### The Impact of Infective Immigrants on the Spread of Dog Rabies

Received: 11 April 2017     Accepted: 2 May 2017     Published: 23 June 2017
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Abstract

In this paper, it is proposed and analyzed a new mathematical model and that is developed on the basis of some reasonable modification made to the standard epidemic model. The impact of immigration, treatment and the effect of vaccination are included in the model. The basic reproduction number is derived using the next generation matrix method. Disease free equilibrium point is found and endemic equilibrium state is identified. Numerical simulation study is conducted using ode 45 of MATLAB. It has been shown that the solution is positive and bounded. Algebraic expression for the reproduction number is constructed. Equilibrium points are identified and their stability analysis is carried out. It is pointed out that the disease dies out if the immigration of the infected dogs is controlled and the vaccination and the treatments are improved. Otherwise, the disease spreads rapidly in the dog population and it becomes an epidemic. Further, it is also pointed out that the impact of infective immigrants on the spread of dog rabies is positive and additive. The details are presented and discussed in the text.

 Published in American Journal of Applied Mathematics (Volume 5, Issue 3) DOI 10.11648/j.ajam.20170503.12 Page(s) 68-77 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), 2017. Published by Science Publishing Group
Keywords

Infective Immigrants, Rabies, Vaccination, Compartmental Model, Stability, Reproductive Number, Simulation

References
 [1] "Tesfaye Tadesse Ega, Living stone S. Luboobi, Modeling the Dynamics of Rabies Transmission with Vaccination and Stability Analysis, Applied and Computational Mathematics, (2015), 4 (6): 409-419. [2] John Bingham, Canine Rabies Ecology in Southern Africa, Emerging Infectious.www.cdc.gov/eid. vol. 11, No. 9, September 2005. [3] Elif Demisrci, A New Mathematical Approach for Rabies Endemic, Applied Mathematical Sciences, Vol. 8, (2015), no. 2, 59-67. [4] Kwaku Marri Addo. An SEIR Mathematical Model for Dog Rabies. Case Study: Bongo District, Ghana, M.Sc. Dissertation Kwame Nkrumah University of science and Technology, 2012. [5] Sandra DW, A. Arsum Arsin, Anwar Mallongi, The Dynamic Model Approach in Estimating Rabies Death in North Toroja Regency, International Journal of Sciences: Basic and Applied Research (IJSAR)(2015) volume 24, No 1. Pp 421-429. [6] Tiffany Ngo Leung, Mathematical Models for dog rabies that include the curtailing effect of human intervention, Australian Mathematical Sciences institute. [7] Demsis Dejene, Purnachandra Rao Koya, Population Dynamics of Dog Rabies Disease, IOSR Journal of Mathematics (IOSRJM) volume 12, Issue 3 ver. IV (may.-Jun 2016), pp 110-120. [8] Bernoulli, D. Blower, S. (2004). An attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it. Reviews in Medical Virology, 14, 275a""288. [9] Dancho Desaleng, Purnachandra Rao Koya. The Role of Polluted Air and Population Density in the Spread of Mycobacterium Tuberculosis Disease, Journal of Multidisciplinary Engineering Science and Technology (JMEST). Vol. 2, Issue 5, May – 2015, Pp 1212 – 20. ISSN: 3159 – 0040. http://www.jmest.org/wp content/uploads/JMESTN42350782.pdf [10] Ethiopian health and nutrition research institute (EHNRI). Rabies case report 1990 – 2010, EHNRI, Addis Ababa, Ethiopia, 2011. [11] Purnachandra Rao Koya and Dejen Ketema Mamo. Ebola Epidemic Disease: Modeling, Stability Analysis, Spread Control Technique, Simulation Study and Data Fitting. Journal of Multidisciplinary Engineering Science and Technology (JMEST), Vol. 2, Issue 3, March 2015, pp 476 – 84. ISSN: 3159 – 0040. http://www.jmest.org/wpcontent/uploads/JMESTN42350548.pdf [12] S. N. Sivanandam and S. N. Deepa. Linear system design using Routh column polynomials, Songklanakarin J. Sci. Technol. 2007, 29 (6), 1651 - 1659. [13] F. Brauer and P. van den Driessche 2001 Models for transmission of disease with immigration of infectives, Math. Biosci, 171 (2): 143–1542. [14] Coffee Megan, Lurie Mark N., and Garnett Geoff P. Modeling the impact of migration on the HIV epidemic in South Africa. AIDS, 21 (3): 343–350, 2007. [15] Tadele Degefa Bedada, Mihretu Nigatu Lemma and Purnachandra Rao Koya. Mathematical Modeling and simulation study of Influenza disease. Journal of Multidisciplinary Engineering Science and Technology (JMEST), Vol. 2, Issue 11, November 2015, Pp 3263 – 69. ISSN: 3159 – 0040. http://www.jmest.org/wp-content/uploads/JMESTN42351208.pdf [16] Tadele Tesfa Tegegne, Purnachandra Rao Koya and Temesgen Tibebu Mekonnen. Impact of Heterosexuality and Homosexuality on the transmission and dynamics of HIV/AIDS, IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 12, Issue 6 Ver. V (Nov. - Dec. 2016), PP 38-49. DOI: 10.9790/5728-1206053849. [17] Fekadu Tadege Kobe, Purnachandra Rao Koya. Controlling the Spread of Malaria Disease Using Intervention Strategies, Journal of Multidisciplinary Engineering Science and Technology (JMEST), Vol. 2, Issue 5, May 2015, pp 1068 – 74. ISSN: 3159 – 0040. http://www.jmest.org/wp-content/uploads/JMESTN42350745.pdf [18] Eti DWI Wiraningsih, Folashade Agusto, Lina Aryati. A Stability Analysis of Rabies Model with Vaccination Effect and Culling in Dogs, Applied Mathematical Sc. Vol. (9): 3805-3817, 2015. "
Cite This Article
• APA Style

