### Modeling the Dynamics of Rabies Transmission with Vaccination and Stability Analysis

Received: 7 September 2015     Accepted: 21 September 2015     Published: 10 October 2015
Abstract

In this paper we formulate a deterministic mathematical model for the transmission dynamics of rabies in human and animal within and around Addis Ababa, Ethiopia. Our model involves vaccination program for dog population. The basic reproduction number and effective reproduction numbers are computed and the results are entirely depending on the parameters of dog population, which shows the responsibility of dog population for human and livestock infection. For a specified set of values of parameters as deduced from the data provided by Ethiopian Public Health Institute of Addis Ababa, the basic reproduction number R0 and the effective reproduction number Re works out to be 2 and 1.6 respectively, which indicates the disease will be endemic. The numerical simulation of reproduction ratio shows that the combination of vaccination, culling of stray dogs and controlling annual crop of new born puppies are the best method to control rabies transmission within and around Adds Ababa. The disease - free equilibrium ε0 is computed. When the effective reproduction number Re<1 it is proved to be globally asymptotically stable in the feasible region Φ. When Re>1 there exists one endemic equilibrium point which is locally asymptotically stable.

 Published in Applied and Computational Mathematics (Volume 4, Issue 6) DOI 10.11648/j.acm.20150406.13 Page(s) 409-419 Creative Commons This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. Copyright Copyright © The Author(s), 2015. Published by Science Publishing Group
Keywords

Rabies, Addis Ababa, Endemic, Reproduction Number, Equilibrium Points

References
 [1] K. M. Addo, An SEIR Mathematical Model for Dog Rabies. Case Study: Bongo District, Ghana, MSc. Dissertation Kwame Nkrumah University of Science and Technology, 2012. [2] S. N. Sivanandam and S. N. Deepa, Linear system design using Routh Column Polynomials. Songklanakarin J. Sci. Technol. 2007, 29 (6): 1651 - 1659. [3] A. Ali, F. Mengistu, K. Hussen, G. Getahun, A. Deressa, E. Yimer and K. Tafese, Overview of Rabies in and around Addis Ababa, in Animals Examined in EHNRI Zoonoses Laboratory Between, 2003 and 2009, Ethiopian Veterinary Journal, 14 (2010), 91 - 101. [4] C. Castillo - Chavez, Z. Feng, and W. Huang, Mathematical Approaches for Emerging and Re - emerging Infectious Diseases, An Introduction. Springer Verlag, (2002). [5] N. Chitnis, J. Hyman, J. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a malaria model, Bull. Math. Biology. 70 (2008), 1272 - 1296. [6] A. Deressa, A. Ali, M. Bayene, N. Selassie, E. Yimer and K. Hussen, The status of rabies in Ethiopia: A retrospective record review, Ethiopian Journal of Health Development. 24 (2010), 1 - 6. [7] H. Hethcote, The mathematics of infectious diseases SIAM Review. 42 (2000), 599 - 653. [8] Q. Hou, Z. Jin and S. Ruan, Dynamics of rabies epidemics and the impact of control efforts in Guangdong Province, China, Journal of theoretical biology, 300 (2012)39 - 47. [9] A. Iggidr, J. Mbang, G. Sallet& J. J. Tewa. Multi - compartment models, Discrete and Continuous Dynamical Systems series S, 2007 (Special), 506 - 519. [10] W. T. Jemberu, W. Molla, G. Almaw and S. Alemu, Incidence of rabies in humans anddomestic animals and people's awareness in North Gondar Zone, Ethiopia, PLoS Negl. Trop. Dis. 7 (2013). [11] S. Khan, Rabies molecular virology, diagnosis, prevention and treatment, 2012. Available oline at http: //www. biomedcentral. com/ content/ pdf/1743 - 422X - 9 - 50. pdf. Retrieved 13 April 2015. [12] A. L. Lloyd and S. Valeika, Network models in epidemiology: an overview, Complex population dynamics: nonlinear modeling in ecology, epidemiology and genetics, 2007. Available online at http: //infoserve. sandia.gov/sand_doc/ 2008 /086044.pdf. Retrieved 13 April 2015 [13] P. C. Parks, A new proof of the Routh – Hurwitz stability criterion using the second method of Liapunov. In Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 58 (1962), 694 - 702. [14] T. Reta, S. Teshale, A. Deresa, A. Ali, F. Mengistu, D. Sifer and C. Freuling, Rabies in animals and humans in and around Addis Ababa, the capital city of Ethiopia: A retrospective and questionnaire based study, Journal of Veterinary Medicine and Animal Health, 6 (2014), 178 - 186. [15] P. Van der Driessche and J. Watmough, Reproduction numbers and sub threshold endemic equilibria for compartmental models of disease transmission, Math. Biosciences. 180 (2002), 29 - 48. [16] World Health Organization, WHO Expert Consultation on Rabies. Second report, World Health Organization technical report series, p. 1, 2013. [17] M. Yousaf, M. Qasim, S. Zia, R. Khan, U. Ashfaq and S. Khan, Rabies molecular virology, diagnosis, prevention and treatment, Virology Journal, 9 (2012), 1 - 5. [18] J. Zhang, Z. Jin, G. Q. Sun, T. Zhou and S. Ruan, Analysis of rabies in China: transmission dynamics and control, PLoS One, 6 (2011), p. e20891.
• APA Style

