In this paper, a Susceptible-Infected-Quarantined-Susceptible (SIQRS) computer virus propagation model with nonlinear infection rate and two-delay is formulated. The local stability of virus-free equilibrium without delay is examined. Furthermore, we also expound and prove that time-delay plays a crucial role in sufficient conditions for the local stability of the virus-existence equilibrium and the occurrence of Hopf bifurcation at the critical value. Especially, direction and stability of the Hopf bifurcation are demonstrated. Finally, some numerical simulations are presented in order to verify the theoretical results.
Citation: Fangfang Yang, Zizhen Zhang. Dynamics of a nonlinear SIQRS computer virus spreading model with two delays[J]. AIMS Mathematics, 2021, 6(4): 4083-4104. doi: 10.3934/math.2021242
In this paper, a Susceptible-Infected-Quarantined-Susceptible (SIQRS) computer virus propagation model with nonlinear infection rate and two-delay is formulated. The local stability of virus-free equilibrium without delay is examined. Furthermore, we also expound and prove that time-delay plays a crucial role in sufficient conditions for the local stability of the virus-existence equilibrium and the occurrence of Hopf bifurcation at the critical value. Especially, direction and stability of the Hopf bifurcation are demonstrated. Finally, some numerical simulations are presented in order to verify the theoretical results.
| [1] | The 46th China Statistical Report on Internet Development, September, 2020. Available from: http://info.hebau.edu.cn/upload/files/2020/10/af9266d1e4e0cc8b.pdf. |
| [2] | R. Zhang, Y. Deng, Analysis of computer virus epidemic situation in October 2019, Netinfo Security, 12 (2019), 93–93. |
| [3] |
P. C. Xiao, Z. Y. Zhang, X. B. Sun, Smoking dynamics with health education effect, AIMS Mathematics, 3 (2018), 584–599. doi: 10.3934/Math.2018.4.584
|
| [4] |
J. L.Dimi, T. Mbaya, Dynamics analysis of stochastic tuberculosis model transmission withimmune response, AIMS Mathematics, 3 (2018), 391–408. doi: 10.3934/Math.2018.3.391
|
| [5] |
M. A. Khan, S. Ullah, M. Farhan, The dynamics of Zika virus with Caputo fractional derivative, AIMS Mathematics, 4 (2019), 134–146. doi: 10.3934/Math.2019.1.134
|
| [6] |
O. Kostylenko, H. S. Rodrigues, D. F. M. Torres, The spread of a financial virus through Europe and beyond, AIMS Mathematics, 4 (2019), 86–98. doi: 10.3934/Math.2019.1.86
|
| [7] | J. O. Kephart, S. R. White, Directed-graph Epidemiological Models of Computer Viruses, Proceedings of the 1991 IEEE Computer Society Symposium on Research in Security and Privacy, 1 (1991), 343–359. |
| [8] |
X. Y. Liang, Y. Z. Pei, Y. F. Lv, Modeling the state dependent impulse control for computer virus propagation under media coverage, Physica A, 491 (2018), 516–527. doi: 10.1016/j.physa.2017.09.058
|
| [9] | W. Pan, Z. Jin, Edge-based modeling of computer virus contagion on atripartite graph, Appl. Math. Comput., 320 (2018), 282–291. |
| [10] |
R. K. Upadhyay, P. Singh, Modeling and control of computer virus attack on a targetednetwork, Physica A, 538 (2020), 122617. doi: 10.1016/j.physa.2019.122617
|
| [11] | J. Singh, D. Kumar, Z. Hammouch, A. Atangana, Afractional epidemiological for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316 (2018), 504–515. |
| [12] |
J. G. Ren, X. F. Yang, L. X. Yang, Y. H. Xu, F. Z. Yang, A delayed computer virus propagation model and its dynamics, Chaos Solitons and Fractals, 45 (2012), 74–79. doi: 10.1016/j.chaos.2011.10.003
|
| [13] | V. MadhuSudanan, R. Greetha, Dynamics of epidemic computer virus spreading model with delays, Wireless Pers. Commun., 8 (2020), 1–14. |
| [14] |
U. Fatima, M. Ali, N. Ahmed, et al., Numerical modeling of susceptible latent breaking-out quarantine computer virus epidemic dynamics, Heliyon, 4 (2018), e00631. doi: 10.1016/j.heliyon.2018.e00631
|
| [15] | Y. H. Zheng, J. H. Zhu, C. A. Lai, A SEIQR Model considering the Effects of Different Quarantined Rates on Worm Propagation in Mobile Internet, Math. Probl. Eng., 2020 (2020), 1–16. |
| [16] | P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2020), 29–48. |
| [17] | K. R. Meyer, Theory and applications of Hopf bifurcation, SIAM Rev., 24 (2006), 498–499. |
| [18] |
Y. Z. Bai, Y. Y. Li, Stability and Hopf bifurcation for a stage-structured predator-prey model incorporating refuge for prey and additional food for predator, Adv. Differ. Equ., 2019 (2019), 1–20. doi: 10.1186/s13662-018-1939-6
|
| [19] |
C. Li, X. F. Liao, The impact of hybrid quarantine strategies and delay factor on viral prevalence in computer networks, Math. Model. Nat. Pheno., 11 (2016), 105–119. doi: 10.1051/mmnp/201611408
|
| [20] | Q. W. Gao, J. Zhuang, Stability analysis and control strategies for worm attack in mobile networks via a VEIQS propagation model, Appl. Math. Comput., 368 (2020), 124584. |
Figures(10) / Tables(1)
Fangfang Yang, Zizhen Zhang. Dynamics of a nonlinear SIQRS computer virus spreading model with two delays[J]. AIMS Mathematics, 2021, 6(4): 4083-4104. doi: 10.3934/math.2021242
DownLoad: