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June  2019, 12(3): 455-474. doi: 10.3934/dcdss.2019030

Modeling the transmission dynamics of avian influenza with saturation and psychological effect

1. 

Department of Mathematics, City University of Science and Information, Technology, Peshawar, KP, 25000, Pakistan

2. 

Department of Mathematics, Abdul Wali Khan, University Mardan, KP, 23200, Pakistan

3. 

Department of Information Technology Education, University of Education, Winneba (Kumasi campus), Ghana

* Corresponding author: altafdir@gmail.com, makhan@cusit.edu.pk

Received  July 2017 Revised  November 2017 Published  September 2018

The present paper describes the mathematical analysis of an avian influenza model with saturation and psychological effect. The virus of avian influenza is not only a risk for birds but the population of human is also not safe from this. We proposed two models, one for birds and the other one for human. We consider saturated incidence rate and psychological effect in the model. The stability results for each model that is birds and human is investigated. The local and global dynamics for the disease free case of each model is proven when the basic reproduction number $ \mathcal{R}_{0b}<1$ and $ \mathcal{R}_0<1$. Further, the local and global stability of each model is investigated in the case when $ \mathcal{R}_{0b}>1$ and $ \mathcal{R}_0>1$. The mathematical results show that the considered saturation effect in population of birds and psychological effect in population of human does not effect the stability of equilibria, if the disease is prevalent then it can affect the number of infected humans. Numerical results are carried out in order to validate the theoretical results. Some numerical results for the proposed parameters are presented which can reduce the number of infective in the population of humans.

Citation: Muhammad Altaf Khan, Muhammad Farhan, Saeed Islam, Ebenezer Bonyah. Modeling the transmission dynamics of avian influenza with saturation and psychological effect. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 455-474. doi: 10.3934/dcdss.2019030
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Centers for Disease Control and Prevention (CDC), Information on Avian Influenza, http://www.cdc.gov/?u/avian?u/index.htm. Google Scholar

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F. B. AgustoS. Bewick and W. F. Fagan, Mathematical model of Zika virus with vertical transmission, Infectious Disease Modelling, 2 (2017), 244-267.  doi: 10.1016/j.idm.2017.05.003.  Google Scholar

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B. S. T. AlkahtaniA. Atangana and I. Koca, Novel analysis of the fractional Zika model using the Adams type predictor-corrector rule for non-singular and non-local fractional operators, J. Nonlinear Sci. Appl., 10 (2017), 3191-3200.  doi: 10.22436/jnsa.010.06.32.  Google Scholar

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B. S. T. AlkahtaniI. Koca and A. Atangana, Analysis of a new model of H1N1 spread: Model obtained via Mittag-Leffler function, Advances in Mechanical Engineering, 9 (2017), 1-8.  doi: 10.1177/1687814017705566.  Google Scholar

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B. S. T. AlkahtaniA. Atangana and I. Koca, A new nonlinear triadic model of predatorrey based on derivative with non-local and non-singular kernel, Advances in Mechanical Engineering, 8 (2016), 1-17.   Google Scholar

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A. Atangana and E. Alabaraoye, Solving a system of fractional partial differential equations arising in the model of HIV infection of CD4+ cells and attractor one-dimensional Keller-Segel equations, Advances in Difference Equations, 2013 (2013), 14pp. doi: 10.1186/1687-1847-2013-94.  Google Scholar

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G. Birkhoff and G. C. Rota, Ordinary differential equations, Introductions to Higher Mathematics Ginn and Company, Boston, Mass.-New York-Toronto, 1962.  Google Scholar

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L. Bourouiba, Lydia, et al. , The interaction of migratory birds and domestic poultry and its role in sustaining avian influenza, SIAM Journal on Applied Mathematics, 71 (2011), 487–516. doi: 10.1137/100803110.  Google Scholar

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V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Mathematical Biosciences, 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

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X. Dongmei and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical Biosciences, 208 (2007), 419-429.  doi: 10.1016/j.mbs.2006.09.025.  Google Scholar

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P. Van den Driessche and J. Watmough, Reproduction number and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Bios, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

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N. Ferguson, Capturing human behaviour, Nature, 446 (2007), 733-733.  doi: 10.1038/446733a.  Google Scholar

