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STABILITY ANALYSIS OF A COMPUTER VIRUS PROPAGATION MODEL WITH ANTIDOTE IN VULNERABLE SYSTEM

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Acta Mathematica Scientia 2016年第1期36卷 49-61页

作者： Nguyen Huu KHANH Nguyen Bich HUY Department of Mathematics College of Natural SciencesCan Tho University Departmet of Mathematics Ho Chi Minh University of Education

We study a proposed model describing the propagation of computer virus in the network with antidote in vulnerable system. Mathematical analysis shows that dynamics of the spread of computer viruses is determined by the threshold Ro. If Ro 〈 1, the virusfree equilibrium is globally asymptotically stable, and if R0 〉 1, the endemic equilibrium is globally asymptotically stable. Lyapunov functional method as well as geometric approach are used for proving the global stability of equilibria. A numerical investigation is carried out to confirm the analytical results. Through parameter analysis, some effective strategies for eliminating viruses are suggested.

关键词： computer virus virus-free equilibrium endemic equilibrium global stability antidote

Two-group SIR Epidemic Model with Stochastic Perturbation

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Acta Mathematica Sinica,English Series 2012年第12期28卷 2545-2560页

作者： Chun Yan JI Da Qing JIANG Ning Zhong SHI School of Mathematics and Statistics Changshu Institute of Technology School of Mathematics and Statistics Northeast Normal University

A stochastic two-group SIR model is presented in this paper. The existence and uniqueness of its nonnegative solution is obtained, and the solution belongs to a positively invariant set. Further- more, the globally asymptotical stability of the disease-free equilibrium is deduced by the stochastic Lyapunov functional method if R0 〈 1, which means the disease will die out. While if R0 〉 1, we show that the solution is fluctuating around a point which is the endemic equilibrium of the deterministic model in time average. In addition, the intensity of the fluctuation is proportional to the intensity of the white noise. When the white noise is small, we consider the disease will prevail. At last, we illustrate the dynamic behavior of the model and their approximations via a range of numerical experiments.

关键词： Stochastic two-group SIR model disease-free equilibrium endemic equilibrium stochastic Lyapunov function asymptotically stable in the large

ANALYSIS OF AN SI EPIDEMIC MODEL WITH NONLINEAR TRANSMISSION AND STAGE STRUCTURE

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Acta Mathematica Scientia 2003年第4期23卷 440-446页

作者：陆忠华 *** 高淑京 *** 陈兰荪 *** Supercomputing Center of Computer Network Information Center Chinese Academy of Sciences Beijing 100080 China E-mail:zhlu Department of Mathematics Guangzhou University Guangzhou 510405 China E-mail:gaosjmath Institute of Mathematics Academy of Mathematics & System Sciences Academia Sinica Beijing 100080 China E-mail:lschen

A disease transmission model of SI type with stage structure is formulated. The stability of disease free equilibrium, the existence and uniqueness of an endemic equilibrium, the existence of a global attractor are in... 详细信息

关键词： SI epidemic model threshold disease free equilibrium endemic equilibrium global attractor

The asymptotic behavior of a stochastic multigroup SIS model

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International Journal of Biomathematics 2018年第3期11卷 115-130页

作者： Chunyan Ji Daqing Jiang School of Mathematics and Statistics Changshu Institute of Technology Changshu 215500 Jiangsu P. R. China School of Mathematics and Statistics Northeast Normal University Changchun 130024 Jilin P. R. China College of Science China University of Petroleum (East China) Qingdao 266580 P. R. China Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group King Abdulaziz University Jeddah Saudi Arabia

In this paper, we explore the long time behavior of a multigroup Susceptible-Infected Susceptible （SIS） model with stochastic perturbations. The conditions for the disease to die out are obtained. Besides, we also show that the disease is fluctuating around the endemic equilibrium under some conditions. Moreover, there is a stationary distribution under stronger conditions. At last, some numerical simulations are applied to support our theoretical results.

关键词： A multigroup Susceptible-Infected-Susceptible (SIS) model disease free equilibrium endemic equilibrium exponential stability prevail stationary distribution.

Hopf bifurcation of nonlinear incidence rates SIR epidemiological models with stage structure

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Communications in Nonlinear Science and Numerical Simulation 2001年第4期6卷 205-209页

作者： Zhonghua LU, Xianning LIU and Lansun CHEN Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China e-mail: lschen@*** Xi'an Statistic College, Xi'an 710061, China e-mail: lzh Institute of Mathematics Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100080 China Xi'an Statistic College Xi'an 710061 China Department of Mathematics Southwest China Normal University Chongging 400715 China

A nonlinear incidence rates SIR epidemiological model with stage structure is considered. Different to bilinear incidence rates models, there exist two positive equilibria and whether or not the disease will eventuall... 详细信息

关键词： Hopf bifurcation SIR epidemiological model. disease-free equilibrium endemic equilibrium

