Citation: |
[1] |
S. Ai and R. Albashaireh, Traveling waves in spatial SIRS models, J. Dyn. Diff. Equat., 26 (2014), 143-164.doi: 10.1007/s10884-014-9348-3. |
[2] |
M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states, Appl. Math. Lett., 25 (2012), 2095-2099.doi: 10.1016/j.aml.2012.05.006. |
[3] |
M. S. Bartlett, Deterministic and stochastic models for recurrent epidemics, inProc. 3rd Berkeley Symp. Mathematical Statistics and Probability, Vol. 4. No. 81. Berkeley: University of California Press, 1956. |
[4] |
C. Briggs and H. Godfray, The dynamics of insect-pathogen interactions in stage-structured populations, Amer. Nat., 145 (1995), 855-887. |
[5] |
K. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Camb. Phil. Soc., 81 (1977), 431-433.doi: 10.1017/S0305004100053494. |
[6] |
V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42 (1978), 41-61.doi: 10.1016/0025-5564(78)90006-8. |
[7] |
Z. Chen and Z. Zhao, Harnack principle for weakly coupled elliptic systems, J. Differential Equations, 139 (1997), 261-282.doi: 10.1006/jdeq.1997.3300. |
[8] |
K. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math., 9 (1979), 31-42.doi: 10.1216/RMJ-1979-9-1-31. |
[9] |
O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130.doi: 10.1007/BF02450783. |
[10] |
W. Ding, W. Huang and S. Kansakar, Traveling wave solutions for a diffusive SIS epidemic model, Discrete Contin. Dyn. Syst. B, 18 (2013), 1291-1304.doi: 10.3934/dcdsb.2013.18.1291. |
[11] |
A. Ducrot, M. Langlais and P. Magal, Qualitative analysis and travelling wave solutions for the SI model with vertical transmission, Commum. Pure Appl. Anal., 11 (2012), 97-113.doi: 10.3934/cpaa.2012.11.97. |
[12] |
A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. R. Soc. Edin. Ser. A Math., 139 (2009), 459-482.doi: 10.1017/S0308210507000455. |
[13] |
A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 24 (2011), 2891-2911.doi: 10.1088/0951-7715/24/10/012. |
[14] |
A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multigroup age-structure epidemic models, Arch. Ration. Mech. Anal., 195 (2010), 311-331.doi: 10.1007/s00205-008-0203-8. |
[15] |
L. Evans, Partial Differential Equaitons, Second edition, Graduate studies in mathematics, American Mathematical Society, 2010. |
[16] |
J. Fang and X. Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Differential Equations, 248 (2010), 2199-2226.doi: 10.1016/j.jde.2010.01.009. |
[17] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Vol. 224. Springer, 2001. |
[18] |
H. Hethcote, The mathematics of infectious diseases, SIAM. Rev., 42 (2000), 599-653.doi: 10.1137/S0036144500371907. |
[19] |
W. Hirsch, H. Hamisch and J. P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1948), 221-246.doi: 10.1002/cpa.3160380607. |
[20] |
Y. Hosono and B. Ilyas, Travelling waves for a simple diffusive epidemic model, Math. Model Meth. Appl. Sci., 5 (1995), 935-966.doi: 10.1142/S0218202595000504. |
[21] |
G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207.doi: 10.1007/s11538-009-9487-6. |
[22] |
W. T. Li, G. Lin, C. Ma and F. Y. Yang, Traveling waves of a nonlocal delayed SIR epidemic model without outbreak threshold, Discrete Contin. Dyn. Syst. B, 19 (2014), 467-484.doi: 10.3934/dcdsb.2014.19.467. |
[23] |
G. Lin, Invasion traveling wave solution of a predator-prey system, Nonlinear Anal., 96 (2014), 47-58.doi: 10.1016/j.na.2013.10.024. |
[24] |
G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to Lotka-Volterra competition-diffusion models with distributed delays, J. Dyn. Diff. Equat., 26 (2014), 583-605.doi: 10.1007/s10884-014-9355-4. |
[25] |
C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. Real Word Appl., 11 (2010), 3106-3109.doi: 10.1016/j.nonrwa.2009.11.005. |
[26] |
J. D. Murray, Mathematical Biology. II. Spatial Models and Biomedical Applications, 3rd edn, Springer-Verlag, New York, 2003. |
[27] |
J. Roughgarden, Theory of Population Genetics and Evolutionary Ecology: An Introduction, New York, Macmillan, 1979. |
[28] |
S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in Mathematics for Life Science and Medicine (eds. Y. Takeuchi, K. Sato and Y. Iwasa), Springer-Verlag, New York, (2007), pp. 97-122. |
[29] |
S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts, in Spatial Ecology, Chapman & Hall/CRC, Boca Raton, FL, (2009), 293-316. |
[30] |
H. Smith and X. Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.doi: 10.1137/S0036141098346785. |
[31] |
H. R. Thieme and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470.doi: 10.1016/S0022-0396(03)00175-X. |
[32] |
H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Nonlinear Sci., 21 (2011), 747-783.doi: 10.1007/s00332-011-9099-9. |
[33] |
X. Wang, H. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst., 32 (2012), 3303-3324.doi: 10.3934/dcds.2012.32.3303. |
[34] |
Z. C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237-261.doi: 10.1098/rspa.2009.0377. |
[35] |
Z. C. Wang, J. Wu and R. Liu, Traveling waves of avian influenza spread, Proc. Amer. Math. Soc., 140 (2012), 3931-3946.doi: 10.1090/S0002-9939-2012-11246-8. |
[36] |
M. Wu and P. Weng, Stability of a stage-structured diffusive SIR model with delays (Chinese), J. South China Normal Univ. Natur. Sci. Ed., 45 (2013), 20-23. |
[37] |
S. L. Wu and P. Weng, Entire solutions for a multi-type SIS nonlocal epidemic model in R or Z, J. Math. Anal. Appl., 394 (2012), 603-615.doi: 10.1016/j.jmaa.2012.05.009. |
[38] |
J. Yang, S. Liang and Y. Zhang, Travelling waves of a delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion, PLoS One, 6 (2011), e21128.doi: 10.1371/journal.pone.0021128. |
[39] |
Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction-diffusion Equations, Science Press, Beijing, 2011. |
[40] |
E. Zeilder, Nonlinear Functional Analysis and Its Applications: I, Fixed-piont Theorems, New York, Springer-Verlag, 1986. |