American Institute of Mathematical Sciences

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doi: 10.3934/dcdsb.2020339

The threshold dynamics of a discrete-time echinococcosis transmission model

Cuicui Li 1, , Lin Zhou 1, , Zhidong Teng 1,, and  Buyu Wen 1,2,

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  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.

Keywords: Echinococcosis, discrete-time model, basic reproduction number, equilibrium, stability, permanence.
Mathematics Subject Classification: Primary: 92D30, 39A60; Secondary: 39A30.
Citation: Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020339
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[4]

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

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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.  Google Scholar

[6]

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

[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.  Google Scholar

[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. Google Scholar

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G. M. Clifford, S. 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.   Google Scholar

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

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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.  Google Scholar

[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.  Google Scholar

[13]

J. Eckert, F. 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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[24]

P. Liu, J. Li, Y. Li and et al., The epidemic situation and causative analysis of echinococcosis (in Chinese), China Anim. Heal. Insp., 33 (2016), 48-51.   Google Scholar

[25]

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

[26]

M. G. Roberts, J. 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.  Google Scholar

[27]

M. G. Roberts, J. 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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

Z. Teng, Y. 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.   Google Scholar

[31]

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

[32]

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

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[36]

L. Wang, Z. 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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[41]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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. Google Scholar

[9]

G. M. Clifford, S. 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.   Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

J. Eckert, F. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[24]

P. Liu, J. Li, Y. Li and et al., The epidemic situation and causative analysis of echinococcosis (in Chinese), China Anim. Heal. Insp., 33 (2016), 48-51.   Google Scholar

[25]

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

[26]

M. G. Roberts, J. 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.  Google Scholar

[27]

M. G. Roberts, J. 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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

Z. Teng, Y. 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.   Google Scholar

[31]

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

[32]

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

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[36]

L. Wang, Z. 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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[41]

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

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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