Abstract
Two SIRS alcoholism models with relapse on networks with fixed and adaptive weight are introduced. The spread of alcoholism threshold {R_0} is calculated by the next generation matrix method. For the model with fixed weight, we prove that when {R_0} < 1, the alcohol free equilibrium is globally asymptotically stable, then the drinking crowd gradually disappear. When {R_0} > 1, the alcoholism equilibrium is global attractivity, then the density of alcoholics will remain in a stable value. For the model with adaptive weight, we only make some numerical simulations. We also give two effective strategies. Our results show that the treatment of recuperator for stopping relapsing and preventing the susceptible people to drink are two effective measures to eliminate alcoholism problem, and preventing the susceptible people to drink is more effective when the proportion of recuperator to accept treatment is equal to the proportion of susceptible people to refuse drinking alcohol.
Highlights
Alcohol use and abuse have been part of human society for centuries
We study the impact of the fixed weight and adaptive weight on the spread of alcoholism
KSk (t) (t) represents the new problem alcoholic individuals per unit time, which is proportional to the degree k, the density of susceptible individuals Sk (t) and the probability that alcoholism transmits through a link (t), while ik is the transmission rate from nodes with degree i to nodes with degree k, φ(i) is the infectivity of the problem alcoholic nodes with degree i
Summary
Alcohol use and abuse have been part of human society for centuries. The College Alcohol Study defines alcoholism as male students who had five or more and female students who had four or more drinks in a row at least once in a 2-week period (Wechsler 2000). As the disease becomes severe, individuals tend to be more cautious in social contacts and make some reflection such as decreasing the out going visits, cutting down the meeting time and reducing the intimacy Such behaviors will change the strengths of nodes and the weights of links, which can be seen as an adaptive weight network. Zhu et al (2013) proposed a modified epidemic SIS model on an adaptive and weighted contact network, they introduced the general forms of the weight function and presented a new weight called “adaptive weight”. KSk (t) (t) represents the new problem alcoholic individuals per unit time, which is proportional to the degree k, the density of susceptible individuals Sk (t) and the probability that alcoholism transmits through a link (t), while ik is the transmission rate from nodes with degree i to nodes with degree k, φ(i) is the infectivity of the problem alcoholic nodes with degree i. − Ak (t) − Ik (t) − Rk (t) at steady-state, it is sufficient to study the limiting systems kg(k) b kg(k) b+μ
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