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Global existence for a two-phase flow model with cross-diffusion
Spreading speed of a degenerate and cooperative epidemic model with free boundaries
| 1. | School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, China |
| 2. | School of Science and Technology, University of New England, Armidale, NSW, 2351, Australia |
This paper deals with the spreading speed of the disease described by a partially degenerate and cooperative epidemic model with free boundaries. We show that the spreading speed is determined by a semi-wave problem. To find such a semi-wave solution, we prove the existence of a monotone solution to a reduced ODE by an upper and lower solution approach. And then we establish the uniqueness of the semi-wave solution via the sliding method. It is demonstrated that the precise asymptotic spreading speed is less than the minimal speed of traveling waves.
References:
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I. Ahn, S. Beak and Z. Lin,
The spreading fronts of an infective environment in a man-environment-man epidemic model, Appl. Math. Model., 40 (2016), 7082-7101.
doi: 10.1016/j.apm.2016.02.038. |
| [2] |
G. Bunting, Y. Du and K. Krakowski,
Spreading speed revisited: analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.
doi: 10.3934/nhm.2012.7.583. |
| [3] |
V. Capasso and L. Maddalena,
Convergence to equilibrium states for a reaction-diffusion system modeling the spatial spread of a class of bacterial and viral diseases, J. Math. Biol., 13 (1981/82), 173-184.
doi: 10.1007/BF00275212. |
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V. Capasso and S. L. Paveri-Fontana,
A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. d'Epidemiol. Sante Publique, 27 (1979), 32-121.
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E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. |
| [6] |
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. |
| [7] |
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: 10.1137/110822608. |
| [8] |
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. |
| [9] |
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. |
| [10] |
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. |
| [11] |
Y. Du, M. Wang and M. Zhou,
Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pures Appl., 107 (2017), 253-287.
doi: 10.1016/j.matpur.2016.06.005. |
| [12] |
Y. Du and C. H. Wu, Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries, Calc. Var. Partial Differential Equations, 57 (2018), Art. 52, 36 pp.
doi: 10.1007/s00526-018-1339-5. |
| [13] |
J. Ge, K. I. 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. |
| [14] |
H. Gu, B. Lou and M. Zhou,
Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768.
doi: 10.1016/j.jfa.2015.07.002. |
| [15] |
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. |
| [16] |
K. I. 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. |
| [17] |
Z. Lin,
A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.
doi: 10.1088/0951-7715/20/8/004. |
| [18] |
Z. Lin and H. Zhu,
Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 1381-1409.
doi: 10.1007/s00285-017-1124-7. |
| [19] |
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. |
| [20] |
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. |
| [21] |
M. Wang and J. Zhao,
Free boundary problems for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672.
doi: 10.1007/s10884-014-9363-4. |
| [22] |
M. Wang and J. Zhao,
A free boundary problem for the predator-prey model with double free boundaries, J. Dynam. Differential Equations, 29 (2017), 957-979.
doi: 10.1007/s10884-015-9503-5. |
| [23] |
M. Wang and Y. Zhang,
Dynamics for a diffusive prey-predator model with different free boundaries, J. Differential Equations, 264 (2018), 3527-3558.
doi: 10.1016/j.jde.2017.11.027. |
| [24] |
Z. Wang, H. Nie and Y. Du,
Asymptotic spreading speed for the weak competition system with a free boundary, Discrete Contin. Dyn. Syst., 39 (2019), 5223-5262.
doi: 10.3934/dcds.2019213. |
| [25] |
Z. Wang, H. Nie and Y. Du,
Spreading speed for a West Nile virus model with free boundary, J. Math. Biol., 79 (2019), 433-466.
doi: 10.1007/s00285-019-01363-2. |
| [26] |
J. Wu and X. Zou,
Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.
doi: 10.1023/A:1016690424892. |
| [27] |
X. Q. Zhao and W. Wang,
Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1117-1128.
doi: 10.3934/dcdsb.2004.4.1117. |
show all references
References:
| [1] |
I. Ahn, S. Beak and Z. Lin,
The spreading fronts of an infective environment in a man-environment-man epidemic model, Appl. Math. Model., 40 (2016), 7082-7101.
doi: 10.1016/j.apm.2016.02.038. |
| [2] |
G. Bunting, Y. Du and K. Krakowski,
Spreading speed revisited: analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.
doi: 10.3934/nhm.2012.7.583. |
| [3] |
V. Capasso and L. Maddalena,
Convergence to equilibrium states for a reaction-diffusion system modeling the spatial spread of a class of bacterial and viral diseases, J. Math. Biol., 13 (1981/82), 173-184.
doi: 10.1007/BF00275212. |
| [4] |
V. Capasso and S. L. Paveri-Fontana,
A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. d'Epidemiol. Sante Publique, 27 (1979), 32-121.
|
| [5] |
E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. |
| [6] |
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. |
| [7] |
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: 10.1137/110822608. |
| [8] |
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. |
| [9] |
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. |
| [10] |
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. |
| [11] |
Y. Du, M. Wang and M. Zhou,
Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pures Appl., 107 (2017), 253-287.
doi: 10.1016/j.matpur.2016.06.005. |
| [12] |
Y. Du and C. H. Wu, Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries, Calc. Var. Partial Differential Equations, 57 (2018), Art. 52, 36 pp.
doi: 10.1007/s00526-018-1339-5. |
| [13] |
J. Ge, K. I. 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. |
| [14] |
H. Gu, B. Lou and M. Zhou,
Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768.
doi: 10.1016/j.jfa.2015.07.002. |
| [15] |
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. |
| [16] |
K. I. 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. |
| [17] |
Z. Lin,
A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.
doi: 10.1088/0951-7715/20/8/004. |
| [18] |
Z. Lin and H. Zhu,
Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 1381-1409.
doi: 10.1007/s00285-017-1124-7. |
| [19] |
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. |
| [20] |
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. |
| [21] |
M. Wang and J. Zhao,
Free boundary problems for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672.
doi: 10.1007/s10884-014-9363-4. |
| [22] |
M. Wang and J. Zhao,
A free boundary problem for the predator-prey model with double free boundaries, J. Dynam. Differential Equations, 29 (2017), 957-979.
doi: 10.1007/s10884-015-9503-5. |
| [23] |
M. Wang and Y. Zhang,
Dynamics for a diffusive prey-predator model with different free boundaries, J. Differential Equations, 264 (2018), 3527-3558.
doi: 10.1016/j.jde.2017.11.027. |
| [24] |
Z. Wang, H. Nie and Y. Du,
Asymptotic spreading speed for the weak competition system with a free boundary, Discrete Contin. Dyn. Syst., 39 (2019), 5223-5262.
doi: 10.3934/dcds.2019213. |
| [25] |
Z. Wang, H. Nie and Y. Du,
Spreading speed for a West Nile virus model with free boundary, J. Math. Biol., 79 (2019), 433-466.
doi: 10.1007/s00285-019-01363-2. |
| [26] |
J. Wu and X. Zou,
Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.
doi: 10.1023/A:1016690424892. |
| [27] |
X. Q. Zhao and W. Wang,
Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1117-1128.
doi: 10.3934/dcdsb.2004.4.1117. |
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