Tesfaye Tadesse Ega, Livingstone S. Luboobi, Dmitry Kuznetsov. (2015). Modeling the Dynamics of Rabies Transmission with Vaccination and Stability Analysis. Applied and Computational Mathematics, 4(6), 409-419. https://doi.org/10.11648/j.acm.20150406.13

ACS Style

Tesfaye Tadesse Ega; Livingstone S. Luboobi; Dmitry Kuznetsov. Modeling the Dynamics of Rabies Transmission with Vaccination and Stability Analysis. Appl. Comput. Math. 2015, 4(6), 409-419. doi: 10.11648/j.acm.20150406.13

AMA Style

Tesfaye Tadesse Ega, Livingstone S. Luboobi, Dmitry Kuznetsov. Modeling the Dynamics of Rabies Transmission with Vaccination and Stability Analysis. Appl Comput Math. 2015;4(6):409-419. doi: 10.11648/j.acm.20150406.13

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author = {Tesfaye Tadesse Ega and Livingstone S. Luboobi and Dmitry Kuznetsov},
title = {Modeling the Dynamics of Rabies Transmission with Vaccination and Stability Analysis},
journal = {Applied and Computational Mathematics},
volume = {4},
number = {6},
pages = {409-419},
doi = {10.11648/j.acm.20150406.13},
url = {https://doi.org/10.11648/j.acm.20150406.13},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150406.13},
abstract = {In this paper we formulate a deterministic mathematical model for the transmission dynamics of rabies in human and animal within and around Addis Ababa, Ethiopia. Our model involves vaccination program for dog population. The   basic reproduction number and effective reproduction numbers are computed and the results are entirely depending on the parameters of dog population, which shows the responsibility of dog population for human and livestock   infection. For a specified set of values of parameters as deduced from the data provided by Ethiopian Public Health Institute of Addis Ababa, the basic reproduction number R0 and the effective reproduction number   Re works out to be 2 and 1.6 respectively, which indicates the disease will be endemic. The numerical simulation of reproduction ratio shows that the combination of vaccination, culling of stray dogs and controlling   annual crop of new born puppies are the best method to control rabies transmission within and around Adds Ababa. The disease - free equilibrium ε0 is computed. When the effective reproduction number   Re<1 it is proved to be globally asymptotically stable in the feasible region Φ. When Re>1 there exists one endemic equilibrium point which is locally asymptotically stable.},
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}
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AB  - In this paper we formulate a deterministic mathematical model for the transmission dynamics of rabies in human and animal within and around Addis Ababa, Ethiopia. Our model involves vaccination program for dog population. The   basic reproduction number and effective reproduction numbers are computed and the results are entirely depending on the parameters of dog population, which shows the responsibility of dog population for human and livestock   infection. For a specified set of values of parameters as deduced from the data provided by Ethiopian Public Health Institute of Addis Ababa, the basic reproduction number R0 and the effective reproduction number   Re works out to be 2 and 1.6 respectively, which indicates the disease will be endemic. The numerical simulation of reproduction ratio shows that the combination of vaccination, culling of stray dogs and controlling   annual crop of new born puppies are the best method to control rabies transmission within and around Adds Ababa. The disease - free equilibrium ε0 is computed. When the effective reproduction number   Re<1 it is proved to be globally asymptotically stable in the feasible region Φ. When Re>1 there exists one endemic equilibrium point which is locally asymptotically stable.
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Author Information
• CoCSE School, Nelson Mandela African Institution of Science and Technology, Arusha, Tanzania

• CoCSE School, Nelson Mandela African Institution of Science and Technology, Arusha, Tanzania

• CoCSE School, Nelson Mandela African Institution of Science and Technology, Arusha, Tanzania

• Sections