• Previous Article
    The Lewy-Stampacchia inequality for the fractional Laplacian and its application to anomalous unidirectional diffusion equations
  • DCDS-B Home
  • This Issue
  • Next Article
    Null controllability for a class of stochastic singular parabolic equations with the convection term
doi: 10.3934/dcdsb.2021174
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Concentration phenomenon of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with spontaneous infection

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu Province, China

* Corresponding author: Chengxia Lei

Received  February 2021 Revised  May 2021 Early access July 2021

Fund Project: The research was partially supported by NSF of China (No. 11971454, 11801232, 11671175), the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Natural Science Foundation of the Jiangsu Province(No. BK20180999), the Foundation of Jiangsu Normal University (No. 17XLR008)

In this paper, we investigate the effect of spontaneous infection and advection for a susceptible-infected-susceptible epidemic reaction-diffusion-advection model in a heterogeneous environment. The existence of the endemic equilibrium is proved, and the asymptotic behaviors of the endemic equilibrium in three cases (large advection; small diffusion of the susceptible population; small diffusion of the infected population) are established. Our results suggest that the advection can cause the concentration of the susceptible and infected populations at the downstream, and the spontaneous infection can enhance the persistence of infectious disease in the entire habitat.

Citation: Chengxia Lei, Xinhui Zhou. Concentration phenomenon of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with spontaneous infection. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021174
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 Contin. Dyn. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.  Google Scholar

[2]

F. Altarelli, A. Braunstein, L. Dall'Asta, J. R. Wakeling and R. Zecchina, Containing epidemic outbreaks by message-passing techniques, Phys. Rev. X, 4 (2014), 021024. Google Scholar

[3]

H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Jpn., 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.  Google Scholar

[4]

Y. Cai and W. Wang, Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 989-1013.  doi: 10.3934/dcdsb.2015.20.989.  Google Scholar

[5]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Ser. Math. Comput. Biol., John Wiley and Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[6]

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

[7]

R. Cui, H. Li, R. Peng and M. Zhou, Concentration behavior of endemic equilibrium for a reaction-diffusion-advection SIS epidemic model with mass action infection mechanism, preprint, (2019). Google Scholar

[8]

R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equations, 261 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.  Google Scholar

[9]

K. A. DahmenD. R. Nelson and N. M. Shnerb, Life and death near a windy oasis, J. Math. Biol., 41 (2000), 1-23.  doi: 10.1007/s002850000025.  Google Scholar

[10]

K. Deng and Y. Wu, Dynamics of an SIS epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929-946.  doi: 10.1017/S0308210515000864.  Google Scholar

[11]

Z. Du and R. Peng, A priori $L^\infty$ estimates for solutions of a class of reaction-diffusion systems, J. Math. Biol., 72 (2016), 1429-1439.  doi: 10.1007/s00285-015-0914-z.  Google Scholar

[12]

L. Dung, Dissipativity and global attractors for a class of quasilinear parabolic systems, Comm. Partial Differential Equations, 22 (1997), 413-433.  doi: 10.1080/03605309708821269.  Google Scholar

[13]

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

[14]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Springer, Berlin, 2010.  Google Scholar

[15]

S. Han and C. Lei, Global stability of equilibria of a diffusive SEIR epidemic model with nonlinear incidence, Appl. Math. Lett., 98 (2019), 114-120.  doi: 10.1016/j.aml.2019.05.045.  Google Scholar

[16]

A. HillD. G. RandM. A. Nowak and N. A. Christakis, Emotions as infectious diseases in a large social network: The SISa model, Proc. R. Soc. B, 277 (2010), 3827-3835.   Google Scholar

[17]

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

[18]

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.  Google Scholar

[19] M. J. Keeling and P. Rohani, Modeling Infectious Disease in Humans and Animals, Princeton University Press, Princeton, NJ, 2016.   Google Scholar
[20]

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

[21]

K.-Y. LamY. Lou and F. Lutscher, Evolution of dispersal in closed advective environment, J. Biol. Dyn., 9 (2015), 188-212.  doi: 10.1080/17513758.2014.969336.  Google Scholar

[22]

C. LeiF. Li and J. Liu, Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4499-4517.  doi: 10.3934/dcdsb.2018173.  Google Scholar

[23]

B. Li and Q. Bie, Long-time dynamics of an SIRS reaction-diffusion epidemic model, J. Math. Anal. Appl., 475 (2019), 1910-1926.  doi: 10.1016/j.jmaa.2019.03.062.  Google Scholar

[24]

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

[25]

H. 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.  Google Scholar

[26]

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

[27]

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.  Google Scholar

[28]

