October 2017, 14(5&6): 1425-1445. doi: 10.3934/mbe.2017074

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)

Received  July 26, 2016 Accepted  March 09, 2017 Published  May 2017

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.

Citation: Kai Wang, Zhidong Teng, Xueliang Zhang. Dynamical behaviors of an Echinococcosis epidemic model with distributed delays. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1425-1445. doi: 10.3934/mbe.2017074
References:
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Baidu, How Long is the Life Expectancy of Dogs? 2011, Available from: http://wenku.baidu.com/view/e7ccf4fec8d376eeaeaa31f9.html.

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P. A. CabreraG. HaranU. benavidezS. ValledorG. PereraS. LloydM. A. GemmellM. BaraibarA. MoranaJ. Maissonave and M. Carballo, Transmission dynamics of Echinococcus granulosus, Taenia hydatigena and Taenia ovis in sheep in Uruguay, Int. J. Parasitol., 25 (1995), 807-813. doi: 10.1016/0020-7519(94)00209-7.

[5]

Y. CaoJ. Wen and Q. Zheng, Analysis of the epidemic status of Echinococcosis in Xinjiang in 2010, J. Ningxia Med. Univ., 33 (2011), 784-788.

[6]

J. Eckert, M. A. Gemmell, F. X. Meslin and Z. S. Pawlowski, WHO/OIE manual on Echinococcosis in humans and animals: A public health problem of global concern, Paris: World Health Organization/World Organization for Animal Health, 2001.

[7]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, NewYork, 1993.

[8]

S. LahmarH. DebbekL. H. ZhangD. P. McManusA. SouissiS. Chelly and P. R. Torgerson, Tansmission dynamics of the Echinococcus granulosus sheep-dog strain(G1 genotype) in camels in Tunisia, Vet. Parasitol., 121 (2004), 151-156. doi: 10.1016/j.vetpar.2004.02.016.

[9]

MalikeNusilaitiZulihumaerLvWaliXiAbudureyimuQiuAbuduaini and Hanati, Investigation of Echinococcus infection in domestic in Xinjiang, Chin. J. Anim.Infect. Dis., 19 (2011), 57-60(in Chinese).

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National Bureau of Statistics of China, China Statistical Yearbook, 2011 Available from: http://www.stats.gov.cn/tjsj/ndsj/2011/indexch.htm.

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Nusilaiti and Zulihumaer, The infection situation of Echinococcosis and countermeasures of some counties and cities in xinjiang, Xinjiang Livestock Industry, 6 (2011), 45-46.

[12]

H. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, 1995.

[13]

K. TakumiA. ViresM. ChuJ. MulderP. Teunis and J. Giessen, Evidence for an increasing presence of Echinococcus multilocularis in foxes in The Netherlands, Inter. J. Parasitol., 38 (2008), 571-578. doi: 10.1016/j.ijpara.2007.09.014.

[14]

The Government of Xinjiang Uygur Autonomous Region of China, The improvement of people's living standard, 2010, Available from: http://www.xinjiang.gov.cn/2011/11/15/64.html.

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H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically automous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267.

[16]

P. R. TorgersonK. K. BurtisurnovB. S. ShaikenovA. T. RysmukhambetovaA. M. Abdybekova and A. E. Ussenbayev, Modelling the transmission dynamics of Echinococcus granulosus in sheep and cattle in Kazakhstan, Vet. Parasitol., 114 (2003), 143-153. doi: 10.1016/S0304-4017(03)00136-5.

[17]

S. Wang and S. Ye, Textbook of Medical Microbiology and Parasitology, 1$^{st}$ edition, Science Press, Beijing, 2006.

[18]

Z. Wang and X.-Q. Zhao, Global dynamics of a time-delayed dengue transmission model, Can. Appl. Math. Q., 20 (2012), 89-113.

[19]

K. WangX. ZhangZ. JinH. MaZ. Teng and L. Wang, Modeling and analysis of the transmission of Echinococcosis with application to Xinjiang Uygur Autonomous Region of China, J. Theor. Biol., 333 (2013), 78-90. doi: 10.1016/j.jtbi.2013.04.020.

[20]

W. Wu, The Chinese National Plan for the Control Echincoccosis, Urumq: ISUOG's World Congress of Hydatidology Final Programme & Abstracts Book, 2011(in Chinese).

[21]

L. WuB. SongW. Du and J. Lou, Mathematical modelling and control of Echinococcus in Qinghai province, China, Math. Biosci. Eng., 10 (2013), 425-444. doi: 10.3934/mbe.2013.10.425.

[22]

Xinjiang CDC, Information Center of Xinjiang Autonomous Region, 2012, Available from: http://www.xjcdc.com/.

[23]

XiSongXueNusilaitiMalike and Zulihumaer, Survey on Echinococcus granulosus infection in shepherd dogs in tianshan mountainous area of Hejing city, Xinjiang province, Chin. J. Anim. Infect. Dis., 18 (2010), 59-62.

[24]

Z. Xu and X.-Q. Zhao, A vector-bias malaria model with incubation period and diffusion, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2015-2034. doi: 10.3934/dcdsb.2012.17.2615.

[25]

S. Yan and Y. Zhang, The control and prevention of livestock Echinococcosis in Xinjiang, Grass-Feeding Livestock, (1994), 45-47.

