Modelling And Research On Alcoholism Dynamics With Media Reports

Alcoholism is one of the dangerous behaviors that affect people's life and human health.It is extremely harmful to individuals,families and society.Alcoholism is a wide spread problem all over the world.It is not only a serious public health problem,but also a social problem.Research,prevention and control of alcoholism-related problems have become a topic of serious concern.At present,many scholars have used mathematical models to study alcoholism-related problems,but most of them have established alcoholism models under the condition of homogeneous mixture of the population.However,the influence of time delay effect,heterogeneity and alcoholism age factors is seldom considered.On the basis of the existing research results and taking into account the influence of media reports,delay effects,heterogeneity and alcoholism age,this paper further establishes alcoholism models and discusses their alcoholism dynamics.The main work of this paper includes the following aspects:?1?Research on alcoholism model with media reports and single delay effectConsidering reporting the harm of alcoholism through the media can enhance people's ideological understanding of the harm of alcoholism,the possibility of alcoholism among the population is reduced.However,if social activities between potential drinkers and alcoholics are frequent,they will be affected and may become new alcoholics.And from full contact to“infection”,there is bound to be a certain delay effect in the process of developing into al-coholics.Therefore,under the condition of homogeneous and uniform mixing of the crowd,media coverage is introduced in the modelling process,and the impact of media has satura-tion effect.At the same time,involving the failure of media influence,a kind of alcoholism dynamics model with the impact of media and single delay is established.Furthermore,using the classical Lyapunov function method,we prove that the model has a alcohol-free equilibrium E0 when the alcoholism basic reproduction number R0<1,and the E0 is glob-ally asymptotically stable,that is,alcoholism disappears in the crowd.When R₀=1,the system generates a critical branch at the alcohol-free equilibrium point E₀.When R₀>1,the model has an alcoholism equilibrium E^*and the E^*is locally asymptotically stable,that is,alcoholism persists in the crowd.In particular,the introduction of delay?leads the Hopf bifurcation occurs at the E^*as the delay increases.Corresponding numerical simulations are given to verify the rationality of the model and the correctness of the theoretical results.?2?Research on alcoholism model with media reports and multiple delay effectsIn order to more accurately describe the law of alcoholism development,it is necessary to consider that other populations also have different delay effects.Based on this,multiple time delays are considered in the modelling process,and a kind of alcoholism model with media reports and multiple time delay effects is established and studied.The introduction of multiple time delays increases the depth and difficulty of system dynamics behavior analysis.Of course,the results are closer to reality.According to the difference of time delays,the research process is divided into four situations.The alcoholism basic reproduction number R₀of model is defined,and the models are discussed in detail respectively.The results show that the multi-time delay alcoholism model has more complex dynamics than the single time delay case.Under the corresponding delay situations I,II and III,for the alcohol-free equilibrium P0,the results show that P0 is globally asymptotically stable when R0<1 and satisfies certain conditions.Otherwise,P₀ is unstable.For the alcoholism equilibrium P^*,it is proved that P^*is locally asymptotically stable when mathematicalr₀>1 and meets certain conditions.Otherwise,the system appears a Hopf branch at P^*.In the case of IV,it is proved that P₀ is globally asymptotically stable when certain conditions are satisfied.Otherwise,the Hopf bifurcation appears at P₀,and the delay damages the stability of the system.At the same time,the impact of media reports and three time delays on alcoholism is discussed in detail.Using Latin Hypercube Sampling method and Partial Rank Matrix Correlation Coefficient method,the sensitivity analysis of each model parameter to the basic reproduction number R₀and the number of alcoholics was studied.Measures to control key parameters with high sensitivity are put forward.Finally,the corresponding numerical simulation is given.?3?Research on multi-group alcoholism model with media reportsBased on the prevalence of heterogeneous factors,the population is divided into n groups according to different regions.Then,a multi-group alcoholism model with media reports is established.Compared with homogeneous single-group model,the large scale of multi-group model brings difficulties in system dynamics analysis,and the research method is different from the discussion of single-group case.In this paper,by using the non-negative matrix inequality and the related knowledge of directed graph and matrix tree in graph the-ory,the suitable Lyapunov functional is constructed to analyze the dynamic behavior of the model.During the study,the alcoholism reproduction number R₀for the multi-group system was also defined.It is proved that when R₀?1,the alcohol-free equilibrium P₀is glob-ally asymptotically stable.When R₀>1,the alcoholism equilibrium point P* is globally asymptotically stable.At the same time,the corresponding numerical simulation is given to verify the correctness of the theoretical results.According to theoretical analysis and numer-ical simulation results,it can be seen that although heterogeneous factors do not lead to more complex dynamic behaviors,the model considered is closer to reality.The research results consider both intra-group?between residents in the same region?and inter-group?between residents in different regions?effects.In addition,heterogeneity affects the threshold value of the system,i.e.,the alcoholism reproduction number R₀is an undiminished function of the number of groups n.?4?Research on multi-group alcoholism model with media reports and age struc-tureThe influence of alcoholism age on the development trend of alcoholism and its control cannot be ignored.Therefore,a multi-group alcoholism model with age structure was estab-lished and studied.The results show that the dynamic behavior of the system is completely determined by the corresponding alcoholism basic regeneration number R₀.When R₀?1,the system has a alcohol-free equilibrium P₀ that is globally asymptotically stable.When R₀>1,the system has an alcoholism equilibrium P^*that is also globally asymptotically stable.At the same time,the influence of media reports and drinking age factors on the pre-vention and control of drinking is analyzed.Finally,the influence of the alcoholism age on the prevention and control of alcoholism problem is analyzed.The corresponding numerical simulations of the main conclusions are given.To sum up,this paper first considers the influence mechanism of the introduction fac-tors and establishes several reasonable alcoholism models.Secondly,the basic properties of the model are studied in detail.The corresponding numerical simulation is made and the practical significance of the results is analyzed.In this paper,more influencing factors are considered,and the mathematical model can more accurately describe the problem of alcoholism and correctly depict people's long-term dynamic behavior of alcoholism.Con-sidering more factors increases the complexity of the model and the difficulty of research.Of course,the research results are more helpful for people to understand the key factors affect-ing the development of alcoholism and grasp the change law of alcoholism population,so as to correctly predict the development trend of alcoholism.Finally,according to the research results,some feasible and effective control measures and suggestions are put forward to help people effectively prevent and control alcohol-related problems.This research has important application value and practical significance.