The impact of nonlinear infection rate on the spread of computer virus

Nonlinear Dynamics

Volumn 82, Issue 1-2, 2015, Pages 85-95

  Yang, Lu Xing ^a   Yang, Xiaofan ^a  

^a CHONGQING UNIVERSITY   (China)

Author keywords

Asymptotically autonomous system; Computer virus propagation model; Generalized Poincare Bendixson theorem; Global stability; Nonlinear infection rate; Saturation effect; Viral equilibrium

Indexed keywords

EPIDEMIOLOGY;

ASYMPTOTICALLY AUTONOMOUS SYSTEM; COMPUTER VIRUS PROPAGATION; GLOBAL STABILITY; NONLINEAR INFECTION RATES; POINCARE; SATURATION EFFECTS;

COMPUTER VIRUSES;

EID: 84943355064     PISSN: 0924090X     EISSN: 1573269X     Source Type: Journal    
DOI: 10.1007/s11071-015-2140-z     Document Type: Article

References (44)

1. Szor, P.: The Art of Computer Virus Research and Defense. Addison-Wesley Education Publishers Inc., Reading (2005)
2. Cohen, F.: Computer viruses: theory and experiments. Comput. Secur. 6(1), 22–35 (1987)
3. Murray, W.H.: The application of epidemiology to computer viruses. Comput. Secur. 7(2), 130–150 (1988)
4. Kephart, J.O., White, S.R.: Directed-graph epidemiological models of computer viruses. In: Proc. 1991 IEEE Comput. Soc. Symp. Res. Secur. Privacy, pp. 343–359 (1991)
5. Kephart, J.O., White, S.R.: Measuring and modeling computer virus prevalence. In: Proc. 1993 IEEE Comput. Soc. Symp. Res. Secur. Privacy, pp. 2–15 (1993)
6. Muroya, Y., Enatsu, Y., Li, H.: Global stability of a delayed SIRS computer virus propagation model. Int. J. Comput. Math. 91(3), 347–367 (2014)
7. Mishra, B.K., Pandey, S.K.: Dynamic model of worms with vertical transmission in computer network. Appl. Math. Comput. 217(21), 8438–8446 (2011)
8. Zhu, Q., Yang, X., Yang, L.-X., Zhang, X.: A mixing propagation model of computer viruses and countermeasures. Nonlinear Dyn. 73(3), 1433–1441 (2013)
9. Mishra, B.K., Saini, D.K.: SEIRS epidemic model with delay for transmission of malicious objects in computer network. Appl. Math. Comput. 188(2), 1476–1482 (2007)
10. Feng, L., Liao, X., Li, H., Han, Q.: Hopf bifurcation analysis of a delayed viral infection model in computer networks. Math. Comput. Model. 56(7–8), 167–179 (2012)
11. Zhang, C., Zhao, Y., Wu, Y., Deng, S.: A stochastic dynamic model of computer viruses. Discrete Dyn. Nat. Soc. (2012). Article ID 264874
12. Pastor-Satorras, R., Vespignani, A.: Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86(14), 3200–3203 (2001)
13. Pastor-Satorras, R., Vespignani, A.: Epidemic dynamics and endemic states in complex networks. Phys. Rev. E 63(6) (2001). Article ID 066117
14. Huang, C.-Y., Lee, C.-L., Wen, T.-H., Sun, C.-T.: A computer virus spreading model based on resource limitations and interaction costs. J. Syst. Softw. 86(3), 801–808 (2013)
15. Yang, L.-X., Yang, X., Wen, L., Liu, J.: A novel computer virus propagation model and its dynamics. Int. J. Comput. Math. 89(17), 2307–2314 (2012)
16. Yang, L.-X., Yang, X.: The spread of computer viruses under the influence of removable storage devices. Appl. Math. Comput. 219(8), 3914–3922 (2012)
17. Yang, M., Zhang, Z., Li, Q., Zhang, G.: An SLBRS model with vertical transmission of computer virus over Internet. Discrete Dyn. Nat. Soc. (2012). Article ID 693695
18. Yang, L.-X., Yang, X., Zhu, Q., Wen, L.: A computer virus model with graded cure rates. Nonlinear Anal. RWA 14(1), 414–422 (2013)
