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Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions
Propagation phenomena for a criss-cross infection model with non-diffusive susceptible population in periodic media
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China |
This paper is concerned with propagation phenomena for an epidemic model describing the circulation of a disease within two populations or two subgroups in periodic media, where the susceptible individuals are assumed to be motionless. The spatial dynamics for the cooperative system obtained by a classical transformation are investigated, including spatially periodic steady state, spreading speeds and pulsating travelling fronts. It is proved that the minimal wave speed is linearly determined and given by a variational formula involving linear eigenvalue problem. Further, we prove that the existence and non-existence of travelling wave solutions of the model are entirely determined by the basic reproduction ratio $ \mathcal{R}_{0} $. As an application, we prove that if the localized amount of infectious individuals are introduced at the beginning, then the solution of such a system has an asymptotic spreading speed in large time and that is exactly coincident with the minimal wave speed.
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K. M. Alanaz, Z. Jackiewicz and H. R. Thieme,
Spreading speeds of rabies with territorial and diffusing rabid foxes, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2143-2183.
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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 Contin. Dyn. Syst., 21 (2008), 1-20.
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B. Ambrosio, A. Ducrot and S. Ruan, Generalized traveling waves for time-dependent reaction-diffusion systems, Math. Ann., (2020).
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D. G. Aronson and H. F. Weinberger,
Multidimensional nonlinear diffusion arising in population genetic, Adv. in Math., 30 (1978), 33-76.
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C. Beaumont, J.-B. Burie, A. Ducrot and P. Zongo,
Propagation of salmonella within an industrial hen house, SIAM J. Appl. Math., 72 (2012), 1113-1148.
doi: 10.1137/110822967. |
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H. Berestycki and F. Hamel,
Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032.
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H. Berestycki, F. Hamel and N. Nadirashvili,
The speed of propagation for KPP type problems. Ⅰ: Periodic framework, J. Eur. Math. Soc., 7 (2005), 173-213.
doi: 10.4171/JEMS/26. |
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H. Berestycki, F. Hamel and N. Nadirashvili,
The speed of propagation for KPP type problems. Ⅱ: Genaral domains, J. Amer. Math. Soc., 23 (2010), 1-34.
doi: 10.1090/S0894-0347-09-00633-X. |
[9] |
H. Berestycki, F. Hamel and L. Roques,
Analysis of the periodically fragmented environment model: Ⅰ-Species persistence, J. Math. Biol., 51 (2005), 75-113.
doi: 10.1007/s00285-004-0313-3. |
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H. Berestycki, F. Hamel and L. Roques,
Analysis of the periodically fragmented environment model: Ⅱ-biological invasions and pulsating travelling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146.
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H. Berestycki, F. Hamel and L. Rossi,
Liouville-type results for semilinear elliptic equations in unbounded domains, Ann. Math. Pura Appl., 186 (2007), 469-507.
doi: 10.1007/s10231-006-0015-0. |
[12] |
A. Ducrot and T. Giletti,
Convergence to a pulsating travelling wave for an epidemic reaction-diffusion system with non-diffusive susceptible population, J. Math. Biol., 69 (2014), 533-552.
doi: 10.1007/s00285-013-0713-3. |
[13] |
A. Ducrot, P. Magal and S. Ruan,
Travelling wave solutions in multigroup age-structured epidemic models, Arch. Ration. Mech. Anal., 195 (2010), 311-331.
doi: 10.1007/s00205-008-0203-8. |
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J. Fang and X.-Q. Zhao,
Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.
doi: 10.1137/140953939. |
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R. A. Fisher,
The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369.
