Dynamical behaviors of an Echinococcosis epidemic model with distributed delays

Kai Wang; Zhidong Teng; Xueliang Zhang; Kai Wang; Zhidong Teng; Xueliang Zhang

doi:10.3934/mbe.2017074

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 5&6: 1425-1445. doi: 10.3934/mbe.2017074

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Dynamical behaviors of an Echinococcosis epidemic model with distributed delays

. Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi, Xinjiang 830054, China

Received: 26 July 2016 Accepted: 09 March 2017 Published: 01 October 2017
MSC : Primary: 37N25, 93D30; Secondary: 92B05

In this paper, a novel spreading dynamical model for Echinococcosis with distributed time delays is proposed. For the model, we firstly give the basic reproduction number $\mathcal{R}_0$ and the existence of a unique endemic equilibrium when $\mathcal{R}_0 \gt 1$. Furthermore, we analyze the dynamical behaviors of the model. The results show that the dynamical properties of the model is completely determined by $\mathcal{R}_0$. That is, if $\mathcal{R}_0 \lt 1$, the disease-free equilibrium is globally asymptotically stable, and if $\mathcal{R}_0 \gt 1$, the model is permanent and the endemic equilibrium is globally asymptotically stable. According to human Echinococcosis cases from January 2004 to December 2011 in Xinjiang, China, we estimate the parameters of the model and study the transmission trend of the disease in Xinjiang, China. The model provides an approximate estimate of the basic reproduction number $\mathcal{R}_0=1.23$ in Xinjiang, China. From theoretic results, we further find that Echinococcosis is endemic in Xinjiang, China. Finally, we perform some sensitivity analysis of several model parameters and give some useful measures on controlling the transmission of Echinococcosis.
- Echinococcosis,
- distributed delays,
- transmission dynamics,
- basic reproduction number,
- global stability
Citation: Kai Wang, Zhidong Teng, Xueliang Zhang. Dynamical behaviors of an Echinococcosis epidemic model with distributed delays[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1425-1445. doi: 10.3934/mbe.2017074

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Abstract

In this paper, a novel spreading dynamical model for Echinococcosis with distributed time delays is proposed. For the model, we firstly give the basic reproduction number $\mathcal{R}_0$ and the existence of a unique endemic equilibrium when $\mathcal{R}_0 \gt 1$. Furthermore, we analyze the dynamical behaviors of the model. The results show that the dynamical properties of the model is completely determined by $\mathcal{R}_0$. That is, if $\mathcal{R}_0 \lt 1$, the disease-free equilibrium is globally asymptotically stable, and if $\mathcal{R}_0 \gt 1$, the model is permanent and the endemic equilibrium is globally asymptotically stable. According to human Echinococcosis cases from January 2004 to December 2011 in Xinjiang, China, we estimate the parameters of the model and study the transmission trend of the disease in Xinjiang, China. The model provides an approximate estimate of the basic reproduction number $\mathcal{R}_0=1.23$ in Xinjiang, China. From theoretic results, we further find that Echinococcosis is endemic in Xinjiang, China. Finally, we perform some sensitivity analysis of several model parameters and give some useful measures on controlling the transmission of Echinococcosis.

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