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Dynamical behaviors of an Echinococcosis epidemic model with distributed delays

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  • 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.

    Mathematics Subject Classification: Primary: 37N25, 93D30; Secondary: 92B05.


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  • Figure 1.  Life cycle of Echinococcus granulosus

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

    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 $S_D(0)=2\times10^6$, $I_D(0)=8\times10^5$, $S_L(0)=8.4\times10^8$, $I_L(0)=5.7\times10^7$, $S_H(0)=1.96\times10^7$, $E_H(0)=1500$, $I_H(0)=4$

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

    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

    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

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

    Region Infection rate
    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%
    Turpan 36% 16.68%
     | Show Table
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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%
    Turpan 36%
     | Show Table
    DownLoad: CSV

    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
     | Show Table
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