October  2021, 26(10): 5197-5216. doi: 10.3934/dcdsb.2020339

The threshold dynamics of a discrete-time echinococcosis transmission model

1. 

College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China

2. 

School of Information Engineering, Eastern Liaoning University, Dandong 108001, China

* Corresponding author: Zhidong Teng

Received  July 2020 Revised  September 2020 Published  October 2021 Early access  November 2020

Fund Project: Research of Zhidong Teng was supported by the NSF of China (Grant No. 11771373, 11861065) and the NSF of Xinjiang, China (Grant No. 2016D03022), Research of Buyu Wen was supported by the Doctoral Research Start-up Fund of Eastern Liaoning University of Liaoning, China(Grant No. 2019BS023)

In this paper, based on the transmission mechanism of echinococcosis in China, we propose a discrete-time dynamical model for the transmission of echinococcosis. The research results indicate that transmission dynamics of this discrete-time model are determined by basic reproduction number $ R_{0} $. It is shown that when $ R_{0}\leq1 $ then the disease-free equilibrium is globally asymptotically stable and when $ R_{0}>1 $ then the model is permanent while the disease-free equilibrium is unstable. Finally, on the basis of the theoretical results established in this paper, we come up with some specific measures to control the transmission of echinococcosis.

Citation: Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5197-5216. doi: 10.3934/dcdsb.2020339
References:
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A. Abdybekova, A. Sultanov, B. Karatayev, et al, Epidemiology of echinococcosis in Kazakhstan: An update, J. Helminthol., 89 (2015), 647-650. doi: 10.1017/S0022149X15000425.

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R. Azlaf, A. Dakkak, A. Chentoufi, et al, Modelling the transmission of echinococcus granulosus in dogs in the northwest and in the southwest of Morocco, Vet. Parasitol., 145 (2007), 297-303. doi: 10.1016/j.vetpar.2006.12.014.

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B. Boufana, J. Qiu, X. Chen, et al, First report of Echinococcus shiquicus in dogs from eastern Qinghai-Tibet plateau region, China, Acta Trop., 127 (2013), 21-24. doi: 10.1016/j.actatropica.2013.02.019.

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D. Carmena and G. A. Cardona, Cnine echinococcosis: Globak epidemiology and genotypic diversity, Acta Trop., 128 (2013), 441-460. 

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M. Chen and H. Wang, Dynamics of a discrete-time stoichiometric optimal foraging model, Disc. Cont. Dyn. Syst. B, (2020). doi: 10.3934/dcdsb.2020264.

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E. Cleary, T. S. Barnes, Y. Xu, et al, Impact of "Grain to Green" programme on echinococcosis infection in Ningxia Autonomous Region Of China, Vet. Parasitol., 205 (2014), 523-531.

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G. M. CliffordS. Gallus and R. Herrero, World wide distribution of human papilkom avirus types in cytologically normal women in the international a gency for research on cancer HPV prevalence surveys: A poolied anslysis, Lancet, 336 (2005), 991-998. 

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P. S. Craig, Epidemioligy of human alveolar echinococcosis in China, Parasitol. Int., 55 (2006), 221-225. 

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P. S. Craig, P. Giraudoux, D. Shi, et al, An epidemiological and ecological study of human alveolar echinococcosis transmission in south Gansu, China, Acta Trop., 77 (2000), 167-177. doi: 10.1016/S0001-706X(00)00134-0.

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O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, John Wiley & Sons Ltd., Chichester, New York, 2000.

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J. EckertF. J. Conraths and K. Tackmann, Echinococcosis: An emerging or re-emerging zoonosis?, Int. J. Parasitol., 30 (2000), 1283-1294.  doi: 10.1016/S0020-7519(00)00130-2.

[14]

Y. Enatsu, Y. Muroya, G. Izzo, et al, Global dynamics of difference equations for SIR epidemic models with a class of nonlinear incidence rates, J. Diff. Equat. Appl., 18 (2012), 1163-1181. doi: 10.1080/10236198.2011.555405.

[15]

Y. Enatsu and Y. Muroya, Global stability for a class of discrete SIR epidemic models, Math. Biosci. Eng., 7 (2010), 347-361.  doi: 10.3934/mbe.2010.7.347.

[16]

J. E. Franke and A. A Yakubu, Discrete-time SIS epidemic model in a seasonal environment, SIAM J. Appl. Math., 66 (2006), 1563-1587.  doi: 10.1137/050638345.

[17]

E. Gascoigne and J. P. Crilly, Control of tapeworms in sheep: A risk-based approach, In Practice, 36 (2014), 285-293.  doi: 10.1136/inp.g2962.

[18]

L. Huang, Y. Huang, Q. Wang, et al, An agent-based model for control strategies of echinococcus granulosus. Vet. Parasitol., 179 (2011), 84-91. doi: 10.1016/j.vetpar.2011.01.047.