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S. A. GourleyR. Liu and J. Wu, Spatiotemporal distributions of migratory birds: Patchy models with delay, SIAM Journal on Applied Dynamical Systems, 9 (2010), 589-610.  doi: 10.1137/090767261.  Google Scholar

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A. B. Gumel, Global dynamics of a two-strain avian influenza model, International Journal of Computer Mathematics, 86 (2009), 85-108.  doi: 10.1080/00207160701769625.  Google Scholar

[24]

H. GuoM. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Am. Math. Soc., 136 (2008), 2793-2802.  doi: 10.1090/S0002-9939-08-09341-6.  Google Scholar

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L. Joseph, M. Roy and M. Martcheva, An avian influenza model and its fit to human avian influenza cases, Advances in Disease Epidemiology, Nova Science Publishers, New York, (2009), 1–30. Google Scholar

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E. Jung, et al. Optimal control strategy for prevention of avian influenza pandemic, Journal of Theoretical Biology, 260 (2009), 220–229. doi: 10.1016/j.jtbi.2009.05.031.  Google Scholar

[27]

J. M. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008.  Google Scholar

[28]

J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematis, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1976.  Google Scholar

[29]

J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, PA, 1976.  Google Scholar

[30]

M.-T. LiZ. JinG. Q. Sun and J. Zhang, Modeling direct and indirect disease transmission using multi-group mode, J. Math. Anal. Appl., 446 (2017), 1292-1309.  doi: 10.1016/j.jmaa.2016.09.043.  Google Scholar

[31]

W. Liping, Human Exposure to Live Poultry and Psychological and Behavioral Responses to Influenza A (H7N9), China-Emerging Infectious Disease journal, 20 (2014). Google Scholar

[32]

S. Liu, L. Pang, S. Ruan and X. Zhang, Global dynamics of avian influenza epidemic models with psychological effect, Comp. Math. Meth. Med., (2015), Art. ID 913726, 12 pp. doi: 10.1155/2015/913726.  Google Scholar

[33]

W.-M. LiuA. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, Journal of Mathematical Biology, 23 (1986), 187-204.  doi: 10.1007/BF00276956.  Google Scholar

[34]

T. Necibe and M. Martcheva, Modeling seasonality in avian influenza H5N1, Journal of Biological Systems, 21 (2013), 1340004, 30 pp. doi: 10.1142/S0218339013400044.  Google Scholar

[35]

X. Nijuan, et al., Knowledge, attitudes and practices (KAP) relating to avian influenza in urban and rural areas of China, BMC Infectious Diseases, 10 (2010), 34. Google Scholar

[36]

K. M. Owolabi and A. Atangana, Spatiotemporal dynamics of fractional predatorrey system with stage structure for the predator, International Journal of Applied and Computational Mathematics, 3 (2017), S903-S924.  doi: 10.1007/s40819-017-0389-2.  Google Scholar

[37]

L. RongsongJ. Wu and H. Zhu, Media/psychological impact on multiple outbreaks of emerging infectious diseases, Computational and Mathematical Methods in Medicine, 8 (2007), 153-164.  doi: 10.1080/17486700701425870.  Google Scholar

[38]

F. Sebastian, et al., The spread of awareness and its impact on epidemic outbreaks, Proceedings of the National Academy of Sciences, 106 (2009), 6872–6877. Google Scholar

[39]

F. Sebastian, M. Salathe and V. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: A review, Journal of the Royal Society Interface, (2010). Google Scholar

[40]

F. Sebastian, et al., Nine challenges in incorporating the dynamics of behaviour in infectious diseases models, Epidemics, 10 (2015), 21–25. Google Scholar

[41]

I. Shingo, et al., A geographical spread of vaccine-resistance in avian influenza epidemics, Journal of Theoretical Biology, 259 (2009), 219–228. doi: 10.1016/j.jtbi.2009.03.040.  Google Scholar

[42]

I. ShingoY. Takeuchi and X. Liu, Avian-human influenza epidemic model, Mathematical Bios., 207 (2007), 1-25.  doi: 10.1016/j.mbs.2006.08.001.  Google Scholar

[43]

I. ShingoY. Takeuchi and X. Liu, Avian flu pandemic: Can we prevent it?, Journal of Theoretical Biology, 257 (2009), 181-190.  doi: 10.1016/j.jtbi.2008.11.011.  Google Scholar

[44]

