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Convergence rate of synchronization of systems with additive noise
Dynamics of a nonlocal SIS epidemic model with free boundary
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China |
This paper is concerned with the spreading or vanishing of a epidemic disease which is characterized by a diffusion SIS model with nonlocal incidence rate and double free boundaries. We get the full information about the sufficient conditions that ensure the disease spreading or vanishing, which exhibits a detailed description of the communicable mechanism of the disease. Our results imply that the nonlocal interaction may enhance the spread of the disease.
References:
[1] |
S. Ai and R. Albashaireh,
Traveling waves in spatial SIRS models, J. Dynam. Differential Equations, 26 (2014), 143-164.
doi: 10.1007/s10884-014-9348-3. |
[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] |
D. Aronson,
The asymptotic speed of propagation of a simple epidemic, Research Notes in Math., 14 (1977), 1-23, Pitman, London.
|
[4] |
R. Cantrell and C. Cosner,
Spatial Ecology via Reaction-Diffusion Equations Wiley Series in Mathematical and Computational Biology, 2003. |
[5] |
J. Cao, W. T. Li, J. Wang and F. Yang, A free boundary problem of a diffusive SIRS model
with nonlinear incidence, submitted, 2015. |
[6] |
X. Chen and A. Friedman,
A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800.
doi: 10.1137/S0036141099351693. |
[7] |
X. Chen and A. Friedman,
A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986.
doi: 10.1137/S0036141002418388. |
[8] |
Y. Du and Z. Guo,
Spreading-Vanishing dichotomy in a diffusive logistic model with a free boundary Ⅱ, J. Differential Equations, 250 (2011), 4336-4366.
doi: 10.1016/j.jde.2011.02.011. |
[9] |
Y. Du, Z. Guo and R. Peng,
A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.
doi: 10.1016/j.jfa.2013.07.016. |
[10] |
Y. Du and Z. Lin,
Spreading-Vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.
doi: 10.1137/090771089. |
[11] |
Y. Du and Z. Lin,
The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132.
doi: 10.3934/dcdsb.2014.19.3105. |
[12] |
Y. Du and B. Lou,
Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.
doi: 10.4171/JEMS/568. |
[13] |
Y. Du, H. Matsuzawa and M. Zhou,
Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396.
doi: 10.1137/130908063. |
[14] |
Y. Du and X. Liang,
Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 279-305.
doi: 10.1016/j.anihpc.2013.11.004. |
[15] |
A. Ducrot and P. Magal,
Travelling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 459-482.
doi: 10.1017/S0308210507000455. |
[16] |
T. Faria, W. Huang and J. Wu,
Travelling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A, 462 (2006), 229-261.
doi: 10.1098/rspa.2005.1554. |
[17] |
J. Ge, K. Kim, Z. 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. |
[18] |
J. S. Guo and C. H. Wu,
On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895.
doi: 10.1007/s10884-012-9267-0. |
[19] |
J. S. Guo and C. H. Wu,
Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27.
doi: 10.1088/0951-7715/28/1/1. |
[20] |
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. |
[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] |
H. Huang and M. Wang,
The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2039-2050.
doi: 10.3934/dcdsb.2015.20.2039. |
[23] |
D. Kendall,
Some problems in theory of dams, J. Roy. Statist. Soc. Ser. B., 19 (1957), 207-212.
|
[24] |
D. Kendall,
Mathematical Models of the Spread of Infection Mathematics and Computer Science in Biology and Medicine. H. M. Stationary Off, London, 1965. |
[25] |
A. Inkyung and Z. Lin, The spreading fronts of an infective environment in a manenvironment-man epidemic model, arXiv: 1408.6326. |
[26] |
K. Kim, Z. Lin and Q. Zhang,
An SIR epidemic model with free boundary, Nonlinear Anal. Real World Appl., 14 (2013), 1992-2001.
doi: 10.1016/j.nonrwa.2013.02.003. |
[27] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva,
Linear and Quasilinear Equations of Parabolic Type Academic Press, New York, London, 1968. |
[28] |
C. Lei, Z. Lin and Q. Zhang,
The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differential Equations, 257 (2014), 145-166.
doi: 10.1016/j.jde.2014.03.015. |
[29] |
Z. Lin,
A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.
doi: 10.1088/0951-7715/20/8/004. |
[30] |
Z. Lin, Y. Zhao and P. Zhou,
The infected frontier in an SEIR epidemic model with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2355-2376.
doi: 10.3934/dcdsb.2013.18.2355. |
[31] |
Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, Tutor. Math. Biosci. Ⅳ: Evol. Ecol. , Springer, 1922 (2008), 171{205. |
[32] |
D. Mollison,
Possible velocities for a simple epidemic, Adv. Appl. Probab., 4 (1972), 233-257.