[19]

W. Iraqi, Canine echinococcosis: The predominance of immature eggs in adult tapeworms of Echinococcus granulosus in stray dogs from Tunisia, J. Helminthol., 91 (2017), 380-383.  doi: 10.1017/S0022149X16000341.

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J. P. LaSalle, The Stability of Dynamical Systems, Society for Industrial and Applied Mathematics, Philadelphia, 1976.

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X. Li, B. Shi, L. Zhao, et al, The epidemic and control situation of hydaid disease in Xinjiang (in Chinese), Grass-Feeding Livest., 157 (2012), 47{52.

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T. Y. Li, J. M. Qiu, W. Yang, et al, Echinococcosis in tibetan populations, western sichuan province, China Emerg. Infect. Dis., 11 (2015), 1866-1873.

[23]

J. Liu, L. Liu, X. Feng, et al, Global dynamics of a time-delayed echinococcosis transmission model, Adv. Diff. Equat., 2015 (2015), 99. doi: 10.1186/s13662-015-0356-3.

[24]

P. LiuJ. LiY. Li and et al., The epidemic situation and causative analysis of echinococcosis (in Chinese), China Anim. Heal. Insp., 33 (2016), 48-51. 

[25]

Z. Ma, Y. Zhou, W. Wang, et al, Mathematical Modelling and Research of Epidemic Dynamical Systems, Science Press, Beijing, 2004.

[26]

M. G. RobertsJ. R. Lawson and M. A. Gemmell, Population dynamics in echinococcosis and cysticercosis: Mathematical model of the life-cycle of Echinococcus granulosus, Parasitology, 92 (1986), 621-641.  doi: 10.1017/S0031182000065495.

[27]

M. G. RobertsJ. R. Lawson and M. A. Gemmell, Population dynamics in echinococcosis and cysticercosis: Mathematical model of the life-cycles of Taenia hydatigena and T. ovis, Parasitology, 94 (1987), 181-197.  doi: 10.1017/S0031182000053555.

[28]

X. Rong, M. Fan, X. Sun, et al, Impact of disposing stray dogs on risk assessment and control of echinococcosis in Inner Mongolia, Math. Biosci., 299 (2018), 85-96. doi: 10.1016/j.mbs.2018.03.008.

[29]

Y. Solitang and L. Jiang, Prevention research progress of echinococcosis in China, J. Parasitol. Dis., 18 (2000), 179-181. 

[30]

Z. TengY. Wang and M. Rehim, On the backward difference scheme for a class of SIRS epidemic models with nonlinear incidence, J. Comput. Anal. Appl., 20 (2016), 1268-1289. 

[31]

P. R. Torgerson, The use of mathematical models to stimuiate control options for echinococcosis, Acta Trop., 85 (2003), 211-221. 

[32]

P. R. Torgerson, Mathematical models for control of cycstic echinococcosis, Parasitol. Int., 55 (2006), 253-258. 

[33]

P. R. Torgerson, The emergence of echinococcosis in central Asia, Parasitology, 140 (2013), 1667-1673.  doi: 10.1017/S0031182013000516.

[34]

P. R. Torgerson, K. K. Burtisurnov, B. S. Shaikenov, et al, Modelling the transmission dynamics of Echinococcus granulosus in sheep and cattle in Kazakhstan, Vet. Parastiol., 114 (2003), 143-153. doi: 10.1016/S0304-4017(03)00136-5.

[35]

P. R. Torgerson, I. Ziadinov, D. Aknazarov, et al, Modelling the age variation of larval protoscoleces of Echinococcus granulosus in sheep, Int. J. Parastiol., 39 (2009), 1031-1035. doi: 10.1016/j.ijpara.2009.01.004.

[36]

L. WangZ. Teng and H. Jiang, Global attractivity of a discrete SIRS epidemic model with standard incidence rate, Math. Meth. Appl. Sci., 36 (2013), 601-619.  doi: 10.1002/mma.2734.

[37] S. Wang and S. Ye, Textbook of Medical Microbiology and Parasitology (in Chinese), 1$^nd$ edition, Science Press, Beijing, 2006. 
[38]

K. Wang, X. Zhang, Z. Jin, et al, 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.

[39]

Y. Xie and Y. Li, Stability analysis and control strategies for a new SIS epidemic model in heterogeneous networks, Appl. Math. Comput., 383 (2020), 125381, 11pp. doi: 10.1016/j.amc.2020.125381.

[40]

Y. Xie, B. Ming and X. Huang, Dynamical analysis for a fractional-order prey-predator model with Holling Ⅲ type functional response and discontinuous harvest, Appl. Math. Letters, 106 (2020), 106342, 8pp. doi: 10.1016/j.aml.2020.106342.

