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

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 0 : if 0 ≤ 1, then the drug-free equilibrium is globally asymptotically stable; if 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 graphtheoretic 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.

Mathematical analysis of an age structured heroin-cocaine epidemic model

“…Since r(Ψ [0]) > 1, from the continuity, without loss of generality, we can assume that and λ are sufficiently small and T is sufficiently large such that r(Ψ ,T,λ ) > 1. By (21), as in the proof of Theorem 5.2, we can prove thatĴ 1 [λ] =Ĵ 2 [λ] = 0. This contradicts to the positivity of the solution for u 0 ∈ Ω 0 .…”

“…In [19], Liu and Zhang studied the global behavior of a heroin epidemic model with distributed time delay: the drug-free equilibrium is globally asymptotically stable if the basic reproduction number R 0 is less than 1, whereas a disease endemic equilibrium is locally asymptotically stable and the system is uniformly persistent if R 0 > 1. For other studies on the drug epidemic models with time delay, see, e.g., [9,13,20,21].…”

Hopf Bifurcation Analysis of a Synthetic Drug Transmission Model with Time Delays

“…Making the substitution v 1 � ω 2 1 , equation 13can be rewritten as v 4 1 + e 13 v 3 1 + e 12 v 2 1 + e 11 v 1 + e 10 � 0.…”

“…Increased use of heroin and other addictive drugs is an issue of concern all over the world. Drug abuse a ects not only life quality of the general public but also the overall situation of social stability and economic development [1][2][3]. e data from the World Drug Report published by the United Nations (U.N.) showed that thirty-ve million people worldwide su er from drug abuse disorders, and only one-seventh has received treatment [4].…”

“…In the subsequent papers, Wang et al [9][10][11][12] studied global stability of a heroin epidemic model with bilinear incidence rate, respectively. Some other works related to the dynamical behaviour of heroin epidemic models with nonlinear incidence rates can be found in [3,[13][14][15], and models with age structure can be found in [13,[15][16][17][18]. For the analytical study of stochastic heroin epidemic models or some other heroin epidemic models, one can also see [1,2,[19][20][21][22].…”