April  2021, 26(4): 2201-2238. doi: 10.3934/dcdsb.2020360

Long-time dynamics of a diffusive epidemic model with free boundaries

School of Science and Technology, University of New England, Armidale, NSW 2351, Australia

* Corresponding author: Yihong Du

Dedicated to Professor Sze-Bi Hsu on the occasion of his 70th birthday

Received  August 2020 Revised  October 2020 Published  April 2021 Early access  December 2020

Fund Project: This work was supported by the Australian Research Council and a PhD scholarship of the University of New England

In this paper, we determine the long-time dynamical behaviour of a reaction-diffusion system with free boundaries, which models the spreading of an epidemic whose moving front is represented by the free boundaries. The system reduces to the epidemic model of Capasso and Maddalena [5] when the boundary is fixed, and it reduces to the model of Ahn et al. [1] if diffusion of the infective host population is ignored. We prove a spreading-vanishing dichotomy and determine exactly when each of the alternatives occurs. If the reproduction number $ R_0 $ obtained from the corresponding ODE model is no larger than 1, then the epidemic modelled here will vanish, while if $ R_0>1 $, then the epidemic may vanish or spread depending on its initial size, determined by the dichotomy criteria. Moreover, when spreading happens, we show that the expanding front of the epidemic has an asymptotic spreading speed, which is determined by an associated semi-wave problem.

Citation: Rong Wang, Yihong Du. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2201-2238. doi: 10.3934/dcdsb.2020360
References:
[1]

I. AhnS. Baek 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.  Google Scholar

[2]

G. BuntingY. 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.  Google Scholar

[3]

J. CaoW. T. Li and F. Yang, Dynamics of a nonlocal SIS epidemic model with free boundary, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 247-266.  doi: 10.3934/dcdsb.2017013.  Google Scholar

[4]

V. Capasso and S. L. Paveri-fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Revue d'epidemiologie et de sante publique, 27 (1979), 121-132.   Google Scholar

[5]

V. Capasso and L. Maddalena, Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases, J. Math. Biol., 13 (1981/82), 173-184.  doi: 10.1007/BF00275212.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. DingY. Du and X. Liang, Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 2: Spreading speed, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1539-1573.  doi: 10.1016/j.anihpc.2019.01.005.  Google Scholar

[9]

Y. DuZ. M. 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.  Google Scholar

[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.  Google Scholar

[11]

Y. Du and Z. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discret. Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132.  doi: 10.3934/dcdsb.2014.19.3105.  Google Scholar

[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.  Google Scholar

[13]

Y. DuM. 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.  Google Scholar

[14]

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), 36 pp. doi: 10.1007/s00526-018-1339-5.  Google Scholar

[15]

J. Fang and X-Q. Zhao, Monotone wavefronts for partially degenerate reaction-diffusion systems, J. Dynam. Differential Equations, 21 (2009), 663-680.  doi: 10.1007/s10884-009-9152-7.  Google Scholar

[16]

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.  Google Scholar

[17]

K. Hadeler, Stefan problem, traveling fronts, and epidemic spread, Discrete Contin. Dyn. Syst. B., 21 (2016), 417-436.  doi: 10.3934/dcdsb.2016.21.417.  Google Scholar

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H. Huang and M. Wang, The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. B., 20 (2015), 2039-2050.  doi: 10.3934/dcdsb.2015.20.2039.  Google Scholar

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Y. KanekoH. Matsuzawa and Y. Yamada, Asymptotic profiles of solutions and propagating terrace for a free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity, SIAM J. Math. Anal., 52 (2020), 65-103.  doi: 10.1137/18M1209970.  Google Scholar

[20]

K. KimZ. 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.  Google Scholar

[21]

C. X. LeiZ. Lin and Q. Y. 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.  Google Scholar

[22]

B. LiH. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosc., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[23]

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.  Google Scholar

[24]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1996.  Google Scholar

[25]

W. Merz and P. Rybka, A free boundary problem describing reaction-diffusion problems in chemical vapor infiltration of pyrolytic carbon, J. Math. Anal. Appl., 292 (2004), 571-588.  doi: 10.1016/j.jmaa.2003.12.025.  Google Scholar

[26]

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.  Google Scholar

[27]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[28]

L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971.  Google Scholar

[29]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995.  Google Scholar

[30]

J.-B. Wang and X.-Q. Zhao, Uniqueness and global stability of forced waves in a shifting environment, Proc. Amer. Math. Soc., 147 (2019), 1467-1481.  doi: 10.1090/proc/14235.  Google Scholar

[31]

M. Wang, Existence and uniqueness of solutions of free boundary problems in heterogeneous environments, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 415-421.  doi: 10.3934/dcdsb.2018179.  Google Scholar