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.  Google Scholar

[29]

F. LutscherE. McCauley and M. A. Lewis, Spatial patterns and coexistence mechanisms in systems with unidirectional flow, Theor. Popul. Biol., 71 (2007), 267-277.   Google Scholar

[30]

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

[31]

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.  Google Scholar

[32]

M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3.  Google Scholar

[33]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conf. Ser. in Appl. Math., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.  Google Scholar

[34]

M. T. O'Regan and J. M. Drake, Theory of early warning signals of disease emergence and leading indicators of elimination, Theor. Econ., 6 (2013), 333-357.   Google Scholar

[35]

R. Peng, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model. I, J. Differential Equations, 247 (2009), 1096-1119.  doi: 10.1016/j.jde.2009.05.002.  Google Scholar

[36]

R. Peng and S. 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.  Google Scholar

[37]

R. Peng and Y. Wu, Global $L^\infty$-bounds and long-time behavior of a diffusive epidemic system in heterogeneous environment, SIAM J. Math. Anal., 53 (2021), 2776-2810.  doi: 10.1137/19M1276030.  Google Scholar

[38]

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.  Google Scholar

[39]

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.  Google Scholar

[40]

H. ShiZ. Duan and G. Chen, An SIS model with infective medium on complex networks, Physica A, 387 (2008), 2133-2144.   Google Scholar

[41]

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

[42]

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

[43]

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

[44]

M. YangG. Chen and X. Fu, A modified SIS model with an infective medium on complex networks and its global stability, Physica A, 390 (2011), 2408-2413.   Google Scholar

[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.   Google Scholar

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 Contin. Dyn. Syst., 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.  Google Scholar

[2]

F. Altarelli, A. Braunstein, L. Dall'Asta, J. R. Wakeling and R. Zecchina, Containing epidemic outbreaks by message-passing techniques, Phys. Rev. X, 4 (2014), 021024. Google Scholar

[3]

H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Jpn., 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.  Google Scholar

[4]

Y. Cai and W. Wang, Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 989-1013.  doi: 10.3934/dcdsb.2015.20.989.  Google Scholar

[5]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Ser. Math. Comput. Biol., John Wiley and Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[6]

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

[7]

R. Cui, H. Li, R. Peng and M. Zhou, Concentration behavior of endemic equilibrium for a reaction-diffusion-advection SIS epidemic model with mass action infection mechanism, preprint, (2019). Google Scholar

[8]

R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equations, 261 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.  Google Scholar

[9]

K. A. DahmenD. R. Nelson and N. M. Shnerb, Life and death near a windy oasis, J. Math. Biol., 41 (2000), 1-23.  doi: 10.1007/s002850000025.  Google Scholar

[10]

K. Deng and Y. Wu, Dynamics of an SIS epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929-946.  doi: 10.1017/S0308210515000864.  Google Scholar

[11]

Z. Du and R. Peng, A priori $L^\infty$ estimates for solutions of a class of reaction-diffusion systems, J. Math. Biol., 72 (2016), 1429-1439.  doi: 10.1007/s00285-015-0914-z.  Google Scholar

[12]

L. Dung, Dissipativity and global attractors for a class of quasilinear parabolic systems, Comm. Partial Differential Equations, 22 (1997), 413-433.  doi: 10.1080/03605309708821269.  Google Scholar

[13]

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

[14]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Springer, Berlin, 2010.  Google Scholar

[15]

S. Han and C. Lei, Global stability of equilibria of a diffusive SEIR epidemic model with nonlinear incidence, Appl. Math. Lett., 98 (2019), 114-120.  doi: 10.1016/j.aml.2019.05.045.  Google Scholar

[16]

A. HillD. G. RandM. A. Nowak and N. A. Christakis, Emotions as infectious diseases in a large social network: The SISa model, Proc. R. Soc. B, 277 (2010), 3827-3835.   Google Scholar

[17]

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

[18]

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.  Google Scholar

[19] M. J. Keeling and P. Rohani, Modeling Infectious Disease in Humans and Animals, Princeton University Press, Princeton, NJ, 2016.   Google Scholar
[20]

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

[21]

K.-Y. LamY. Lou and F. Lutscher, Evolution of dispersal in closed advective environment, J. Biol. Dyn., 9 (2015), 188-212.  doi: 10.1080/17513758.2014.969336.  Google Scholar

[22]

C. LeiF. Li and J. Liu, Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4499-4517.  doi: 10.3934/dcdsb.2018173.  Google Scholar

[23]

B. Li and Q. Bie, Long-time dynamics of an SIRS reaction-diffusion epidemic model, J. Math. Anal. Appl., 475 (2019), 1910-1926.  doi: 10.1016/j.jmaa.2019.03.062.  Google Scholar

