Mathematical Researches

fa اعتیاد به اینترنت، مدلسازی و کنترل Addiction to Internet, Modeling and Control هندسه سیستم های دینامیکی Geometry of dynamical systems مقاله مستقل Original Manuscript در این مقاله، مدل ریاضی اعتیاد به اینترنت را ارائه میکنیم. تاثیر برنامههای آموزشی را روی کنترل این اعتیاد بررسی میکنیم. عدد باز تولید اساسی را به دست میآوریم.  با استفاده از قضیه چاوز-سانگ وقوع انشعاب عقبگرد را نشان میدهیم. پایداری کلی نقاط تعادلی را با استفاده از روش پایداری هندسی و تابع لیاپانوف ثابت میکنیم. Introduction In recent years, the use of the Internet has been increasing. Attractions and many educational and recreational applications, etc. have caused the emergence of a new phenomenon called Internet addiction.   Internet addiction is a new and interesting topic that arises from social changes including the cellularization of society and the family. Poor family support of their members, poor social skills, and easy access to Internet has contributed to this addiction. Today, there are Internet addiction treatment clinics around the world. In addition to the treatment, the person in these clinics receives the necessary training to use the Internet properly. The effects of Internet addiction are similar to other addictions. These effects include mental disorders, euphoria and desire to consume more, etc. Because of the great similarities between Internet addiction and other addictions, in this article we have developed the epidemic model of heroin addiction in White and Comiskey [1]. Material and methods In the proposed model, we have studied the effect of education and prevention. To examine this issue, we have considered five classes: 1- Susceptible individuals, 2- Internet users without training, 3- Educated Internet users and fully aware of the harms of the Internet addiction, 4- Internet addicts 5- Addicts under treatment and education. We examine the dynamic behavior of the model such as backward bifurcation, local and global stability equilibrium points. We investigate the boundary and obtain the basic reproduction number of the system. We study the existence of endemic equilibrium points and, using the Chavez-Sang theorem, show that the backward bifurcation occurs. We obtain sufficient conditions for the global stability, the addiction-free equilibrium point, and the endemic equilibrium point using the Lyapunov function and the geometric stability method. Results and discussion The epidemic model of heroin addiction was first introduced by White and Comiskey. In this article, we have developed this model. First, we have considered the age group of 15 to 65 years old as people who are inclined to use the Internet. Then some of these people become Internet users and a group of these people may become addicted to the Internet due to the excessive use of the Internet. At a certain rate, these addicts are treated and educated. On the one hand, Internet users may not become addicted and receive the necessary training and become professional Internet users. Conclusion In this paper, we have conducted a complete qualitative study of the model including the existence and evaluation of local and global stability of the equilibrium points of the model. We have shown that under certain conditions the addiction-free equilibrium point is local and global asymptotical stable. Using the compound matrix, we have obtained the conditions for the global stability of the endemic equilibrium point. We have shown the occurrence of backward bifurcation. This bifurcation indicates that when<img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:41px; height:24px" >, addiction remains in the society. To control addiction, we need to get<img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image004.gif" style="width:20px; height:24px" >. Reduce <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image006.gif" style="width:20px; height:24px" > to less than<img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image004.gif" style="width:20px; height:24px" >. انشعاب عقب گرد, اعتیاد به اینترنت , پایداری Global Stability, Backward Bifurcation, Addiction to Internet 184 202 http://mmr.khu.ac.ir/browse.php?a_code=A-10-1206-1&slc_lang=fa&sid=1 Atena Ghasemabadi آتنا قاسم آبادی ghasemabadi.math@gmail.com 10031947532846005478 10031947532846005478 Yes Esfarayen University of technology مجتمع آموزش عالی فنی و مهندسی اسفراین