Mathematical Modelling of The Dynamics of Poverty, Crime and Imprisonment

Abstract: This study explores the evolution and application of mathematical modelling to complex social issues such as poverty, crime, and terrorism. Traditionally rooted in epidemiology, compartmental models have been successfully adopted in criminology and public health to capture the dynamics of addiction, ideological radicalization, and recidivism. The model was derived from a five–compartment representation from which a set of five ordinary differential equations (ODEs) was developed to capture the dynamism of poverty, crime, and imprisonment in a deterministic SCJ₁J₂R compartmental model using the next generation matrix to obtain the reproduction number R₀. This is used to analyze the local stability of the crime-free equilibrium of the SCJ₁J₂R model. The crime-free equilibrium and the local stability of the endemic equilibrium show that R₀ is asymptotically stable if R₀ < 1. We use hypothetical data to simulate the sensitivity of the parameters of the basic reproduction number (R₀) so as to obtain R₀ < 1 for a crime-free society. The SCJ₁J₂R compartmental model differentiates incarceration based on criminal evidence and accounts for both natural and crime–induced mortality as well as reintegration processes. This review highlights the growing role of mathematical approaches in policy-relevant analysis of community safety, systemic intervention, and stability of the models.