doi: 10.3934/dcdsb.2022014
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A reaction-diffusion-advection SIS epidemic model with linear external source and open advective environments

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Guohong Zhang

Received  September 2021 Revised  December 2021 Early access January 2022

Fund Project: G.-H. Zhang is partially supported by grants from National Science Foundation of China(11871403); X.-L. Wang is partially supported by grants from Fundamental Research Funds for the Central Universities (XDJK2020B050)

In this paper, we propose a reaction-diffusion-advection SIS epidemic model with linear external source to study the effects of open advective environments on the persistence and extinction of infectious diseases. Threshold-type results on the global dynamics in terms of the basic reproduction number $ \mathcal{R}_{0} $ are established. It is found that the introduction of open advective environments leads to different monotonicity and asymptotic properties of the basic reproduction number $ \mathcal{R}_0 $ with respect to the diffusion rate $ d_I $ and advection speed $ q $. Our analytical results suggest that increasing the advection speed or decreasing the diffusion rate of infected individuals helps to eradicate the diseases in open advective environments.

Citation: Xu Rao, Guohong Zhang, Xiaoli Wang. A reaction-diffusion-advection SIS epidemic model with linear external source and open advective environments. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022014
References:
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L. J. S. AllenB. M. BolkerY. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Continuous Dynam. Systems, 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.

[2]

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 J. Appl. Math., 67 (2007) 1283–1309. doi: 10.1137/060672522.

[3]

M. BallykL. DungD. A. Jones and H. Smith, Effects of random motility on microbial growth and competition in a flow reactor, SIAM J. Appl. Math., 59 (1999), 573-596.  doi: 10.1137/S0036139997325345.

[4]

R. H. CuiK. Y. Lam and Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Differ. Equ., 263 (2017), 2343-2373.  doi: 10.1016/j.jde.2017.03.045.

[5]

R. H. Cui and Y. Lou, A spatial sis model in advective heterogeneous environments, J. Differ. Equ., 261 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.

[6]

K. Deng and Y. X. Wu, Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh Sect., 146 (2016), 929-946.  doi: 10.1017/S0308210515000864.

[7]

M. M. Desai and D. R. Nelson, A quasispecies on a moving oasis, Theor. Pop. Biol., 67 (2005), 33-45.  doi: 10.1016/j.tpb.2004.07.005.

[8]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.

[9]

J. GeK. I. KimZ. G. Lin and H. P. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differ. Equ., 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035.

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A. HillD. G. RandM. A. Nowak and N. A. Christakis, Emotions as infectious diseases in a large social network: the SIS model, Proc. R. Soc. B, 277 (2010), 3827-3835.  doi: 10.1098/rspb.2010.1217.

[13]

A. L. Hill, D. G. Rand, M. A. Nowak, N. A. Christakis and C. T. Bergstrom, Infectious disease modeling of social contagion in networks, PLoS Comput. Biol., 6 (2010), e1000968, 15 pp. doi: 10.1371/journal.pcbi.1000968.

[14]

W. HuangM. Han and K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Math. Biosci. Eng., 7 (2010), 51-66.  doi: 10.3934/mbe.2010.7.51.

[15]

D. JiangZ. Wang and L. Zhang, A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment, Discrete Continuous Dynam. Systems-B, 23 (2018), 4557-4578.  doi: 10.3934/dcdsb.2018176.

[16]

K. Kuto, H. Matsuzawa and R. Peng, Concentration profile of endemic equilibrium of a reaction-diffusion-advection sis epidemic model, Calc. Var. Partial Dif., 56 (2017), Paper No. 112, 28 pp. doi: 10.1007/s00526-017-1207-8.

[17]

K. Y. LamS. Liu and Y. Lou, Selected topics on reaction-diffusion-advection models from spatial ecology, Math. Appl. Sci. Eng., 1 (2020), 91-206.  doi: 10.5206/mase/10644.

