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Long-time dynamics and semi-wave of a delayed nonlocal epidemic model with free boundaries

Part of: Miscellaneous topics - Partial differential equations Parabolic equations and systems Functional-differential and differential-difference equations

Published online by Cambridge University Press:  05 October 2023

Qiaoling Chen ,
Sanyi Tang ,
Zhidong Teng  and
Feng Wang
Qiaoling Chen
Affiliation:
School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710119, PR China School of Science, Xi'an Polytechnic University, Xi'an 710048, PR China (qiaolingf@126.com)
Sanyi Tang
Affiliation:
School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710119, PR China (sytang@snnu.edu.cn)
Zhidong Teng
Affiliation:
College of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830017, PR China (zhidong@xju.edu.cn)
Feng Wang
Affiliation:
School of Mathematics and Statistics, Xidian University, Xi'an 710071, PR China (wangfeng@xidian.edu.cn)
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Abstract

This paper is concerned with a nonlocal reaction–diffusion system with double free boundaries and two time delays. The free boundary problem describes the evolution of faecally–orally transmitted diseases. We first show the well-posedness of global solution, and then establish the monotonicity and asymptotic property of basic reproduction number for the epidemic model without delays, which is defined by spectral radius of the next infection operator. By introducing the generalized principal eigenvalue defined in general domain, we obtain an upper bound of the limit value of basic reproduction number. We discuss the spreading and vanishing phenomena in terms of the basic production number. By employing the perturbed approximation method and monotone iteration method, we establish the existence, uniqueness and monotonicity of solution to semi-wave problem. When spreading occurs, we determine the asymptotic spreading speeds of free boundaries by constructing suitable upper and lower solutions from the semi-wave solutions. Moreover, spreading speeds for partially degenerate diffusion case are provided in a similar way.

