September  2020, 19(9): 4599-4620. doi: 10.3934/cpaa.2020208

The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and 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

* Corresponding author

Received  January 2019 Revised  April 2020 Published  June 2020

Fund Project: M. Zhao was supported by a scholarship from the China Scholarship Council (201806180022), W.T. Li was supported by NSF of China (11731005, 11671180) and Y. Du was supported by the Australian Research Council (DP190103757)

In this paper, we examine an epidemic model which is described by a system of two equations with nonlocal diffusion on the equation for the infectious agents $ u $, while no dispersal is assumed in the other equation for the infective humans $ v $. The underlying spatial region $ [g(t), h(t)] $ (i.e., the infected region) is assumed to change with time, governed by a set of free boundary conditions. In the recent work [33], such a model was considered where the growth rate of $ u $ due to the contribution from $ v $ is given by $ cv $ for some positive constant $ c $. Here this term is replaced by a nonlocal reaction function of $ v $ in the form $ c\int_{g(t)}^{h(t)}K(x-y)v(t,y)dy $ with a suitable kernel function $ K $, to represent the nonlocal effect of $ v $ on the growth of $ u $. We first show that this problem has a unique solution for all $ t>0 $, and then we show that its longtime behaviour is determined by a spreading-vanishing dichotomy, which indicates that the long-time dynamics of the model is not vastly altered by this change of the term $ cv $. We also obtain sharp criteria for spreading and vanishing, which reveal that changes do occur in these criteria from the earlier model in [33] where the term $ cv $ was used; in particular, small nonlocal dispersal rate of $ u $ alone no longer guarantees successful spreading of the disease as in the model of [33].

Citation: Meng Zhao, Wantong Li, Yihong Du. The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4599-4620. doi: 10.3934/cpaa.2020208
References:
[1]

I. AhnS. 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.  Google Scholar

[2]

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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

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J. F. CaoY. DuF. Li and W. T. Li, The dynamics of a nonlocal diffusion model with free boundary, J. Funct. Anal., 277 (2019), 2772-2814.  doi: 10.1016/j.jfa.2019.02.013.  Google Scholar

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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), 173-184.  doi: 10.1007/BF00275212.  Google Scholar

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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. Henri Poincare Anal. Non Lineaire, 36 (2019), 1539-1573.  doi: 10.1016/j.anihpc.2019.01.005.  Google Scholar

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Y. Du, F. Li and M. Zhou, Semi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundaries, preprint, arXiv: 1909.03711. Google Scholar

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Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 45 (2013), 1995-1996.  doi: 10.1137/090771089.  Google Scholar

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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.  Google Scholar

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Y. Du, M. Wang and M. Zhao, Two species nonlocal diffusion systems with free boundaries, preprint, arXiv: 1907.04542. Google Scholar

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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

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Y. DuL. Wei and L. Zhou, Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary, J. Dyn. Differ. Equ., 30 (2018), 1389-1426.  doi: 10.1007/s10884-017-9614-2.  Google Scholar

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J. GeK. I. KimZ. Lin and H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differ. Equ., 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035.  Google Scholar

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H. GuB. 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.  Google Scholar

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J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dyn. Differ. Equ., 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0.  Google Scholar

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Y. Kaneko and H. Matsuzawa, Spreading and vanishing in a free boundary problem for nonlinear diffusion equations with a given forced moving boundary, J. Differ. Equ., 265 (2018), 1000-1043.  doi: 10.1016/j.jde.2018.03.026.  Google Scholar

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Y. Kawai and Y. Yamada, Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity, J. Differ. Equ., 261 (2016), 538-572.  doi: 10.1016/j.jde.2016.03.017.  Google Scholar

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K. I. 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

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F. LiX. Liang and W. Shen, Diffusive KPP equations with free boundaries in time almost periodic environments: Ⅱ. Spreading speeds and semi-wave solutions, J. Differ. Equ., 261 (2016), 2403-2445.  doi: 10.1016/j.jde.2016.04.035.  Google Scholar

[23]

L. Li, W. Sheng and M. Wang, Systems with nonlocal vs. local diffusions and free boundaries, J. Math. Anal. Appl., 483 (2020), Art. 123646. doi: 10.1016/j.jmaa.2019.123646.  Google Scholar

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W. T. LiY. J. Sun and Z. C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.  doi: 10.1016/j.nonrwa.2009.07.005.  Google Scholar

[25]

W. T. Li, M. Zhao and J. Wang, Spreading fronts in a partially degenerate integro-differential reaction-diffusion system, Z. Angew. Math. Phys., 68 (2017), Art. 109, 28 pp. doi: 10.1007/s00033-017-0858-9.  Google Scholar

[26]

X. Liang, Semi-wave solutions of KPP-Fisher equations with free boundaries in spatially almost periodic media, J. Math. Pures Appl., 127 (2019), 299-308.  doi: 10.1016/j.matpur.2018.09.007.  Google Scholar

[27]

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

[28]

M. Wang, On some free boundary problems of the prey-predator model, J. Differ. Equ., 256 (2014), 3365-3394.  doi: 10.1016/j.jde.2014.02.013.  Google Scholar

[29]

M. Wang and J. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dyn. Differ. Equ., 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.  Google Scholar

[30]

