Dynamical modelling of street protests using the Yellow Vest Movement and Khabarovsk as case studies

Abstract

Social protests, in particular in the form of street protests, are a frequent phenomenon of modern world often making a significant disruptive effect on the society. Understanding the factors that can affect their duration and intensity is therefore an important problem. In this paper, we consider a mathematical model of protests dynamics describing how the number of protesters change with time. We apply the model to two events such as the Yellow Vest Movement 2018-2019 in France and Khabarovsk protests 2019-2020 in Russia. We show that in both cases our model provides a good description of the protests dynamics. We consider how the model parameters can be estimated by solving the inverse problem based on the available data on protesters number at different time. The analysis of parameter sensitivity then allows for determining which factor(s) may have the strongest effect on the protests dynamics.

Figure 1

Number of people participating in the street protests vs time in the two motivating examples: (a) Yellow Vests Movement 2018–2019 (France), (b) Khabarovsk 2019–2020 (Russia). The dotted line is added for convenience of visualization only. For the YVM, the data were taken from Ref.; for the Khabarovsk protests, the data were collected over 2020–2021 from relevant publications in the media.

Figure 2

Flow-chart of the model of social protests. The boxes correspond to different groups, the arrows show transitions between the groups, the labels at the arrows show the transition rates. The active protesters $(I+C)$ are assumed to consist of newly engaged ones (I) and the experienced or mature ones (C), the former are recruited (as a result of the interaction with the active protesters) from a pool of potential protesters (S). Both new and experienced protesters can retire from the protests, e.g. after becoming tired or disillusioned by the protests, but can eventually join the pool of potential protesters again.

Figure 3

Positive steady state values of $S,I,C,\text{and}R$ (in (a–d), respectively) for the system (6)–(9) as a function of parameter $\gamma$. Other parameter values are given in the text. Here the lower and upper steady states are stable (cf. A and B in (c) as an example) and the ‘intermediate’ steady state (cf. D) is unstable. The sigmoidal shape of the curves indicates that the number of the steady states can change in response to a change in $\gamma$ (i.e. when the vertical line in (c) moves left or right).

Figure 4

Positive steady state values of $S,I,C,\text{and}R$ (in (a–d), respectively) for the system (6)–(9) as a function of parameter ${\delta }_2$. Other parameter values are given in the text. The shape of the curves indicates that the number of the steady states can change in response to a change in ${\delta }_2$ (e.g. when the vertical line in (c) moves sufficiently far to the right).

Figure 5

Number of protesters $I+C$ (in thousands) vs time (in days) obtained from Eqs. (6)–(9) for (a) different values of $\gamma$ and (b) for different values of ${\delta }_2$. Other parameter values are given in the text. The initial conditions are given by Set A (see Eqs. (17)). We readily observe that an increase in $\gamma$ or decrease in ${\delta }_2$ change the protest pattern making it persistent over time.

Figure 6

Number of protesters $I+C$ (in thousands) vs time (in days) obtained from Eqs. (6)–(9) for different values of $\gamma$ and the initial conditions as in Set B (see Eq. (18)). Other parameter values are given in the text. We readily observe that an increase in $\gamma$ tends to change the protest pattern resulting in a much larger number of protesters.

Figure 7

The steady state values of (a) I and (b) C for the system (6)–(9) obtained as a function of $\gamma$ for $n=3$. In this case, the number of steady states does not change with a change in $\gamma$ (no bifurcation occurs); there is always just one steady state. Other parameter values are the same as in Fig. 3.

Figure 8

The two stable branches (the unstable branch is not shown for the sake of visualization clarity) of the steady state dependence on $\gamma$ obtained for different values of n between 4 and 10; (a) for I, (b) for C. Thus, in all cases the system exhibits bistability. Other parameter values are the same as in Fig. 3.

Figure 9

The two stable branches (the unstable branch is not shown for the sake of visualization clarity) of the steady state dependence on ${\delta }_2$ obtained for different values of n between 4 and 10; (a) for I, (b) for C. Thus, in all cases the system exhibits bistability. Other parameter values are the same as in Fig. 4.

Figure 10

(a) Solution of the model (6)–(9) obtained for parameter sets ${\mathbf{q}}^{(1)}$ and ${\mathbf{q}}^{(2)}$ (cf. Table 1) shown in comparison with the data on the Yellow Vest Movement. The initial conditions are chosen as $S(0)=55,212,000$, $I(0)=288,000$ and $C(0)=R(0)=0$. (b) Dynamics of one of the ‘hidden variables’: the number of retired protesters R vs time as predicted by the model (6)–(9) obtained for parameter set ${\mathbf{q}}^{(2)}$.

Figure 11

Solutions of the model (6)–(9) obtained for parameter sets (a) ${\mathbf{q}}^{(1)}$ and (b) ${\mathbf{q}}^{(2)}$ (see Table 2) shown in comparison with the data on the Khabarovsk protests. We readily observe that, for the two parameter sets, the solution fits the data equally well. We therefore conclude that the accuracy of the model to describe the data is robust with regard to the choice of functional.

Figure 12

Sensitivity functions for all parameters of the system (6)–(9) obtained for the ‘measured’ variable $(I+C)$, with simulations performed for parameter set ${\mathbf{q}}^{(2)}$ (cf. Table 1).

Figure 13

The norm of the perpendiculars (in logarithmic scale) calculated at each iteration of the orthogonal method. The horizontal axis shows the succession of algorithm’s iterations (six altogether). Note that, after six steps, only parameters ${\delta }_1$ and ${\delta }_2$ survive, which indicates their strong inter-dependence.

Figure 14

Solutions of the model (6)–(9) shown in comparison with the data on the Yellow Vest Movement for a different noise level, (a) for $\delta =5\%$ and (b) for $\delta =10\%$. In both cases, the black crosses show the original (undisturbed) data and the coloured crosses (green in (a) and blue in (b)) show the noisy (randomly disturbed) data. The dashed black curve (that in both cases lays very close to the coloured curve) shows the solution obtained for parameters restored from the true (undisturbed) data (cf. the left column and ${\mathbf{q}}^{(2)}$ in Table 1).

Figure 15

Solutions of the model (6)–(9) shown in comparison with the data on the Khabarovsk protests for a different noise level, (a) for $\delta =5\%$ and (b) for $\delta =10\%$. In both cases, the black crosses show the original (undisturbed) data and the coloured crosses (green in (a) and blue in (b)) show the noisy (randomly disturbed) data. The dashed black curve (that lays very close to the coloured curve) shows the solution obtained for parameters restored from the true (undisturbed) data (cf. the left column of this table and ${\mathbf{q}}^{(2)}$ in Table 2).