# Epidemiology Models: SIR Modeling

*The review was prepared by Mila Nezdoimyshapko, Alla Loseva*

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:Sfor susceptible (not immune),Ifor infectious (spreading the virus),Rfor 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 calleddeterministic.

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.