Study Of Alcohol Models With Stage Structure And Relapse On Complex Networks

In recent years, college students’ drinking problem is increasingly serious, and there are often traffic accidents or violence caused by drinking. So, it is of great practical significance to study alcoholism transmission mechanism and then take effective measures to prevent and control it. At present, researches on alcoholism mainly based on differential equation to establish mathematical model, then analysis the alcoholism propagation and give out relevant control strategies. These models are very intuitive but unable to describe the heterogeneity of the individuals in the population. Alcoholism mainly spread by groups of social networks, therefore the model with complex networks theory has more practical significance. As the maturation of complex networks theory, network-based epidemic research has become one of the main trend of dynamics modeling of infectious diseases. Based on existing research works, this paper establishes two different types of alcoholism propagation models via mean field theory, and carries on a thorough research, by means of epidemic dynamics, complex networks theory and the theory of ordinary differential equation. Our research mainly includes the following two aspects:First, an SAIRS alcoholism model with stage structure on the scale-free networks is proposed. Alcoholics in the model is divided into two groups, light problem alcoholics and heavy problem alcoholics, and we introduced the empty nodes in the model to consider the individual’s birth and death. The basic reproductive number R₀ of the alcoholism spread is derived by the next generation matrix method. When R₀< 1, the model has an alcohol free equilibrium. Globally asymptotically stable of the alcohol free equilibrium is proved by the comparison theorem. When R₀> 1,there also has an alcoholism equilibrium. By persistence theorem and iterative rule,global attractivity of the alcoholism equilibrium is proved. At the same time, the correctness of the theories are validated by numerical simulation.Second, as is known to all that the spread of alcoholism is influenced by the intimacy between individuals（weight） and the relapse of the recovery people. So,in this paper, an SIRS alcoholism model with relapse on the scale-free networks with fixed weight and adaptive weight is introduced. The basic reproductive number R₀ of the alcoholism spread is also derived by the next generation matrix method.This model is discussed in two different conditions, fixed weight model and adaptive weight model. When it is fixed weight model, if R₀< 1, the alcohol free equilibrium is globally asymptotically stable, if R₀> 1, the alcoholism equilibrium is globally attractive. When it is adaptive weight model, this paper only carries on some numerical simulations. By numerical simulations, this paper also compares the similarities and differences of the model without weight, with fixed weight and with adaptive weight. In addition, this paper also puts forward two effective control alcoholism strategies.