Dynamical Analysis of a Cocaine-Heroin Epidemiological Model with Spatial Distributions

TL;DR

This work formulates a spatially explicit Cocaine-Heroin SCHR model as a reaction-diffusion system with diffusion on a bounded domain and no-flux boundaries. It establishes well-posedness (existence, uniqueness, nonnegativity, and boundedness) and analyzes threshold behavior through the basic reproduction number $\mathcal{R}_0$, proving global and local stability results for drug-free and drug-addicted equilibria via Lyapunov functionals. An extended model with treatment compartments is developed, with analogous existence and stability analyses and supporting numerical simulations. Numerical experiments using 1D diffusion demonstrate the theoretical stability results and provide visual validation. The study lays groundwork for incorporating more complex features (delays, fractional dynamics, stochasticity) and for calibrating to real-world data to guide public health interventions.

Abstract

This article 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, non-negativity, and boundedness of solutions for the proposed model. In addition, the local stability of both a drug-free equilibrium and a drug-addiction equilibrium are analyzed by studying the corresponding characteristic equations. The research provides conclusive evidence that when the basic reproductive number $\mathcal{R}_0$ exceeds 1, the drug-addiction equilibrium is globally asymptotically stable. Conversely, using comparative arguments, it is shown that if $\mathcal{R}_0$ is less than 1, the drug-free equilibrium is globally asymptotically stable. Furthermore, the article includes a series of numerical simulations to visually convey and support the analytical results.

Figures (6)

• Figure 1: Transfer diagram for the proposed SCHR model.
• Figure 2: Transfer diagram for the extended cocaine-heroin SCHR model.
• Figure 3: Numerical results of \ref{['E2.1']}--\ref{['E2.2']} when $\eta_1 = \eta_2 = 3\times 10^{-2}$ and $\sigma = \beta = 10^{-3}$ ($\mathcal{R}_0 < 1$).
• Figure 4: Numerical results of \ref{['E7.1']}--\ref{['E7.3']} when $\eta_1 = \eta_2 = 3\times 10^{-2}$, $\mu_1 = 0.05$ and $\sigma = \beta = 10^{-3}$ ($\mathcal{R}_0 < 1$).
• Figure 5: Numerical results of \ref{['E2.1']}--\ref{['E2.2']} when $\eta_1 = \eta_2 = 10^{-2}$, $\sigma = 0.2$, and $\beta = 2 \times 10^{-3}$ ($\mathcal{R}_0 > 1$).

Theorems & Definitions (8)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• Theorem 5