R. Shigui and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, Journal of Differential Equations, 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X.  Google Scholar

[45]

O. Sonja, Poultry-handling Practices during Avian Influenza Outbreak, Thailand, Emerging Infectious Disease journal, 11 (2005). Google Scholar

[46]

C. Wolf, A mathematical model for the propagation of a hantavirus in structured populations, Discrete Continuous Dynam. Systems - B, 4 (2004), 1065-1089.  doi: 10.3934/dcdsb.2004.4.1065.  Google Scholar

[47]

M. Xinling and W. Wang, A discrete model of avian influenza with seasonal reproduction and transmission, Journal of Biological Dynamics, 4 (2010), 296-314.  doi: 10.1080/17513751003793009.  Google Scholar

[48]

X. Yanni, et al., Transmission potential of the novel avian influenza A (H7N9) infection in mainland China, Journal of Theoretical Biology, 352 (2014), 1–5. doi: 10.1016/j.jtbi.2014.02.038.  Google Scholar

[49]

H. Ying-Hen, et al. Quantification of bird-to-bird and bird-to-human infections during 2013 novel H7N9 avian influenza outbreak in China, PloS One, 9 (2014), e111834. Google Scholar

[50]

J. Zhang, et al., Determination of original infection source of H7N9 avian influenza by dynamical model, Scientific Reports, 4 (2014), 48–46. doi: 10.1038/srep04846.  Google Scholar

show all references

References:
[1]

Centers for Disease Control and Prevention (CDC), Types of influenza virus, http://www.cdc.gov/?u/about/viruses/types.htm Google Scholar

[2]

Centers for Disease Control and Prevention (CDC), Information on Avian Influenza, http://www.cdc.gov/?u/avian?u/index.htm. Google Scholar

[3]

Centers for Disease Control and Prevention (CDC), Influenza Type A Viruses, http://www.cdc.gov/flu/avianflu/influenza-avirus-subtypes.htm. Google Scholar

[4]

F. B. Agusto, Optimal isolation control strategies and cost-effectiveness analysis of a two-strain avian influenza model, Biosystems, 113 (2013), 155-164.  doi: 10.1016/j.biosystems.2013.06.004.  Google Scholar

[5]

F. B. AgustoS. Bewick and W. F. Fagan, Mathematical model of Zika virus with vertical transmission, Infectious Disease Modelling, 2 (2017), 244-267.  doi: 10.1016/j.idm.2017.05.003.  Google Scholar

[6]

J. D. Alexander, An overview of the epidemiology of avian influenza, Vaccine, 25 (2007), 5637-5644.   Google Scholar

[7]

B. S. T. AlkahtaniA. Atangana and I. Koca, Novel analysis of the fractional Zika model using the Adams type predictor-corrector rule for non-singular and non-local fractional operators, J. Nonlinear Sci. Appl., 10 (2017), 3191-3200.  doi: 10.22436/jnsa.010.06.32.  Google Scholar

[8]

B. S. T. AlkahtaniI. Koca and A. Atangana, Analysis of a new model of H1N1 spread: Model obtained via Mittag-Leffler function, Advances in Mechanical Engineering, 9 (2017), 1-8.  doi: 10.1177/1687814017705566.  Google Scholar

[9]

B. S. T. AlkahtaniA. Atangana and I. Koca, A new nonlinear triadic model of predatorrey based on derivative with non-local and non-singular kernel, Advances in Mechanical Engineering, 8 (2016), 1-17.   Google Scholar

[10]

M. R. Anderson, M. M. Robert and B. Anderson, Infectious diseases of humans: Dynamics and control, Oxford: Oxford University Press, 28 (1992). Google Scholar

[11]

A. Atangana and E. Alabaraoye, Solving a system of fractional partial differential equations arising in the model of HIV infection of CD4+ cells and attractor one-dimensional Keller-Segel equations, Advances in Difference Equations, 2013 (2013), 14pp. doi: 10.1186/1687-1847-2013-94.  Google Scholar

[12]

A. Atangana and B. S. T. Alkahtani, Modeling the spread of Rubella disease using the concept of with local derivative with fractional parameter, Complexity, 21 (2016), 442-451.  doi: 10.1002/cplx.21704.  Google Scholar

[13]

