dcyphr | The anatomy of the 2019 Chilean social unrest


In this study, the researchers analyzed the unrest that occurred in Chile in 2019 through the lens of an epidemic-like mathematical model. They adjusted the conditions of the model using data about the unrest, and observed that the number of violent events follow well-defined patterns of social unrests that have been observed in other countries since the 1960s. The researchers also added disturbances to the model that alters the model to mimic the chaotic nature of the 2019 Chilean social unrest, as epidemic models peak at a point and start to decrease, which is not what occurred in Chile. The model provides interesting insights into the dynamics of social unrest.


The researchers wanted to create a mathematical model to understand patterns of social unrest using data from the 2019 Chilean social unrest.


Social unrest was commonplace all over the world in 2019, with a recent report showing that almost a quarter of all countries saw a significant increase in social unrest.

Although the causes of civil unrest may differ from country to country, all episodes disrupt daily life and can result in violence, causing injuries and even death. In addition, civil unrest may cause destruction to transport systems, private and public property, and supply chains. If civil unrest progresses to the nationwide level, it may even result in political instability, coup, or government overthrow.

In 2019, rising subway fares triggered social unrest in Chile. Protestors turned violent and unrest spread throughout the country, and authorities declared a state of emergency and imposed curfews on several cities. The first month of unrest is reported to have caused $4.6 billion in infrastructure damage and $3 billion to the Chilean economy.

The ability to describe and predict the course of social unrest may provide insights into how governments can make decisions to control and reduce events of social unrest. In this way, mathematical models may have an advantage over simple observation, as they can describe how and what affects the dynamics of social unrest. Mathematical modeling can also allow different scenarios to be explored, and testing different strategies to quell social unrest at a low monetary and ethical cost.

Epidemic-like models have been used to describe social phenomena in previous studies, including the 2005 French riots and 2011 disorder in London. In these models, one case of infectious disease can become an epidemic if the individual comes into contact with a large number of susceptible individuals. In less dense populations, the infected individual comes into contact with less people and an epidemic is less likely. In one study, the researchers found that epidemic models described riots in the U.S that occurred throughout the 1960s, which required susceptibility of individuals to be exposed to riots and get involved, among other factors.

In this paper, the researchers pursued the question, “Can we model the temporal evolution of the 2019 Chilean riot using epidemic-like mathematical models?” To answer it, the researchers compared available data about the riot with the mathematical model they developed, as well as characteristics of the Chilean riot with social unrest in other countries. Because the Chilean social unrest saw many rebounds that are not accounted for in epidemic-like models, they must be revamped to fully describe the social unrest in Chile.

The 2019 Chilean Social Unrest

Economic and Social Context

The Chilean economy has been praised as a successful case in Latin America, and since 1990 the economy has resulted in many social improvements in the country including decreasing the percentage of the country living in poverty and increasing life expectancy. In 2010, Chile became the first South American OECD country. However, even though per capita GDP grew almost sixfold, Chile is still one of the most unequal countries among OECD countries--the richest 1% have almost 27% of the nation’s entire wealth. This income inequality may have contributed to the feeling of discontent and unfairness in the Chilean public leading up to October 2019, when the unrest began. As the Chilean economy has grown, Chile has seen a growth in social movements including the Student Movement, Mapuche Movement, Labor Movement, Feminist Movement, and Environmental Movement, among others that have shaped the national politics.

Figure 1 summarizes the key events as well as number of serious events that occurred during the Chilean social unrest from October to November 2019, with more detail in Appendix A. 

Social crisis and catastrophe theory

The unrest in Chile followed a pattern that is common with previous events of similar nature in U.S cities, Paris, and London: a “warming up” period over an extended period, followed by a sudden outbreak of unrest that reaches a peak of disorder caused by a triggering event, and a relaxation phase where where disorder returns to a low level. These events suggest that triggering events are necessary for social unrest to occur, but it is unclear whether they are sufficient. Such events have occurred without generating great unrest, which means that other factors are necessary to trigger an “outbreak,” including economic factors, food scarcity, racial tensions, and reputation of police. In Chile, unfulfilled expectations and negative perceptions of the governing class seemed to play a factor.

Economic theories explain countries’ economies as systems that can be stable or unstable, exhibiting exponential growth, collapse, or oscillations. Relevantly, Catastrophe Theory describes the evolution of social unrest in a set of parameters, which can change dramatically due to small disturbances and create social instability, or an abrupt change in the social behavior of the people.

Empirical behavior of the social unrest

Data that measures social unrest in Chile was taken from the Undersecretary for Human Rights of the Chilean Ministry of Justice and Human rights, which includes events like looting, fire, destruction of private or public property, and other similar events. The data is illustrated in Figure 2, with the number of serious events on the y-axis and time on the x-axis. The behavior of these events is described as follows:

  1. Event (1) was the most prominent, reaching 350 serious events by October 21st.

  2. Events (3) and (4) correspond to spatially separated riots in different cities.

  3. Serious events occurred almost on a weekly basis, which may be explained by most violent acts happening on Friday afternoons, becoming an important day for the social movement.

  4. The Chilean unrest follows patterns of previous episodes of unrest, with every major event being followed by a diffusion process in which serious events decrease. This decrease is exponential, and is depicted in Figure 3.