[41]

X. Zhao, Dynamical Systems in Population Biology, Springer Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.

show all references

References:
[1]

A. Abdybekova, A. Sultanov, B. Karatayev, et al, Epidemiology of echinococcosis in Kazakhstan: An update, J. Helminthol., 89 (2015), 647-650. doi: 10.1017/S0022149X15000425.

[2]

J. M. Atkinson, G. M. Williams, L. Yakob, et al., Synthesising 30 years of mathematical modelling of echinococcus transmission, Plos Negl. Trop. Dis., 7 (2013), e2386. doi: 10.1371/journal.pntd.0002386.

[3]

R. Azlaf, A. Dakkak, A. Chentoufi, et al, Modelling the transmission of echinococcus granulosus in dogs in the northwest and in the southwest of Morocco, Vet. Parasitol., 145 (2007), 297-303. doi: 10.1016/j.vetpar.2006.12.014.

[4]

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

[5]

B. Boufana, J. Qiu, X. Chen, et al, First report of Echinococcus shiquicus in dogs from eastern Qinghai-Tibet plateau region, China, Acta Trop., 127 (2013), 21-24. doi: 10.1016/j.actatropica.2013.02.019.

[6]

D. Carmena and G. A. Cardona, Cnine echinococcosis: Globak epidemiology and genotypic diversity, Acta Trop., 128 (2013), 441-460. 

[7]

M. Chen and H. Wang, Dynamics of a discrete-time stoichiometric optimal foraging model, Disc. Cont. Dyn. Syst. B, (2020). doi: 10.3934/dcdsb.2020264.

[8]

E. Cleary, T. S. Barnes, Y. Xu, et al, Impact of "Grain to Green" programme on echinococcosis infection in Ningxia Autonomous Region Of China, Vet. Parasitol., 205 (2014), 523-531.

[9]

G. M. CliffordS. Gallus and R. Herrero, World wide distribution of human papilkom avirus types in cytologically normal women in the international a gency for research on cancer HPV prevalence surveys: A poolied anslysis, Lancet, 336 (2005), 991-998. 

[10]

P. S. Craig, Epidemioligy of human alveolar echinococcosis in China, Parasitol. Int., 55 (2006), 221-225. 

[11]

P. S. Craig, P. Giraudoux, D. Shi, et al, An epidemiological and ecological study of human alveolar echinococcosis transmission in south Gansu, China, Acta Trop., 77 (2000), 167-177. doi: 10.1016/S0001-706X(00)00134-0.

[12]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, John Wiley & Sons Ltd., Chichester, New York, 2000.

[13]

J. EckertF. J. Conraths and K. Tackmann, Echinococcosis: An emerging or re-emerging zoonosis?, Int. J. Parasitol., 30 (2000), 1283-1294.  doi: 10.1016/S0020-7519(00)00130-2.

[14]

Y. Enatsu, Y. Muroya, G. Izzo, et al, Global dynamics of difference equations for SIR epidemic models with a class of nonlinear incidence rates, J. Diff. Equat. Appl., 18 (2012), 1163-1181. doi: 10.1080/10236198.2011.555405.

[15]

Y. Enatsu and Y. Muroya, Global stability for a class of discrete SIR epidemic models, Math. Biosci. Eng., 7 (2010), 347-361.  doi: 10.3934/mbe.2010.7.347.

[16]

J. E. Franke and A. A Yakubu, Discrete-time SIS epidemic model in a seasonal environment, SIAM J. Appl. Math., 66 (2006), 1563-1587.  doi: 10.1137/050638345.

[17]

E. Gascoigne and J. P. Crilly, Control of tapeworms in sheep: A risk-based approach, In Practice, 36 (2014), 285-293.  doi: 10.1136/inp.g2962.

[18]

L. Huang, Y. Huang, Q. Wang, et al, An agent-based model for control strategies of echinococcus granulosus. Vet. Parasitol., 179 (2011), 84-91. doi: 10.1016/j.vetpar.2011.01.047.

[19]

W. Iraqi, Canine echinococcosis: The predominance of immature eggs in adult tapeworms of Echinococcus granulosus in stray dogs from Tunisia, J. Helminthol., 91 (2017), 380-383.  doi: 10.1017/S0022149X16000341.

[20]

J. P. LaSalle, The Stability of Dynamical Systems, Society for Industrial and Applied Mathematics, Philadelphia, 1976.

[21]

X. Li, B. Shi, L. Zhao, et al, The epidemic and control situation of hydaid disease in Xinjiang (in Chinese), Grass-Feeding Livest., 157 (2012), 47{52.

[22]

T. Y. Li, J. M. Qiu, W. Yang, et al, Echinococcosis in tibetan populations, western sichuan province, China Emerg. Infect. Dis., 11 (2015), 1866-1873.

