Powiadomienia systemowe
- Sesja wygasła!
Identyfikatory
Warianty tytułu
Języki publikacji
Abstrakty
In this paper, we present a vaccination model with multiple transmission ways and derive the control reproduction number. The stability analysis of both the disease-free and endemic equilibria is carried out, and bifurcation theory is applied to explore a variety of dynamics of this model. In addition, we present numerical simulations to verify the model predictions. Mathematical results suggest that vaccination is helpful for disease control by decreasing the control reproduction number below unity.
Słowa kluczowe
Rocznik
Tom
Strony
761--772
Opis fizyczny
Bibliogr. 42 poz., tab., wykr.
Twórcy
autor
- School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China
autor
- School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China
Bibliografia
- [1] Anderson, R.M. and May, R.M.(1990). Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford.
- [2] Arino, J.C., McCluskey, C. and van den Driessche P. (2003). Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM Journal on Applied Mathematics, 64(1): 260–276.
- [3] Blayneh, K.W., Gumel, A.B., Lenhart, S. and Clayton, T. (2010). Backward bifurcation and optimal control in transmission dynamics of West Nile Virus, Bulletin of Mathematical Biology 72(4): 1006–1028.
- [4] Brauer, F. (2004). Backward bifurcations in simple vaccination models, Journal of Mathematical Analysis and Application 298(2): 418–431.
- [5] Blower, S.M. and Dowlatabadi, H. (1994). Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example, International Statistical Review 62(2): 229–243.
- [6] Buonomo, B. and Lacitignola, D. (2011). On the backward bifurcation of a vaccination model with nonlinear incidence, Nonlinear Analysis: Modelling and Control 16(1): 30–46.
- [7] Buonomo, B. and Lacitignola, D. (2008). On the dynamics of an SEIR epidemic model with a convex incidence rate, Ricerche di Matematica 57(2): 261–281.
- [8] Buonomo, B. and Lacitignola, D. (2010). Analysis of a tuberculosis model with a case study in Uganda, Journal of Biological Dynamics 4(6): 571–593.
- [9] Castillo-Chavez, C. and Song, B.(2004). Dynamical models of tubercolosis and their applications, Mathematical Biosciences and Engineering 1(2): 361–404.
- [10] Capasso, V. and Paveri-Fontana, S.L. (1979). A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Revue D’Epidémiologie de Santé Publique 27(2): 121–132.
- [11] Chitnis, N., Cushing, J.M. and Hyman, J.M.(2006). Bifurcation analysis of a mathematical model for malaria transmission, SIAM Journal of Applied Mathematics 67(1): 24–45.
- [12] Chitnis, N., Cushing, J.M. and Cushing, J.M. (2008). Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bulletin of Mathematical Biology 79(5): 1272–1296.
- [13] Codec¸o, C.T. (2001). Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infectious Diseases, 1:1.
- [14] Dietz, K. and Schenzle, D.(1985). Mathematical models for infectious disease statistics, in A.C. Atkinson and S.E. Fienberg (Eds.), Centenary Volume of the International Statistical Institute, Springer-Verlag, Berlin, pp. 167–204.
- [15] Feckan, M. (2001). Criteria on the nonexistence of invariant Lipschitz submanifolds for dynamical systems, Journal of Differential Equations 174(2): 392–419.
- [16] Gani, J., Yakowitz, S. and Blount, M. (1997). The spread and quarantine of HIV infection in a prison system, SIAM Journal on Applied Mathematics 57(6): 1510–1530.
- [17] Gumel, A.B. and Moghadas, S.M. (2003). A qualitative study of a vaccination model with non-linear incidence, Applied Mathematics and Computation 143(2–3): 409–419.
- [18] Hartley, D.M., Morris, J.G. and Smith, D.L. (2006). Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLoS Medicine 3(1): 63–69.
- [19] Hethcote, H.W. (2000). The mathematics of infectious diseases, SIAM Review 42(4):599–653.
- [20] Korn, G.A. and Korn, T.M. (2000). Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for References and Review, Dover Publications, Mineola, NY.