Aberu Mengistu Tulu, Purnachandra Rao Koya. (2017). The Impact of Infective Immigrants on the Spread of Dog Rabies. American Journal of Applied Mathematics, 5(3), 68-77. https://doi.org/10.11648/j.ajam.20170503.12

ACS Style

Aberu Mengistu Tulu; Purnachandra Rao Koya. The Impact of Infective Immigrants on the Spread of Dog Rabies. Am. J. Appl. Math. 2017, 5(3), 68-77. doi: 10.11648/j.ajam.20170503.12

AMA Style

Aberu Mengistu Tulu, Purnachandra Rao Koya. The Impact of Infective Immigrants on the Spread of Dog Rabies. Am J Appl Math. 2017;5(3):68-77. doi: 10.11648/j.ajam.20170503.12

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title = {The Impact of Infective Immigrants on the Spread of Dog Rabies},
journal = {American Journal of Applied Mathematics},
volume = {5},
number = {3},
pages = {68-77},
doi = {10.11648/j.ajam.20170503.12},
url = {https://doi.org/10.11648/j.ajam.20170503.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20170503.12},
abstract = {In this paper, it is proposed and analyzed a new mathematical model and that is developed on the basis of some reasonable modification made to the standard epidemic model. The impact of immigration, treatment and the effect of vaccination are included in the model. The basic reproduction number is derived using the next generation matrix method. Disease free equilibrium point is found and endemic equilibrium state is identified. Numerical simulation study is conducted using ode 45 of MATLAB. It has been shown that the solution is positive and bounded. Algebraic expression for the reproduction number is constructed. Equilibrium points are identified and their stability analysis is carried out. It is pointed out that the disease dies out if the immigration of the infected dogs is controlled and the vaccination and the treatments are improved. Otherwise, the disease spreads rapidly in the dog population and it becomes an epidemic. Further, it is also pointed out that the impact of infective immigrants on the spread of dog rabies is positive and additive. The details are presented and discussed in the text.},
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AB  - In this paper, it is proposed and analyzed a new mathematical model and that is developed on the basis of some reasonable modification made to the standard epidemic model. The impact of immigration, treatment and the effect of vaccination are included in the model. The basic reproduction number is derived using the next generation matrix method. Disease free equilibrium point is found and endemic equilibrium state is identified. Numerical simulation study is conducted using ode 45 of MATLAB. It has been shown that the solution is positive and bounded. Algebraic expression for the reproduction number is constructed. Equilibrium points are identified and their stability analysis is carried out. It is pointed out that the disease dies out if the immigration of the infected dogs is controlled and the vaccination and the treatments are improved. Otherwise, the disease spreads rapidly in the dog population and it becomes an epidemic. Further, it is also pointed out that the impact of infective immigrants on the spread of dog rabies is positive and additive. The details are presented and discussed in the text.
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Author Information
• School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia

• School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia

• Sections