[18]

B. Li, H. C. Li and Y. C. Tong, Analysis on a diffusive sis epidemic model with logistic source, Z. Angew. Math. Phys., 68 (2017), Paper No. 96, 25 pp. doi: 10.1007/s00033-017-0845-1.

[19]

H. C. LiR. Peng and F. B. Wang, Varying total population enhances disease persistence: Qualitative analysis on a diffusive sis epidemic model, J. Differ. Equ., 262 (2017), 885-913.  doi: 10.1016/j.jde.2016.09.044.

[20]

H. C. LiR. Peng and Z. A. Wang, On a diffusive susceptible-infected-susceptible epidemic model with mass action mechanism and birth-death effect: analysis, simulations, and comparison with other mechanisms, SIAM J. Appl. Math., 78 (2018), 2129-2153.  doi: 10.1137/18M1167863.

[21]

H. C. LiR. Peng and T. Xiang, Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion, Euro. J. Appl. Math., 31 (2020), 26-56.  doi: 10.1017/S0956792518000463.

[22]

Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342.  doi: 10.1007/s00285-013-0730-2.

[23]

F. LutscherM. A. Lewis and E. Mccauley, Effects of heterogeneity on spread and persistence in rivers, Bull. Math. Biol., 68 (2006), 2129-2160.  doi: 10.1007/s11538-006-9100-1.

[24]

F. LutscherM. A. Lewis and E. Pachepsky, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749-772.  doi: 10.1137/050636152.

[25]

F. LutscherE. McCauley and M. A. Lewis, Effects of heterogeneity on spread and persistence in rivers, Bull. Math. Biol., 68 (2006), 2129-2160.  doi: 10.1007/s11538-006-9100-1.

[26]

P. Magal and X. Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.

[27]

R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model, Part I, J. Differ. Equ., 247 (2009), 1096-1119.  doi: 10.1016/j.jde.2009.05.002.

[28]

R. Peng and S. Q. Liu, Global stability of the steady states of an sis epidemic reaction-diffusion model, Nonlinear Anal., 71 (2009), 239-247.  doi: 10.1016/j.na.2008.10.043.

[29]

R. Peng and F. Q. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement, Phys. D, 259 (2013), 8-25.  doi: 10.1016/j.physd.2013.05.006.

[30]

R. Peng and X. Q. Zhao, A reaction-diffusion sis epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451.

[31]

N. M. ShnerbK. A. Dahmen and D. R. Nelson, Life and death near a windy oasis, J. Math. Biol., 41 (2000), 1-23.  doi: 10.1007/s002850000025.

[32]

P. F. SongY. Lou and Y. N. Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment, J. Differ. Equ., 267 (2019), 5084-5114.  doi: 10.1016/j.jde.2019.05.022.

[33]

X. Y. Sun and R. H. Cui, Analysis on a diffusive SIS epidemic model with saturated incidence rate and linear source in a heterogeneous environment, J. Math. Anal. Appl., 490 (2020), 124212, 22 pp. doi: 10.1016/j.jmaa.2020.124212.

[34]

Y. C. Tong and C. X. Lei, An SIS epidemic reaction-diffusion model with spontaneous infection in a spatially heterogeneous environment, Nonlinear Anal-Real., 41 (2018), 443-460.  doi: 10.1016/j.nonrwa.2017.11.002.

[35]

W. D. Wang and X. Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942.

[36]

X. W. WenJ. P. Ji and B. Li, Asymptotic profiles of the endemic equilibrium to a diffusive sis epidemic model with mass action infection mechanism, J. Math. Anal. Appl., 458 (2017), 715-729.  doi: 10.1016/j.jmaa.2017.08.016.

[37]

Y. X. Wu and X. F. Zou, Asymptotic profiles of steady states for a diffusive sis epidemic model with mass action infection mechanism, J. Differ. Equ., 261 (2016), 4424-4447.  doi: 10.1016/j.jde.2016.06.028.

