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.
- brucellosis,
- dynamic modeling,
- global dynamical behavior,
- stability
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

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.

References

[1]	M. J. Corbel, Brucellosis: an overview, Emerg. Infect. Dis., 3(1997), 213–221.
[2]	G. Pappas, N. Akritidis, M. Bosilkovski, et al., Brucellosis, N. Engl. J. Med., 352(2005), 2325–2536.
[3]	M. T. Li, G. Q. Sun, J. Zhang, et al., Transmission dynamics and control for a Brucellosis Model in Hinggan League of Inner Mongolia, China, Math. Biosci. Eng., 11(2014), 1115–1137.
[4]	M. L. Boschiroli, V. Foulongne and D. O'Callaghan, Brucellosis: a worldwide zoonosis, Curr. Opin. Microbiol., 4(2001), 58–64.
[5]	G. Pappas, P. Papadimitriou, N. Akritidis, et al., The new global map of human brucellosis, Lancet Infect. Dis., 6(2006), 91–99.
[6]	M. P. Franco, M. Mulder, R. H. Gilman, et al., Human brucellosis, Lancet Infect. Dis., 7(2007), 775–786.
[7]	M. N. Seleem, S. M. Boyle and N. Sriranganathan, Brucellosis: A re-emerging zoonosis, Vet. Microbiol., 140 (2010), 392–398.
[8]	H. Heesterbeek, R. M. Anderson, V. Andreasen, et al., Modeling infectious disease dynamics in the complex landscape of global health, Science, 6227 (2015), aaa4339.
[9]	M. T. Li, G. Q. Sun, J. Zhang, et al., Transmission dynamics of a multi-group brucellosis model with mixed cross infection in public farm, Appl. Math. Comput., 237(2014), 582–594.
[10]	G. G. Jorge and N. Raul, Analysis of a model of bovine brucellosis using singular perturbations, J. Math. Biol., 33(1994), 211–223.
[11]	J. Zinsstag, F. Roth, D. Orkhon, et al., A model of animalChuman brucellosis transmission in Mongolia, Prev. Vet. Med., 69(2005), 77–95.
[12]	B. Alnseba, B. Chahrazed and M. Pierre, A model for ovine brucellosis incorporating direct and indirect transmission, J. Biol. Dyn., 4(2010), 2–11.
[13]	Q. Hou, X. D. Sun, J. Zhang, et al., Modeling the transmission dynamics of brucellosis in Inner Mongolia Autonomous Region, China, Math. Biosci., 242(2013), 51–58.
[14]	Q. Hou, X. D. Sun, Y. M. Wang, et al., Global properties of a general dynamic model for animal diseases: A case study of brucellosis and tuberculosis transmission, J. Math. Anal. Appl., 414 (2014), 424–433.
[15]	W. Beauvais, I. Musallam and J. Guitian, Vaccination control programs for multiple livestock host species: An age-stratified, seasonal transmission model for brucellosis control in endemic settings, Paras. Vector, 9(2016), 55.
[16]	P. Lou, L. Wang, X. Zhang, et al., Modelling Seasonal Brucellosis Epidemics in Bayingolin Mongol Autonomous Prefecture of Xinjiang, China, 2010-2014, BioMed Res. Int., 2016(2016), 5103718.
[17]	D. O. Montiel, M. Bruce, K. Frankena, et al., Financial analysis of brucellosis control for small-scale goat farming in the Bajio region, Mexico, Prev. Vet. Med., 118(2015), 247–259.
[18]	L. Yang, Z. W. Bi, Z. Q. Kou, et al., Time-series analysis on human brucellosis during 2004-2013 in Shandong province, China, Zoonoses Public Health, 62(2015), 228–235.
[19]	M. T. Li, G. Q. Sun, W. Y. Zhang, et al., Model-based evaluation of strategies to control brucellosis in China, Int. J. Env. Res. Pub. He., 14(2017), 295.
[20]	M. T. Li, Z. Jin, G. Q. Sun, et al., Modeling direct and indirect disease transmission using multi-group model, J. Math. Anal. Appl., 446(2017), 1292–1309.
[21]	O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28(1990), 365–382.
[22]	P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equi-libria for compartmental models of disease transmission, Math. Biosci., 180(2002), 29–48.
[23]	O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface, 7(2010), 873–885.
[24]	M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Diff. Equat., 248(2010), 1–20.
[25]	J. P. Lasalle, The stability of dynamical dystems, in: Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976.
[26]	M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canad. Appl. Math. Quart., 14(2006), 259–284.
[27]	H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global lyapunov functions, Proc. Amer. Math. Soc., 136(2008), 2793–2802.
[28]	S. Marino, I. B. Hogue, C. J. Ray, et al., A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theoret. Biol., 254(2008), 178–196.

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