On the dynamics of a vaccination model with multiple transmission ways

Shu Liao ; Weiming Yang

International Journal of Applied Mathematics and Computer Science, Tome 23 (2013), p. 761-772 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2013-01-01

Zbl 1285.93043

EUDML-ID : urn:eudml:doc:262342

Bibliographie

[000] Anderson, R.M. and May, R.M.(1990). Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford.

[001] 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. | Zbl 1034.92025

[002] 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. | Zbl 1191.92024

[003] Brauer, F. (2004). Backward bifurcations in simple vaccination models, Journal of Mathematical Analysis and Application 298(2): 418-431. | Zbl 1063.92037

[004] 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. | Zbl 0825.62860

[005] 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. | Zbl 1271.34045

[006] 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. | Zbl 1232.34061

[007] 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.

[008] Castillo-Chavez, C. and Song, B.(2004). Dynamical models of tubercolosis and their applications, Mathematical Biosciences and Engineering 1(2): 361-404. | Zbl 1060.92041

[009] 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.

[010] 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. | Zbl 1107.92047

[011] 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. | Zbl 1142.92025

[012] Codeço, C.T. (2001). Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infectious Diseases, 1:1.

[013] 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. | Zbl 0586.92017

[014] Feckan, M. (2001). Criteria on the nonexistence of invariant Lipschitz submanifolds for dynamical systems, Journal of Differential Equations 174(2): 392-419. | Zbl 1002.34033

[015] 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. | Zbl 0892.92021

[016] 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. | Zbl 1018.92029

[017] 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.

[018] Hethcote, H.W. (2000). The mathematics of infectious diseases, SIAM Review 42(4):599-653. | Zbl 0993.92033

[019] 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. | Zbl 1326.00020

[020] Kribs-Zaleta, C.M. (1999). Structured models for heterosexual disease transmission, Mathematical Biosciences 160(1): 83-108. | Zbl 0980.92031

[021] 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. | Zbl 0998.92033

[022] Kribs-Zaleta, C.M. and Velasco-Hernandez, J.X. (2000). A simple vaccination model with multiple endemic states, Mathematical Biosciences 164(2): 183-201. | Zbl 0954.92023

[023] Li, G. and Zhen, J. (2005). Global stability of an SEI epidemic model with general contact rate, Chaos Solitons & Fractals 23(3): 997-1004. | Zbl 1062.92062

[024] Liao, S. and Wang, J. (2011). Stability analysis and application of a mathematical cholera model, Mathematical Biosciences and Engineering 8(3):733-752. | Zbl 1260.92063

[025] Liu, X. and Wang, C. (2010). Bifurcation of a predator-prey model with disease in the prey, Nonlinear Dynamics 62(4):841-850. | Zbl 1215.37053

[026] 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.

[027] 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. | Zbl 1005.92031

[028] 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.

[029] 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. | Zbl 1291.92103

[030] 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. | Zbl 1168.92326

[031] 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. | Zbl 1035.92023

[032] Sanchez, L.A. (2010). Existence of periodic orbits for high-dimensional autonomous systems, Journal of Mathematical Analysis and Applications 363(2): 409-418. | Zbl 1203.34051

[033] 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. | Zbl 1198.92030

[034] 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. | Zbl 1015.92036

[035] 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. | Zbl 0961.92029

[036] 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.

[037] 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. | Zbl 1192.65115

[038] 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. | Zbl 1198.34171

[039] 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. | Zbl 1167.34338

[040] Zhang, J. and Ma, Z. (2003). Global dynamics of an SEIR epidemic model with saturating contact rate, Mathematical Biosciences 185(1): 15-32. | Zbl 1021.92040

[041] World Health Organization (2010). Zimbabwe, http://www.who.int/countries/zwe/en/.