A. Atangana and R. T. Alqahtani, Modelling the spread of river blindness disease via the caputo fractional derivative and the beta-derivative, Entropy, 18 (2016), p40. doi: 10.3390/e18020040.  Google Scholar

[14]

C. T. Bauch and A. P. Galvani, Social factors in epidemiology, Science, 342 (2013), 47-49.  doi: 10.1126/science.1244492.  Google Scholar

[15]

G. Birkhoff and G. C. Rota, Ordinary differential equations, Introductions to Higher Mathematics Ginn and Company, Boston, Mass.-New York-Toronto, 1962.  Google Scholar

[16]

L. Bourouiba, Lydia, et al. , The interaction of migratory birds and domestic poultry and its role in sustaining avian influenza, SIAM Journal on Applied Mathematics, 71 (2011), 487–516. doi: 10.1137/100803110.  Google Scholar

[17]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Mathematical Biosciences, 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[18]

C. Castillo-Chavez, et al, Mathematical approaches for emerging and reemerging infectious diseases: models, methods, and theory, Springer, New York, 2002. Google Scholar

[19]

X. Dongmei and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical Biosciences, 208 (2007), 419-429.  doi: 10.1016/j.mbs.2006.09.025.  Google Scholar

[20]

P. Van den Driessche and J. Watmough, Reproduction number and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Bios, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[21]

N. Ferguson, Capturing human behaviour, Nature, 446 (2007), 733-733.  doi: 10.1038/446733a.  Google Scholar

[22]

S. A. GourleyR. Liu and J. Wu, Spatiotemporal distributions of migratory birds: Patchy models with delay, SIAM Journal on Applied Dynamical Systems, 9 (2010), 589-610.  doi: 10.1137/090767261.  Google Scholar

[23]

A. B. Gumel, Global dynamics of a two-strain avian influenza model, International Journal of Computer Mathematics, 86 (2009), 85-108.  doi: 10.1080/00207160701769625.  Google Scholar

[24]

H. GuoM. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Am. Math. Soc., 136 (2008), 2793-2802.  doi: 10.1090/S0002-9939-08-09341-6.  Google Scholar

[25]

L. Joseph, M. Roy and M. Martcheva, An avian influenza model and its fit to human avian influenza cases, Advances in Disease Epidemiology, Nova Science Publishers, New York, (2009), 1–30. Google Scholar

[26]

E. Jung, et al. Optimal control strategy for prevention of avian influenza pandemic, Journal of Theoretical Biology, 260 (2009), 220–229. doi: 10.1016/j.jtbi.2009.05.031.  Google Scholar

[27]

J. M. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008.  Google Scholar

[28]

J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematis, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1976.  Google Scholar

[29]

J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, PA, 1976.  Google Scholar

[30]

M.-T. LiZ. JinG. Q. Sun and J. Zhang, Modeling direct and indirect disease transmission using multi-group mode, J. Math. Anal. Appl., 446 (2017), 1292-1309.  doi: 10.1016/j.jmaa.2016.09.043.  Google Scholar

[31]

W. Liping, Human Exposure to Live Poultry and Psychological and Behavioral Responses to Influenza A (H7N9), China-Emerging Infectious Disease journal, 20 (2014). Google Scholar

[32]

S. Liu, L. Pang, S. Ruan and X. Zhang, Global dynamics of avian influenza epidemic models with psychological effect, Comp. Math. Meth. Med., (2015), Art. ID 913726, 12 pp. doi: 10.1155/2015/913726.  Google Scholar

[33]

W.-M. LiuA. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, Journal of Mathematical Biology, 23 (1986), 187-204.  doi: 10.1007/BF00276956.  Google Scholar

[34]

T. Necibe and M. Martcheva, Modeling seasonality in avian influenza H5N1, Journal of Biological Systems, 21 (2013), 1340004, 30 pp. doi: 10.1142/S0218339013400044.  Google Scholar

[35]

X. Nijuan, et al., Knowledge, attitudes and practices (KAP) relating to avian influenza in urban and rural areas of China, BMC Infectious Diseases, 10 (2010), 34. Google Scholar

[36]

K. M. Owolabi and A. Atangana, Spatiotemporal dynamics of fractional predatorrey system with stage structure for the predator, International Journal of Applied and Computational Mathematics, 3 (2017), S903-S924.  doi: 10.1007/s40819-017-0389-2.  Google Scholar

[37]