doi: 10.2307/1425997. |
[33] |
D. Mottoni, P. Orlandi and A. Tesei,
Asymptotic behavior for a system describing epidemics with migration and spatial spread of infection, Nonlinear Anal., 3 (1979), 663-675.
doi: 10.1016/0362-546X(79)90095-6. |
[34] |
R. Peng and X. Q. Zhao,
The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031.
doi: 10.3934/dcds.2013.33.2007. |
[35] |
L. I. Rubinstein,
The Stefan Problem American Mathematical Society, Providence, RI, 1971. |
[36] |
J. Wang and L. Zhang,
Invasion by an inferior or superior competitor: A diffusive competition model with a free boundary in a heterogeneous environment, J. Math. Anal. Appl., 423 (2015), 377-398.
doi: 10.1016/j.jmaa.2014.09.055. |
[37] |
M. Wang,
On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394.
doi: 10.1016/j.jde.2014.02.013. |
[38] |
M. Wang,
The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266.
doi: 10.1016/j.jde.2014.10.022. |
[39] |
M. Wang and J. Zhao,
Free boundary problem for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672.
doi: 10.1007/s10884-014-9363-4. |
[40] |
M. Wang and J. Zhao,
A free boundary problem for a predator-prey model with double free boundaries, J. Dynam. Differential Equations, (2015), 1-23.
doi: 10.1007/s10884-015-9503-5. |
[41] |
M. Wang,
A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483-508.
doi: 10.1016/j.jfa.2015.10.014. |
[42] |
Z. C. Wang and J. Wu,
Traveling waves of a diffusive Kermack-McKendrick epidemic model with nonlocal delayed transmission, Proc. R. Soc. Lond. Ser. A, 466 (2010), 237-261.
doi: 10.1098/rspa.2009.0377. |
[43] |
C. H. Wu,
Spreading speed and traveling waves for a two-species weak competition system with free boundary, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2441-2455.
doi: 10.3934/dcdsb.2013.18.2441. |
[44] |
J. Yang and B. Lou,
Traveling wave solutions of competitive models with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 817-826.
doi: 10.3934/dcdsb.2014.19.817. |
[45] |
J. Zhao and M. Wang,
A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal. Real World Appl., 16 (2014), 250-263.
doi: 10.1016/j.nonrwa.2013.10.003. |
[46] |
P. Zhou and D. Xiao,
The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954.
doi: 10.1016/j.jde.2013.12.008. |
show all references
References:
[1] |
S. Ai and R. Albashaireh,
Traveling waves in spatial SIRS models, J. Dynam. Differential Equations, 26 (2014), 143-164.
doi: 10.1007/s10884-014-9348-3. |
[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] |
D. Aronson,
The asymptotic speed of propagation of a simple epidemic, Research Notes in Math., 14 (1977), 1-23, Pitman, London.
|
[4] |
R. Cantrell and C. Cosner,
Spatial Ecology via Reaction-Diffusion Equations Wiley Series in Mathematical and Computational Biology, 2003. |
[5] |
J. Cao, W. T. Li, J. Wang and F. Yang, A free boundary problem of a diffusive SIRS model
with nonlinear incidence, submitted, 2015. |
[6] |
X. Chen and A. Friedman,
A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800.
doi: 10.1137/S0036141099351693. |
[7] |
X. Chen and A. Friedman,
A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986.
doi: 10.1137/S0036141002418388. |
[8] |
Y. Du and Z. Guo,
Spreading-Vanishing dichotomy in a diffusive logistic model with a free boundary Ⅱ, J. Differential Equations, 250 (2011), 4336-4366.
doi: 10.1016/j.jde.2011.02.011. |
[9] |
Y. Du, Z. Guo and R. Peng,
A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.
doi: 10.1016/j.jfa.2013.07.016. |
[10] |
Y. Du and Z. Lin,
Spreading-Vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.
doi: 10.1137/090771089. |
[11] |
Y. Du and Z. Lin,
The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132.
doi: 10.3934/dcdsb.2014.19.3105. |
[12] |
Y. Du and B. Lou,
Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.
doi: 10.4171/JEMS/568. |
[13] |
Y. Du, H. Matsuzawa and M. Zhou,
Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396.
doi: 10.1137/130908063. |
[14] |
Y. Du and X. Liang,
Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 279-305.
doi: 10.1016/j.anihpc.2013.11.004. |
[15] |
A. Ducrot and P. Magal,
Travelling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 459-482.