[32]

Z. WangH. 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.  Google Scholar

[33]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Diffrential Equations, 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.  Google Scholar

[34]

Y. Tao and M. Chen, An elliptic-hyperbolic free boundary problem modelling cancer therapy, Nonlinearity, 19 (2006), 419-440.  doi: 10.1088/0951-7715/19/2/010.  Google Scholar

[35]

M. ZhaoW. Li and W. J. Ni, Spreading speed of a degenerate and cooperative epidemic model with free boundaries, Discrete Contin. Dyn. Syst. B, 25 (2020), 981-999.  doi: 10.3934/dcdsb.2019199.  Google Scholar

[36]

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.  Google Scholar

show all references

References:
[1]

I. AhnS. Baek 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.  Google Scholar

[2]

G. BuntingY. 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.  Google Scholar

[3]

J. CaoW. T. Li and F. Yang, Dynamics of a nonlocal SIS epidemic model with free boundary, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 247-266.  doi: 10.3934/dcdsb.2017013.  Google Scholar

[4]

V. Capasso and S. L. Paveri-fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Revue d'epidemiologie et de sante publique, 27 (1979), 121-132.   Google Scholar

[5]

V. Capasso and L. Maddalena, Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases, J. Math. Biol., 13 (1981/82), 173-184.  doi: 10.1007/BF00275212.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. DingY. Du and X. Liang, Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 2: Spreading speed, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1539-1573.  doi: 10.1016/j.anihpc.2019.01.005.  Google Scholar

[9]

Y. DuZ. M. 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.  Google Scholar

[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.  Google Scholar

[11]

Y. Du and Z. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discret. Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132.  doi: 10.3934/dcdsb.2014.19.3105.  Google Scholar

[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.  Google Scholar

[13]

Y. DuM. 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.  Google Scholar

[14]

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), 36 pp. doi: 10.1007/s00526-018-1339-5.  Google Scholar

[15]

J. Fang and X-Q. Zhao, Monotone wavefronts for partially degenerate reaction-diffusion systems, J. Dynam. Differential Equations, 21 (2009), 663-680.  doi: 10.1007/s10884-009-9152-7.  Google Scholar

[16]

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.  Google Scholar

[17]

K. Hadeler, Stefan problem, traveling fronts, and epidemic spread, Discrete Contin. Dyn. Syst. B., 21 (2016), 417-436.  doi: 10.3934/dcdsb.2016.21.417.  Google Scholar

[18]

H. Huang and M. Wang, The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. B., 20 (2015), 2039-2050.  doi: 10.3934/dcdsb.2015.20.2039.  Google Scholar

[19]

Y. KanekoH. Matsuzawa and Y. Yamada, Asymptotic profiles of solutions and propagating terrace for a free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity, SIAM J. Math. Anal., 52 (2020), 65-103.  doi: 10.1137/18M1209970.  Google Scholar

[20]

K. KimZ. 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.  Google Scholar

[21]

C. X. LeiZ. Lin and Q. Y. 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.  Google Scholar

[22]

B. LiH. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosc., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[23]

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.  Google Scholar

[24]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1996.  Google Scholar

[25]

W. Merz and P. Rybka, A free boundary problem describing reaction-diffusion problems in chemical vapor infiltration of pyrolytic carbon, J. Math. Anal. Appl., 292 (2004), 571-588.  doi: 10.1016/j.jmaa.2003.12.025.  Google Scholar

[26]

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.  Google Scholar

[27]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.  Google Scholar

[28]

L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971.  Google Scholar

[29]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995.  Google Scholar

[30]

J.-B. Wang and X.-Q. Zhao, Uniqueness and global stability of forced waves in a shifting environment, Proc. Amer. Math. Soc., 147 (2019), 1467-1481.  doi: 10.1090/proc/14235.  Google Scholar

[31]

M. Wang, Existence and uniqueness of solutions of free boundary problems in heterogeneous environments, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 415-421.  doi: 10.3934/dcdsb.2018179.  Google Scholar

[32]

Z. WangH. 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.  Google Scholar

[33]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Diffrential Equations, 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.  Google Scholar

[34]

Y. Tao and M. Chen, An elliptic-hyperbolic free boundary problem modelling cancer therapy, Nonlinearity, 19 (2006), 419-440.  doi: 10.1088/0951-7715/19/2/010.  Google Scholar

[35]

M. ZhaoW. Li and W. J. Ni, Spreading speed of a degenerate and cooperative epidemic model with free boundaries, Discrete Contin. Dyn. Syst. B, 25 (2020), 981-999.  doi: 10.3934/dcdsb.2019199.  Google Scholar

[36]

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.  Google Scholar

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