L. RongsongJ. Wu and H. Zhu, Media/psychological impact on multiple outbreaks of emerging infectious diseases, Computational and Mathematical Methods in Medicine, 8 (2007), 153-164.  doi: 10.1080/17486700701425870.  Google Scholar

[38]

F. Sebastian, et al., The spread of awareness and its impact on epidemic outbreaks, Proceedings of the National Academy of Sciences, 106 (2009), 6872–6877. Google Scholar

[39]

F. Sebastian, M. Salathe and V. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: A review, Journal of the Royal Society Interface, (2010). Google Scholar

[40]

F. Sebastian, et al., Nine challenges in incorporating the dynamics of behaviour in infectious diseases models, Epidemics, 10 (2015), 21–25. Google Scholar

[41]

I. Shingo, et al., A geographical spread of vaccine-resistance in avian influenza epidemics, Journal of Theoretical Biology, 259 (2009), 219–228. doi: 10.1016/j.jtbi.2009.03.040.  Google Scholar

[42]

I. ShingoY. Takeuchi and X. Liu, Avian-human influenza epidemic model, Mathematical Bios., 207 (2007), 1-25.  doi: 10.1016/j.mbs.2006.08.001.  Google Scholar

[43]

I. ShingoY. Takeuchi and X. Liu, Avian flu pandemic: Can we prevent it?, Journal of Theoretical Biology, 257 (2009), 181-190.  doi: 10.1016/j.jtbi.2008.11.011.  Google Scholar

[44]

R. Shigui and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, Journal of Differential Equations, 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X.  Google Scholar

[45]

O. Sonja, Poultry-handling Practices during Avian Influenza Outbreak, Thailand, Emerging Infectious Disease journal, 11 (2005). Google Scholar

[46]

C. Wolf, A mathematical model for the propagation of a hantavirus in structured populations, Discrete Continuous Dynam. Systems - B, 4 (2004), 1065-1089.  doi: 10.3934/dcdsb.2004.4.1065.  Google Scholar

[47]

M. Xinling and W. Wang, A discrete model of avian influenza with seasonal reproduction and transmission, Journal of Biological Dynamics, 4 (2010), 296-314.  doi: 10.1080/17513751003793009.  Google Scholar

[48]

X. Yanni, et al., Transmission potential of the novel avian influenza A (H7N9) infection in mainland China, Journal of Theoretical Biology, 352 (2014), 1–5. doi: 10.1016/j.jtbi.2014.02.038.  Google Scholar

[49]

H. Ying-Hen, et al. Quantification of bird-to-bird and bird-to-human infections during 2013 novel H7N9 avian influenza outbreak in China, PloS One, 9 (2014), e111834. Google Scholar

[50]

J. Zhang, et al., Determination of original infection source of H7N9 avian influenza by dynamical model, Scientific Reports, 4 (2014), 48–46. doi: 10.1038/srep04846.  Google Scholar

Figure 1.  The behavior of infected individuals $I_h$, keeping $\alpha = m = 0.001$. Figure 1(a): varying $\beta_a$ and $\beta_h = 8\times 10^{-7}$ is fixed. Figure 1(b): varying $\beta_h$ and $\beta_a = 3\times 10^{-6}$ is fixed
Figure 2.  The behavior of infected individuals $I_h$ when $ \mathcal{R}_{0}>1$. Figure 2(a): $\alpha = m = 0$, Figure 2(b): $\alpha = m = 0.001$
Figure 3.  The behavior of infected individuals $I_h$ and $ \mathcal{R}_{0}>1$: Figure 3(a) when $\alpha = 0.001,~0.001,~0.01 $ and $m = 0.001$ fixed. Figure 3(b) when $m = 0.001,~0.001,~0.01 $ and $\alpha = 0.001$ fixed
Figure 4.  The behavior of infected individuals $I_h$ and $ \mathcal{R}_{0}<1$: Figure 4(a) when $\alpha = 0.001,~0.001,~0.01 $ and $m = 0.001$ fixed. Figure 3(b) when $m = 0.001,~0.001,~0.01 $ and $\alpha = 0.001$ fixed
Figure 5.  The behavior of infected individuals $I_h$ and $ \mathcal{R}_{0}<1$: $\alpha = 0.001,~0.01,~0.1$, $m = 0.001,~0.01,~0.1$
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