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T. Giletti,
Convergence to pulsating traveling waves with minimal speed in some KPP heterogeneous problems, Calc. Var. Partial Differ. Equ., 51 (2014), 265-289.
doi: 10.1007/s00526-013-0674-9. |
[19] |
A. Källén, P. Arcuri and J. D. Murray,
A simple model for the spatial spread and control of rabies, J. Theor. Biol., 116 (1985), 377-393.
doi: 10.1016/S0022-5193(85)80276-9. |
[20] |
W. O. Kermack and A. G. McKendrick,
A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721.
doi: 10.1098/rspa.1927.0118. |
[21] |
A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bulletin Université d'État à Moscou(Bjul. Moskowskogo Gos. Univ.), Série internationale A, 1 (1937), 1–26. Google Scholar |
[22] |
K.-Y. Lam and Y. Lou,
Asymptotic behavior of the principal eigenvalue for cooperative elliptic systems and applications, J. Dyn. Differ. Equ., 28 (2016), 29-48.
doi: 10.1007/s10884-015-9504-4. |
[23] |
X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1–40; Comm. Pure Appl. Math., 61 (2008), 137–138 (Erratum).
doi: 10.1002/cpa.20221. |
[24] |
X. Liang and X.-Q. Zhao,
Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903.
doi: 10.1016/j.jfa.2010.04.018. |
[25] |
Y. Lou and X.-Q. Zhao,
A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.
doi: 10.1007/s00285-010-0346-8. |
[26] |
P. Magal and C. McCluskey,
Two group infection age model including an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095.
doi: 10.1137/120882056. |
[27] |
R. H. Martin Jr., Nonlinear Operators and Differential Equations in Banach Spaces, Wiley-Interscience, New York, 1976. |
[28] |
J. D. Murray, Mathematical Biology I: An Introduction and II: Spatial Models and Biomedical Applications, 3rd ed., Springer, New York, 2002.
doi: 10.1007/b98868. |
[29] |
G. Nadin,
Some dependence results between the spreading speed and the coefficients of the space-time periodic Fisher-KPP equation, European J. Appl. Math., 22 (2011), 169-185.
doi: 10.1017/S0956792511000027. |
[30] |
G. Nadin,
The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator, SIAM J. Math. Anal., 41 (2009/10), 2388-2406.
doi: 10.1137/080743597. |
[31] |
N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford Univ. Press, Oxford, 1997.
doi: 10.2307/6013. |
[32] |
N. Shigesada, K. Kawasaki and E. Teramoto,
Traveling periodic waves in heterogeneous environments, Theor. Population Biol., 30 (1986), 143-160.
doi: 10.1016/0040-5809(86)90029-8. |
[33] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. AMS, Providence, RI, 1995.
doi: 10.1090/surv/041. |
[34] |
D. L. Smith, J. Dushoff and F. E. McKenzie,
The risk of a mosquito-borne infection in a heterogeneous environment, PLoS Biol., 2 (2004), 1957-1964.
doi: 10.1371/journal.pbio.0020368. |
[35] |
G. Sweers,
Strong positivity in $C(\overline{\Omega})$ for elliptic systems, Math. Z., 209 (1992), 251-271.
doi: 10.1007/BF02570833. |
[36] |
H. R. Thieme,
Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.
doi: 10.1137/080732870. |
[37] |
W. Wang and X.-Q. Zhao,
A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168.
doi: 10.1137/090775890. |
[38] |
X.-S. Wang and X.-Q. Zhao,
Pulsating waves of a partially degenerate reaction-diffusion system in a periodic habitat, J. Differ. Equ., 259 (2015), 7238-7259.
|
[39] |
Z.-C. Wang, L. Zhang and X.-Q. Zhao,
Time periodic traveling waves for a periodic and diffusive SIR epidemic model, J. Dyn. Differ. Equ., 30 (2018), 379-403.
doi: 10.1007/s10884-016-9546-2. |
[40] |
H. F. Weinberger,
On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.
doi: 10.1007/s00285-002-0169-3. |
[41] |
P. Weng and X.-Q. Zhao,
Spatial dynamics of a nonlocal and delayed population model in a periodic habitat, Discrete Contin. Dyn. Syst., 29 (2011), 343-366.