Keywords

Eepidemic modelfree boundarytime delaybasic reproduction numbersemi-wave

MSC classification

Primary: 35K57: Reaction-diffusion equations
Secondary: 35K61: Nonlinear initial-boundary value problems for nonlinear parabolic equations 35R35: Free boundary problems 34K30: Equations in abstract spaces
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 47
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Ahn, I., Beak, S. and Lin, Z. G.. The spreading fronts of an infective environment in a man–environment–man epidemic model. Appl. Math. Model. 40 (2016), 7082–7101.CrossRefGoogle Scholar
Berestycki, H., Ducasse, R. and Rossi, L.. Generalized principal eigenvalues for heterogeneous road-field systems. Commun. Contemp. Math. 22 (2020), 1950013.CrossRefGoogle Scholar
Berestycki, H., Hamel, F. and Rossi, L.. Liouville-type results for semilinear elliptic equations in unbounded domains. Ann. Mater. Pura Appl. 186 (2007), 469–507.CrossRefGoogle Scholar
Berestycki, H. and Rossi, L.. On the principal eigenvalue of elliptic operators in $\mathbb {R}^N$ and applications. J. Eur. Math. Soc. 8 (2006), 195–215.CrossRefGoogle Scholar
Burlando, L.. Monotonicity of spectral radius for positive operators on ordered Banach spaces. Arch. Math. 56 (1991), 49–57.CrossRefGoogle Scholar
Cao, J. F., Du, Y. H., Li, F. and Li, W. T.. The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries. J. Funct. Anal. 277 (2019), 2772–2814.CrossRefGoogle Scholar
Cao, J. F., Li, W. T. and Yang, F. Y.. Dynamics of a nonlocal SIS epidemic model with free boundary. Discrete Contin. Dyn. Syst. Ser. B 22 (2017), 247–266.Google Scholar
Capasso, V.. Asymptotic stability for an integro-differential reaction–diffusion system. J. Math. Anal. Appl. 103 (1984), 575–588.CrossRefGoogle Scholar
Capasso, V. and Maddalena, L., A nonlinear diffusion system modelling the spread of orofaecal diseases. In: Nonlinear Phenomena in Mathematical Sciences (ed. Lakshmikantham, V.) (Academic Press, New York, 1981). pp 207–217.Google Scholar
Chen, Q. L., Li, F. Q., Teng, Z. D. and Wang, F.. Global dynamics and asymptotic spreading speeds for a partially degenerate epidemic model with time delay and free boundaries. J. Dynam. Differ. Equ. 34 (2022), 1209–1236.CrossRefGoogle Scholar
Chen, Q. L., Tang, S. Y., Teng, Z. D. and Wang, F.. Spreading dynamics of a diffusive epidemic model with free boundaries and two delays. Eur. J. Appl. Math. (2023), 1–37. https://doi.org/10.1017/S0956792523000220.Google Scholar
Du, Y. H., Fang, J. and Sun, N. K.. A delay induced nonlocal free boundary problem. Math. Ann 386 (2023), 2061–2106.CrossRefGoogle Scholar
Du, Y. H., Guo, Z. M. and Peng, R.. A diffusive logistic model with a free boundary in time-periodic environment. J. Funct. Anal. 265 (2013), 2089–2142.CrossRefGoogle Scholar
Du, Y. H., Li, F. and Zhou, M. L.. Semi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundaries. J. Math. Pures Appl. 154 (2021), 30–66.CrossRefGoogle Scholar
Du, Y. H. and Lin, Z. G.. Spreading–vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 42 (2010), 377–405.CrossRefGoogle Scholar
Du, Y. H. and Lou, B. D.. Spreading and vanishing in nonlinear diffusion problems with free boundaries. J. Eur. Math. Soc. 17 (2015), 2673–2724.CrossRefGoogle Scholar
Du, Y. H. and Ni, W. J.. Spreading speed for some cooperative systems with nonlocal diffusion and free boundaries, part 1: semi-wave and a threshold condition. J. Differ. Equ. 308 (2022), 369–420.CrossRefGoogle Scholar
Huang, H. M. and Wang, M. X.. A nonlocal SIS epidemic problem with double free boundaries. Z. Angew. Math. Phys. 70 (2019), 109.CrossRefGoogle Scholar
Li, W. T., Zhao, M. and Wang, J.. Spreading fronts in a partially degenerate integro-differential reaction–diffusion system. Z. Angew. Math. Phys. 68 (2017), 109.CrossRefGoogle Scholar
Liang, X., Zhang, L. and Zhao, X. Q.. Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for Lyme disease). J. Dynam. Differ. Equ. 31 (2019), 1247–1278.CrossRefGoogle Scholar
Liang, X. and Zhao, X. Q.. Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Comm. Pure Appl. Math. 60 (2007), 1–40.CrossRefGoogle Scholar
Lin, Z. G. and Zhu, H. P.. Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary. J. Math. Biol. 75 (2017), 1381–1409.CrossRefGoogle ScholarPubMed
Smith, H. L.. Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems (American Mathematical Society, Providence, RI, 1995).Google Scholar
Sun, N. K. and Fang, J.. Propagation dynamics of Fisher-KPP equation with time delay and free boundaries. Calc. Var. Partial Differ. Equ. 58 (2019), 148.CrossRefGoogle Scholar
Tang, Y. B., Dai, B. X. and Li, Z. Z.. Dynamics of a Lotka-Volterra weak competition model with time delays and free boundaries. Z. Angew. Math. Phys. 73 (2022), 143.CrossRefGoogle Scholar
Tarboush, A. K., Lin, Z. G. and Zhang, M. Y.. Spreading and vanishing in a West Nile virus model with expanding fronts. Sci. China Math. 60 (2017), 841–860.CrossRefGoogle Scholar
Thieme, H. R. and Zhao, X. Q.. Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction–diffusion models. J. Differ. Equ. 195 (2003), 430–470.CrossRefGoogle Scholar
Wang, M. X.. A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment. J. Funct. Anal. 270 (2016), 483–508.CrossRefGoogle Scholar
Wang, M. X. and Zhang, Y.. The time-periodic diffusive competition models with a free boundary and sign-changing growth rates. Z. Angew. Math. Phys. 67 (2016), 132.CrossRefGoogle Scholar
Wang, R. and Du, Y. H.. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete Contin. Dyn. Syst. Ser. B 26 (2021), 2201–2238.CrossRefGoogle Scholar
Wang, R. and Du, Y. H.. Long-time dynamics of a nonlocal epidemic model with free boundaries: Spreading–vanishing dichotomy. J. Differ. Equ. 327 (2022), 322–381.CrossRefGoogle Scholar
Wang, R. and Du, Y. H.. Long-time dynamics of an epidemic model with nonlocal diffusion and free boundaries: Spreading speed. Discrete Contin. Dyn. Syst. 43 (2023), 121–161.CrossRefGoogle Scholar
Wang, W. D. and Zhao, X. Q.. A nonlocal and time-delayed reaction–diffusion model of dengue transmission. SIAM J. Appl. Math. 71 (2011), 147–168.CrossRefGoogle Scholar
Wang, W. D. and Zhao, X. Q.. Basic reproduction numbers for reaction–diffusion epidemic models. SIAM J. Appl. Dyn. Syst. 11 (2012), 1652–1673.CrossRefGoogle Scholar
Wang, Z. G., Nie, H. and Du, Y. H.. Spreading speed for a West Nile virus model with free Boundary. J. Math. Biol. 79 (2019), 433–466.CrossRefGoogle Scholar
Wu, J. H. and Zou, X. F.. Traveling wave fronts of reaction–diffusion systems with delay. J. Dynam. Differ. Equ. 13 (2001), 651–687.CrossRefGoogle Scholar
Wu, S. L. and Hsu, C. H.. Existence of entire solutions for delayed monostable epidemic models. Trans. Am. Math. Soc. 368 (2016), 6033–6062.CrossRefGoogle Scholar
Wu, S. L., Hsu, C. H. and Xiao, Y. Y.. Global attractivity, spreading speeds and traveling waves of delayed nonlocal reaction–diffusion systems. J. Differ. Equ. 258 (2015), 1058–1105.CrossRefGoogle Scholar
Xu, D. S. and Zhao, X. Q.. Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discrete Contin. Dyn. Syst. B 5 (2005), 1043–1056.Google Scholar
Zhao, M., Li, W. T. and Ni, W. J.. Spreading speed of a degenerate and cooperative epidemic model with free boundaries. Discrete Contin. Dyn. Syst. B 25 (2020), 981–999.Google Scholar
Zhao, M., Zhang, Y., Li, W. T. and Du, Y. H.. The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries. J. Differ. Equ. 269 (2020), 3347–3386.CrossRefGoogle Scholar
Zhao, X. Q. and Jing, Z. J.. Global asymptotic behavior in some cooperative systems of functional differential equations. Canad Appl. Math. Quart. 4 (1996), 421–444.Google Scholar