Deterministic and Stochastic Study for an Infected Computer Network Model Powered by a System of Antivirus Programs
We investigate the various conditions that control the extinction and stability of a nonlinear mathematical spread model with stochastic perturbations. This model describes the spread of viruses into an infected computer network which is powered by a system of antivirus software. The system is analy...
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| Format: | Article |
| Language: | English |
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Wiley
2017-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2017/3540278 |
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| author | Youness El Ansari Ali El Myr Lahcen Omari |
| author_facet | Youness El Ansari Ali El Myr Lahcen Omari |
| author_sort | Youness El Ansari |
| collection | DOAJ |
| description | We investigate the various conditions that control the extinction and stability of a nonlinear mathematical spread model with stochastic perturbations. This model describes the spread of viruses into an infected computer network which is powered by a system of antivirus software. The system is analyzed by using the stability theory of stochastic differential equations and the computer simulations. First, we study the global stability of the virus-free equilibrium state and the virus-epidemic equilibrium state. Furthermore, we use the Itô formula and some other theoretical theorems of stochastic differential equation to discuss the extinction and the stationary distribution of our system. The analysis gives a sufficient condition for the infection to be extinct (i.e., the number of viruses tends exponentially to zero). The ergodicity of the solution and the stationary distribution can be obtained if the basic reproduction number Rp is bigger than 1, and the intensities of stochastic fluctuations are small enough. Numerical simulations are carried out to illustrate the theoretical results. |
| format | Article |
| id | doaj-art-6aed34d9af03415cbacdbce6244b9a8f |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-6aed34d9af03415cbacdbce6244b9a8f2025-02-03T06:13:48ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2017-01-01201710.1155/2017/35402783540278Deterministic and Stochastic Study for an Infected Computer Network Model Powered by a System of Antivirus ProgramsYouness El Ansari0Ali El Myr1Lahcen Omari2Laboratory of Computer Sciences, Modeling and Systems, Department of Mathematics, Faculty of Sciences, Sidi Mohamed Ben Abdellah University, Dhar-Mahraz, BP 1796, Atlas, Fez, MoroccoLaboratory of Computer Sciences, Modeling and Systems, Department of Mathematics, Faculty of Sciences, Sidi Mohamed Ben Abdellah University, Dhar-Mahraz, BP 1796, Atlas, Fez, MoroccoLaboratory of Computer Sciences, Modeling and Systems, Department of Mathematics, Faculty of Sciences, Sidi Mohamed Ben Abdellah University, Dhar-Mahraz, BP 1796, Atlas, Fez, MoroccoWe investigate the various conditions that control the extinction and stability of a nonlinear mathematical spread model with stochastic perturbations. This model describes the spread of viruses into an infected computer network which is powered by a system of antivirus software. The system is analyzed by using the stability theory of stochastic differential equations and the computer simulations. First, we study the global stability of the virus-free equilibrium state and the virus-epidemic equilibrium state. Furthermore, we use the Itô formula and some other theoretical theorems of stochastic differential equation to discuss the extinction and the stationary distribution of our system. The analysis gives a sufficient condition for the infection to be extinct (i.e., the number of viruses tends exponentially to zero). The ergodicity of the solution and the stationary distribution can be obtained if the basic reproduction number Rp is bigger than 1, and the intensities of stochastic fluctuations are small enough. Numerical simulations are carried out to illustrate the theoretical results.http://dx.doi.org/10.1155/2017/3540278 |
| spellingShingle | Youness El Ansari Ali El Myr Lahcen Omari Deterministic and Stochastic Study for an Infected Computer Network Model Powered by a System of Antivirus Programs Discrete Dynamics in Nature and Society |
| title | Deterministic and Stochastic Study for an Infected Computer Network Model Powered by a System of Antivirus Programs |
| title_full | Deterministic and Stochastic Study for an Infected Computer Network Model Powered by a System of Antivirus Programs |
| title_fullStr | Deterministic and Stochastic Study for an Infected Computer Network Model Powered by a System of Antivirus Programs |
| title_full_unstemmed | Deterministic and Stochastic Study for an Infected Computer Network Model Powered by a System of Antivirus Programs |
| title_short | Deterministic and Stochastic Study for an Infected Computer Network Model Powered by a System of Antivirus Programs |
| title_sort | deterministic and stochastic study for an infected computer network model powered by a system of antivirus programs |
| url | http://dx.doi.org/10.1155/2017/3540278 |
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