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Abstract The effect that different police protest management methods have on protesters’ physical and mental trauma is still not well understood and is a matter of debate. In this paper, we take a two-pronged approach to gain insight into this issue. First, we perform statistical analysis on time series data of protests provided by ACLED and spanning the period of time from January 1, 2020, until March 13, 2021. After observing the data, it becomes apparent that employing kinetic impact projectiles is correlated with an increase in protests in the following days. Moreover, it serves as a more accurate indicator of the subsequent death toll compared to the mere number of protests. This leads to the conclusion that the utilization of less-lethal weapons appears to provoke rather than quell protests, exhibiting an inflammatory effect. Next, we provide a mathematical framework to model modern, but well-established social psychology research on compliance theory and crowd dynamics. Our results show that understanding the heterogeneity of the crowd is key for protests that lead to a reduction of social tension and minimization of physical and mental trauma in protesters. 

Ukraine’s tug-of-war between Russia and the West has had significant and lasting consequences for the country. In 2013, Viktor Yanukovych, the Ukrainian president aligned with Russia, opted against signing an association agreement with the European Union. This agreement aimed to facilitate trade and travel between the EU and Ukraine. This decision sparked widespread protests that coalesced in Kyiv’s Maidan Square, eventually becoming known as the Euromaidan protests. In this study, we analyze the protest data from 2013, sourced from Ukraine’s Center for Social and Labor Research. Despite the dataset’s limitations and occasional inconsistencies, we demonstrate the extraction of valuable insights and the construction of a descriptive model from such data. Our investigation reveals a pre-existing state of self-excitation within the system even before the onset of the Euromaidan protests. This self-excitation intensified during the Euromaidan protests. A statistical analysis indicates that the government’s utilization of force correlates with increased future protests, exacerbating rather than quelling the protest movement. Furthermore, we introduce the implementation of Hawkes process models to comprehend the spatiotemporal dynamics of the protest activity. Our findings highlight that, while protest activities spread across the entire country, the driving force behind the dynamics of these protests was the level of activity in Kyiv. Furthermore, in contrast to prior research that emphasized geographical proximity as a key predictor of event propagation, our study illustrates that the political alignment among oblasts, which are the distinct municipalities comprising Ukraine, had a more profound impact than mere geographic distance. This underscores the significance of social and cultural factors in molding the trajectory of political movements. 

Partial differential equations were developed in the 18th century to model physical systems. Their inception has led to the continued development of a beautiful mathematical theory with an ever-increasing range of applications. In 1890, Poincar´e observed that its encompassing framework can allow us to see similarities in a wide range of physical applications. We now know that the similarities extend far beyond physical applications to other fields such as chemistry, biology, ecology, and even sociology. We provide a brief history of the applications of partial differential equations and showcase some recent works with applications in ecology and sociology. 

It is well known that relocation strategies in ecology can make the difference between extinction and persistence. We consider a reaction-advection-diffusion framework to analyze movement strategies in the context of species which are subject to a strong Allee effect. The movement strategies we consider are a combination of random Brownian motion and directed movement through the use of an environmental signal. We prove that a population can overcome the strong Allee effect when the signals are super-harmonic. In other words, an initially small population can survive in the long term if they aggregate sufficiently fast. A sharp result is provided for a specific signal that can be related to the Fokker-Planck equation for the Orstein-Uhlenbeck process. We also explore the case of pure diffusion and pure aggregation and discuss their benefits and drawbacks, making the case for a suitable combination of the two as a better strategy.