Nonlinear Dynamics of Violence Diffusion: The Fractional Derivative-Based Numerical Solutions and Stability Analysis

The proliferation of violence has emerged as a significant concern in contemporary society, posing a complex challenge that demands comprehensive intervention strategies. To enhance existing approaches, this investigation proposes a deterministically defined nonlinear mathematical model, to further explain the dynamics of the diffusion within a population. In the model, the population is divided into five distinct compartments that contain the Caputo-Fabrizio fractional derivative. The compartments comprise the Susceptible, Violence-exposed, Violently infectious, Negotiation, and Reconciled classes, which are used to investigate the model. This study contains both qualitative and quantitative analyses: in the qualitative analysis, we validate the model by identifying endemic equilibrium points, determining the basic reproduction number, and assessing stability properties using established stability theory. In the quantitative analysis, the analytical solution is derived by using the Laplace transform with the Adomian decomposition method (LADM) and the Homotopy perturbation method (HPM). The numerical simulations are performed using MATLAB for varying arbitrary fractional orders and provide graphical representations. The validity of the violence issue within the defined population is addressed through the results found by simulation, which reveals that the dynamics of the compartments are sensitive to different fractional orders, underscoring the efficacy of the fractional approach.