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Research in Applied Mathematics: Vol. 1
Research Article
Research in Applied Mathematics
Vol. 1 (2017), Article ID 101263, 16 pages
doi:10.11131/2017/101263

Mathematical Analysis of Visceral Leishmaniasis Model

F. Boukhalfa, M. Helal, and A. Lakmeche

Laboratory of Biomathematics, Department of Mathematics, Univ. Sidi-Bel-Abbes, P.B. 89, Sidi-Bel-Abbes, 22000, Algeria

Received 13 November 2016; Accepted 9 February 2017

Editor: Paul Bracken

Copyright © 2017 F. Boukhalfa 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

In this work, we consider a mathematical model describing the dynamics of visceral leishmaniasis in a population of dogs D. First, we consider the case of constant total population D, this is the case where birth and death rates are equal, in this case transcritical bifurcation occurs when the basic reproduction number 0 is equal to one, and global stability is shown by the mean of suitable Lyapunov functions. After that, we consider the case where the birth and death rates are different, if the birth rate is great than death rate the total dog population increases exponentially, while the infectious dogs I dies out if the basic reproduction number is less than one, if it is great than one then D goes to infinity. We also prove that the total population D will extinct for birth rate less than death rate. Finally we give numerical simulations.

Research Article
Research in Applied Mathematics
Vol. 1 (2017), Article ID 101263, 16 pages
doi:10.11131/2017/101263

Mathematical Analysis of Visceral Leishmaniasis Model

F. Boukhalfa, M. Helal, and A. Lakmeche

Laboratory of Biomathematics, Department of Mathematics, Univ. Sidi-Bel-Abbes, P.B. 89, Sidi-Bel-Abbes, 22000, Algeria

Received 13 November 2016; Accepted 9 February 2017

Editor: Paul Bracken

Copyright © 2017 F. Boukhalfa 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

In this work, we consider a mathematical model describing the dynamics of visceral leishmaniasis in a population of dogs D. First, we consider the case of constant total population D, this is the case where birth and death rates are equal, in this case transcritical bifurcation occurs when the basic reproduction number 0 is equal to one, and global stability is shown by the mean of suitable Lyapunov functions. After that, we consider the case where the birth and death rates are different, if the birth rate is great than death rate the total dog population increases exponentially, while the infectious dogs I dies out if the basic reproduction number is less than one, if it is great than one then D goes to infinity. We also prove that the total population D will extinct for birth rate less than death rate. Finally we give numerical simulations.