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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 |
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.
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. 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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. |
[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. 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.
|
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. |
[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. 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.
|
[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. |
[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. 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. |
[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. |
[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. |

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