June  2015, 20(4): 989-1013. doi: 10.3934/dcdsb.2015.20.989

Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment

1. 

Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275

2. 

College of Mathematics and Information Science, Wenzhou University, Wenzhou, 325035

Received  April 2014 Revised  July 2014 Published  February 2015

In this paper, we explore a parasite-host epidemiological model incorporating demographic and epidemiological processes in a spatially heterogeneous environment in which the individuals are subject to a random movement. We show the global stability of the extinction equilibrium in three different cases, and prove the existence, uniqueness and the global stability of the disease--free equilibrium. When the death rate in the model becomes a constant, we give the existence of the endemic equilibrium and the global stability of the endemic equilibrium in a special case. Furthermore, we perform a series of numerical simulations to display the effects of the movement of hosts and the heterogeneous environment on the disease dynamics. Our analytical and numerical results reveal that the disease extinction/outbreak can be ignited by both individual mobility and the environmental heterogeneity.
Citation: Yongli Cai, Weiming Wang. Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 989-1013. doi: 10.3934/dcdsb.2015.20.989
References:
[1]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model,, SIAM Journal on Applied Mathematics, 67 (2007), 1283. doi: 10.1137/060672522. Google Scholar

[2]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model,, Discrete and Continuous Dynamical Systems, 21 (2008), 1. doi: 10.3934/dcds.2008.21.1. Google Scholar

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M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence,, Mathematical Biosciences, 189 (2004), 75. doi: 10.1016/j.mbs.2004.01.003. Google Scholar

[4]

R. M. Anderson, R. M. May and B. Anderson, Infectious Diseases of Humans: Dynamics and Control,, Wiley Online Library, (1992). Google Scholar

[5]

S. Anita and V. Capasso, A stabilizability problem for a reaction-diffusion system modelling a class of spatially structured epidemic systems,, Nonlinear Analysis: Real World Applications, 3 (2002), 453. doi: 10.1016/S1468-1218(01)00025-6. Google Scholar

[6]

C. Bain, Applied mathematical ecology,, Journal of Epidemiology and Community Health, 44 (1990). doi: 10.1136/jech.44.3.254-b. Google Scholar

[7]

F. Berezovsky, G. Karev, B. Song and C. Castillo-Chavez, A simple epidemic model with surprising dynamics,, Mathematical Biosciences and Engineering, 2 (2005), 133. Google Scholar

[8]

E. Beretta and Y. Takeuchi, Global stability of an SIR epidemic model with time delays,, Journal of Mathematical Biology, 33 (1995), 250. doi: 10.1007/BF00169563. Google Scholar

[9]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations,, John Wiley & Sons, (2003). doi: 10.1002/0470871296. Google Scholar

[10]

V. Capasso, Mathematical Structures of Epidemic Systems,, Springer, (1993). doi: 10.1007/978-3-540-70514-7. Google Scholar

[11]

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

[12]

C. Castillo-Chavez and A.-A. Yakubu, Dispersal, disease and life-history evolution,, Mathematical Biosciences, 173 (2001), 35. doi: 10.1016/S0025-5564(01)00065-7. Google Scholar

[13]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, Journal of Functional Analysis, 8 (1971), 321. doi: 10.1016/0022-1236(71)90015-2. Google Scholar

[14]

D. Ebert, M. Lipsitch and K. L. Mangin, The effect of parasites on host population density and extinction: experimental epidemiology with daphnia and six microparasites,, The American Naturalist, 156 (2000), 459. Google Scholar

[15]

W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model of the spread of feline leukemia virus (FeLV) through a highly heterogeneous spatial domain,, SIAM Journal on Mathematical Analysis, 33 (2001), 570. doi: 10.1137/S0036141000371757. Google Scholar

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W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A reaction-diffusion system modeling direct and indirect transmission of diseases,, Discrete and Continuous Dynamical Systems-B, 4 (2004), 893. doi: 10.3934/dcdsb.2004.4.893. Google Scholar

[17]

W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model for indirectly transmitted diseases,, Mathematical Biosciences, 206 (2007), 233. doi: 10.1016/j.mbs.2005.07.005. Google Scholar

[18]

B. Fred and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology(Second Edition),, Springer, (2012). doi: 10.1007/978-1-4614-1686-9. Google Scholar

[19]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840., Springer-Verlag Berlin, (1981). Google Scholar

[20]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar

[21]

H. W. Hethcote and S. A. Levin, Periodicity in epidemiological models,, In Applied Mathematical Ecology, (1989), 193. doi: 10.1007/978-3-642-61317-3_8. Google Scholar

[22]

