BibTex RIS Kaynak Göster
Yıl 2019, Cilt: 9 Sayı: 2, 206 - 219, 01.06.2019

Öz

Kaynakça

  • Kermack, W. O. and McKendrick, A. G., (1927), A Contribution to the Mathematical Theory of Epidemics, Proc. Roy. Soc. Lond. A, 115, pp. 700-721.
  • Anderson R. M. and May, R. M., (1979), Population biology of infectious diseases: Part I, Nature, 280, pp. 105111.
  • Anderson R. M. and May, R. M., (1991), Infectious Diseases of Humans: Dynamics and Control, Oxford University Press.
  • Hethcote, H. W., Stech, H. W. and vandenDriessche, P., Periodicity and stability in epidemiological models: A survey, In Differential Equations and Applications in Ecology, Epidemiology and Population Problems Academic Press New York (1981) pp. 65-85.
  • Hethcote H. W. and Levin, S. A., (1989), Periodicity in epidemiological models, In Applied Mathe- matical Ecology, Springer, New York, pp. 193-211.
  • Smith, H. L., (1983), Subharmonic bifurcation in an S-I-R epidemic model, Journal of Mathematical Biology, 17, (2), pp. 163177.
  • Kuznetsov, Yu. A. and Piccardi, C., (1994), Bifurcation analysis of periodic SEIR and SIR epidemic models, Journal of Mathematical Biology, 32, (2), pp. 109121.
  • Huang, W. Z., Cooke K. L. and Castillo-Chavez, C., (1992), Stability and Bifurcation for a Multiple- Group Model for the Dynamics of HIV/AIDS Transmission, SIAM J. Appl. Math., 52, (3), pp. 835854. [9] Shulgin, B., Stone L. and Agur, Z., (1998), Pulse vaccination strategy in the SIR epidemic model, Bulletin of Mathematical Biology, 60, (6), pp. 11231148.
  • Korobeinikov, A. and Wake, G. C., (2002), Lyapunov Functions and Global Stability for SIR, SIRS, and SIS Epidemiological Models, Applied Mathematics Letters, 15, pp. 955-960.
  • Li, C., Hu, W. and Huang, T., (2014), Stability and Bifurcation Analysis of a Modified Epidemic Model for Computer Viruses Mathematical Problems in Engineering, 14, pp. 784684.
  • Zhang, Z. and Yang, H., (2015), Dynamical Analysis of a Viral Infection Model with Delays in Com- puter Networks, Applied Mathematics and Mathematical Problems in Engineering, 15, pp. 280856.
  • Zhao, T. and Bi, D., (2017), Hopf bifurcation of a computer virus spreading model in the network with limited anti-virus ability, Advances in Difference Equations, pp. 183.
  • Wang, C. and Chai, S., (2016), Hopf bifurcation of an SEIRS epidemic model with delays and vertical transmission in the network, Advances in Difference Equations, pp.100.
  • Zhang, X., Li, C. and Huang, T., (2017), Bifurcation Analysis for an SEIRS-V Model with Delays on the Transmission of Worms in a Wireless Sensor Network, Mathematical Problems in Engineering, 15, pp. 9898726.
  • Zhang, Z., Wang, Y., Bi, D. and Guerrini, L., (2017), Stability and Hopf Bifurcation Analysis for a Computer Virus Propagation Model with Two Delays and Vaccination, Discrete Dynamics in Nature and Society, 17, pp. 3536125.
  • Zhang, X., Li, C. and Huang, T., (2016), Impact of impulsive detoxication on the spread of computer virus, Advances in Difference Equations, pp. 218.
  • Piqueira, J. R. C. and Araujo, V. O., (2009), A modified epidemiological model for computer viruses, Applied Mathematics and Computation, 213 (2) pp. 355-360.
  • Han, X. and Tan, Q., (2010), Dynamical behavior of computer virus on Internet, Applied Mathematics and Computation, 217 (6), pp. 2520-2526.
  • Li, J., Teng, Z., Wang, G., Zhang, L. and Hu, C., (2017), Stability and bifurcation analysis of an SIR epidemic model with logistic growth and saturated treatment, Chaos, Solitons and Fractals, 99, pp. 6371.
  • Beretta, E. and Takeuchi, Y., (1995), Global stability of an SIR epidemic model with time delays, J. Math. Biol., 33 (3), pp. 250-260.
  • Freedman, H. I. and Ruan, S., (1995), Uniform persistence in functional differential equations, J. Differential Equations, 115, pp. 173-192.
  • Ma, W., Song, M. and Takeuchi, Y., (2004), Global Stability of an SIR Epidemic Model with Time Delay, Applied Mathematics Letters, 17, pp. 1141-1145.
  • Xu, R. and Ma, Z., (2009), Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Analysis: Real World Applications, 10, pp. 3175-3189.
  • McCluskey, C. C., (2010), Global stability for an SIR epidemic model with delay and nonlinear incidence Nonlinear Analysis: Real World Applications, 11, pp. 3106-3109.
  • Sun, X., Wei, J. and Spechler, J. A., (2015), Stability and bifurcation analysis in a viral infection model with delays, Advances in Difference Equations, pp. 332.
  • Nowak, M. A. and Bangham, C. R. M., (1996), Population dynamics of immune responses to persistent viruses, Science, 272, pp. 74-79.
  • Beddington, J. R., (1975), Mutual interference between parasites or predators and its effect on search- ing efficiency, J. Animal Ecol., 44, pp. 331-340.
  • DeAngelis, D. L., Goldstein, R. A. and O’Neill, R.V., (1975), A model for trophic interaction, Ecology, 56, pp. 881-892.
  • Perelson, A., Neumann, A., Markowitz, M., Leonard, J. and Ho, D., (1996), HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science, 271, pp. 1582-1586.
  • Perelson A. and Nelson, P., (1999), Mathematical models of HIV dynamics in vivo, SIAM Rev., 41, pp. 3-44.
  • Hurwitz, A., (1895), Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt, Math. Ann. 46 pp. 273284.
  • Romanovski, V. G. and Shafer, D. S., (2009), The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhauser, Boston-Basel-Berlin.
  • Aybar, I. K., Aybar, O. O., Fercec, B., Romanovski, V. G., Samal, S. S. and Weber, A., (2015), Investigation of invariants of a chemical reaction system with algorithms of computer algebra, MATCH Commun. Math. Comput. Chem., 74 (3), pp. 465-480.
  • Aybar, I. K., Aybar, O. O., Dukaric, M., Fercec, B., (2018), Dynamical analysis of a two prey-one predator system with quadratic self interaction, Appl Math Comput., 333, pp. 118-132.
  • Antonov, V., Dolicanin, D., Romanovski, V. G. and Toth, J., (2016), Invariant Planes and Periodic Oscillations in the MayLeonard Asymmetric Model, MATCH Commun. Math. Comput. Chem., 76, pp. 455-474.
  • Boulierd, F., Hana, M., Lemaired, F. and Romanovski, V. G., (2015), Qualitative investigation of a gene model using computer algebra algorithms, Program Comput Soft+, 41 (2), pp. 105111.