doi: 10.3934/dcds.2011.29.343. |
[42] |
C. Wu, D. Xiao and X.-Q. Zhao,
Spreading speeds of a partially degenerate reaction-diffusion system in a periodic habitat, J. Differ. Equ., 255 (2013), 3983-4011.
doi: 10.1016/j.jde.2013.07.058. |
[43] |
J. Xin,
Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.
doi: 10.1137/S0036144599364296. |
[44] |
X. Yu and X.-Q. Zhao,
Propagation phenomena for a reaction-advection-diffusion competition model in a periodic habitat, J. Dyn. Differ. Equ., 29 (2017), 41-66.
doi: 10.1007/s10884-015-9426-1. |
[45] |
L. Zhang, Z.-C. Wang and X.-Q. Zhao,
Time periodic traveling wave solutions for a Kermack-McKendrick epidemic model with diffusion and seasonality, J. Evol. Equ., 20 (2020), 1029-1059.
doi: 10.1007/s00028-019-00544-2. |
[46] |
G. Zhao and S. Ruan,
Spatial and temporal dynamics of a nonlocal viral infection model, SIAM J. Appl. Math., 78 (2018), 1954-1980.
doi: 10.1137/17M1144106. |
[47] |
L. Zhao, Z.-C. Wang and S. Ruan,
Traveling wave solutions in a two-group epidemic model with latent period, Nonlinearity, 30 (2017), 1287-1325.
doi: 10.1088/1361-6544/aa59ae. |
[48] |
L. Zhao, Z.-C. Wang and S. Ruan,
Traveling wave solutions in a two-group SIR epidemic model with constant recruitment, J. Math. Biol., 77 (2018), 1871-1915.
doi: 10.1007/s00285-018-1227-9. |
[49] |
X.-Q. Zhao, Dynamical Systems in Population Biology, CMS Books in Mathematics (Ouvrages de Mathématiques de la SMC), 2$^{nd}$ edition, Springer, Cham, 2017.
doi: 10.1007/978-3-319-56433-3. |
show all references
References:
[1] |
K. M. Alanaz, Z. Jackiewicz and H. R. Thieme,
Spreading speeds of rabies with territorial and diffusing rabid foxes, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2143-2183.
doi: 10.3934/dcdsb.2019222. |
[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 Contin. Dyn. Syst., 21 (2008), 1-20.
doi: 10.3934/dcds.2008.21.1. |
[3] |
B. Ambrosio, A. Ducrot and S. Ruan, Generalized traveling waves for time-dependent reaction-diffusion systems, Math. Ann., (2020).
doi: 10.1007/s00208-020-01998-3. |
[4] |
D. G. Aronson and H. F. Weinberger,
Multidimensional nonlinear diffusion arising in population genetic, Adv. in Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[5] |
C. Beaumont, J.-B. Burie, A. Ducrot and P. Zongo,
Propagation of salmonella within an industrial hen house, SIAM J. Appl. Math., 72 (2012), 1113-1148.
doi: 10.1137/110822967. |
[6] |
H. Berestycki and F. Hamel,
Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032.
doi: 10.1002/cpa.3022. |
[7] |
H. Berestycki, F. Hamel and N. Nadirashvili,
The speed of propagation for KPP type problems. Ⅰ: Periodic framework, J. Eur. Math. Soc., 7 (2005), 173-213.
doi: 10.4171/JEMS/26. |
[8] |
H. Berestycki, F. Hamel and N. Nadirashvili,
The speed of propagation for KPP type problems. Ⅱ: Genaral domains, J. Amer. Math. Soc., 23 (2010), 1-34.
doi: 10.1090/S0894-0347-09-00633-X. |
[9] |
H. Berestycki, F. Hamel and L. Roques,
Analysis of the periodically fragmented environment model: Ⅰ-Species persistence, J. Math. Biol., 51 (2005), 75-113.
doi: 10.1007/s00285-004-0313-3. |
[10] |
H. Berestycki, F. Hamel and L. Roques,
Analysis of the periodically fragmented environment model: Ⅱ-biological invasions and pulsating travelling fronts, J. Math. Pures Appl., 84 (2005), 1101-1146.