W. Huang, M. Han and K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission,, Mathematical Biosciences and Engineering, 7 (2010), 51. doi: 10.3934/mbe.2010.7.51. Google Scholar

[23]

T. W. Hwang and Y. Kuang, Deterministic extinction effect of parasites on host populations,, Journal of Mathematical Biology, 46 (2003), 17. doi: 10.1007/s00285-002-0165-7. Google Scholar

[24]

T. W. Hwang and Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model,, Mathematical Biosciences and Engineering, 2 (2005), 743. doi: 10.3934/mbe.2005.2.743. Google Scholar

[25]

Y. Kang and C. Castillo-Chavez, A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness,, Discrete and Continuous Dynamical Systems-B, 19 (2014), 89. Google Scholar

[26]

Y. Kang, S. K. Sasmal, A. R. Bhowmick and J. Chattopadhyay, Dynamics of a predator-prey system with prey subject to Allee effects and disease,, Mathematical Biosciences and Engineering, 11 (2014), 877. doi: 10.3934/mbe.2014.11.877. Google Scholar

[27]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics-I,, Bulletin of Mathematical Biology, 53 (1991), 33. Google Scholar

[28]

K. I. Kim, Z. Lin and Q. Zhang, An SIR epidemic model with free boundary,, Nonlinear Analysis: Real World Applications, 14 (2013), 1992. doi: 10.1016/j.nonrwa.2013.02.003. Google Scholar

[29]

A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models,, Mathematical Medicine and Biology, 22 (2005), 113. doi: 10.1093/imammb/dqi001. Google Scholar

[30]

X. Z. Li, W. S. Li and M. Ghosh, Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment,, Applied Mathematics and Computation, 210 (2009), 141. doi: 10.1016/j.amc.2008.12.085. Google Scholar

[31]

W. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of sirs epidemiological models,, Journal of Mathematical Biology, 23 (1986), 187. doi: 10.1007/BF00276956. Google Scholar

[32]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis,, CRC Press, (2010). Google Scholar

[33]

J. Ma and Z. Ma, Epidemic threshold conditions for seasonally forced seir models,, Mathematical Biosciences and Engineering, 3 (2006), 161. doi: 10.3934/mbe.2006.3.161. Google Scholar

[34]

Z. Ma, Y. Zhou and J. Wu, Modeling and Dynamics of Infectious Diseases,, Higher Education Press, (2009). doi: 10.1142/7223. Google Scholar

[35]

C. Neuhauser, Mathematical challenges in spatial ecology,, Notices of the AMS, 48 (2001), 1304. Google Scholar

[36]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, New York: Springer, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[37]

R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. part I,, Journal of Differential Equations, 247 (2009), 1096. doi: 10.1016/j.jde.2009.05.002. Google Scholar

[38]

R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model,, Nonlinear Analysis: Theory, 71 (2009), 239. doi: 10.1016/j.na.2008.10.043. Google Scholar

[39]

R. Peng and F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement,, Physica D: Nonlinear Phenomena, 259 (2013), 8. doi: 10.1016/j.physd.2013.05.006. Google Scholar

[40]

R. Peng and X. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment,, Nonlinearity, 25 (2012), 1451. doi: 10.1088/0951-7715/25/5/1451. Google Scholar

[41]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, Journal of Functional Analysis, 7 (1971), 487. doi: 10.1016/0022-1236(71)90030-9. Google Scholar

[42]

M. Robinson, N. I. Stilianakis and Y. Drossinos, Spatial dynamics of airborne infectious diseases,, Journal of Theoretical Biology, 297 (2012), 116. doi: 10.1016/j.jtbi.2011.12.015. Google Scholar

[43]

R. Ross, The Prevention of Malaria(2nd ed.),, Murray, (1911). Google Scholar

[44]

S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models,, In Mathematics for Life Science and Medicine, (2007), 97. Google Scholar

[45]

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

[46]

J. Shi, Persistence and bifurcation of degenerate solutions,, Journal of Functional Analysis, 169 (1999), 494. doi: 10.1006/jfan.1999.3483. Google Scholar

[47]

H. L. Smith, Subharmonic bifurcation in an SIR epidemic model,, Journal of Mathematical Biology, 17 (1983), 163. doi: 10.1007/BF00305757. Google Scholar

[48]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs, (1995). Google Scholar

[49]

N. Tuncer and M. Martcheva, Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion,, Journal of Biological Dynamics, 6 (2012), 406. doi: 10.1080/17513758.2011.614697. Google Scholar

[50]