A MODEL FOR CONTROL OF HIV/AIDS WITH PARENTAL CARE

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International Journal of Biomathematics 2013年第2期6卷 131-145页

作者： GBENGA JACOB ABIODUN NIZAR MARCUS KAZEEM OARE OKOSUN PETER JOSEPH WITBOOI Department of Mathematics and Applied Mathematics University of the Western Cape Private Bag X17 Bellville 7535 South Africa Department of Mathematics Vaal University of Technology Private Bag X021 Vanderbijlpark 1900 South Africa Department oJ Mathematzcs and Applied Mathematzes University of the Western Cape Private Bag X17 BeUviUe 7535 South Africa

In this study we investigate the HIV/AIDS epidemic in a population which experiences a significant flow of immigrants. We derive and analyze a mathematical model that describes the dynamics of HIV infection among the immigrant youths and how parental care can minimize or prevent the spread of the disease in the population. We analyze the model with both screening control and parental care, then investigate its stability and sensitivity behavior. We also conduct both qualitative and quantitative analyses. It is observed that in the absence of infected youths, disease-free equilibrium is achievable and is globally asymptotically stable. We establish optimal strategies for the control of the disease with screening and parental care, and provide numerical simulations to illustrate the analytic results.

关键词： HIV/AIDS infected immigrants basic reproduction number disease-freeequilibrium endemic equilibrium parental care.

Modeling transmission dynamics of Ebola virus disease

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International Journal of Biomathematics 2017年第4期10卷 297-331页

作者： Mudassar Imran Adnan Khan Ali R. Ansari Syed Touqeer Hussain Shah International Center for Applied Mathematics and Computational Bioengineering Department of Mathematics and Natural Sciences Gulf University for Science and Technology Hawally Kuwait Department of Mathematics Lahore University of Management Sciences Lahore Pakistan

Ebola virus disease （EVD） has emerged as a rapidly spreading potentially fatal disease. Several studies have been performed recently to investigate the dynamics of EVD. In this paper, we study the transmission dynamics of EVD by formulating an SEIR-type transmission model that includes isolated individuals as well as dead individuals that are not yet buried. Dynamical systems analysis of the model is performed, and it is consequently shown that the disease-free steady state is globally asymptotically stable when the basic reproduction number, R0 is less than unity. It is also shown that there exists a unique endemic equilibrium when R0 〉 1. Using optimal control theory, we propose control strategies, which will help to eliminate the Ebola disease. We use data fitting on models, with and without isolation, to estimate the basic reproductive numbers for the 2014 outbreak of EVD in Liberia and Sierra Leone.

关键词： Ebola virus disease endemic equilibrium basic reproduction number hos-pitalization infectiousness optimal control transmission dynamics time series datafitting.

GLOBAL DYNAMICS OF A DIFFERENTIAL SUSCEPTIBILITY MODEL

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International Journal of Biomathematics 2012年第5期5卷 219-238页

作者： M. R. RAZVAN S. YASAMAN Sharif University of TechnologyDepartment of Mathematical SciencesP. O. Box 11155-9315 Tehran Iran School of MathematicsInstitute for Research in Fundamental Sciences ( IPM) P. O. Box 19395-5746 Tehran Iran

An SIS epidemiological model in a population of varying size with two dissimilar groups of susceptible individuals has been analyzed. We prove that all the solutions tend to the equilibria of the system. Then we use the Poincar~ Index theorem to determine the number of the rest points and their stability properties. It has been shown that bistability occurs for suitable values of the involved parameters. We use the perturbations of the pitchfork bifurcation points to give examples of all possible dynamics of the system. Some numerical examples of bistability and hysteresis behavior of the systeIn has been also provided.

关键词： Epidemiological model differential susceptibility disease-free equilibrium endemic equilibrium Poincar~ index bifurcation.

AN SIRS EPIDEMIC MODEL

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Applied Mathematics(A Journal of Chinese Universities) 2004年第1期19卷 101-108页

作者： ChenJunjie ZhejiangUniv Hangzhou310027China

This paper considers an SIRS epidemic model that incorporates constant immigrati on rate, a general population size dependent contact rate and proportional tran sfer rate from the infective class to susceptible class.A threshold parameter σ is identified. If σ≤1, the disease free equilibrium is globally stab le. If σ>1, a unique endemic equilibrium is locally asymptotically stable. For two important special cases of mass action incidence and standard incidence, global stability of the endemic equilibrium is proved provided the threshold is larger than unity. Some previous results are extended and improved.

关键词： epidemic model threshold endemic equilibrium global stability. ON 3 CHOOSABILITY OF PLANE GRAPHS WITHOUT 6 ,7 AND 9 CYCLES$$$$ Zhang Haihui 1,2 Xu Baogang 11School of Math. and Comput. Sci. Nanjing Normal Univ. Nanji ng 21009