Asymptotic analysis of endemic equilibrium to a brucellosis model

Mingtao Li; Xin Pei; Juan Zhang; Li Li; Mingtao Li; Xin Pei; Juan Zhang; Li Li

doi:10.3934/mbe.2019291

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 5836-5850. doi: 10.3934/mbe.2019291

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Asymptotic analysis of endemic equilibrium to a brucellosis model

Mingtao Li ^{1
,
,},
Xin Pei ²,
Juan Zhang ³,
Li Li ^{4
,
,}

1.
School of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, P. R. China
2.
Data Science And Technology, North University of China, Taiyuan, Shanxi 030051, P. R. China
3.
Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi, China
4.
School of Computer and Information Technology, Shanxi University, Taiyuan, Shanxi 030006, P. R. China

Received: 30 March 2019 Accepted: 13 June 2019 Published: 22 June 2019

Brucellosis is one of the worlds major infectious and contagious bacterial disease. In order to study different types of brucellosis transmission models among sheep, we propose a deterministic model to investigate the transmission dynamics of brucellosis with the flock of sheep divided into basic ewes and other sheep. The global dynamical behavior of this model is given: including the basic repro-duction number, the existence and uniqueness of positive equilibrium, the global asymptotic stability of the equilibrium. We prove the uniqueness of positive endemic equilibrium through using proof by contradiction, and the global stability of endemic equilibrium by using Lyapunov function. Especially, we give the specific coefficients of global Lyapunov function, and show the calculation method of these specific coefficients. By running numerical simulations for the cases with the basic reproduction number to demonstrate the global stability of the equilibria and the unique endemic equilibrium, re-spectively. By some sensitivity analysis of the basic reproduction number on parameters, we find that vaccination rate of sheep and seropositive detection rate of recessive infected sheep are very important factor for brucellosis.

Keywords:

Citation: Mingtao Li, Xin Pei, Juan Zhang, Li Li. Asymptotic analysis of endemic equilibrium to a brucellosis model[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5836-5850. doi: 10.3934/mbe.2019291

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Abstract

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