[38]

O. Vasilyeva and F. Lutscher, Population dynamics in rivers: Analysis of steady states, Can. Appl. Math. Quart., 18 (2010), 439-469. 

[39]

W. H. Xie, G. Q. Liang, W. Wang and Y. H. She, A spatial SIS model with Holling II incidence rate, Int. J. Biomath., 12 (2019), 1950092, 27 pp. doi: 10.1142/S179352451950092X.

[40]

F. Y. Yang and W. T. Li, Dynamics of a nonlocal dispersal sis epidemic model, Commun. Pur. Appl. Anal., 16 (2017), 781-797.  doi: 10.3934/cpaa.2017037.

[41]

F. Y. YangW. T. Li and S. G. Ruan, Dynamics of a nonlocal dispersal SIS epidemic model with Neumann boundary conditions, J. Differ. Equ., 267 (2019), 2011-2051.  doi: 10.1016/j.jde.2019.03.001.

[42]

J. L. Zhang and R. H. Cui, Qualitative analysis on a diffusive SIS epidemic system with logistic source and spontaneous infection in a heterogeneous environment, Nonlinear Anal-Real., 55 (2020), 103115, 20pp. doi: 10.1016/j.nonrwa.2020.103115.

[43]

M. ZhaoW. T. Li and Y. H. Du, The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries, Commun. Pur. Appl. Anal., 19 (2020), 4599-4620.  doi: 10.3934/cpaa.2020208.

[44]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer, 2003. doi: 10.1007/978-0-387-21761-1.

[45]

X. Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Canad. Appl. Math. Quart., 3 (1995), 473-495. 

[46]

X. ZhouC. Lei and J. Xiong, Qualitative analysis on an sis epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment, Discrete Continuous Dynam. Systems-B, 25 (2020), 81-98.  doi: 10.3934/dcdsb.2019173.

[47]

S. Y. Zhu and J. L. Wang, Asymptotic profiles of steady states for a diffusive SIS epidemic model with spontaneous infection and a logistic source, Commun. Pur. Appl. Anal., 19 (2020), 3323-3340.  doi: 10.3934/cpaa.2020147.

show all references

References:
[1]

L. J. S. AllenB. M. BolkerY. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Continuous Dynam. Systems, 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.

[2]

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 J. Appl. Math., 67 (2007) 1283–1309. doi: 10.1137/060672522.

[3]

M. BallykL. DungD. A. Jones and H. Smith, Effects of random motility on microbial growth and competition in a flow reactor, SIAM J. Appl. Math., 59 (1999), 573-596.  doi: 10.1137/S0036139997325345.

[4]

R. H. CuiK. Y. Lam and Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Differ. Equ., 263 (2017), 2343-2373.  doi: 10.1016/j.jde.2017.03.045.

[5]

R. H. Cui and Y. Lou, A spatial sis model in advective heterogeneous environments, J. Differ. Equ., 261 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.

[6]

K. Deng and Y. X. Wu, Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh Sect., 146 (2016), 929-946.  doi: 10.1017/S0308210515000864.

[7]

M. M. Desai and D. R. Nelson, A quasispecies on a moving oasis, Theor. Pop. Biol., 67 (2005), 33-45.  doi: 10.1016/j.tpb.2004.07.005.

[8]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.

[9]

J. GeK. I. KimZ. G. Lin and H. P. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differ. Equ., 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035.

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, $2^{nd}$ edition, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[11]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.

[12]

A. HillD. G. RandM. A. Nowak and N. A. Christakis, Emotions as infectious diseases in a large social network: the SIS model, Proc. R. Soc. B, 277 (2010), 3827-3835.  doi: 10.1098/rspb.2010.1217.

[13]

A. L. Hill, D. G. Rand, M. A. Nowak, N. A. Christakis and C. T. Bergstrom, Infectious disease modeling of social contagion in networks, PLoS Comput. Biol., 6 (2010), e1000968, 15 pp. doi: 10.1371/journal.pcbi.1000968.