doi: 10.1016/j.matpur.2004.10.006. |
[11] |
H. Berestycki, F. Hamel and L. Rossi,
Liouville-type results for semilinear elliptic equations in unbounded domains, Ann. Math. Pura Appl., 186 (2007), 469-507.
doi: 10.1007/s10231-006-0015-0. |
[12] |
A. Ducrot and T. Giletti,
Convergence to a pulsating travelling wave for an epidemic reaction-diffusion system with non-diffusive susceptible population, J. Math. Biol., 69 (2014), 533-552.
doi: 10.1007/s00285-013-0713-3. |
[13] |
A. Ducrot, P. Magal and S. Ruan,
Travelling wave solutions in multigroup age-structured epidemic models, Arch. Ration. Mech. Anal., 195 (2010), 311-331.
doi: 10.1007/s00205-008-0203-8. |
[14] |
J. Fang and X.-Q. Zhao,
Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.
doi: 10.1137/140953939. |
[15] |
R. A. Fisher,
The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[16] |
W. E. Fitzgibbon, C. B. Martin and J. J. Morgan,
A diffusive epidemic model with criss-cross dynamics, J. Math. Anal. Appl., 184 (1994), 399-414.
doi: 10.1006/jmaa.1994.1209. |
[17] |
W. E. Fitzgibbon, J. J. Morgan and G. F. Webb, An outbreak vector-host epidemic model with spatial structure: The 2015-2016 Zika outbreak in Rio De Janeiro, Theor. Biol. Med. Modell., 14 (2017), 7. Google Scholar |
[18] |
T. Giletti,
Convergence to pulsating traveling waves with minimal speed in some KPP heterogeneous problems, Calc. Var. Partial Differ. Equ., 51 (2014), 265-289.
doi: 10.1007/s00526-013-0674-9. |
[19] |
A. Källén, P. Arcuri and J. D. Murray,
A simple model for the spatial spread and control of rabies, J. Theor. Biol., 116 (1985), 377-393.
doi: 10.1016/S0022-5193(85)80276-9. |
[20] |
W. O. Kermack and A. G. McKendrick,
A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721.
doi: 10.1098/rspa.1927.0118. |
[21] |
A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bulletin Université d'État à Moscou(Bjul. Moskowskogo Gos. Univ.), Série internationale A, 1 (1937), 1–26. Google Scholar |
[22] |
K.-Y. Lam and Y. Lou,
Asymptotic behavior of the principal eigenvalue for cooperative elliptic systems and applications, J. Dyn. Differ. Equ., 28 (2016), 29-48.
doi: 10.1007/s10884-015-9504-4. |
[23] |
X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1–40; Comm. Pure Appl. Math., 61 (2008), 137–138 (Erratum).
doi: 10.1002/cpa.20221. |
[24] |
X. Liang and X.-Q. Zhao,
Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903.
doi: 10.1016/j.jfa.2010.04.018. |
[25] |
Y. Lou and X.-Q. Zhao,
A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.
doi: 10.1007/s00285-010-0346-8. |
[26] |
P. Magal and C. McCluskey,
Two group infection age model including an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095.
doi: 10.1137/120882056. |
[27] |
R. H. Martin Jr., Nonlinear Operators and Differential Equations in Banach Spaces, Wiley-Interscience, New York, 1976. |
[28] |
J. D. Murray, Mathematical Biology I: An Introduction and II: Spatial Models and Biomedical Applications, 3rd ed., Springer, New York, 2002.
doi: 10.1007/b98868. |
[29] |
G. Nadin,
Some dependence results between the spreading speed and the coefficients of the space-time periodic Fisher-KPP equation, European J. Appl. Math., 22 (2011), 169-185.
doi: 10.1017/S0956792511000027. |
[30] |
G. Nadin,
The effect of the Schwarz rearrangement on the periodic principal eigenvalue of a nonsymmetric operator, SIAM J. Math. Anal., 41 (2009/10), 2388-2406.
