A West Nile virus nonlocal model with free boundaries and seasonal succession. (English) Zbl 1506.35283
Summary: The paper deals with a West Nile virus (WNv) model, in which the nonlocal diffusion characterizes the long-range movement of birds and mosquitoes, the free boundaries describe their spreading fronts, and the seasonal succession accounts for the effect of the warm and cold seasons. The well-posedness of the mathematical model is established, and its long-term dynamical behaviours, which depend upon the generalized eigenvalues of the corresponding linearized differential operator, are investigated. For both spatially independent and nonlocal WNv models with seasonal successions, the generalized eigenvalues are studied and applied to determine whether the spreading or vanishing occurs. Our results extend those for the case with nonlocal diffusion but no free boundary and those for the case with free boundary but local diffusion, respectively. The generalized eigenvalues reveal that there exists positive correlation between the duration of the warm season and the risk of infection. Moreover, the initial infection length, the initial infection scale and the spreading ability to new areas all play important roles for the long time behaviors of the time dependent solutions.
MSC:
| 35R35 | Free boundary problems for PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K57 | Reaction-diffusion equations |
| 92D30 | Epidemiology |
References:
| [1] | Bates P (2006) On some nonlocal evolution equations arising in materials science. Am Math Soc 48:13-52 · Zbl 1101.35073 |
| [2] | Bao, X.; Shen, W., Criteria for the existence of principal eigenvalues of time periodic cooperative linear systems with nonlocal dispersal, Proc Am Math Soc, 145, 2881-2894 (2017) · Zbl 1401.35304 · doi:10.1090/proc/13602 |
| [3] | Beck, C.; Jimenezclavero, M.; Leblond, A., Flaviviruses in Europe: complex circulation patterns and their consequences for the diagnosis and control of West Nile disease, Int J Environ Res Public Health, 10, 6049-6083 (2013) · doi:10.3390/ijerph10116049 |
| [4] | Bodnar, M.; Velazquez, J., An integro-differential equation arising as a limit of individual cell-based models, J Differ Equ, 222, 341-380 (2006) · Zbl 1089.45002 · doi:10.1016/j.jde.2005.07.025 |
| [5] | Bowman, C.; Gumel, A.; Driessche, P., A mathematical model for assessing control strategies against West Nile virus, Bull Math Biol, 67, 1107-1133 (2005) · Zbl 1334.92392 · doi:10.1016/j.bulm.2005.01.002 |
| [6] | Cao, J.; Du, Y.; Li, F.; Li, W., The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J Funct Anal, 277, 2772-2814 (2019) · Zbl 1418.35229 · doi:10.1016/j.jfa.2019.02.013 |
| [7] | Diekmann, O.; Heesterbeek, J.; Metz, J., On the definition and the computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations, J Math Biol, 28, 365-382 (1990) · Zbl 0726.92018 · doi:10.1007/BF00178324 |
| [8] | Du, Y.; Lin, Z., Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J Math Anal, 42, 377-405 (2010) · Zbl 1219.35373 · doi:10.1137/090771089 |
| [9] | Du, Y.; Ni, W., Analysis of a West Nile virus model with nonlocal diffusion and free boundaries, Nonlinearity, 33, 4407-4448 (2020) · Zbl 1439.35220 · doi:10.1088/1361-6544/ab8bb2 |
| [10] | Du, Y.; Li, F.; Zhou, M., Semi-wave and spreading speed of the nonlocal Fisher-KPP equation with free boundaries, J Math Pure Appl, 154, 30-66 (2021) · Zbl 1473.35665 · doi:10.1016/j.matpur.2021.08.008 |
| [11] | Du, Y.; Wang, M.; Zhao, M., Two species nonlocal diffusion systems with free boundaries, Discrete Contin Dyn Syst A, 42, 1127-1162 (2022) · Zbl 1483.35347 · doi:10.3934/dcds.2021149 |
| [12] | Fournier, N.; Laurencot, P., Well-posedness of Smoluchowski’s coagulation equation for a class of homogeneous kernels, J Funct Anal, 233, 351-379 (2006) · Zbl 1106.45003 · doi:10.1016/j.jfa.2005.07.013 |
| [13] | Gilboa, G.; Osher, S., Nonlocal linear image regularization and supervised segmentation, Multiscale Model Simul, 6, 595-630 (2007) · Zbl 1140.68517 · doi:10.1137/060669358 |