- [21] Kribs-Zaleta, C.M. (1999). Structured models for heterosexual disease transmission, Mathematical Biosciences 160(1): 83–108.
- [22] Kribs-Zaleta, C.M. and Martchevab, M. (2002). Vaccination strategies and backward bifurcation in an age-since-infection structured model, Mathematical Biosciences 177/178: 317–332.
- [23] Kribs-Zaleta, C.M. and Velasco-Hernandez, J.X. (2000). A simple vaccination model with multiple endemic states, Mathematical Biosciences 164(2): 183–201.
- [24] Li, G. and Zhen, J. (2005). Global stability of an SEI epidemic model with general contact rate, Chaos Solitons & Fractals 23(3): 997–1004.
- [25] Liao, S. and Wang, J. (2011). Stability analysis and application of a mathematical cholera model, Mathematical Biosciences and Engineering 8(3):733–752.
- [26] Liu, X. and Wang, C. (2010). Bifurcation of a predator-prey model with disease in the prey, Nonlinear Dynamics 62(4):841–850.
- [27] Marino, S., Hogue, I., Ray, C.J. and Kirschner, D.E.(2008). A methodology for performing global uncertainty and sensitivity analysis in system biology, Journal of Theoretical Biology 254(1): 178–196.
- [28] Moghadas, S.M. and Gumel, A.B. (2002). Global stability of a two-stage epidemic model with generalized non-linear incidence, Mathematics and Computers in Simulation 60(1–2): 107–118.
- [29] Mukandavire, Z., Liao, S., Wang, J., Gaff, H., Smith, D.L. and Morris, J.G. (2011). Estimating the reproductive numbers for the 2008–2009 cholera outbreaks in Zimbabwe, Proceedings of the National Academy of Sciences of the United States of America 108(21): 8767–8772.
- [30] Samsuzzoha, M.D., Singh, M. and Lucy, D. (2012). A numerical study on an influenza epidemic model with vaccination and diffusion, Applied Mathematics and Computation 219(1): 122–141.
- [31] Song, X., Jiang, Y. and Wei, H.(2009). Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays, Applied Mathematics and Computation 214(2): 381–390.
- [32] Szymańska, Z. (2013). Analysis of immunotherapy models in the context of cancer dynamics, International Journal of Applied Mathematics and Computer Science 13(3): 407–418.
- [33] Sanchez, L.A. (2010). Existence of periodic orbits for high-dimensional autonomous systems, Journal of Mathematical Analysis and Applications 363(2): 409–418.
- [34] Tien, J.H. and Earn, D.J.D. (2010). Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bulletin of Mathematical Biology 72(6): 1502–1533.
- [35] Van den Driessche, P. and Watmough, J.(2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences 180(1–2): 29–48.
- [36] Van den Driessche, P. and Watmough, J. (2000). A simple SIS epidemic model with a backward bifurcation, Journal of Mathematical Biology 40(6): 525–540.
- [37] Vynnycky, E., Trindall, A. and Mangtani, P.(2007). Estimates of the reproduction numbers of Spanish influenza using morbidity data, International Journal of Epidemiology 36(4): 881–889.
- [38] Yildirim, A. and Cherruault, Y.(2009). Analytical approximate solution of a SIR epidemic model with constant vaccination strategy by homotopy perturbation method, Kybernetes 38(9): 1566–1575.
- [39] Yu, H.G., Zhong, S.M., Agarwal, R.P. and Xiong L.L. (2010). Species permanence and dynamical behaviour analysis of an impulsively controlled ecological system with distributed time delay, Computers and Mathematics with Applications 59(2): 3824–3835.
- [40] Zhang, X. and Liu, X. (2009). Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment, Nonlinear Analysis: Real World Applications 10(2): 565–575.
- [41] Zhang, J. and Ma, Z. (2003). Global dynamics of an SEIR epidemic model with saturating contact rate, Mathematical Biosciences 185(1): 15–32.
- [42] World Health Organization (2010). Zimbabwe, http://www.who.int/countries/zwe/en/.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-d2128a3e-553c-436b-8977-2fb237431cdc