it is a bit too simplified, as people do not become sick forever: if the disease is not fatal, at some point they recover. It would be more realistic, then, to include a third group in our model: the recovered. They would be different from the healthy in that when an infected person contacts them, they can not become sick, as now they have an immunity against the disease.

The models that require splitting the population into groups, or classes, are called compartmental models and widely used in epidemiology. Our model is a basic compartmental model. It is called

SIR model: S for susceptible (not immune), I for infectious (spreading the virus), R for recovered (immune, at least temporarily). People transit between the classes like this:
S → I → R.
This sequence is logical and fixed, therefore the model is called deterministic.

Knowing how many people are in each class at each time point helps researchers to predict the spread of the disease and the duration of the epidemic. If we make the model even more sophisticated, we can demonstrate how various factors affect the outcome: for example, how quarantine or physical distancing reduces the number of the sick at the peak of the epidemic. (See also such estimations based on historical data on Spanish flu pandemic in 1918–1919.)

With this post, we start a series of publications on epidemiology models. Today we will look at the map of publications that use the SIR model explained above. 

For the review, we have performed a systematic search in the scientific literature database Scopus and have built a map of publications based on their reference lists (Figure 1). Proximity in this map and belonging to the same cluster mean that the papers cite the same publications, therefore the papers are likely to consider similar issues. The map is built using VOSviewer software.

The studies using SIR models can be split into five clusters:

• navy, top center: dynamics in SIR models,
• light-blue, on the left: global stability and vaccination influence,
• purple, center: nuanced SIR models,
• yellow, on the right: social networks,
• blue, bottom: travelling waves.

Cluster description

Navy cluster: dynamics in SIR models

Publications of this cluster build the models that include the time dimension. For example, the most notable study here models measles spread in Britain (Bjørnstad, Finkenstädt, and Grenfell 2002). Before the start of mass vaccination campaigns, in large cities, the disease was ever-present with varying intensity. At the same time, in smaller communities there were occasional outbreaks, and after each of them, the virus temporarily became extinct. The researchers captured this dynamic, as well as changing seasons that contribute to the spread, by modeling the epidemic through two-week intervals and this way developing a new type of SIR model – TSIR model (T stands for time-series). It appeared, for example, that the number of cases varied proportional to the city size, and transmission rates had a strong seasonal variation.

The cluster, in general, describes the dynamics of diseases. Lloyd (2001) considers the varying chances to recover depending on the time since infection. Tien and Earn (2010) introduce another transmission pathway (besides direct contact), with the infectiousness that decays with time.

The studies also consider the seasonality that affects the spatial and temporal dynamics of epidemics (Keeling, Rohani, and Grenfell 2001). These publications consider not only the outbreak and the infectious period but also the post-epidemic dynamics (Stone, Olinky, and Huppert 2007).

Light-blue cluster: global stability

For such a dynamical system as a society, global stability exists when from any state the system is currently in, it is moving towards a stable state. While at first, the state of the system is “zero infected, all healthy”, with an outbreak of epidemic this state changes. If after such a disturbance the society moves back to the initial state, this state can be considered globally stable. 

The studies in this cluster investigate such phenomena. For example, Beretta and Takeuchi (1995) and McCluskey (2010) consider the situation where the disease is ever-present with varying intensity – that is, the steady state is not a point of “no infected”, but a fluctuating number of the sick. When the outbreaks of the disease arise, after them this number returns to the previous amplitude of oscillations. The authors conclude that among all the possible patterns of the epidemic, this pattern is as globally stable as the situation without any disease at all. 

Some papers discuss how the global stability emerges from pulse vaccination – a method of repeatedly vaccinating a group at risk until the pathogen stops spreading (d’Onofrio 2005; Stone, Shulgin, and Agur 2000). Shulgin, Stone, and Agur (1998) show that even with the complex dynamics of the system, pulse vaccination can still lead to epidemic eradication.

Purple cluster: nuanced SIR models

Studies here consider how SIR models can be fitted to specific cases by adding relevant characteristics of the society. For instance, Dangbé et al. (2017) model cholera spread and integrate such factors as socioeconomic status of the population, its behavior (in particular, good hygiene practices), and environmental factors. In turn, Miller Neilan et al. (2010), also studying cholera, introduce additional components in the model that indicate how infected is the drinking water and whether the infection can remain asymptomatic. 

Hyman and Li (2007) split the class of the infected people in several classes, according to how long they are infected. The authors presume that some people change their behavior as they develop the disease, either communicating less actively or seeking treatment, and thus become less infectious and recover at higher rates.

Finally, the chapter by Allen (2008) reviews several possible extensions of SIR and suchlike models, which integrate probabilities (e.g., of disease extinction or outbreak) and therefore are called stochastic.

Yellow cluster: social networks

This cluster shows how the disease can be transmitted through the networks of communication. We could think that the most well-connected nodes in the network infect the most people; but if their communication circle is isolated, the infection would not spread from their immediate neighbors. A more realistic assumption is that the nodes that are connecting different communities are the most effective transmitters. However, in very large networks it requires too much calculation to identify such nodes. One of the most cited papers here by Chen et al. (2012) presents a new, more easily computable metric to find the most infectious nodes. Another method is suggested by Li et al. (2014).

