# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
SEIQRS model for the transmission of malicious objects in computer network. (English) Zbl 1185.68042
Summary: Susceptible $\left(S\right)$ - exposed $\left(E\right)$ - infectious $\left(I\right)$ - quarantined $\left(Q\right)$ - recovered $\left(R\right)$ model for the transmission of malicious objects in computer network is formulated. Thresholds, equilibria, and their stability are also found with cyber mass action incidence. Threshold $Rcq$ determines the outcome of the disease. If $R\le 1$, the infected fraction of the nodes disappear so the disease die out, while if $Rcq>1$, the infected fraction persists and the feasible region is an asymptotic stability region for the endemic equilibrium state. Numerical methods are employed to solve and simulate the system of equations developed. The effect of quarantine on recovered nodes is analyzed. We have also analyzed the behavior of the susceptible, exposed, infected, quarantine, and recovered nodes in the computer network.
##### MSC:
 68M10 Network design and communication of computer systems 34D20 Stability of ODE 92D30 Epidemiology
##### References:
 [1] Newman, M. E. J.; Forrest, Stephanie; Balthrop, Justin: Email networks and the spread of computer viruses, Phys. rev. E 66, 035101-035104 (2002) [2] Anderson, R. M.; May, R. M.: Infectious diseases of humans, dynamics and control, (1992) [3] Anderson, R. M.; May, R. M.: Population biology of infectious disease I, Nature 180, 361-367 (1999) [4] Mishra, Bimal Kumar; Saini, D. K.: SEIRS epidemic model with delay for transmission of malicious objects in computer network, Appl. math. Comput. 188, No. 2, 1476-1482 (2007) · Zbl 1118.68014 · doi:10.1016/j.amc.2006.11.012 [5] Mishra, Bimal Kumar; Saini, Dinesh: Mathematical models on computer virus, Appl. math. Comput. 187, No. 2, 929-936 (2007) · Zbl 1120.68041 · doi:10.1016/j.amc.2006.09.062 [6] Mishra, Bimal Kumar; Jha, Navnit: Fixed period of temporary immunity after run of anti-malicious software on computer nodes, Appl. math. Comput. 190, No. 2, 1207-1212 (2007) · Zbl 1117.92052 · doi:10.1016/j.amc.2007.02.004 [7] Gelenbe, E.: Dealing with software viruses: a biological paradigm, Inform. security technical rep. 12, No. 4, 242-250 (2007) [8] Erol Gelenbe, Keeping viruses under control, in: Computer and Information Sciences – ISCIS 2005, 20th International Symposium, vol. 3733, Lecture Notes in Computer Science, Springer, October 2005. [9] Erol Gelenbe, Varol Kaptan, Yu Wang, Biological metaphors for agent behaviour, in: Computer and Information Sciences – ISCIS 2004, 19th International Symposium, vol. 3280, Lecture Notes in Computer Science, Springer-Verlag, October 2004, pp. 667 – 675. [10] J.R.C. Piqueira, F.B. Cesar, Dynamical models for computer virus propagation, Math. Prob. Eng., doi:10.1155/2008/940526. · Zbl 1189.68036 · doi:10.1155/2008/940526 [11] Piqueira, J. R. C.; Navarro, B. F.; Monteiro, L. H. A.: Epidemiological models applied to virus in computer networks, J. comput. Sci. 1, No. 1, 31-34 (2005) [12] S. Forest, S. Hofmeyr, A. Somayaji, T. Longstaff, Self – nonself discrimination in a computer, in: Proceedings of IEEE Symposium on Computer Security and Privacy, 1994, pp. 202 – 212. [13] Y. Wang, C.X. Wang, Modeling the effects of timing parameters on virus propagation, in: 2003 ACM Workshop on Rapid Malcode, ACM, October 2003, pp. 61 – 66. [14] Kermack, W. O.; Mckendrick, A. G.: Contributions of mathematical theory to epidemics, Proc. royal soc. London – series A 115, 700-721 (1927) · Zbl 53.0517.01 · doi:10.1098/rspa.1927.0118 [15] Kermack, W. O.; Mckendrick, A. G.: Contributions of mathematical theory to epidemics, Proc. roy. Soc. London – series A 138, 55-83 (1932) · Zbl 0005.30501 · doi:10.1098/rspa.1932.0171 [16] Kermack, W. O.; Mckendrick, A. G.: Contributions of mathematical theory to epidemics, Proc. roy. Soc. London – series A 141, 94-122 (1933) [17] Zou, C. C.; Gong, W. B.; Towsley, D.; Gao, L. X.: The monitoring and early detection of Internet worms, IEEE/ACM trans. Network. 13, No. 5, 961-974 (2005) [18] Kephart, J. O.; White, S. R.; Chess, D. M.: Computers and epidemiology, IEEE spectrum, 20-26 (1993) [19] Keeling, M. J.; Eames, K. T. D.: Networks and epidemic models, J. roy. Soc. interf. 