Dynamical behaviors of an Echinococcosis epidemic model with distributed delays

Kai Wang; Zhidong Teng; Xueliang Zhang

doi:10.3934/mbe.2017074

Article Contents

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

a.
Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi, Xinjiang 830054, China
b.
College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046, China

* Corresponding author: Kai Wang (E-mail:wangkaimath@sina.com)
* Corresponding author: Kai Wang (E-mail:wangkaimath@sina.com)

Received: July 25, 2016

Accepted: March 08, 2017

Published: November 2017

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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 $\mathcal{R}_0>1$. 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 $\mathcal{R}_0<1$, the disease-free equilibrium is globally asymptotically stable, and if $\mathcal{R}_0>1$, 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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Mathematics Subject Classification: Primary: 37N25, 93D30; Secondary: 92B05.

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Figure 1. Life cycle of Echinococcus granulosus

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Figure 2. Monthly new reported Echinococcosis cases in Xinjinag from 2004 to 2011

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Figure 3. The comparison between the reported human Echinococcosis cases in Xinjiang, China from January 2004 to December 2011 and the simulation of $I_H(t)$ from the model. The initial values used in the simulations were $S_D(0)=2\times10^6$, $I_D(0)=8\times10^5$, $S_L(0)=8.4\times10^8$, $I_L(0)=5.7\times10^7$, $S_H(0)=1.96\times10^7$, $E_H(0)=1500$, $I_H(0)=4$

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Figure 4. The tendency of the human Echinococcosis cases $I_H(t)$ in short and long times

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Figure 5. The tendency of the human Echinococcosis cases $I_H(t)$ with different values of $\mathcal{R}_0$. When $\beta_1=3.3\times10^{-10}$(lower curve) and $6.3\times10^{-10}$, and the values of other parameters in Table 3 do not change, $\mathcal{R}_0=0.9932$ and $1.2321$, respectively

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Figure 6. The influence of parameters on $\mathcal{R}_0$. (a) versus $A_1$; (b) versus $d_1$; (c) versus $\beta_1$; (d) versus $\sigma$. Other parameter values in Table 3 do not change

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Table 1. Infection of cattle and sheep liver/lung in Xinjiang, China

Region		Infection rate
Region		Sheep	Cattle
Northern Xinjiang	Ili	70%	41%
	Tacheng	63%	25%
	Altay	65.8%	27%
	Changji	50.78%	9.23%
Southern Xinjiang	Kashgar	47%	-
	Hotan	25%	18%
	Kezilesu Kirgiz Autonomous Prefecture	38%	6.2%
	Aksu	50.3%	6.29%
	Bayingolin Mongolia Autonomous Prefecture	60.3%	12.3%
Eastern Xinjiang	Hami	42%	17%
Eastern Xinjiang	Turpan	36%	16.68%

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Table 2. Infection of dog in Xinjiang, China

Region		Infection rate
Northern Xinjiang	Ili	70%
	Tacheng	63%
	Altay	65.8%
	Changji	50.78%
Southern Xinjiang	Kashgar	47%
	Hotan	25%
	Kezilesu Kirgiz Autonomous Prefecture	38%
	Aksu	50.3%
	Bayingolin Mongolia Autonomous Prefecture	60.3%
Eastern Xinjiang	Hami	42%
Eastern Xinjiang	Turpan	36%

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Table 3. Parameters and their values (unit: month⁻¹)

Parameters	Value	Comments	Source
$A_1$	$1.34\times 10^4$	recruitment rate for dog	[25]
$d_1$	0.0067	dog natural mortality rate	[1]
$\beta_1$	$6.3\times10^{-10}$	livestock to dog transmission rate	fitting
$\sigma$	1/6	recovery rate from infected to non-infected dogs	[13]
$A_2$	$8.7\times10^{6}$	recruitment rate for livestock	[10]
$d_2$	0.0275	livestock mortality rate	assumption
$\beta_2$	$2.8\times10^{-8}$	parasite egg-to-livestock transmission rate	fitting
$h_1$	1/3	survival time of larval cysts into the infection offal	[6]
$h_2$	1.17	average life expectancy for Echinococcus eggs	[17]
$A_3$	$2\times10^{4}$	human annual birth population	[2]
$d_3$	0.0012	human natural mortality rate	[14]
$\omega$	$1/(14\times12)$	human incubation period	[13]
$\mu$	0.0793%	human disease-related death rate	[5]
$\gamma$	0.0625	treatment/recovery rate	assumption
$\beta_3$	$2.96\times10^{-12}$	parasite egg-to-human transmission rate	fitting

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