Dynamical analysis of a heroin–cocaine epidemic model with nonlinear incidence and spatial heterogeneity

Abstract

In this paper, we investigated a new heroin–cocaine epidemic model which incorporates spatial heterogeneity and nonlinear incidence rate. The main project of this paper is to explore the threshold dynamics in terms of the basic reproduction number ${\mathcal{R}}_0$, which was defined by applying the next-generation operator. The threshold type results shown that if ${\mathcal{R}}_0<1$, then the drug-free steady state is globally asymptotically stable. If ${\mathcal{R}}_0>1$, then heroin–cocaine spread is uniformly persistent. Furthermore, the globally asymptotic stability of the drug-free steady state has been established for the critical case of ${\mathcal{R}}_0=1$ by analysing the local asymptotic stability and global attractivity.

Keywords:

• Heroin–cocaine spread
• spatial heterogeneity
• the basic reproduction number
• uniformly persistent
• global asymptotically stable

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China (#11701445, #11971379, #11801439), by Natural Science Basic Research Program in Shaanxi Province of China (2022JM-042, 2020JQ-831, 2021JM-320).