Mathematical Modeling of the Role of Imitation in Crime Dynamics

Abstract

Crime remains one of the significant problems that countries are grappling with globally. With shrinking economies and increasing poverty, crime has been on the rise in many countries. In this paper, we propose a system of non-linear ordinary differential equations to model crime dynamics in the presence of imitation. The model consists of four independent compartments: individuals who are not at risk of committing a crime, individuals at risk of committing a crime, individuals committing a crime, and individuals convicted and jailed for a crime. The model is analyzed using the basic reproduction number. The analysis shows the system has a locally asymptotically stable crime-free equilibrium when the basic reproduction number is less than unity. The model exhibits a backward bifurcation in which two endemic equilibria coexist with the crime-free equilibrium. When the basic reproduction number exceeds unity, the system has a locally asymptotically stable endemic equilibrium, and the crime-free becomes unstable. Numerical simulations are carried out to verify the analytical results. The sensitivity analysis shows that the relapse rate highly influences the basic reproduction number of our model. This indicates that the proportion of individuals leaving prisons and becoming criminals should be minimized to minimize crime.

Figures (6)

• Figure 1: A flow diagram showing the schematic illustration of the model considered in this work.
• Figure 2: Backward bifurcation curve in the plane $(\mathcal{R}_0,C)$. The estimated parameter values are, $\pi=13820,\beta=8.5\times 10^{-6},\alpha=0.0002,\mu=0.01316,\varepsilon=0.88,\theta=0.01,\gamma=0.9,\sigma=0.6,p=0.2$, and $q=0.5$. The threshold value is $\mathcal{R}_{0}^c = 0.3815$. It shows the occurrence of a backward bifurcation of system \ref{['eq:1']}. This implies that when $\mathcal{R}_0<1$, a small positive unstable endemic equilibrium (dotted curve) appears while a crime-free and another positive endemic equilibrium (in red) are locally asymptotically stable. It can also be observed that the two endemic equilibria disappear when $\mathcal{R}_0$ is decreased below the critical value $\mathcal{R}_{0}^c<1$.
• Figure 3: Time series of the state variables $S_1(t), S_2(t), C(t)$, and $R(t)$ showing local stability of the crime-free equilibrium $E^0$. The system \ref{['eq:1']} was simulated numerically for time $500 \ \text{months}$ with $0.01$ time-step starting from randomly and uniformly chosen initial conditions $(S_1(0), S_2(0), C(0), R(0))$ in the interval $[0, 10^6]$. We considered the parameter values in Table \ref{['tab:model-params-a']}. These parameters correspond to $\mathcal{R}_0= 0.6462$.
• Figure 4: Time series of the state variables $S_1(t), S_2(t), C(t)$, and $R(t)$ showing local stability of the crime-free equilibrium $E^*$. The system \ref{['eq:1']} was simulated numerically for time $500 \ \text{months}$ with $0.01$ time-step starting from randomly and uniformly chosen initial conditions $(S_1(0), S_2(0), C(0), R(0))$ in the interval $[0, 10^6]$. We considered the parameter values in Table \ref{['tab:model-params-b']}. These parameters correspond to $\mathcal{R}_0=1.5108$.
• Figure 5: A contour plot how model parameters $\sigma$ and $\gamma$ affect $\mathcal{R}_0$. The parameters $\sigma$ and $\gamma$ are varied in the intervals $[0.1, 0.5]$ and $[0.05, 0.4]$, and the remaining parameters are as provided in Tables \ref{['tab:model-params-a']} and \ref{['tab:model-params-b']}.

Theorems & Definitions (8)

• Theorem 3.1: Existence of crime-free equilibrium
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• Definition 3.5