A DYNAMICAL ANALYSIS OF THE VIRUS REPLICATION EPIDEMIC MODEL

Yıl 2019, Cilt: 9 Sayı: 2, 206 - 219, 01.06.2019

Öz

In this article, the stability and the computational algebraic properties of a virus replication epidemic model is investigated. The model is represented by a three dimensional dynamical system with six parameters. The conditions for the existence of Hopf bifurcation in the system are given. Then, the model with the Beddington- DeAngelis functional response instead of the original nonlinear response function has been studied in order to understand the e ect of the Beddington-DeAngelis functional response on the qualitative properties of the system. The stability of the systems at the singular points is investigated and the conditions for the systems to have the analytic rst integrals and Hopf bifurcation are given. Finally, the results are illustrated by giving numerical examples.

Kaynakça

  • Kermack, W. O. and McKendrick, A. G., (1927), A Contribution to the Mathematical Theory of Epidemics, Proc. Roy. Soc. Lond. A, 115, pp. 700-721.
  • Anderson R. M. and May, R. M., (1979), Population biology of infectious diseases: Part I, Nature, 280, pp. 105111.
  • Anderson R. M. and May, R. M., (1991), Infectious Diseases of Humans: Dynamics and Control, Oxford University Press.
  • Hethcote, H. W., Stech, H. W. and vandenDriessche, P., Periodicity and stability in epidemiological models: A survey, In Differential Equations and Applications in Ecology, Epidemiology and Population Problems Academic Press New York (1981) pp. 65-85.
  • Hethcote H. W. and Levin, S. A., (1989), Periodicity in epidemiological models, In Applied Mathe- matical Ecology, Springer, New York, pp. 193-211.
  • Smith, H. L., (1983), Subharmonic bifurcation in an S-I-R epidemic model, Journal of Mathematical Biology, 17, (2), pp. 163177.
  • Kuznetsov, Yu. A. and Piccardi, C., (1994), Bifurcation analysis of periodic SEIR and SIR epidemic models, Journal of Mathematical Biology, 32, (2), pp. 109121.
  • Huang, W. Z., Cooke K. L. and Castillo-Chavez, C., (1992), Stability and Bifurcation for a Multiple- Group Model for the Dynamics of HIV/AIDS Transmission, SIAM J. Appl. Math., 52, (3), pp. 835854. [9] Shulgin, B., Stone L. and Agur, Z., (1998), Pulse vaccination strategy in the SIR epidemic model, Bulletin of Mathematical Biology, 60, (6), pp. 11231148.
  • Korobeinikov, A. and Wake, G. C., (2002), Lyapunov Functions and Global Stability for SIR, SIRS, and SIS Epidemiological Models, Applied Mathematics Letters, 15, pp. 955-960.
  • Li, C., Hu, W. and Huang, T., (2014), Stability and Bifurcation Analysis of a Modified Epidemic Model for Computer Viruses Mathematical Problems in Engineering, 14, pp. 784684.
  • Zhang, Z. and Yang, H., (2015), Dynamical Analysis of a Viral Infection Model with Delays in Com- puter Networks, Applied Mathematics and Mathematical Problems in Engineering, 15, pp. 280856.
  • Zhao, T. and Bi, D., (2017), Hopf bifurcation of a computer virus spreading model in the network with limited anti-virus ability, Advances in Difference Equations, pp. 183.
  • Wang, C. and Chai, S., (2016), Hopf bifurcation of an SEIRS epidemic model with delays and vertical transmission in the network, Advances in Difference Equations, pp.100.
  • Zhang, X., Li, C. and Huang, T., (2017), Bifurcation Analysis for an SEIRS-V Model with Delays on the Transmission of Worms in a Wireless Sensor Network, Mathematical Problems in Engineering, 15, pp. 9898726.
  • Zhang, Z., Wang, Y., Bi, D. and Guerrini, L., (2017), Stability and Hopf Bifurcation Analysis for a Computer Virus Propagation Model with Two Delays and Vaccination, Discrete Dynamics in Nature and Society, 17, pp. 3536125.