19. Zhang, C., Liu, W., Xiao, J., Zhao, Y.: Hopf bifurcation of an improved SLBS model under the influence of latent period. Math. Probl. Eng. 2013 (2013). Article ID 196214
20. Muroya, Y., Li, H., Kuniya, T.: On global stability of a nonresident computer virus model. Acta Math. Sci. 19(6), 1935–1944 (2014)
21. Yang, L.-X., Yang, X., Wen, L., Liu, J.: Propagation behavior of virus codes in the situation that infected computers are connected to the Internet with positive probability, Discrete Dyn. Nat. Soc. (2012). Article ID 693695
22. Yang, L.-X., Yang, X.: A new epidemic model of computer viruses. Commun. Nonlinear Sci. Numer. Simul. 19(6), 1935–1944 (2014)
23. Capasso, V.: Mathematical Structures of Epidemic Systems. Lecture Notes in Biomathematics, vol. 97. Springer, Berlin (1993)
24. Levin, S.A., Hallam, T.G., Gross, L.J.: Applied Mathematical Ecology. Springer, New York (1989)
25. Anderson, R.M., May, R.M.: Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford (1991)
26. Capasso, V., Serio, G.: A generalization of the Kermack-McKendrick deterministic epidemic model. Math. Biosci. 42(1–2), 43–61 (1978)
27. Korobeinikov, A.: Global properties of infectious disease models with nonlinear incidence. Bull. Math. Biol. 69(6), 1871–1886 (2007)
28. Korobeinikov, A.: Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol. 68(3), 615–626 (2006)
29. Liu, W.M., Levin, S.A., Iwasa, Y.: Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23(2), 187–204 (1986)
30. Liu, W.M., Hethcote, H.W., Levin, S.A.: Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25(4), 359–380 (1987)
31. Hethcote, H.W., van den Driessche, P.: Some epidemiological models with nonlinear incidence. J. Math. Biol. 29(3), 271–287 (1991)
32. Derrick, W.R., van den Driessche, P.: Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population. Discrete Contin. Dyn. Syst. B 2, 299–309 (2003)
33. Hethcote, H.W., Levin, S.A.: Periodicity in epidemiological models, in Applied Mathematical Ecology (Trieste, 1986), Biomathematics 18. Springer, Berlin, pp. 193–211 (1989)
34. Ruan, S., Wang, W.: Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. Differ. Equ. 188(1), 135–163 (2003)
35. Xiao, D., Ruan, S.: Global analysis of an epidemic model with nonmonotone incidence rate. Math. Biosci. 208(2), 419–429 (2007)
36. Tang, Y., Huang, D., Ruan, S., Zhang, W.: Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate. SIAM J. Appl. Math. 69(2), 621–639 (2008)
37. Li, G., Wang, W.: Bifurcation analysis of an epidemic model with nonlinear incidence. Appl. Math. Comput. 214(2), 411–423 (2009)
38. Yuan, H., Guan, L., Guan, C.: On modeling the crowding and psychological effects in network-virus prevalence with nonlinear epidemic model. Appl. Math. Comput. 219, 2387–2397 (2012)
39. Gan, C., Yang, X., Liu, W., Zhu, Q., Zhang, X.: An epidemic model of computer viruses with vaccination and generalized nonlinear incidence rate. Appl. Math. Comput. 222, 265–274 (2013)
40. Thieme, H.R.: Convergence results and a Poincare–Bendixson trichotomy for asymptotically autonomous differential equations. J. Math. Biol. 30(7), 755–763 (1992)
41. Robinson, R.C.: An Introduction to Dynamical Systems: Continuous and Discrete. Prentice Hall, Englewood Cliffs (2004)
42. http://anti-virus-software-review.toptenreviews.com/how-infected-are-we.html
43. http://tech.163.com/14/0917/06/A6ASUBAU000915BF.html
44. http://pcedu.pconline.com.cn/pingce/pingcesystem/1109/2527365_all.html