Epidemic dynamics on a delayed multi-group heroin epidemic model with nonlinear incidence rate

Volume 9, Issue 5, pp 2149--2160 http://dx.doi.org/10.22436/jnsa.009.05.20

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Authors

Xianning Liu - Key Laboratory of Eco-environments in Three Gorges Reservoir Region (Ministry of Education), School of Mathematics and Statistics, Southwest University, Chongqing 400715, China. Jinliang Wang - School of Mathematical Science, Heilongjiang University, Harbin 150080, China.

Abstract

For a multi-group Heroin epidemic model with nonlinear incidence rate and distributed delays, we study some aspects of its global dynamics. By a rigorous analysis of the model, we establish that the model demonstrates a sharp threshold property, completely determined by the values of \(\Re_0\): if \(\Re_0 \leq 1\), then the drug-free equilibrium is globally asymptotically stable; if \(\Re_0 > 1\), then there exists a unique endemic equilibrium and it is globally asymptotically stable. A matrix-theoretic method based on the Perron eigenvector is used to prove the global asymptotic stability of the drug-free equilibrium and a graph- theoretic method based on Kirchhoff's matrix tree theorem was used to guide the construction of Lyapunov functionals for the global asymptotic stability of the endemic equilibrium.

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Keywords

• Heroin epidemic model
• multi-group
• global stability
• Lyapunov functionals.

MSC

•  34D30
•  92D30

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