2, No. 4, 295-307 (2005) [20] Ma. M. Williamson, J. Leill, An Epidemiological Model of Virus Spread and Cleanup, 2003, lt;http://www.hpl.hp.com/techreports/gt;. [21] Newman, M. E. J.; Forrest, S.; Balthrop, J.: Email networks and the spread of computer virus, Phys. rev. E 66, 035101-1-035101-4 (2002) [22] Draief, M.; Ganesh, A.; Massouili, L.: Thresholds for virus spread on networks, Ann. appl. Prob. 18, No. 2, 359-378 (2008) · Zbl 1137.60051 · doi:10.1214/07-AAP470 [23] W.T. Richard, J.C. Mark, Modeling virus propagation in peer-to-peer networks, in: IEEE International Conference on Information, Communications and Signal Processing (ICICS 2005), pp. 981 – 985. [24] Yan, Ping; Liu, Shengqiang: SEIR epidemic model with delay, J. aust. Math. soc. Series B – appl. Math. 48, No. 1, 119-134 (2006) · Zbl 1100.92058 · doi:10.1017/S144618110000345X [25] J.O. Kephart, A biologically inspired immune system for computers, in: Proceedings of International Joint Conference on Artificial Intelligence, 1995. [26] Madar, N.; Kalisky, T.; Cohen, R.; Ben Avraham, D.; Havlin, S.: Immunization and epidemic dynamics in complex networks, Eur. phys. J. B 38, 269-276 (2004) [27] Pastor-Satorras, R.; Vespignani, A.: Epidemics and immunization in scale-free networks, handbook of graphs and networks: from the genome to the Internet, (2002) [28] May, R. M.; Lloyd, A. L.: Infection dynamics on scale-free networks, Phys. rev. E 64, No. 066112, 1-3 (2001) [29] S. Datta, H. Wang, The effectiveness of vaccinations on the spread of email-borne computer virus, in: IEEE CCECE/CCGEI, IEEE, May 2005, pp. 219 – 223. [30] C.C. Zou, W. Gong, D. Towsley, Worm propagation modeling and analysis under dynamic quarantine defense, in: Proceedings of the ACM CCS Workshop on Rapid Malcode, ACM, 2003, pp. 51 – 60. [31] D. Moore, C. Shannon, G.M. Voelker, S. Savage, Internet quarantine: requirements for containing self-propagating code, in: Proceedings of IEEE INFOCOM2003, IEEE, April, 2003. [32] T. Chen, N. Jamil, Effectiveness of quarantine in worm epidemics, in: IEEE International Conference on Communications 2006, IEEE, June 2006, pp. 2142 – 2147. [33] Hethcote, H. W.; Den Driessche, P. Van: Some epidemiological models with nonlinear incidence, J. math. Biol. 29, 271-287 (1991) · Zbl 0722.92015 · doi:10.1007/BF00160539 [34] Ruan, S.; Wang, W.: Dynamical behavior of an epidemic model with a non linear incidence rate, Journal of differential equations 188, 135-163 (2003) · Zbl 1028.34046 · doi:10.1016/S0022-0396(02)00089-X [35] Hethcote, H.; Zhein, M.; Shengbing, L.: Effects of quarantine in six endemic models for infectious diseases, Math. biosci. 180, 141-160 (2002) · Zbl 1019.92030 · doi:10.1016/S0025-5564(02)00111-6 [36] Feng, Z.; Thieme, H. R.: Endemic models with arbitrarily distributed periods of infection, I: General theory, SIAM J. Appl. math. 61, 803 (2000) · Zbl 0991.92028 · doi:10.1137/S0036139998347834 [37] Feng, Z.; Thieme, H. R.: Endemic models with arbitrarily distributed periods of infection, II: Fast disease dynamics and permanent recovery, SIAM J. Appl. math. 61, 983 (2000) · Zbl 1016.92035 · doi:10.1137/S0036139998347846 [38] Feng, Z.; Thieme, H. R.: Recurrent outbreaks of childhood disease revisited: the impact of isolation, Math. biosci. 128, 93 (1995) · Zbl 0833.92017 · doi:10.1016/0025-5564(94)00069-C [39] Wu, L. I.; Feng, Z.: Homoclinic bifurcation in an SIQR model for childhood disease, J. diff. Eqn. 168, 150 (2000) · Zbl 0969.34042 · doi:10.1006/jdeq.2000.3882 [40] Greenhalgh, D.: Hopf bifurcation in epidemic models with a latent period and non-permanent immunity, Math. comp. Modell. 25, 85-107 (1997) · Zbl 0877.92023 · doi:10.1016/S0895-7177(97)00009-5 [41] Hethcote, H. W.; Stech, H. W.; Den Driessche, P. Van: Periodicity and stability in epidemic models: a survey, Epidemics and population problems, 65-85 (1981) · Zbl 0477.92014 [42] Cook, K. L.; Den Driessche, P. Van: Analysis of SEIRS epidemic model with two delays, J. math. Biol. 35, 240-260 (1996) · Zbl 0865.92019 · doi:10.1007/s002850050051 [43] Li, M. Y.; Graff, J. R.; Wang, L. C.; Karsai, J.: Global dynamics of a SEIR model with a varying total population size, Math. biosci. 160, 191-213 (1999) · Zbl 0974.92029 · doi:10.1016/S0025-5564(99)00030-9 [44] Li, M. Y.; Muldowney, J. S.: Global stability for the SEIR model in epidemiology, Math. biosci. 125, 155-164 (1995) · Zbl 0821.92022 · doi:10.1016/0025-5564(95)92756-5 [45] Li, M. Y.; Wang, L.: Global stability in some SEIR epidemic models, Mathematical approaches for emerging and reemerging infectious diseases: models, methods and theory 126, 295-312 (2002) · Zbl 1022.92035 [46] Michael, Y.; Smith, H.; Wang, L.: Global dynamics of SIER epidemic model with vertical transmission, SIAM journal of applied mathematics 62, No. 1, 58-69 (2001) · Zbl 0991.92029 · doi:10.1137/S0036139999359860 [47] Hale, J. K.: Ordinary differential equations, (1980)