[14]

W. HuangM. Han and K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Math. Biosci. Eng., 7 (2010), 51-66.  doi: 10.3934/mbe.2010.7.51.

[15]

D. JiangZ. Wang and L. Zhang, A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment, Discrete Continuous Dynam. Systems-B, 23 (2018), 4557-4578.  doi: 10.3934/dcdsb.2018176.

[16]

K. Kuto, H. Matsuzawa and R. Peng, Concentration profile of endemic equilibrium of a reaction-diffusion-advection sis epidemic model, Calc. Var. Partial Dif., 56 (2017), Paper No. 112, 28 pp. doi: 10.1007/s00526-017-1207-8.

[17]

K. Y. LamS. Liu and Y. Lou, Selected topics on reaction-diffusion-advection models from spatial ecology, Math. Appl. Sci. Eng., 1 (2020), 91-206.  doi: 10.5206/mase/10644.

[18]

B. Li, H. C. Li and Y. C. Tong, Analysis on a diffusive sis epidemic model with logistic source, Z. Angew. Math. Phys., 68 (2017), Paper No. 96, 25 pp. doi: 10.1007/s00033-017-0845-1.

[19]

H. C. LiR. Peng and F. B. Wang, Varying total population enhances disease persistence: Qualitative analysis on a diffusive sis epidemic model, J. Differ. Equ., 262 (2017), 885-913.  doi: 10.1016/j.jde.2016.09.044.

[20]

H. C. LiR. Peng and Z. A. Wang, On a diffusive susceptible-infected-susceptible epidemic model with mass action mechanism and birth-death effect: analysis, simulations, and comparison with other mechanisms, SIAM J. Appl. Math., 78 (2018), 2129-2153.  doi: 10.1137/18M1167863.

[21]

H. C. LiR. Peng and T. Xiang, Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion, Euro. J. Appl. Math., 31 (2020), 26-56.  doi: 10.1017/S0956792518000463.

[22]

Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342.  doi: 10.1007/s00285-013-0730-2.

[23]

F. LutscherM. A. Lewis and E. Mccauley, Effects of heterogeneity on spread and persistence in rivers, Bull. Math. Biol., 68 (2006), 2129-2160.  doi: 10.1007/s11538-006-9100-1.

[24]

F. LutscherM. A. Lewis and E. Pachepsky, The effect of dispersal patterns on stream populations, SIAM Rev., 47 (2005), 749-772.  doi: 10.1137/050636152.

[25]

F. LutscherE. McCauley and M. A. Lewis, Effects of heterogeneity on spread and persistence in rivers, Bull. Math. Biol., 68 (2006), 2129-2160.  doi: 10.1007/s11538-006-9100-1.

[26]

P. Magal and X. Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.

[27]

R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model, Part I, J. Differ. Equ., 247 (2009), 1096-1119.  doi: 10.1016/j.jde.2009.05.002.

[28]

R. Peng and S. Q. Liu, Global stability of the steady states of an sis epidemic reaction-diffusion model, Nonlinear Anal., 71 (2009), 239-247.  doi: 10.1016/j.na.2008.10.043.

[29]

R. Peng and F. Q. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement, Phys. D, 259 (2013), 8-25.  doi: 10.1016/j.physd.2013.05.006.

[30]

R. Peng and X. Q. Zhao, A reaction-diffusion sis epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451.

[31]

N. M. ShnerbK. A. Dahmen and D. R. Nelson, Life and death near a windy oasis, J. Math. Biol., 41 (2000), 1-23.  doi: 10.1007/s002850000025.

[32]

P. F. SongY. Lou and Y. N. Xiao, A spatial SEIRS reaction-diffusion model in heterogeneous environment, J. Differ. Equ., 267 (2019), 5084-5114.  doi: 10.1016/j.jde.2019.05.022.