[24]

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

[25]

H. 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.  Google Scholar

[26]

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

[27]

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.  Google Scholar

[28]

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.  Google Scholar

[29]

F. LutscherE. McCauley and M. A. Lewis, Spatial patterns and coexistence mechanisms in systems with unidirectional flow, Theor. Popul. Biol., 71 (2007), 267-277.   Google Scholar

[30]

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

[31]

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.  Google Scholar

[32]

M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3.  Google Scholar

[33]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conf. Ser. in Appl. Math., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.  Google Scholar

[34]

M. T. O'Regan and J. M. Drake, Theory of early warning signals of disease emergence and leading indicators of elimination, Theor. Econ., 6 (2013), 333-357.   Google Scholar

[35]

R. Peng, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model. I, J. Differential Equations, 247 (2009), 1096-1119.  doi: 10.1016/j.jde.2009.05.002.  Google Scholar

[36]

R. Peng and S. 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.  Google Scholar

[37]

R. Peng and Y. Wu, Global $L^\infty$-bounds and long-time behavior of a diffusive epidemic system in heterogeneous environment, SIAM J. Math. Anal., 53 (2021), 2776-2810.  doi: 10.1137/19M1276030.  Google Scholar

[38]

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.  Google Scholar

[39]

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.  Google Scholar

[40]

H. ShiZ. Duan and G. Chen, An SIS model with infective medium on complex networks, Physica A, 387 (2008), 2133-2144.   Google Scholar

[41]

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

[42]

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

[43]

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

[44]

M. YangG. Chen and X. Fu, A modified SIS model with an infective medium on complex networks and its global stability, Physica A, 390 (2011), 2408-2413.   Google Scholar

[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.   Google Scholar

[1]

Renhao Cui. Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2997-3022. doi: 10.3934/dcdsb.2020217

[2]

Chengxia Lei, Jie Xiong, Xinhui Zhou. Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 81-98. doi: 10.3934/dcdsb.2019173

[3]

Danhua Jiang, Zhi-Cheng Wang, Liang Zhang. A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4557-4578. doi: 10.3934/dcdsb.2018176

[4]

Yachun Tong, Inkyung Ahn, Zhigui Lin. Effect of diffusion in a spatial SIS epidemic model with spontaneous infection. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4045-4057. doi: 10.3934/dcdsb.2020273

[5]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Corrigendum: Dynamics of a reaction-diffusion-advection model for two competing species. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4989-4995. doi: 10.3934/dcds.2014.34.4989

[6]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Dynamics of a reaction-diffusion-advection model for two competing species. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 3841-3859. doi: 10.3934/dcds.2012.32.3841

[7]

Siyao Zhu, Jinliang Wang. Asymptotic profiles of steady states for a diffusive SIS epidemic model with spontaneous infection and a logistic source. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3323-3340. doi: 10.3934/cpaa.2020147

[8]

Siyao Zhu, Jinliang Wang. Analysis of a diffusive SIS epidemic model with spontaneous infection and a linear source in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1999-2019. doi: 10.3934/dcdsb.2020013

[9]

Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1

[10]

Wenzhang Huang, Maoan Han, Kaiyu Liu. Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. Mathematical Biosciences & Engineering, 2010, 7 (1) : 51-66. doi: 10.3934/mbe.2010.7.51

[11]

Chris Cosner. Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1701-1745. doi: 10.3934/dcds.2014.34.1701

[12]

Bo Duan, Zhengce Zhang. A two-species weak competition system of reaction-diffusion-advection with double free boundaries. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 801-829. doi: 10.3934/dcdsb.2018208

[13]

Mostafa Bendahmane, Kenneth H. Karlsen. Renormalized solutions of an anisotropic reaction-diffusion-advection system with $L^1$ data. Communications on Pure & Applied Analysis, 2006, 5 (4) : 733-762. doi: 10.3934/cpaa.2006.5.733

[14]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116

[15]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017

[16]

Bo Duan, Zhengce Zhang. A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021067

[17]

Linfeng Mei, Xiaoyan Zhang. On a nonlocal reaction-diffusion-advection system modeling phytoplankton growth with light and nutrients. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 221-243. doi: 10.3934/dcdsb.2012.17.221

[18]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

[19]

Jisheng Kou, Huangxin Chen, Xiuhua Wang, Shuyu Sun. A linear, decoupled and positivity-preserving numerical scheme for an epidemic model with advection and diffusion. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021094

[20]

Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039

2020 Impact Factor: 1.327

Article outline

[Back to Top]