Mathematical Stability Analysis of Bullying’s Impact on Student's Mental Health

Bullying is a problem that takes root from the previous generation to the next. The most common bullying cases occur in junior high school students or equivalent with an age range of 13-15 years. Bullying can have an impact on student's mental health. This study develops and analysis a SEIRS-type mathematical model to understand the dynamics of mental health disorders due to bullying among junior high school students. The model includes four subpopulations: subpopulations are vulnerable to mental health disorders due to bullying (S), subpopulations who experience bullying but have not shown any mental health disorders (E), subpopulations that experience mental health disorders due to bullying (I), and subpopulations who have recovered from mental health disorders due to bullying (R). Based on the results of the analysis, two equilibrium points were obtained, namely the mental disorder-free equilibrium point (P_1^*) and the endemic equilibrium point (P_2^*). Next, determine the basic reproduction number using the next matrix generation method. Based on the results of numerical simulation using Matlab R2013a software, it was obtained that if R_01 then the mental disorder-free equilibrium point is stable local asymptotic and mental health disorders due to bullying cannot spread in the student population. Meanwhile, if R_01 then the endemic equilibrium point is stable, local asymptotic and mental health disorders due to bullying can spread in the student population. The rate of interaction between subpopulations vulnerable to mental health disorders due to bullying and subpopulations that experience mental health disorders due to bullying (\alpha) has a significant influence on the number of individuals in each subpopulation.