Modeling and optimal control: The role of media on the dynamics of the spread of crime

Abstract

A non-linear mathematical model is developed and analyzed in order to investigate the interactions between the criminal and non-criminal populations while taking into account the influence of media on the non-criminal class. We obtained the basic reproduction number of the model while taking into account constant recruitment and death-type demography. The model's equilibria have been examined, and their stabilities have been thoroughly addressed. The backward bifurcation is presented for a basic reproduction number less than one. Sensitivity analysis is conducted on the proposed model to determine the key parameters of the basic reproduction number of the model. This model is further expanded to an optimal control problem to investigate the effects of time-dependent crime control rate on the size of the criminal population using Pontryagin’s maximum principle. To demonstrate our analytical findings, we present numerical simulations.