Mathematical modeling and analysis of the co-dynamics of crime and drug abuse

Abstract

This study explores the dynamics of crime and substance abuse within a population by developing a novel mathematical model that integrates social interactions, rehabilitation efforts, and relapse probabilities. The model introduces a critical metric, the control reproduction number \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\mathscr{R}_j)$$\\end{document}, to quantify the invasion threshold for these behaviors. The findings reveal that the crime/substance-free equilibrium is globally asymptotically stable when \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {R}_j\\le 1$$\\end{document}. At the same time, entrenched equilibria become stable where \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {R}_j >1$$\\end{document}. Additionally, the model predicts the potential for a co-existent equilibrium where crime/substance abuse and a free state can coexist if all reproduction numbers exceed unity. Sensitivity analysis identifies key factors influencing \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {R}_j$$\\end{document}, including behavioral transmission, internal progression rates, intervention efficacy, and recovery/relapse probabilities. Numerical simulations validate theoretical predictions regarding the stability of different equilibria, highlighting the critical importance of interventions targeting transmission reduction and rehabilitation efficiency. The research underscores the significance of understanding invasion dynamics for the coexistence of behaviors. It demonstrates the utility of mathematical modeling in elucidating the spread of social phenomena and informing effective control strategies.