In general, publications here discuss different approaches to modeling epidemic spread through networks (Kenah and Robins 2007; Lindquist et al. 2011), including the situations where the virus can mutate or more then one pathogen is present (Masuda and Konno 2006). A distinct topic is modeling the spread of sexually transmitted diseases (Rocha, Liljeros, and Holme 2011).

Blue cluster: travelling waves

Waves “travel” if they move in space, as sound moves from us to a person we speak with. If we speak loud enough, and the person does not have hearing problems, they will hear what we say. Now if instead of sound we consider the disease, using this approach we can model whether it will spread (Bai and Zhang 2015; Li and Yang 2014; Wang and Wang 2016). As Wang and Wu (2009) show, this depends solely on the basic reproduction number (the number of people that a sick person can infect when everyone is susceptible). The speed of the disease spreading, however, depends also on whether people travel long distances and other factors.

Please proceed to page 2 to see general reviews on epidemic modeling and the description of our data.

We have turned to the publications on human-animal interfaces. For the review, we have performed a systematic search in the scientific literature database Scopus and have built a map of publications based on their reference lists (Figure 1). Proximity in this map and belonging to the same cluster mean that the papers cite the same publications, therefore the papers are likely to consider similar issues. The map is built using VOSviewer software.

The papers roughly split into five clusters:

• navy, center: epidemiology, virology and genetic analyses (coronaviruses and other viruses),
• purple, top right: swine viruses,
• yellow, bottom right: avian viruses,
• blue, second from the left: opportunistic infections.
• light blue, on the left: hepeviridae viruses (hepatitis E viruses).

The majority of publications are located on the right side of the map. Purple cluster (top right) and yellow cluster (bottom right) are proximately located because both of them are about influenza. They are located close to the navy cluster (center) because it contains publications that describe pathogens with the greatest zoonotic potential, such as influenza virus A. Blue (second from the left) and light blue (on the left) clusters are located distantly from purple and yellow clusters because they are about different types of infections with different subsequent diseases, therefore their reference lists rarely overlap.

Cluster description

Navy cluster reviews different types of viruses such as henipavirus, rotaviruses, and coronaviruses. A lot of publications are about virology and genetic diversity of viruses, especially about their genomes. Many works also describe the emergence and resurgence of viruses, their epidemiology, and how they are transmitted. Particular attention is paid to viruses transmitted by bats. These works are located at the bottom of the cluster. As for the coronaviruses, a lot of the works trace the transmission from camels to humans.

Purple cluster devoted to swine viruses contains a lot of publications that tell about the need for increased surveillance and further research. Many publications are connected to the recent pandemic in 2009.

Yellow cluster on avian viruses covers their epidemiology, origins, transmission, causes, and their potential. Some publications also describe genetics and molecular structure.

Light-blue cluster represents the family of viruses hepeviridae and the main disease that they cause – hepatitis E. Publications cover the origins, characteristics, modes of transmission, consequences, and prevention. Most of the publications study swine hosts. Most of the research in this cluster was published in the 1990s. The most cited publication describes the connection between novel swine hepatitis E virus and the human hepatitis E virus (Meng et al. 1997).

Blue cluster is about opportunistic infections that are caused by pathogens and usually only affect people with a weakened immune system (taking the opportunity of it – hence the name). The majority of publications study people with HIV and cancer. Publications also describe molecular epidemiology and genetic characteristics of these infections.

General reviews on human-animal interfaces

• Greger, Michael. 2007. ‘The Human/Animal Interface: Emergence and Resurgence of Zoonotic Infectious Diseases’. Critical Reviews in Microbiology 33(4):243–99.
• Jones, Kate E., Nikkita G. Patel, Marc A. Levy, Adam Storeygard, Deborah Balk, John L. Gittleman, and Peter Daszak. 2008. “Global Trends in Emerging Infectious Diseases.” Nature 451(7181):990–93.
• Lloyd-Smith, James O., Dylan George, Kim M. Pepin, Virginia E. Pitzer, Juliet R. C. Pulliam, Andrew P. Dobson, Peter J. Hudson, and Bryan T. Grenfell. 2009. ‘Epidemic Dynamics at the Human-Animal Interface’. Science 326(5958):1362–67.
• Morens, David M., Gregory K. Folkers, and Anthony S. Fauci. 2004. ‘The Challenge of Emerging and Re-Emerging Infectious Diseases’. Nature 430(6996):242–49.
• Taylor, Louise H., Sophia M. Latham, and Mark E. J. Woolhouse. 2001. “Risk Factors for Human Disease Emergence.” Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences 356(1411):983–89.
• Woolhouse, Mark E. J., and Sonya Gowtage-Sequeria. 2005. “Host Range and Emerging and Reemerging Pathogens.” Emerging Infectious Diseases 11(12):1842–47.

On page 2 we provide links to the key papers in each cluster.