| [14] | Hale, J., Ordinary differential equations (1980), New York: Wiley, New York · Zbl 0433.34003 |
| [15] | Hu, S.; Tessier, A., Seasonal succession and the strength of intra- and interspecific competition in a Daphnia assemblage, Ecology, 76, 2278-2294 (1995) · doi:10.2307/1941702 |
| [16] | Hsu, S.; Zhao, X., A Lotka-Volterra competition model with seasonal succession, J Math Biol, 64, 109-130 (2012) · Zbl 1284.34054 · doi:10.1007/s00285-011-0408-6 |
| [17] | Klausmeier, C., Successional state dynamics: a novel approach to modeling nonequilibrium foodweb dynamics, J Theoret Biol, 262, 584-595 (2010) · Zbl 1403.92330 · doi:10.1016/j.jtbi.2009.10.018 |
| [18] | Lewis, M.; Renclawowicz, J.; Driessche, P., Traveling waves and spread rates for a West Nile virus model, Bull Math Biol, 68, 3-23 (2006) · Zbl 1334.92414 · doi:10.1007/s11538-005-9018-z |
| [19] | Lin, Z.; Zhu, H., Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J Math Biol, 75, 1381-1409 (2017) · Zbl 1373.35321 · doi:10.1007/s00285-017-1124-7 |
| [20] | Liu, S.; Huang, H.; Wang, M., Asymptotic spreading of a diffusive competition model with different free boundaries, J Differ Equ, 266, 4769-4799 (2019) · Zbl 1412.35166 · doi:10.1016/j.jde.2018.10.009 |
| [21] | Malthus, T., An essay on the principle of population, 1798 (1998), Pauls Church-Yard: J. Johnson in St, Pauls Church-Yard |
| [22] | Murray, J., Mathematical biology (1998), Berlin: Springer, Berlin · Zbl 0682.92001 |
| [23] | Nadin, G., The principal eigenvalue of a space-time periodic parabolic operator, Annali di Matematica, 188, 269-295 (2009) · Zbl 1193.35113 · doi:10.1007/s10231-008-0075-4 |
| [24] | Natan R, Klein E, Robledo-Arnuncio J, Revilla E (2012) 2012 Dispersal kernels: review dispersal ecology and evolution oxford. Oxford University Press, pp 187-210 |
| [25] | Peng, R.; Zhao, X., The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin Dyn Syst, 33, 2007-2031 (2013) · Zbl 1273.35327 · doi:10.3934/dcds.2013.33.2007 |
| [26] | Steiner, C.; Schwaderer, A.; Huber, V.; Klausmeier, C.; Litch, E., Periodically forced food-chain dynamics: model predictions and experimental validation, Ecology, 90, 3099-3107 (2009) · doi:10.1890/08-2377.1 |
| [27] | van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math Biosci, 180, 29-48 (2002) · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6 |
| [28] | Wan, H.; Zhu, H., The backward bifurcation in compartmental models for West Nile virus, Math Biosci, 272, 20-28 (2010) · Zbl 1194.92067 · doi:10.1016/j.mbs.2010.05.006 |
| [29] | Wang, M., Existence and uniqueness of solutions of free boundary problems in heterogeneous environments, Discrete Contin Dyn Syst Ser B, 33, 415-421 (2019) · Zbl 1404.35230 |
| [30] | Wang J, Wang M (2020a) Free boundary problems with nonlocal and local diffusions I: global solution. J Math Anal Appl 490:123974 · Zbl 1439.35581 |
| [31] | Wang J, Wang M (2020b) Free boundary problems with nonlocal and local diffusions II: spreading-vanishing and long-time behavior. Discrete Contin Dyn Syst Ser B 25:4721-4736 · Zbl 1465.35421 |
| [32] | Wang, Z.; Nie, H.; Du, Y., Spreading speed for a West Nile virus model with free boundary, J Math Biol, 79, 433-466 (2019) · Zbl 1421.35391 · doi:10.1007/s00285-019-01363-2 |
| [33] | Wang, M.; Zhang, Q.; Zhao, X., Dynamics for a diffusive competition model with seasonal succession and different free boundaries, J Differ Equ, 285, 536-582 (2021) · Zbl 1461.35126 · doi:10.1016/j.jde.2021.03.006 |
| [34] | Wonham, M.; Beck, T.; Lewis, M., An epidemiology model for West Nile virus: invansion analysis and control applications, Proc R Soc Lond, 271, 501-507 (2004) · doi:10.1098/rspb.2003.2608 |
| [35] | Zhao M, Li W, Du Y (2020a) The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries. Commun Pure Appl Anal 19:4599-4620 · Zbl 1460.35400 |
| [36] | Zhao M, Zhang Y, Li W (2020b) The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries. J Differ Equ 269:3347-3386 · Zbl 1442.35486 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