V. Volpert and S. Petrovskii, Reaction-diffusion waves in biology,, Physics of life reviews, 6 (2009), 267. doi: 10.1016/j.plrev.2009.10.002. Google Scholar

[51]

W. D. Wang, Epidemic models with nonlinear infection forces,, Mathematical Biosciences and Engineering, 3 (2006), 267. doi: 10.3934/mbe.2006.3.267. Google Scholar

[52]

W. D. Wang and X. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models,, SIAM Journal on Applied Dynamical Systems, 11 (2012), 1652. doi: 10.1137/120872942. Google Scholar

[53]

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

show all references

References:
[1]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model,, SIAM Journal on Applied Mathematics, 67 (2007), 1283. doi: 10.1137/060672522. Google Scholar

[2]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model,, Discrete and Continuous Dynamical Systems, 21 (2008), 1. doi: 10.3934/dcds.2008.21.1. Google Scholar

[3]

M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence,, Mathematical Biosciences, 189 (2004), 75. doi: 10.1016/j.mbs.2004.01.003. Google Scholar

[4]

R. M. Anderson, R. M. May and B. Anderson, Infectious Diseases of Humans: Dynamics and Control,, Wiley Online Library, (1992). Google Scholar

[5]

S. Anita and V. Capasso, A stabilizability problem for a reaction-diffusion system modelling a class of spatially structured epidemic systems,, Nonlinear Analysis: Real World Applications, 3 (2002), 453. doi: 10.1016/S1468-1218(01)00025-6. Google Scholar

[6]

C. Bain, Applied mathematical ecology,, Journal of Epidemiology and Community Health, 44 (1990). doi: 10.1136/jech.44.3.254-b. Google Scholar

[7]

F. Berezovsky, G. Karev, B. Song and C. Castillo-Chavez, A simple epidemic model with surprising dynamics,, Mathematical Biosciences and Engineering, 2 (2005), 133. Google Scholar

[8]

E. Beretta and Y. Takeuchi, Global stability of an SIR epidemic model with time delays,, Journal of Mathematical Biology, 33 (1995), 250. doi: 10.1007/BF00169563. Google Scholar

[9]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations,, John Wiley & Sons, (2003). doi: 10.1002/0470871296. Google Scholar

[10]

V. Capasso, Mathematical Structures of Epidemic Systems,, Springer, (1993). doi: 10.1007/978-3-540-70514-7. Google Scholar

[11]

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

[12]

C. Castillo-Chavez and A.-A. Yakubu, Dispersal, disease and life-history evolution,, Mathematical Biosciences, 173 (2001), 35. doi: 10.1016/S0025-5564(01)00065-7. Google Scholar

[13]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, Journal of Functional Analysis, 8 (1971), 321. doi: 10.1016/0022-1236(71)90015-2. Google Scholar

[14]

D. Ebert, M. Lipsitch and K. L. Mangin, The effect of parasites on host population density and extinction: experimental epidemiology with daphnia and six microparasites,, The American Naturalist, 156 (2000), 459. Google Scholar

[15]

W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model of the spread of feline leukemia virus (FeLV) through a highly heterogeneous spatial domain,, SIAM Journal on Mathematical Analysis, 33 (2001), 570. doi: 10.1137/S0036141000371757. Google Scholar

[16]

W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A reaction-diffusion system modeling direct and indirect transmission of diseases,, Discrete and Continuous Dynamical Systems-B, 4 (2004), 893. doi: 10.3934/dcdsb.2004.4.893. Google Scholar

[17]

W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model for indirectly transmitted diseases,, Mathematical Biosciences, 206 (2007), 233. doi: 10.1016/j.mbs.2005.07.005. Google Scholar

[18]

B. Fred and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology(Second Edition),, Springer, (2012). doi: 10.1007/978-1-4614-1686-9. Google Scholar

[19]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840., Springer-Verlag Berlin, (1981). Google Scholar

[20]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar

[21]

H. W. Hethcote and S. A. Levin, Periodicity in epidemiological models,, In Applied Mathematical Ecology, (1989), 193. doi: 10.1007/978-3-642-61317-3_8. Google Scholar

[22]

W. Huang, M. Han and K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission,, Mathematical Biosciences and Engineering, 7 (2010), 51. doi: 10.3934/mbe.2010.7.51. Google Scholar

[23]

T. W. Hwang and Y. Kuang, Deterministic extinction effect of parasites on host populations,, Journal of Mathematical Biology, 46 (2003), 17. doi: 10.1007/s00285-002-0165-7. Google Scholar

[24]

T. W. Hwang and Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model,, Mathematical Biosciences and Engineering, 2 (2005), 743. doi: 10.3934/mbe.2005.2.743. Google Scholar