  5. The behavior follows a very nonlinear pattern; there is a sharp increase in serious events prior to the peak of disorder, followed by a slower exponential decrease.

  6. A unique feature of the 2019 Chilean unrest is the number of sizeable events--in other cases, rioting is characterized by culmination of unrest in a single, huge event then dissipates.

Epidemic-like model for riots

Unlike infectious disease models, where infected individuals must come into close contact with susceptible individuals for infection to occur, social media may affect how protests and rioting events spread. So, modeling approaches take into account that every individual has the same probability of interacting with every other individual, as social media bridges gaps between individuals who otherwise might not have interacted with each other.

Model, definitions, and basic properties

The SIR model is the most basic and commonly used model to describe infectious disease spread. I(t) represents the number of infected individuals, S(t) represents the number of susceptible individuals, and R(t) is the number of “returned” individuals, who have been infected and recovered, and are no longer susceptible.

Previous studies have introduced an analogous model for riots, in which I(t) represents the total number of active rioters and S(t) is the number of individuals susceptible to join the riots. The number of riot events, λ(t), is proportional to the number of rioters. The potential “supply” of riot events, σ(t), is defined in terms of S(t).  Based on this, the model proposes equations (1) and (2), with ω and Β representing the rate at which rioters exit riots and the “transmission” rate respectively. Integrating the two equations gives equation (4), and further integration gives equations (5) and (6). λ(t) is calculated through equation (5).

Activation of rioting epidemics and their prevention

The researchers asked, if a small number of rioters (λ) are introduced into a susceptible population (σ), will there be an epidemic of riot events? They found that the dynamics depend on the initial value of σ, and whether σ is greater than the parameter v=ω/B. As long as σ > v, λ increases with to a maximum of σ = v, but since σ is decreasing, eventually λ will decrease and approach zero. This corresponds with the occurrence of an epidemic of riot events. If σ < 0, λ strictly decreases that there will be no epidemic of riot events. These dynamics are represented in Figure 4. Regardless of σ, λ decreases to 0 by equation (7).

σ/v represents the basic reproduction number, R0, which represents how many susceptible individuals one infected person would infect in an epidemic model. The interpretation of R0 for riot activity is the average number of secondary riot events a single riot event introduces into a population of “susceptible” individuals. In the case of riot epidemics, an epidemic would occur when σ/v > 1, and an epidemic will not occur when σ/v < 1.

B is the riot “transmission rate.” One riot event then produces Bσ new riot events in a population with σ/α individuals, where α equals number of events/number of individuals. ω represents the “exit rate” from riot events, such that τ=1/ω is the lifetime of each riot event. In this model, riot epidemics are only reliant on the initial value of potential riot events (σ), the riot “transmission” rate (B), and the duration of a riot event τ.

R0=Bστ determines riot epidemic dynamics, in order to prevent riot events, a public policy would need to implement measures to reduce τ or B such that R0 < 1, and a riot epidemic is prevented. Figure 5(a) shows the threshold curve for R0=1 for σ = 489, in which the region above the curve represents when riot epidemics occur and the region below the curve represents when riot epidemics do not occur. Whether riot epidemics occur or do not occur depends on B and τ.

Strategies to prevent outbreaks include targeting B by reducing the contact rate between rioters and susceptible individuals, or by lowering the probability of infection.

Hamiltonian dynamics

The system described above may be written in terms of Hamilton dynamics. The epidemic model with a Hamiltonian structure is described in equations (13) and (14). 

Parametric periodic and stochastic forced variations of Burbeck, et. al epidemic model

Equations (1) and (2) cannot predict more than one riot event, because σ decreases over time without a returning point. The researchers propose a modification to the original model to account for the increase in potential rioters, σ, which is summarized in equation (15). Then, equations (1) and (15) can be written as (16) and (17) as a set of Hamilton equations.

The Chilean social unrest has a natural weekly period, represented by equation (18), where f0 is the forcing intensity, T is the period, and the δ-function represents a periodic “kick” in σ, which triggers the riot epidemic event. The equations for this “kick-epidemic” model are (19) and (20). Equation (21) represent the jump conditions, or the conditions required to re-start riot epidemics. Equation (22) provides a mathematical model for the transition from one kick to another.

Data analysis

The researchers fit the data from the Undersecretary for Human Rights to equations (1) and (2), which gave equation (23), which provides a relationship between R0 and v for the set of parameters t0, tmax, λ0, and λmax. Combined with a second relationship, equation (24), the researchers found equation (25) as a function that includes all parameters. This equation can be solved using given parameters and can be used to predict the creation/destruction of riot events. The researchers compared these solutions to the actual data from the Undersecretary for Human Rights, which are summarized in Table I. Notably, the researchers found that R0 and ω do not vary substantially across observed riot events.


In this study, the researchers analyzed the 2019 Chilean social unrest using mathematical modeling, and compared findings with public data to understand how unrest evolves over time. They showed that dynamics of social unrest can be understood by analyzing them through epidemic modeling, and that modeling could guide policymakers to make informed decisions towards ensuring public safety. In addition, this model better reflects social unrest than traditional epidemic modeling because of its Hamiltonian nature.

However, the researchers recognize the necessity of interdisciplinary efforts to understand the 2019 Chilean social unrest. In addition, the researchers emphasize that modeling should be used complementarily to other methods of preventing social disorder.