Dynamical analysis of a cocaine–heroin epidemiological model with spatial distributions

Abstract This paper conducts an in-depth investigation of a new spatio-temporal model for the cocaine–heroin epidemiological model with vital dynamics incorporating the Laplacian operator. The study rigorously establishes the existence, uniqueness, nonnegativity, and boundedness of solutions of the proposed model. In addition, the local stability of both drug-free and drug-addiction equilibria is analyzed by studying the corresponding characteristic equations. The research provides conclusive evidence that when the basic reproductive number $\mathcal{R}_{0}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> exceeds 1, then the drug-addiction equilibrium is globally asymptotically stable. Conversely, using comparative arguments, it is shown that if $\mathcal{R}_{0}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> is less than 1, then the drug-free equilibrium is globally asymptotically stable. Furthermore, the paper includes a series of numerical simulations to visually convey and support the analytical results.