doi: 10.1137/080743597. |
[31] |
N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford Univ. Press, Oxford, 1997.
doi: 10.2307/6013. |
[32] |
N. Shigesada, K. Kawasaki and E. Teramoto,
Traveling periodic waves in heterogeneous environments, Theor. Population Biol., 30 (1986), 143-160.
doi: 10.1016/0040-5809(86)90029-8. |
[33] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. AMS, Providence, RI, 1995.
doi: 10.1090/surv/041. |
[34] |
D. L. Smith, J. Dushoff and F. E. McKenzie,
The risk of a mosquito-borne infection in a heterogeneous environment, PLoS Biol., 2 (2004), 1957-1964.
doi: 10.1371/journal.pbio.0020368. |
[35] |
G. Sweers,
Strong positivity in $C(\overline{\Omega})$ for elliptic systems, Math. Z., 209 (1992), 251-271.
doi: 10.1007/BF02570833. |
[36] |
H. R. Thieme,
Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.
doi: 10.1137/080732870. |
[37] |
W. Wang and X.-Q. Zhao,
A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168.
doi: 10.1137/090775890. |
[38] |
X.-S. Wang and X.-Q. Zhao,
Pulsating waves of a partially degenerate reaction-diffusion system in a periodic habitat, J. Differ. Equ., 259 (2015), 7238-7259.
|
[39] |
Z.-C. Wang, L. Zhang and X.-Q. Zhao,
Time periodic traveling waves for a periodic and diffusive SIR epidemic model, J. Dyn. Differ. Equ., 30 (2018), 379-403.
doi: 10.1007/s10884-016-9546-2. |
[40] |
H. F. Weinberger,
On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.
doi: 10.1007/s00285-002-0169-3. |
[41] |
P. Weng and X.-Q. Zhao,
Spatial dynamics of a nonlocal and delayed population model in a periodic habitat, Discrete Contin. Dyn. Syst., 29 (2011), 343-366.
doi: 10.3934/dcds.2011.29.343. |
[42] |
C. Wu, D. Xiao and X.-Q. Zhao,
Spreading speeds of a partially degenerate reaction-diffusion system in a periodic habitat, J. Differ. Equ., 255 (2013), 3983-4011.
doi: 10.1016/j.jde.2013.07.058. |
[43] |
J. Xin,
Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.
doi: 10.1137/S0036144599364296. |
[44] |
X. Yu and X.-Q. Zhao,
Propagation phenomena for a reaction-advection-diffusion competition model in a periodic habitat, J. Dyn. Differ. Equ., 29 (2017), 41-66.
doi: 10.1007/s10884-015-9426-1. |
[45] |
L. Zhang, Z.-C. Wang and X.-Q. Zhao,
Time periodic traveling wave solutions for a Kermack-McKendrick epidemic model with diffusion and seasonality, J. Evol. Equ., 20 (2020), 1029-1059.
doi: 10.1007/s00028-019-00544-2. |
[46] |
G. Zhao and S. Ruan,
Spatial and temporal dynamics of a nonlocal viral infection model, SIAM J. Appl. Math., 78 (2018), 1954-1980.
doi: 10.1137/17M1144106. |
[47] |
L. Zhao, Z.-C. Wang and S. Ruan,
Traveling wave solutions in a two-group epidemic model with latent period, Nonlinearity, 30 (2017), 1287-1325.
doi: 10.1088/1361-6544/aa59ae. |
[48] |
L. Zhao, Z.-C. Wang and S. Ruan,
Traveling wave solutions in a two-group SIR epidemic model with constant recruitment, J. Math. Biol., 77 (2018), 1871-1915.
doi: 10.1007/s00285-018-1227-9. |
[49] |
X.-Q. Zhao, Dynamical Systems in Population Biology, CMS Books in Mathematics (Ouvrages de Mathématiques de la SMC), 2$^{nd}$ edition, Springer, Cham, 2017.
doi: 10.1007/978-3-319-56433-3. |
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