  • Zhang, X., Li, C. and Huang, T., (2016), Impact of impulsive detoxication on the spread of computer virus, Advances in Difference Equations, pp. 218.
  • Piqueira, J. R. C. and Araujo, V. O., (2009), A modified epidemiological model for computer viruses, Applied Mathematics and Computation, 213 (2) pp. 355-360.
  • Han, X. and Tan, Q., (2010), Dynamical behavior of computer virus on Internet, Applied Mathematics and Computation, 217 (6), pp. 2520-2526.
  • Li, J., Teng, Z., Wang, G., Zhang, L. and Hu, C., (2017), Stability and bifurcation analysis of an SIR epidemic model with logistic growth and saturated treatment, Chaos, Solitons and Fractals, 99, pp. 6371.
  • Beretta, E. and Takeuchi, Y., (1995), Global stability of an SIR epidemic model with time delays, J. Math. Biol., 33 (3), pp. 250-260.
  • Freedman, H. I. and Ruan, S., (1995), Uniform persistence in functional differential equations, J. Differential Equations, 115, pp. 173-192.
  • Ma, W., Song, M. and Takeuchi, Y., (2004), Global Stability of an SIR Epidemic Model with Time Delay, Applied Mathematics Letters, 17, pp. 1141-1145.
  • Xu, R. and Ma, Z., (2009), Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Analysis: Real World Applications, 10, pp. 3175-3189.
  • McCluskey, C. C., (2010), Global stability for an SIR epidemic model with delay and nonlinear incidence Nonlinear Analysis: Real World Applications, 11, pp. 3106-3109.
  • Sun, X., Wei, J. and Spechler, J. A., (2015), Stability and bifurcation analysis in a viral infection model with delays, Advances in Difference Equations, pp. 332.
  • Nowak, M. A. and Bangham, C. R. M., (1996), Population dynamics of immune responses to persistent viruses, Science, 272, pp. 74-79.
  • Beddington, J. R., (1975), Mutual interference between parasites or predators and its effect on search- ing efficiency, J. Animal Ecol., 44, pp. 331-340.
  • DeAngelis, D. L., Goldstein, R. A. and O’Neill, R.V., (1975), A model for trophic interaction, Ecology, 56, pp. 881-892.
  • Perelson, A., Neumann, A., Markowitz, M., Leonard, J. and Ho, D., (1996), HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science, 271, pp. 1582-1586.
  • Perelson A. and Nelson, P., (1999), Mathematical models of HIV dynamics in vivo, SIAM Rev., 41, pp. 3-44.
  • Hurwitz, A., (1895), Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt, Math. Ann. 46 pp. 273284.
  • Romanovski, V. G. and Shafer, D. S., (2009), The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhauser, Boston-Basel-Berlin.
  • Aybar, I. K., Aybar, O. O., Fercec, B., Romanovski, V. G., Samal, S. S. and Weber, A., (2015), Investigation of invariants of a chemical reaction system with algorithms of computer algebra, MATCH Commun. Math. Comput. Chem., 74 (3), pp. 465-480.
  • Aybar, I. K., Aybar, O. O., Dukaric, M., Fercec, B., (2018), Dynamical analysis of a two prey-one predator system with quadratic self interaction, Appl Math Comput., 333, pp. 118-132.
  • Antonov, V., Dolicanin, D., Romanovski, V. G. and Toth, J., (2016), Invariant Planes and Periodic Oscillations in the MayLeonard Asymmetric Model, MATCH Commun. Math. Comput. Chem., 76, pp. 455-474.
  • Boulierd, F., Hana, M., Lemaired, F. and Romanovski, V. G., (2015), Qualitative investigation of a gene model using computer algebra algorithms, Program Comput Soft+, 41 (2), pp. 105111.
Toplam 36 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Research Article
Yazarlar

I. Kusbeyzı Aybar Bu kişi benim

Yayımlanma Tarihi 1 Haziran 2019
Yayımlandığı Sayı Yıl 2019 Cilt: 9 Sayı: 2

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