The Dynamics of an Interactional Model of Rabies Transmitted between Human and Dogs

Wei Yang; Jie Lou

Bollettino dell'Unione Matematica Italiana (2009)

  • Volume: 2, Issue: 3, page 591-605
  • ISSN: 0392-4041

Abstract

top
Assuming that the population of dogs is constant and the population of human satisfies the Logistical model, an interactional model of rabies transmitted between human and dogs is formulated. Two thresholds R 0 and R 1 which determine the outcome of the disease are identified. Utilizing the method of Lyapunov function and the property of the cooperative systems, we get the global asymptotic stability for both the disease-free equilibrium and the endemic equilibrium. A critical vaccination rate is obtained, which determines whether the dog rabies dies out or becomes endemic. Some suggestions are provided to the prevention and control of rabies according to the results of analysis and simulations.

How to cite

top

Yang, Wei, and Lou, Jie. "The Dynamics of an Interactional Model of Rabies Transmitted between Human and Dogs." Bollettino dell'Unione Matematica Italiana 2.3 (2009): 591-605. <http://eudml.org/doc/290571>.

@article{Yang2009,
abstract = {Assuming that the population of dogs is constant and the population of human satisfies the Logistical model, an interactional model of rabies transmitted between human and dogs is formulated. Two thresholds $R_\{0\}$ and $R_\{1\}$ which determine the outcome of the disease are identified. Utilizing the method of Lyapunov function and the property of the cooperative systems, we get the global asymptotic stability for both the disease-free equilibrium and the endemic equilibrium. A critical vaccination rate is obtained, which determines whether the dog rabies dies out or becomes endemic. Some suggestions are provided to the prevention and control of rabies according to the results of analysis and simulations.},
author = {Yang, Wei, Lou, Jie},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {591-605},
publisher = {Unione Matematica Italiana},
title = {The Dynamics of an Interactional Model of Rabies Transmitted between Human and Dogs},
url = {http://eudml.org/doc/290571},
volume = {2},
year = {2009},
}

TY - JOUR
AU - Yang, Wei
AU - Lou, Jie
TI - The Dynamics of an Interactional Model of Rabies Transmitted between Human and Dogs
JO - Bollettino dell'Unione Matematica Italiana
DA - 2009/10//
PB - Unione Matematica Italiana
VL - 2
IS - 3
SP - 591
EP - 605
AB - Assuming that the population of dogs is constant and the population of human satisfies the Logistical model, an interactional model of rabies transmitted between human and dogs is formulated. Two thresholds $R_{0}$ and $R_{1}$ which determine the outcome of the disease are identified. Utilizing the method of Lyapunov function and the property of the cooperative systems, we get the global asymptotic stability for both the disease-free equilibrium and the endemic equilibrium. A critical vaccination rate is obtained, which determines whether the dog rabies dies out or becomes endemic. Some suggestions are provided to the prevention and control of rabies according to the results of analysis and simulations.
LA - eng
UR - http://eudml.org/doc/290571
ER -

References

top
  1. ZHANG, Y. Z., The epidemiology of rabies in China, Chinese Journal of vaccines and Immulization, 11 (2005), 140-143 (in Chinese). 
  2. ZHANG, Y. Z. - XIONG, C. L. - XIAO, D. L. - JIANG, R. J. - WANG, Z. X. - ZHANG, L. Z. - FU, Z. F., Human rabies in China, Emerg. Infect. Dis., 11 (2005), 12. 
  3. YU, Y. X., Rabies and the Vaccines, Chinese Medical Techology Press, Beijing, 2001. 
  4. ANDERSON, R. M. - JACKSON, H. C. - MAY, R. M. - SMITH, A. M., Population dynamics of fox rabies in Europe, Nature, 289 (1981), 765-771. 
  5. RHODES, C. J. - ATKINSON, R. P. D. - ANDERSON, R. M. - MACDONALD, D. W., Rabies in Zimbabwe: reservior dogs and the implications for disease control, Philos. Trans. R. Soc. Lond. B Biol. Sci., 353 (1998), 999-1010. 
  6. KALLEN, A. - ARCURI, P. - MURRAY, J. D., A simple model for the spatial spread and control of rabies, J. Theor. Biol., 116 (1985), 377-393. MR809307DOI10.1016/S0022-5193(85)80276-9
  7. ALLEN, L. J. S. - FLORES, D. A. - RATNAYAKE, R. K. - HERBOLD, J. R., Discrete-time deterministic and stochastic models for the spread of rabies, A. M. C., 132 (2002), 271-292. Zbl1017.92027MR1920484DOI10.1016/S0096-3003(01)00192-8
  8. WANG, X. W. - LOU, J., Two dynamical models about rabies between dogs and human, J. Biol. Sys., 16 (2008), 519-529. Zbl1355.92132
  9. HIRSCH, M. W., Systems of differential equations that are competivitive or cooperative. V. Convergence in 3-dimensional systems, J. D. E., 80 (1989), 94-106. Zbl0712.34045MR1003252DOI10.1016/0022-0396(89)90097-1
  10. JIANG, J. F., A note on a global stability theorem of M. W. Hirsch, Proc. A. M. C., 112 (1991), 803-806. Zbl0753.34034MR1043411DOI10.2307/2048704
  11. ZHAO, X., Dynamical systems in population biology, C. M. S., Springer, (2003), 15-20. MR1980821DOI10.1007/978-0-387-21761-1
  12. MA, Z. E. - ZHOU, Y. C. - WANG, W. D. - JIN, Z., The Mathematical Modeling and Research of the Endemiology Dynamics, Science Press, Beijing, (2004), 58-59 (in Chinese). 
  13. COSTILL-CHACEZ, C. - THIEME, H. R., Asymptotically autonomous epidemic models. In: O. Arino et al.(Eds.). Math. Population Dynamics: Analysis of Heterogenieity I Theory of Epidemics, Wuerz, (1995), 33. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.