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Abstract and Applied Analysis
Volume 2015, Article ID 729894, 10 pages
http://dx.doi.org/10.1155/2015/729894
Research Article

A Fractional-Order Epidemic Model for Bovine Babesiosis Disease and Tick Populations

1Instituto de Ciências Exatas, Universidade Federal de Alfenas, 37130-000 Alfenas, MG, Brazil
2Instituto de Química, Universidade Federal de Alfenas, 37130-000 Alfenas, MG, Brazil

Received 7 April 2015; Revised 19 June 2015; Accepted 23 June 2015

Academic Editor: Jinde Cao

Copyright © 2015 José Paulo Carvalho dos Santos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper shows that the epidemic model, previously proposed under ordinary differential equation theory, can be generalized to fractional order on a consistent framework of biological behavior. The domain set for the model in which all variables are restricted is established. Moreover, the existence and stability of equilibrium points are studied. We present the proof that endemic equilibrium point when reproduction number is locally asymptotically stable. This result is achieved using the linearization theorem for fractional differential equations. The global asymptotic stability of disease-free point, when , is also proven by comparison theory for fractional differential equations. The numeric simulations for different scenarios are carried out and data obtained are in good agreement with theoretical results, showing important insight about the use of the fractional coupled differential equations set to model babesiosis disease and tick populations.