A mathematical model for the bullying dynamics in schools

TL;DR

This work develops a four-state compartmental model for bullying dynamics in schools, capturing transitions among susceptible $S$, bully $B$, exposed $E$, and violent $V$ individuals. Using mean-field ODE analysis and $L\times L$ grid simulations, it derives a basic reproduction number $R_0=\alpha/\gamma$ that governs outbreak potential, and provides explicit expressions for the endemic equilibrium: $s_{end}=\gamma/\alpha$, $b_{end}=\epsilon\lambda(\alpha-\gamma)/(\epsilon\alpha\lambda+\beta\gamma(\lambda+\delta))$, $e_{end}=\beta\gamma b_{end}/(\epsilon\alpha)$, $v_{end}=\beta\gamma\delta b_{end}/(\epsilon\alpha\lambda)$; and shows bullying-free stability when $\alpha<\gamma$, with grid results corroborating the dynamics and revealing a spatial threshold shift. The study demonstrates how anti-bullying programs (increasing $\gamma$) and reduced contagion ($\alpha$) can suppress bullying prevalence, and it highlights the role of family and school interventions. The framework provides a quantitative tool for evaluating intervention strategies and supports future work on model extensions and policy optimization.

Abstract

We analyze a mathematical model to understand the dynamics of bullying in schools. The model considers a population divided into four groups: susceptible individuals, bullies, individuals exposed to bullying, and violent individuals. Transitions between these states occur at rates designed to capture the complex interactions among students, influenced by factors such as romantic rejection, conflicts with peers and teachers, and other school-related challenges. These interactions can escalate into bullying and violent behavior. The model also incorporates the role of parents and school administrators in mitigating bullying through intervention strategies. The results suggest that bullying can be effectively controlled if anti-bullying programs implemented by schools are sufficiently robust. Additionally, the conditions under which bullying persists are explored.

Figures (5)

• Figure 1: (Color online) Schematic representation of the model's transitions, showing the four possible states: Susceptible individuals (S), Bully individuals (B), Exposed individuals (E) and Violent individuals (V). The parameters responsible for each transition are also shown.
• Figure 2: Time evolution of the four densities of agents $s(t), b(t), e(t)$ and $v(t)$, obtained from the numerical integration of Eqs. (\ref{['eq7']}) - (\ref{['eq10']}). The parameters are $\beta=0.30, \gamma=0.20, \delta=0.05, \lambda=0.07$ and $\epsilon=0.10$, and we varied $\alpha$: (a) $\alpha=0.05$, (b) $\alpha=0.15$, (c) $\alpha=0.25$ and (d) $\alpha=0.40$. All parameters are given in unities year$^{-1}$. For the considered parameters, we have from Eq. (\ref{['eq13']}): (a) $R_o = 0.25$, (b) $R_o = 0.75$, (c) $R_o = 1.25$, (d) $R_o = 2.00$.
• Figure 3: (Color online) Equilibrium values $s$ (full line), $b$ (dashed line), $e$ (dotted line) and $v$ (dotted-dashed line) as functions of $\alpha$. The lines were obtained from the numerical integration of Eqs. \ref{['eq7']} - \ref{['eq10']}. Both solutions, bullying-free and endemic ones, given by Eqs. \ref{['eq14']} and \ref{['eq15']}, respectively, are shown in the figure. The parameters are $\beta=0.30, \gamma=0.20, \delta=0.05, \lambda=0.07$ and $\epsilon=0.10$. It is important to note that these behaviors are present also for other parameter values, and what is shown here works as a pattern.
• Figure 4: (Color online) Equilibrium values $s$ (squares), $b$ (circles), $e$ (up triangles) and $v$ (down triangles) as functions of $\alpha$ obtained from numerical simulations of the model on a grid of linear size $L=60$. The parameters are $\beta=0.30, \gamma=0.20, \delta=0.05, \lambda=0.07$ and $\epsilon=0.10$. Results are averaged over $100$ independent simulations.
• Figure 5: Snapshots of the population on a grid of linear size $L=60$ at stationary states. The fixed parameters are $\beta=0.30, \gamma=0.20, \delta=0.05, \lambda=0.07$ and $\epsilon=0.10$, and we varied $\alpha$: (a) $\alpha=0.45$, (b) $\alpha=0.55$, (c) $\alpha=0.65$ and (d) $\alpha=0.85$. The squares' colors represent distinct subpopulations, namely $S$ (black), $B$ (red), $E$ (blue) and $V$ (orange).