Fractional-order crime propagation model with non-linear transmission rate

Abstract

Various studies present different mathematical models of ordinary and fractional differential equations to reduce delinquent behavior and encourage prosocial growth. However, these models do not consider the non-linear transmission rate, which depicts reality better than the linear transmission rate, as the relationship between non-criminals and criminals is not linear. In light of this, a novel fractional-order mathematical crime propagation model with a non-linear Beddington–DeAngelis transmission rate is proposed that divides the entire population into three clusters. The present study also compares the crime transmission models for various transmission rates, followed by an analytical investigation. The model shows two equilibrium points (criminal-free and crime-persistence equilibrium). The criminal-free equilibrium is locally and globally asymptotically stable when the criminal generation number is less than one. The crime-persistence equilibrium point does not appear until the criminal generation number exceeds one. In addition, this research investigates the incidence of transcritical bifurcation at the criminal-free equilibrium point. Furthermore, numerical simulations are performed to demonstrate the analytical results. In summary, the finding of this research suggests that as the order of derivative increases, the population approaches equilibrium more swiftly, and criminals decline with time for the different order of derivative.