A modified school bullying model with recovery compartment: stability analysis and optimal control

Document Type : Research Article

Authors

Ayomi Sasmito ¹

Camelia K H ¹

Rosa Maeni ²

¹ Business Mathematics, School of STEM, Prasetiya Mulya University, Tangerang, Indonesia.

² Business Mathematics, School of STEM, Prasetiya Mulya University, Tangerang, Indonesia

10.22067/ijnao.2026.99035.1883

Abstract

This study proposes a compartmental dynamical model of school bullying transmission incorporating an explicit recovery mechanism through counselling and institutional intervention. The population is divided into susceptible, bully, exposed, violent, and recovered subgroups. The model’s qualitative properties are established, including positivity, boundedness, equilibrium analysis, and stability conditions derived using the next-generation matrix approach. The bullying-free equilibrium is shown to be both locally and globally asymptotically stable whenever $\mathcal{R}_0 < 1$, and the endemic equilibrium is shown to exist and be locally asymptotically stable when $\mathcal{R}_0 > 1$ under explicit Routh--Hurwitz conditions. Sensitivity analysis reveals that the perpetrator-generation rate $\alpha$ is the primary driver of bullying persistence, whereas recovery-related interventions substantially reduce long-term prevalence. An optimal control framework based on Pontryagin's Maximum Principle is then formulated using disciplinary and victim-protection interventions. Numerical simulations show that combining both interventions yields the largest reductions in bullying and in overall cost, indicating that integrated institutional responses outperform isolated policies.

Keywords

School bullying

Compartmental model

Basic reproduction number

Sensitivity analysis

Optimal control

Subjects

• Optimal Control

---