[25]

Y. Kang and C. Castillo-Chavez, A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness,, Discrete and Continuous Dynamical Systems-B, 19 (2014), 89. Google Scholar

[26]

Y. Kang, S. K. Sasmal, A. R. Bhowmick and J. Chattopadhyay, Dynamics of a predator-prey system with prey subject to Allee effects and disease,, Mathematical Biosciences and Engineering, 11 (2014), 877. doi: 10.3934/mbe.2014.11.877. Google Scholar

[27]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics-I,, Bulletin of Mathematical Biology, 53 (1991), 33. Google Scholar

[28]

K. I. Kim, Z. Lin and Q. Zhang, An SIR epidemic model with free boundary,, Nonlinear Analysis: Real World Applications, 14 (2013), 1992. doi: 10.1016/j.nonrwa.2013.02.003. Google Scholar

[29]

A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models,, Mathematical Medicine and Biology, 22 (2005), 113. doi: 10.1093/imammb/dqi001. Google Scholar

[30]

X. Z. Li, W. S. Li and M. Ghosh, Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment,, Applied Mathematics and Computation, 210 (2009), 141. doi: 10.1016/j.amc.2008.12.085. Google Scholar

[31]

W. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of sirs epidemiological models,, Journal of Mathematical Biology, 23 (1986), 187. doi: 10.1007/BF00276956. Google Scholar

[32]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis,, CRC Press, (2010). Google Scholar

[33]

J. Ma and Z. Ma, Epidemic threshold conditions for seasonally forced seir models,, Mathematical Biosciences and Engineering, 3 (2006), 161. doi: 10.3934/mbe.2006.3.161. Google Scholar

[34]

Z. Ma, Y. Zhou and J. Wu, Modeling and Dynamics of Infectious Diseases,, Higher Education Press, (2009). doi: 10.1142/7223. Google Scholar

[35]

C. Neuhauser, Mathematical challenges in spatial ecology,, Notices of the AMS, 48 (2001), 1304. Google Scholar

[36]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, New York: Springer, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[37]

R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. part I,, Journal of Differential Equations, 247 (2009), 1096. doi: 10.1016/j.jde.2009.05.002. Google Scholar

[38]

R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model,, Nonlinear Analysis: Theory, 71 (2009), 239. doi: 10.1016/j.na.2008.10.043. Google Scholar

[39]

R. Peng and F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement,, Physica D: Nonlinear Phenomena, 259 (2013), 8. doi: 10.1016/j.physd.2013.05.006. Google Scholar

[40]

R. Peng and X. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment,, Nonlinearity, 25 (2012), 1451. doi: 10.1088/0951-7715/25/5/1451. Google Scholar

[41]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, Journal of Functional Analysis, 7 (1971), 487. doi: 10.1016/0022-1236(71)90030-9. Google Scholar

[42]

M. Robinson, N. I. Stilianakis and Y. Drossinos, Spatial dynamics of airborne infectious diseases,, Journal of Theoretical Biology, 297 (2012), 116. doi: 10.1016/j.jtbi.2011.12.015. Google Scholar

[43]

R. Ross, The Prevention of Malaria(2nd ed.),, Murray, (1911). Google Scholar

[44]

S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models,, In Mathematics for Life Science and Medicine, (2007), 97. Google Scholar

[45]

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

[46]

J. Shi, Persistence and bifurcation of degenerate solutions,, Journal of Functional Analysis, 169 (1999), 494. doi: 10.1006/jfan.1999.3483. Google Scholar

[47]

H. L. Smith, Subharmonic bifurcation in an SIR epidemic model,, Journal of Mathematical Biology, 17 (1983), 163. doi: 10.1007/BF00305757. Google Scholar

[48]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs, (1995). Google Scholar

[49]

N. Tuncer and M. Martcheva, Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion,, Journal of Biological Dynamics, 6 (2012), 406. doi: 10.1080/17513758.2011.614697. Google Scholar

[50]

V. Volpert and S. Petrovskii, Reaction-diffusion waves in biology,, Physics of life reviews, 6 (2009), 267. doi: 10.1016/j.plrev.2009.10.002. Google Scholar

[51]

W. D. Wang, Epidemic models with nonlinear infection forces,, Mathematical Biosciences and Engineering, 3 (2006), 267. doi: 10.3934/mbe.2006.3.267. Google Scholar

[52]

W. D. Wang and X. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models,, SIAM Journal on Applied Dynamical Systems, 11 (2012), 1652. doi: 10.1137/120872942. Google Scholar

[53]

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

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