In the recent paper [
Citation: |
[1] |
N. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equation, Commun. Partial Diff. Eqns., 4 (1979), 827-868.
doi: 10.1080/03605307908820113.![]() ![]() ![]() |
[2] |
L. Allen, B. Bolker, Y. Lou and A. Nevai, Asymptotic profiles of the steady states for an SIS epidemic disease patch model, SIAM J. Appl. Math., 67 (2007), 1283-1309.
doi: 10.1137/060672522.![]() ![]() ![]() |
[3] |
L. Allen, B. Bolker, Y. Lou and A. 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.![]() ![]() ![]() |
[4] |
F. Altarelli, A. Braunstein, L. Dall'Asta, J. Wakeling and R. Zecchina, Containing epidemic outbreaks by message-passing techniques, Physical Review X, 4 (2014), 021024.
doi: 10.1103/PhysRevX.4.021024.![]() ![]() |
[5] |
R. Anderson and R. May, Population biology of infectious diseases, Nature, 280 (1979), 361-367.
doi: 10.1007/978-3-642-68635-1.![]() ![]() |
[6] |
H. Brezis and W. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590.
doi: 10.2969/jmsj/02540565.![]() ![]() ![]() |
[7] |
K. Brown, P. Dunne and R. Gardner, A semilinear parabolic system arising in the theory of superconductivity, J. Differential Equations, 40 (1981), 232-252.
doi: 10.1016/0022-0396(81)90020-6.![]() ![]() ![]() |
[8] |
R. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, Ser. Math. Comput. Biology, Wiley, Chichester, UK, 2003.
doi: 10.1002/0470871296.![]() ![]() ![]() |
[9] |
J. Cui, X. Tao and H. Zhu, An SIS infection model incorporating media coverage, Rocky Mount. J. Math., 38 (2008), 1323-1334.
doi: 10.1216/RMJ-2008-38-5-1323.![]() ![]() ![]() |
[10] |
R. Cui, K.-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.![]() ![]() ![]() |
[11] |
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.![]() ![]() ![]() |
[12] |
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.![]() ![]() ![]() |
[13] |
W. Ding, W. Huang and S. Kansakar, Traveling wave solutions for a diffusive SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1291-1304.
doi: 10.3934/dcdsb.2013.18.1291.![]() ![]() ![]() |
[14] |
Y. Du, R. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956.
doi: 10.1016/j.jde.2008.11.007.![]() ![]() ![]() |
[15] |
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.![]() ![]() ![]() |
[16] |
D. Gao and S. Ruan, An SIS patch model with variable transmission coefficients, Math. Biosci., 232 (2011), 110-115.
doi: 10.1016/j.mbs.2011.05.001.![]() ![]() ![]() |
[17] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equation of Second Order, Springer, 2001.
![]() ![]() |
[18] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.
![]() ![]() |
[19] |
H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.
doi: 10.1137/S0036144500371907.![]() ![]() ![]() |
[20] |
H. Hethcote, Epidemiology models with variable population size, Mathematical understanding of infectious disease dynamics, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., World Sci. Publ., Hackensack, NJ, 16 (2009), 63–89.
doi: 10.1142/9789812834836_0002.![]() ![]() ![]() |
[21] |
A. Hill, D. Rand, M. Nowak and N. Christakis, Emotions as infectious diseases in a large social network: The SISa model, Proceedings of the Royal Society B, 277 (2010), 3827-3835.
doi: 10.1098/rspb.2010.1217.![]() ![]() |
[22] |
A. Hill, D. Rand, M. Nowak and N. Christakis, Infectious disease modeling of social contagion in networks, Plos Computational Biology, 6 (2010), e1000968, 15pp.
doi: 10.1371/journal.pcbi.1000968.![]() ![]() ![]() |
[23] |
W. Huang, M. 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.![]() ![]() ![]() |
[24] |
M. Keeling and P. Rohani, Modeling Infectious Disease in Humans and Animals,, Princeton University Press, 2008.
![]() ![]() |
[25] |
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), Art. 112, 28 pp.
doi: 10.1007/s00526-017-1207-8.![]() ![]() ![]() |
[26] |
C. Lei, F. 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.![]() ![]() ![]() |
[27] |
B. Li, H. Li and Y. Tong, Analysis on a diffusive SIS epidemic model with logistic source, Z. Angew. Math. Phys., 68 (2017), Art. 96, 25 pp.
doi: 10.1007/s00033-017-0845-1.![]() ![]() ![]() |
[28] |
H. Li, R. Peng and F.-B. Wang, Vary 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.![]() ![]() ![]() |
[29] |
H. Li, R. 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.![]() ![]() ![]() |
[30] |
H. Li, R. Peng and T. Xiang, Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion, Eur. J. Appl. Math..
doi: 10.1017/S0956792518000463.![]() ![]() |
[31] |
G. M. Lieberman, Bounds for the steady-state Sel'kov model for arbitrary $p$ in any number of dimensions, SIAM J. Math. Anal., 36 (2005), 1400-1406.
doi: 10.1137/S003614100343651X.![]() ![]() ![]() |
[32] |
C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations, 72 (1988), 1-27.
doi: 10.1016/0022-0396(88)90147-7.![]() ![]() ![]() |
[33] |
Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.
doi: 10.1006/jdeq.1996.0157.![]() ![]() ![]() |
[34] |
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.![]() ![]() ![]() |
[35] |
S. O'Regan and J. Drake, Theory of early warning signals of disease emergence and leading indicators of elimination, Theoretical Econogy, 6 (2013), 333-357.
![]() |
[36] |
R. Peng, Qualitative analysis on a diffusive and ratio-dependent predator-prey model, IMA J. Appl. Math., 78 (2013), 566-586.
doi: 10.1093/imamat/hxr066.![]() ![]() ![]() |
[37] |
R. Peng, J. Shi and M. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488.
doi: 10.1088/0951-7715/21/7/006.![]() ![]() ![]() |
[38] |
R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. Part Ⅰ, J. Differential Equations, 247 (2009), 1096-1119.
doi: 10.1016/j.jde.2009.05.002.![]() ![]() ![]() |
[39] |
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.![]() ![]() ![]() |
[40] |
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.![]() ![]() ![]() |
[41] |
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.![]() ![]() ![]() |
[42] |
H. Shi, Z. Duan and G. Chen, An SIS model with infective medium on complex networks, Physica A, 387 (2008), 2133-2144.
![]() |
[43] |
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.![]() ![]() ![]() |
[44] |
X. Wen, J. 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 (2018), 715-729.
doi: 10.1016/j.jmaa.2017.08.016.![]() ![]() ![]() |
[45] |
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.![]() ![]() ![]() |
[46] |
M. Yang, G. Chen and X. Fu, A modeling SIS model with an infective medium on complex networks and its global stability, Physica A, 390 (2011), 2408-2413.
![]() |
[47] |
X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Can. Appl. Math. Q., 3 (1995), 473-495.
![]() ![]() |