Numerical solution and stability analysis of a nonlinear vaccination model with historical e ects
Numerical solution and stability analysis of a nonlinear vaccination model with historical e ects
In this paper, we extend the classical vaccination epidemic model from a deterministic framework to a model with historical eects by formulating it as a system of fractional-order dierential equations (FDEs). The basic reproduction number R0 of the resulting fractional model is computed and it is shown that if R0 is less than one, the disease-free equilibrium is locally asymptotically stable. Particularly, we analytically calculate a certain threshold-value for R0 and present the existence conditions of endemic equilibrium. By using stability analysis, we prove stability and α-stability of the endemic equilibrium points. The proposed model is applied on Pertussis disease and the fractional nonlinear system of the model is solved by applying multi-step generalized dierential transform method (MSGDTM). Our results show that historical eects play an important role on the disease spreading.
___
- Abuteen, E., Momani, S. and Alawneh, A. Solving the fractional nonlinear Bloch system using the multi-step generalized di erential transform method , Comput. Math. Appl., 68 (12), 2124-2132, 2014.
- Alipour, M., Beghin, L. and Rostamy, D. Generalized Fractional Nonlinear Birth Processes , Methodol. Comput. Appl. Probab., 1-16, 2013.
- Allen, L. J. and Driessche, P. Stochastic epidemic models with a backward bifurcation , Math. Biosci. Eng., 3 (3), 445-458, 2006.
- Anguelov, R., Garba, S. M. and Usaini, S. Backward bifurcation analysis of epidemiological model with partial immunity , Comput. Math. Appl., 68 (9), 931-940, 2014.
- Ansari, M. A, Arora, D. and Ansari, S. P. Chaos control and synchronization of fractional order delay-varying computer virus propagation model , Math. Methods Appl. Sci., Jan 1, 2015.
- Arino, J., McCluske, C. C. and van den Driessche P. Global results for an epidemic model with vaccination that exhibits backward bifurcation , SIAM J. Appl. Math., 64 (1), 260-276, 2003.
- Boguna, M., Lafuerza, L. F., Toral, R. and Serrano, M. A. Simulating non-Markovian stochastic processes , Phys. Rev. E, 90 (4), 042108, 2014.
- [Caputo, M. Linear model of dissipation whose Q is almost frequency independent-II , Geo-physical J. International, 13 (5), 529-539, 1967.
- Demirci, E. and Ozalp, N., A method for solving di erential equations of fractional order . J. Comput. Appl. Math., 236(11), 2754-2762, 2012.
- Demirci, E., Unal, A. and Ozalp, N., A Fractional Order SEIR Model with Density Depen-dent Death Rate, Hacet. J. Math. Stat., 40(2), 2011.
- Ding, Y. and Ye, H. A fractional-order di erential equation model of HIV infection of CD4 +
T-cells, Math. Comput. Modelling, 50 (3), 386-392, 2009.
- Duan, X., Yuan, S., Qiu, Z. and Ma, J. Global stability of an SVEIR epidemic model with ages of vaccination and latency , Comput. Math. Appl., 68 (3), 288-308, 2014.
- Erturk, V., Momani, S. and Odibat, Z. Application of generalized di erential transform method to multi-order fractional di erential equations , Commun. Nonlinear Sci. Numer. Simul., 13 (8), 1642 1654, 2008.
- Erturk, V., Odibat, Z. and Momani, S. An approximate solution of a fractional order dif-ferential equation model of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells, Comput. Math. Appl., 62, 992 1002, 2011.
- Erturk, V. S., Zaman, G. and Momani, S. A numeric analytic method for approximating a giving up smoking model containing fractional derivatives , Comput. Math. Appl., 64 (10), 3065-3074, 2012.
- Frederico, G. S. F. and Torres, D. F. M. Fractional Noether's theorem in the Riesz Caputo sense, Appl. Math. Comput., 217 (3), 1023-1033, 2010.
- Hanert, E., Schumacher, E. and Deleersnijder, E. Front dynamics in fractional-order epi-demic models, J. Theoret. Biol., 279 (1), 9-16, 2011.
- Hethcote, H. W. An age-structured model for pertussis transmission , Math. Biosci., 145
(2), 89-136, 1997.
- Hethcote, H. W. The mathematics of infectious diseases, SIAM review , 42 (4), 599-653, 2000.
- Kermack, W. O. and McKendrick, A. G. Contributions to the mathematical theory of epi-demics I, Bull. Math. Biol., 53 (1), 33-55, 1991.
- Kribs-Zaleta, C. M., Velasco-Hernandez and J. X. A simple vaccination model with multiple endemic states, Math. Biosci., 164, 183-201, 2000.
- Lakshmikantham, V., Theory of fractional dynamic systems , Cambridge Scientifc Publ, 2009.
- Linkenkaer-Hansen, K., Nikouline, V. V., Palva, J. M.,R. and Ilmoniemi, J. Long-range temporal correlations and scaling behavior in human brain oscillations , J. Neuroscience, 21
(4), 1370-1377, 2001.
- Matignon, D. Stability results for fractional di erential equations with applications to control processing, Comput. Eng. Syst. Appl., 2, 963-968, 1996.
- Miller, K. S. and Ross, B. An Introduction to the Fractional Calculus and Fractional Dif-ferential Equations , Wiley, New York 1993.
- Odibat, Z., Bertelle, C., Aziz-Alaoui, M.A., Duchamp and G. A multi-step di erential transform method and application to non-chaotic or chaotic systems , Comput. Math. Appl., 59 (4), 1462 1472, 2010.
- Odibat, Z., Momani,S. and Erturk, V. Generalized di erential transform method: Applica-tion to di erential equations of fractional order , Appl. Math. Comput., 197 (2), 467 477, 2008.
- Ozalp, N. and Demirci, E., A fractional order SEIR model with vertical transmission , Math. Comput. Modelling, 54(1), 1-6, 2011.
- Peng, C. K., Havlin, S., Stanley, H. E. and Goldberger, A. L. Quanti cation of scaling exponents and crossover phenomena in nonstationary heartbeat time series , Chaos, 5 (1), 82-87, 1995.
- Podlubny, I. Fractional Di erential Equations , Academic Press, San Diego, 1999.
- Porco, T. C. and Blower, S. M. Designing HIV vaccination policies: subtypes and cross-immunity, Interfaces, 28 (3), 167-190, 1998.
- Safdari, H., Kamali, M. Z., Shirazi, A. H., Khaliqi, M., Jafari, G. History e ects on network growth, arXiv preprint, 2015, arXiv:1505.06450.
- Van Mieghem, P. and Van de Bovenkamp, R. Non-Markovian infection spread dramatically alters the susceptible-infected-susceptible epidemic threshold in networks , Phys. Rev. Lett., 110 (10), 108701, 2013.
- Wang, P. Y., Lin, S. D. and Srivastava, H. M. Remarks on a simple fractional-calculus approach to the solutions of the Bessel di erential equation of general order and some of its applications, Comput. Math. Appl., 51 (1), 105-114, 2006.
- Yulmetyev, R. M., Emelyanova, N. A., Demin, S. A., Gafarov, F. M., Hanggi, P. and Yulmetyeva, D. G. Non-Markov stochastic dynamics of real epidemic process of respiratory infections, Phys. A, 331 (1), 300-318, 2004.
- Zeng, G.Z., Chen, L.S. and Sun, L.H. Complexity of an SIR epidemic dynamics model with impulsive vaccination control , Chaos, Solitons & Fractals, 26 (2), 495-505, 2005