[23]

J. Liu, L. Liu, X. Feng, et al, Global dynamics of a time-delayed echinococcosis transmission model, Adv. Diff. Equat., 2015 (2015), 99. doi: 10.1186/s13662-015-0356-3.

[24]

P. LiuJ. LiY. Li and et al., The epidemic situation and causative analysis of echinococcosis (in Chinese), China Anim. Heal. Insp., 33 (2016), 48-51. 

[25]

Z. Ma, Y. Zhou, W. Wang, et al, Mathematical Modelling and Research of Epidemic Dynamical Systems, Science Press, Beijing, 2004.

[26]

M. G. RobertsJ. R. Lawson and M. A. Gemmell, Population dynamics in echinococcosis and cysticercosis: Mathematical model of the life-cycle of Echinococcus granulosus, Parasitology, 92 (1986), 621-641.  doi: 10.1017/S0031182000065495.

[27]

M. G. RobertsJ. R. Lawson and M. A. Gemmell, Population dynamics in echinococcosis and cysticercosis: Mathematical model of the life-cycles of Taenia hydatigena and T. ovis, Parasitology, 94 (1987), 181-197.  doi: 10.1017/S0031182000053555.

[28]

X. Rong, M. Fan, X. Sun, et al, Impact of disposing stray dogs on risk assessment and control of echinococcosis in Inner Mongolia, Math. Biosci., 299 (2018), 85-96. doi: 10.1016/j.mbs.2018.03.008.

[29]

Y. Solitang and L. Jiang, Prevention research progress of echinococcosis in China, J. Parasitol. Dis., 18 (2000), 179-181. 

[30]

Z. TengY. Wang and M. Rehim, On the backward difference scheme for a class of SIRS epidemic models with nonlinear incidence, J. Comput. Anal. Appl., 20 (2016), 1268-1289. 

[31]

P. R. Torgerson, The use of mathematical models to stimuiate control options for echinococcosis, Acta Trop., 85 (2003), 211-221. 

[32]

P. R. Torgerson, Mathematical models for control of cycstic echinococcosis, Parasitol. Int., 55 (2006), 253-258. 

[33]

P. R. Torgerson, The emergence of echinococcosis in central Asia, Parasitology, 140 (2013), 1667-1673.  doi: 10.1017/S0031182013000516.

[34]

P. R. Torgerson, K. K. Burtisurnov, B. S. Shaikenov, et al, Modelling the transmission dynamics of Echinococcus granulosus in sheep and cattle in Kazakhstan, Vet. Parastiol., 114 (2003), 143-153. doi: 10.1016/S0304-4017(03)00136-5.

[35]

P. R. Torgerson, I. Ziadinov, D. Aknazarov, et al, Modelling the age variation of larval protoscoleces of Echinococcus granulosus in sheep, Int. J. Parastiol., 39 (2009), 1031-1035. doi: 10.1016/j.ijpara.2009.01.004.

[36]

L. WangZ. Teng and H. Jiang, Global attractivity of a discrete SIRS epidemic model with standard incidence rate, Math. Meth. Appl. Sci., 36 (2013), 601-619.  doi: 10.1002/mma.2734.

[37] S. Wang and S. Ye, Textbook of Medical Microbiology and Parasitology (in Chinese), 1$^nd$ edition, Science Press, Beijing, 2006. 
[38]

K. Wang, X. Zhang, Z. Jin, et al, 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.

[39]

Y. Xie and Y. Li, Stability analysis and control strategies for a new SIS epidemic model in heterogeneous networks, Appl. Math. Comput., 383 (2020), 125381, 11pp. doi: 10.1016/j.amc.2020.125381.

[40]

Y. Xie, B. Ming and X. Huang, Dynamical analysis for a fractional-order prey-predator model with Holling Ⅲ type functional response and discontinuous harvest, Appl. Math. Letters, 106 (2020), 106342, 8pp. doi: 10.1016/j.aml.2020.106342.

[41]

X. Zhao, Dynamical Systems in Population Biology, Springer Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.

Figure 1.  Numerical simulations of solution $ (S_D(t),I_D(t),S_L(t), $ $ I_L(t),x(t),S_H(t),E_H(t),I_H(t)) $ with initial value $ (S_{D}(0),I_{D}(0), $ $ S_{L}(0),I_{L}(0),x(0),S_{H}(0),E_{H}(0),I_{H}(0)) = (2\times10^6,8\times10^5, $ $ 8.4\times10^8, 5.7\times10^7, 1.44\times10^7, 3\times10^7, 9\times10^3, 8\times10^4), (5\times10^4, 4\times10^6, 0.1\times10^8, 1\times10^8, 3\times10^7, 5\times10^6, 1\times10^3, 1\times10^5), (0.4\times10^6, 2\times10^6, 1.2\times10^8, 2.2\times10^8, 1\times10^7, 1.5\times10^7, 7\times10^3, 6\times10^4) $, respectively
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