Global dynamics of a multi-stage brucellosis model with distributed delays and indirect transmission

Qiang Hou; Haiyan Qin; Qiang Hou; Haiyan Qin

doi:10.3934/mbe.2019154

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 4: 3111-3129. doi: 10.3934/mbe.2019154

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Research article Special Issues

Global dynamics of a multi-stage brucellosis model with distributed delays and indirect transmission

Qiang Hou ^,,
Haiyan Qin

Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China

Received: 23 January 2019 Accepted: 19 March 2019 Published: 10 April 2019

The mechanisms of brucellosis transmission are diverse and complex, especially the role of young animals in the spread of brucellosis has not been well studied. In this article, a new deterministic system that incorporates various stages of susceptible individuals and time delay of infection is proposed. Under general biological assumptions, the qualitative properties and stability of the system are studied, the results illustrate that the global dynamics of equilibrium points depend on the basic reproduction number $R_0$: If $R_0\leq 1$, animal brucellosis will eventually die out; and if $R_0> 1$, animal brucellosis is persistent and eventually tends to the endemic steady state. These results suggest that distributed time delay is harmless for the dynamics of the spread of brucellosis when $R_0$ is greater than one or less than or equal to one. Finally, periodic phenomena are found by numerical analysis if the assumptions are not true.
- brucellosis,
- multi-stage model,
- indirect transmission,
- global stability,
- Lyapunov functional
Citation: Qiang Hou, Haiyan Qin. Global dynamics of a multi-stage brucellosis model with distributed delays and indirect transmission[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 3111-3129. doi: 10.3934/mbe.2019154

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Abstract

The mechanisms of brucellosis transmission are diverse and complex, especially the role of young animals in the spread of brucellosis has not been well studied. In this article, a new deterministic system that incorporates various stages of susceptible individuals and time delay of infection is proposed. Under general biological assumptions, the qualitative properties and stability of the system are studied, the results illustrate that the global dynamics of equilibrium points depend on the basic reproduction number $R_0$: If $R_0\leq 1$, animal brucellosis will eventually die out; and if $R_0> 1$, animal brucellosis is persistent and eventually tends to the endemic steady state. These results suggest that distributed time delay is harmless for the dynamics of the spread of brucellosis when $R_0$ is greater than one or less than or equal to one. Finally, periodic phenomena are found by numerical analysis if the assumptions are not true.

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