Soft drug epidemic in deterministic and stochastic case studies

Through the application of mathematical epidemiology principles, the formulation of the soft drugs model is established, and the complete dynamics of this deterministic model are contingent upon the crucial parameter called the basic reproduction number, denoted as \(R_{0}^{d}\). The stochastic soft drug epidemic models are developed by considering the parametric and non-parametric stochastic perturbation techniques. Dynamics of the two different stochastic models are determined using the analog stochastic thresholds \(R_0^{s_{1}}\), \(R_0^{s_{2}}\), respectively. Introducing suitable Lyapunov functionals enables us to establish sufficient axioms for the extinction and permanence of soft drug users in both deterministic and stochastic models. Moreover, the sensitivity of the deterministic and stochastic thresholds to some important parameters involved in the models is illustrated. For the verification of our theoretical results, we develop some numerical simulations using the Euler–Maruyama scheme.