## Abstract

We set up and analyse a mathematical model, the Serious Crime Model, which describes the interaction of mild and serious offenders and potential criminals. However we get more complete results for a simpler version of this model, the Mild Crime Model, with no serious offenders.

For the full Serious Crime Model there are two key parameters R10 and R20 corresponding to the basic reproduction number in the mathematics of infectious diseases, which determine the behaviour of the system. For the Simpler Mild Crime Model there is just one such parameter R10. Both forward and backward bifurcation can occur for this second model with two subcritical non-trivial equilibria possible for R10 < 1 in the backwards case. For backwards bifurcation there is another threshold value R∗0 such that the upper non-trivial equilibrium is unstable for R10 < R∗0 and stable for R10 > R∗0. For forwards bifurcation there is a second additional threshold value R∗∗0 such that the stability of the unique non-trivial equilibrium switches from unstable to stable as R10 passes through R∗∗0. At the end we return to the full Serious Crime Model and discuss the behaviour of this model.

The results are meaningful and interesting because in all of the other epidemiological and sociological models of which we are aware, analogous thresholds to R∗0 and R∗∗0 do not exist. For forwards bifurcation the unique non-trivial

equilibrium, and for backwards bifurcation with two subcritical endemic equilibria the higher non-trivial equilibrium, are also usually always locally asymptotically stable. So our models exhibit unusual and interesting behaviour.

For the full Serious Crime Model there are two key parameters R10 and R20 corresponding to the basic reproduction number in the mathematics of infectious diseases, which determine the behaviour of the system. For the Simpler Mild Crime Model there is just one such parameter R10. Both forward and backward bifurcation can occur for this second model with two subcritical non-trivial equilibria possible for R10 < 1 in the backwards case. For backwards bifurcation there is another threshold value R∗0 such that the upper non-trivial equilibrium is unstable for R10 < R∗0 and stable for R10 > R∗0. For forwards bifurcation there is a second additional threshold value R∗∗0 such that the stability of the unique non-trivial equilibrium switches from unstable to stable as R10 passes through R∗∗0. At the end we return to the full Serious Crime Model and discuss the behaviour of this model.

The results are meaningful and interesting because in all of the other epidemiological and sociological models of which we are aware, analogous thresholds to R∗0 and R∗∗0 do not exist. For forwards bifurcation the unique non-trivial

equilibrium, and for backwards bifurcation with two subcritical endemic equilibria the higher non-trivial equilibrium, are also usually always locally asymptotically stable. So our models exhibit unusual and interesting behaviour.

Original language | English |
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Article number | 128073 |

Number of pages | 38 |

Journal | Applied Mathematics and Computation |

Volume | 453 |

Early online date | 8 May 2023 |

DOIs | |

Publication status | Published - 15 Sept 2023 |

## Keywords

- mathematical models
- bifurcation
- limit cycle
- contagious crime
- violence
- criminal careers