[26]

X.-Q. Zhao and Z. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Canad. Appl. Math. Q., 4 (1996), 421-444.

show all references

References:
[1]

Baidu, How Long is the Life Expectancy of Dogs? 2011, Available from: http://wenku.baidu.com/view/e7ccf4fec8d376eeaeaa31f9.html.

[2]

Baidu, The Sixth Census Data Announced by Xinjiang 2011, Available from: http://wenku.baidu.com/view/83e6d26cb84ae45c3b358cf1.html.

[3]

S. A. Berger and J. S. Marr, Human Parasitic Diseases Sourcebook, 1$^{st}$ edition, Jones and Bartlett Publishers: Sudbury, Massachusetts, 2006.

[4]

P. A. CabreraG. HaranU. benavidezS. ValledorG. PereraS. LloydM. A. GemmellM. BaraibarA. MoranaJ. Maissonave and M. Carballo, Transmission dynamics of Echinococcus granulosus, Taenia hydatigena and Taenia ovis in sheep in Uruguay, Int. J. Parasitol., 25 (1995), 807-813. doi: 10.1016/0020-7519(94)00209-7.

[5]

Y. CaoJ. Wen and Q. Zheng, Analysis of the epidemic status of Echinococcosis in Xinjiang in 2010, J. Ningxia Med. Univ., 33 (2011), 784-788.

[6]

J. Eckert, M. A. Gemmell, F. X. Meslin and Z. S. Pawlowski, WHO/OIE manual on Echinococcosis in humans and animals: A public health problem of global concern, Paris: World Health Organization/World Organization for Animal Health, 2001.

[7]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, NewYork, 1993.

[8]

S. LahmarH. DebbekL. H. ZhangD. P. McManusA. SouissiS. Chelly and P. R. Torgerson, Tansmission dynamics of the Echinococcus granulosus sheep-dog strain(G1 genotype) in camels in Tunisia, Vet. Parasitol., 121 (2004), 151-156. doi: 10.1016/j.vetpar.2004.02.016.

[9]

MalikeNusilaitiZulihumaerLvWaliXiAbudureyimuQiuAbuduaini and Hanati, Investigation of Echinococcus infection in domestic in Xinjiang, Chin. J. Anim.Infect. Dis., 19 (2011), 57-60(in Chinese).

[10]

National Bureau of Statistics of China, China Statistical Yearbook, 2011 Available from: http://www.stats.gov.cn/tjsj/ndsj/2011/indexch.htm.

[11]

Nusilaiti and Zulihumaer, The infection situation of Echinococcosis and countermeasures of some counties and cities in xinjiang, Xinjiang Livestock Industry, 6 (2011), 45-46.

[12]

H. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, 1995.

[13]

K. TakumiA. ViresM. ChuJ. MulderP. Teunis and J. Giessen, Evidence for an increasing presence of Echinococcus multilocularis in foxes in The Netherlands, Inter. J. Parasitol., 38 (2008), 571-578. doi: 10.1016/j.ijpara.2007.09.014.

[14]

The Government of Xinjiang Uygur Autonomous Region of China, The improvement of people's living standard, 2010, Available from: http://www.xinjiang.gov.cn/2011/11/15/64.html.

[15]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically automous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267.

[16]

P. R. TorgersonK. K. BurtisurnovB. S. ShaikenovA. T. RysmukhambetovaA. M. Abdybekova and A. E. Ussenbayev, Modelling the transmission dynamics of Echinococcus granulosus in sheep and cattle in Kazakhstan, Vet. Parasitol., 114 (2003), 143-153. doi: 10.1016/S0304-4017(03)00136-5.

[17]

S. Wang and S. Ye, Textbook of Medical Microbiology and Parasitology, 1$^{st}$ edition, Science Press, Beijing, 2006.

[18]

Z. Wang and X.-Q. Zhao, Global dynamics of a time-delayed dengue transmission model, Can. Appl. Math. Q., 20 (2012), 89-113.

[19]

K. WangX. ZhangZ. JinH. MaZ. Teng and L. Wang, Modeling and analysis of the transmission of Echinococcosis with application to Xinjiang Uygur Autonomous Region of China, J. Theor. Biol., 333 (2013), 78-90. doi: 10.1016/j.jtbi.2013.04.020.

[20]

W. Wu, The Chinese National Plan for the Control Echincoccosis, Urumq: ISUOG's World Congress of Hydatidology Final Programme & Abstracts Book, 2011(in Chinese).

[21]

L. WuB. SongW. Du and J. Lou, Mathematical modelling and control of Echinococcus in Qinghai province, China, Math. Biosci. Eng., 10 (2013), 425-444. doi: 10.3934/mbe.2013.10.425.

[22]

Xinjiang CDC, Information Center of Xinjiang Autonomous Region, 2012, Available from: http://www.xjcdc.com/.

[23]

XiSongXueNusilaitiMalike and Zulihumaer, Survey on Echinococcus granulosus infection in shepherd dogs in tianshan mountainous area of Hejing city, Xinjiang province, Chin. J. Anim. Infect. Dis., 18 (2010), 59-62.

[24]

Z. Xu and X.-Q. Zhao, A vector-bias malaria model with incubation period and diffusion, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2015-2034. doi: 10.3934/dcdsb.2012.17.2615.

[25]

S. Yan and Y. Zhang, The control and prevention of livestock Echinococcosis in Xinjiang, Grass-Feeding Livestock, (1994), 45-47.

[26]

X.-Q. Zhao and Z. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Canad. Appl. Math. Q., 4 (1996), 421-444.

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%
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%
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%
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%
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
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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