Mathematical Modeling of Serial Killer Dynamics and its Prevention Using Machine Learning Based Physics Informed Neural Networks

Abstract

Serial killer activity is a complex social threat that traditional criminology often struggles to predict quantitatively. To better understand this phenomenon, we propose a mathematical modeling framework inspired by epidemiology. The population is divided into four compartments: at-risk individuals (A), active serial killers (S), prevention programs (P), and law enforcement (L). Their interactions are described by a system of nonlinear differential equations representing the spread and control of criminal behavior. We first establish fundamental analytical
properties of the model, including positivity and boundedness of solutions. Two equilibrium states are identified: a crime-free equilibrium and a persistent (endemic) crime state. A threshold parameter, the basic reproduction number R_0, determines stability; the crime-free state is stable when R₀ < 1. We further apply Physics-Informed Neural Networks (PINNs) to estimate model parameters from data, demonstrating accurate recovery with errors below 5%. This combined analytical-computational approach provides a novel framework for studying criminal dynamics.