M. Wang and J. Zhao, A free boundary problem for the predator-prey model with double free boundaries, J. Dyn. Differ. Equ., 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.  Google Scholar

[31]

M. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differ. Equ., 264 (2018), 3527-3558.  doi: 10.1016/j.jde.2017.11.027.  Google Scholar

[32]

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

[33]

M. ZhaoY. ZhangW. T. Li and Y. Du, The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries, J. Differ. Equ., 269 (2020), 3347-3386.  doi: 10.1016/j.jde.2020.02.029.  Google Scholar

show all references

References:
[1]

I. AhnS. 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.  Google Scholar

[2]

F. Andreu-Vaillo, J. M. Maz$\acute{o}$n, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2010. Google Scholar

[3]

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

[4]

J. F. CaoY. DuF. Li and W. T. Li, The dynamics of a nonlocal diffusion model with free boundary, J. Funct. Anal., 277 (2019), 2772-2814.  doi: 10.1016/j.jfa.2019.02.013.  Google Scholar

[5]

V. Capasso, Asymptotic stability for an integro-differential reaction-diffusion system, J. Math. Anal. Appl., 103 (1984), 575-588.  doi: 10.1016/0022-247X(84)90147-1.  Google Scholar

[6]

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), 173-184.  doi: 10.1007/BF00275212.  Google Scholar

[7]

J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differ. Equ., 249 (2010), 2921-2953.  doi: 10.1016/j.jde.2010.07.003.  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. Henri Poincare Anal. Non Lineaire, 36 (2019), 1539-1573.  doi: 10.1016/j.anihpc.2019.01.005.  Google Scholar

[9]

Y. Du, F. Li and M. Zhou, Semi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundaries, preprint, arXiv: 1909.03711. Google Scholar

[10]

Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 45 (2013), 1995-1996.  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, Discrete 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. Du, M. Wang and M. Zhao, Two species nonlocal diffusion systems with free boundaries, preprint, arXiv: 1907.04542. Google Scholar

[14]

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

[15]

Y. DuL. Wei and L. Zhou, Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary, J. Dyn. Differ. Equ., 30 (2018), 1389-1426.  doi: 10.1007/s10884-017-9614-2.  Google Scholar

[16]

J. GeK. I. KimZ. Lin and H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differ. Equ., 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035.  Google Scholar

[17]

H. GuB. 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.  Google Scholar

[18]

J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dyn. Differ. Equ., 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0.  Google Scholar

[19]

Y. Kaneko and H. Matsuzawa, Spreading and vanishing in a free boundary problem for nonlinear diffusion equations with a given forced moving boundary, J. Differ. Equ., 265 (2018), 1000-1043.  doi: 10.1016/j.jde.2018.03.026.  Google Scholar

[20]

Y. Kawai and Y. Yamada, Multiple spreading phenomena for a free boundary problem of a reaction-diffusion equation with a certain class of bistable nonlinearity, J. Differ. Equ., 261 (2016), 538-572.  doi: 10.1016/j.jde.2016.03.017.  Google Scholar

[21]

K. I. 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

[22]

F. LiX. Liang and W. Shen, Diffusive KPP equations with free boundaries in time almost periodic environments: Ⅱ. Spreading speeds and semi-wave solutions, J. Differ. Equ., 261 (2016), 2403-2445.  doi: 10.1016/j.jde.2016.04.035.  Google Scholar

[23]

L. Li, W. Sheng and M. Wang, Systems with nonlocal vs. local diffusions and free boundaries, J. Math. Anal. Appl., 483 (2020), Art. 123646. doi: 10.1016/j.jmaa.2019.123646.  Google Scholar

[24]

W. T. LiY. J. Sun and Z. C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.  doi: 10.1016/j.nonrwa.2009.07.005.  Google Scholar

[25]

W. T. Li, M. Zhao and J. Wang, Spreading fronts in a partially degenerate integro-differential reaction-diffusion system, Z. Angew. Math. Phys., 68 (2017), Art. 109, 28 pp. doi: 10.1007/s00033-017-0858-9.  Google Scholar

[26]

X. Liang, Semi-wave solutions of KPP-Fisher equations with free boundaries in spatially almost periodic media, J. Math. Pures Appl., 127 (2019), 299-308.  doi: 10.1016/j.matpur.2018.09.007.  Google Scholar

[27]

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

[28]

M. Wang, On some free boundary problems of the prey-predator model, J. Differ. Equ., 256 (2014), 3365-3394.  doi: 10.1016/j.jde.2014.02.013.  Google Scholar

[29]

M. Wang and J. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dyn. Differ. Equ., 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.  Google Scholar

[30]

M. Wang and J. Zhao, A free boundary problem for the predator-prey model with double free boundaries, J. Dyn. Differ. Equ., 29 (2017), 957-979.  doi: 10.1007/s10884-015-9503-5.  Google Scholar

[31]

M. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differ. Equ., 264 (2018), 3527-3558.  doi: 10.1016/j.jde.2017.11.027.  Google Scholar

[32]

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

[33]

M. ZhaoY. ZhangW. T. Li and Y. Du, The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries, J. Differ. Equ., 269 (2020), 3347-3386.  doi: 10.1016/j.jde.2020.02.029.  Google Scholar

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