doi: 10.1017/S0308210507000455. |
[16] |
T. Faria, W. Huang and J. Wu,
Travelling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A, 462 (2006), 229-261.
doi: 10.1098/rspa.2005.1554. |
[17] |
J. Ge, K. Kim, Z. 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. |
[18] |
J. S. Guo and C. H. Wu,
On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895.
doi: 10.1007/s10884-012-9267-0. |
[19] |
J. S. Guo and C. H. Wu,
Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27.
doi: 10.1088/0951-7715/28/1/1. |
[20] |
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. |
[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] |
H. Huang and M. Wang,
The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2039-2050.
doi: 10.3934/dcdsb.2015.20.2039. |
[23] |
D. Kendall,
Some problems in theory of dams, J. Roy. Statist. Soc. Ser. B., 19 (1957), 207-212.
|
[24] |
D. Kendall,
Mathematical Models of the Spread of Infection Mathematics and Computer Science in Biology and Medicine. H. M. Stationary Off, London, 1965. |
[25] |
A. Inkyung and Z. Lin, The spreading fronts of an infective environment in a manenvironment-man epidemic model, arXiv: 1408.6326. |
[26] |
K. Kim, Z. Lin and Q. Zhang,
An SIR epidemic model with free boundary, Nonlinear Anal. Real World Appl., 14 (2013), 1992-2001.
doi: 10.1016/j.nonrwa.2013.02.003. |
[27] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva,
Linear and Quasilinear Equations of Parabolic Type Academic Press, New York, London, 1968. |
[28] |
C. Lei, Z. Lin and Q. Zhang,
The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differential Equations, 257 (2014), 145-166.
doi: 10.1016/j.jde.2014.03.015. |
[29] |
Z. Lin,
A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.
doi: 10.1088/0951-7715/20/8/004. |
[30] |
Z. Lin, Y. Zhao and P. Zhou,
The infected frontier in an SEIR epidemic model with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2355-2376.
doi: 10.3934/dcdsb.2013.18.2355. |
[31] |
Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, Tutor. Math. Biosci. Ⅳ: Evol. Ecol. , Springer, 1922 (2008), 171{205. |
[32] |
D. Mollison,
Possible velocities for a simple epidemic, Adv. Appl. Probab., 4 (1972), 233-257.
doi: 10.2307/1425997. |
[33] |
D. Mottoni, P. Orlandi and A. Tesei,
Asymptotic behavior for a system describing epidemics with migration and spatial spread of infection, Nonlinear Anal., 3 (1979), 663-675.
doi: 10.1016/0362-546X(79)90095-6. |
[34] |
R. Peng and X. Q. Zhao,
The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031.
doi: 10.3934/dcds.2013.33.2007. |
[35] |
L. I. Rubinstein,
The Stefan Problem American Mathematical Society, Providence, RI, 1971. |
[36] |
J. Wang and L. Zhang,
Invasion by an inferior or superior competitor: A diffusive competition model with a free boundary in a heterogeneous environment, J. Math. Anal. Appl., 423 (2015), 377-398.
doi: 10.1016/j.jmaa.2014.09.055. |
[37] |
M. Wang,
On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394.
doi: 10.1016/j.jde.2014.02.013. |
[38] |
M. Wang,
The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266.
doi: 10.1016/j.jde.2014.10.022. |
[39] |
M. Wang and J. Zhao,
Free boundary problem for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672.
doi: 10.1007/s10884-014-9363-4. |
[40] |
M. Wang and J. Zhao,
A free boundary problem for a predator-prey model with double free boundaries, J. Dynam. Differential Equations, (2015), 1-23.
doi: 10.1007/s10884-015-9503-5. |
[41] |
M. Wang,
A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483-508.
doi: 10.1016/j.jfa.2015.10.014. |
[42] |
Z. C. Wang and J. Wu,
Traveling waves of a diffusive Kermack-McKendrick epidemic model with nonlocal delayed transmission, Proc. R. Soc. Lond. Ser. A, 466 (2010), 237-261.
doi: 10.1098/rspa.2009.0377. |
[43] |
C. H. Wu,
Spreading speed and traveling waves for a two-species weak competition system with free boundary, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2441-2455.
doi: 10.3934/dcdsb.2013.18.2441. |
[44] |
J. Yang and B. Lou,
Traveling wave solutions of competitive models with free boundaries, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 817-826.
doi: 10.3934/dcdsb.2014.19.817. |
[45] |
J. Zhao and M. Wang,
A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal. Real World Appl., 16 (2014), 250-263.
doi: 10.1016/j.nonrwa.2013.10.003. |
[46] |
P. Zhou and D. Xiao,
The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954.
doi: 10.1016/j.jde.2013.12.008. |
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