Finance

Sociology

Pure mathematics

Population

Consumption (sociology)

Mathematics

Social science

Kernel (algebra)

Mathematical optimization

Psychiatry

Demography

Fractional calculus

Order (exchange)

Economics

Epidemic model

Narcotic

Heroin

Drug

Applied mathematics

Medicine

Results in Physics

Intravenous substance consumption is on the upswing all over the globe, especially in Europe and Asia. It is extremely harmful to society; excessive substance consumption is the leading cause of death. Beyond all prohibited narcotics, heroin is a narcotic that has a substantial negative impact on society and the world at large. In this paper, a heroin epidemic model is developed via an Atangana–Baleanu fractional-order derivative in the Caputo sense describe accurately real world problems, equipped with recovery and persistent immunity. Meanwhile, we have established a globally asymptotically stable equilibrium for both the drug-free and drug-addiction equilibriums. Additionally, we apply a novel scheme that is mingled with the two-step Lagrange polynomial and the basic principle of fractional calculus. The simulation results for various fractional values indicate that as the fractional order decreases from 1, the growth of the epidemic diminishes. The modelling data demonstrates that the suggested containment technique is effective in minimizing the incidence of instances in various categories. Furthermore, modelling the ideal configuration indicated that lowering the fractional-order from 1 necessitates a swift commencement of the implementation of the suggested regulatory technique at the maximum rate and sustaining it throughout a significant proportion of the pandemic time frame.

New numerical dynamics of the heroin epidemic model using a fractional derivative with Mittag-Leffler kernel and consequences for control mechanisms