[33]

X. Y. Sun and R. H. Cui, Analysis on a diffusive SIS epidemic model with saturated incidence rate and linear source in a heterogeneous environment, J. Math. Anal. Appl., 490 (2020), 124212, 22 pp. doi: 10.1016/j.jmaa.2020.124212.

[34]

Y. C. Tong and C. X. Lei, An SIS epidemic reaction-diffusion model with spontaneous infection in a spatially heterogeneous environment, Nonlinear Anal-Real., 41 (2018), 443-460.  doi: 10.1016/j.nonrwa.2017.11.002.

[35]

W. D. Wang and X. Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942.

[36]

X. W. WenJ. P. Ji and B. Li, Asymptotic profiles of the endemic equilibrium to a diffusive sis epidemic model with mass action infection mechanism, J. Math. Anal. Appl., 458 (2017), 715-729.  doi: 10.1016/j.jmaa.2017.08.016.

[37]

Y. X. Wu and X. F. Zou, Asymptotic profiles of steady states for a diffusive sis epidemic model with mass action infection mechanism, J. Differ. Equ., 261 (2016), 4424-4447.  doi: 10.1016/j.jde.2016.06.028.

[38]

O. Vasilyeva and F. Lutscher, Population dynamics in rivers: Analysis of steady states, Can. Appl. Math. Quart., 18 (2010), 439-469. 

[39]

W. H. Xie, G. Q. Liang, W. Wang and Y. H. She, A spatial SIS model with Holling II incidence rate, Int. J. Biomath., 12 (2019), 1950092, 27 pp. doi: 10.1142/S179352451950092X.

[40]

F. Y. Yang and W. T. Li, Dynamics of a nonlocal dispersal sis epidemic model, Commun. Pur. Appl. Anal., 16 (2017), 781-797.  doi: 10.3934/cpaa.2017037.

[41]

F. Y. YangW. T. Li and S. G. Ruan, Dynamics of a nonlocal dispersal SIS epidemic model with Neumann boundary conditions, J. Differ. Equ., 267 (2019), 2011-2051.  doi: 10.1016/j.jde.2019.03.001.

[42]

J. L. Zhang and R. H. Cui, Qualitative analysis on a diffusive SIS epidemic system with logistic source and spontaneous infection in a heterogeneous environment, Nonlinear Anal-Real., 55 (2020), 103115, 20pp. doi: 10.1016/j.nonrwa.2020.103115.

[43]

M. ZhaoW. T. Li and Y. H. Du, The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries, Commun. Pur. Appl. Anal., 19 (2020), 4599-4620.  doi: 10.3934/cpaa.2020208.

[44]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer, 2003. doi: 10.1007/978-0-387-21761-1.

[45]

X. Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Canad. Appl. Math. Quart., 3 (1995), 473-495. 

[46]

X. ZhouC. Lei and J. Xiong, Qualitative analysis on an sis epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment, Discrete Continuous Dynam. Systems-B, 25 (2020), 81-98.  doi: 10.3934/dcdsb.2019173.

[47]

S. Y. Zhu and J. L. Wang, Asymptotic profiles of steady states for a diffusive SIS epidemic model with spontaneous infection and a logistic source, Commun. Pur. Appl. Anal., 19 (2020), 3323-3340.  doi: 10.3934/cpaa.2020147.

Figure 1.  Illustration of the parameter regions of $ (d_{I}, q) $ in Theorem 1.4 and Theorem 1.5 on condition that $ \int_{0}^{L}\beta(x)\;dx>\int_{0}^{L}\gamma(x)\;dx $ and $ \int_{0}^{L}\beta(x)\;dx<\int_{0}^{L}\gamma(x)\;dx $, respectively. The curve is determined by $ \mathcal{R}_0(d_I, q_1(d_I)) = 1, d_I \in (0, \infty) $ and $ \mathcal{R}_0(d_I, q_2(d_I)) = 1, d_I \in (0, d_I^*) $
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