Traveling waves in nonlocal dispersal SIR epidemic model with nonlinear incidence and distributed latent delay. (English) Zbl 1486.92298
Summary: This paper studies the traveling waves in a nonlocal dispersal SIR epidemic model with nonlinear incidence and distributed latent delay. It is found that the traveling waves connecting the disease-free equilibrium with endemic equilibrium are determined by the basic reproduction number \(\mathcal{R}_0\) and the minimal wave speed \(c^\ast\). When \(\mathcal{R}_0>1\) and \(c>c^\ast\), the existence of traveling waves is established by using the upper-lower solutions, auxiliary system, constructing the solution map, and then the fixed point theorem, limiting argument, diagonal extraction method, and Lyapunov functions. When \(\mathcal{R}_0>1\) and \(0< c< c^*\ast\), the nonexistence result is also obtained by using the reduction to absurdity and the theory of asymptotic spreading.
MSC:
| 92D30 | Epidemiology |
| 35K57 | Reaction-diffusion equations |
| 35C07 | Traveling wave solutions |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
Keywords:
nonlocal dispersal epidemic model; nonlinear incidence; distributed latent delay; traveling waves; upper-lower solutions; limiting argumentReferences:
| [1] | Saccomandi, G., The spatial diffusion of diseases, Math. Comput. Model., 25, 12, 83-95 (1997) · Zbl 0933.92030 · doi:10.1016/S0895-7177(97)00096-4 |
| [2] | Dong, F.; Li, W.; Wang, J., Propagation dynamics in a three-species competition model with nonlocal anisotropic dispersal, Nonlinear Anal., Real World Appl., 48, 232-266 (2019) · Zbl 1425.92153 · doi:10.1016/j.nonrwa.2019.01.012 |
| [3] | Tomás, C.; Fatini, M.; Pettersson, R., A stochastic SIRI epidemic model with relapse and media coverage, Discrete Contin. Dyn. Syst., 23, 8, 3483-3501 (2018) · Zbl 1404.92172 |
| [4] | Cai, Y.; Wang, W., Fish-hook bifurcation branch in a spatial heterogeneous epidemic model with cross-diffusion, Nonlinear Anal., Real World Appl., 30, 99-125 (2016) · Zbl 1339.35329 · doi:10.1016/j.nonrwa.2015.12.002 |
| [5] | Thieme, H.; Zhao, X., Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differ. Equ., 195, 2, 430-470 (2003) · Zbl 1045.45009 · doi:10.1016/S0022-0396(03)00175-X |
| [6] | Crooks, E.; Dancer, E.; Hilhorst, D.; Mimura, M.; Ninomiya, H., Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions, Nonlinear Anal., Real World Appl., 5, 4, 645-665 (2004) · Zbl 1079.35008 · doi:10.1016/j.nonrwa.2004.01.004 |
| [7] | Postnikov, E.; Sokolov, I., Continuum description of a contact infection spread in a SIR model, Math. Biosci., 208, 1, 205-215 (2007) · Zbl 1116.92056 · doi:10.1016/j.mbs.2006.10.004 |
| [8] | Duan, X.; Li, X.; Martcheva, M., Qualitative analysis on a diffusive age-structured heroin transmission model, Nonlinear Anal., Real World Appl., 54 (2020) · Zbl 1437.92118 · doi:10.1016/j.nonrwa.2020.103105 |
| [9] | Duan, X.; Yin, J.; Li, X.; Martcheva, M., Competitive exclusion in a multi-strain virus model with spatial diffusion and age of infection, J. Math. Anal. Appl., 459, 2, 717-742 (2018) · Zbl 1381.92076 · doi:10.1016/j.jmaa.2017.10.074 |
| [10] | Qiao, S.; Yang, F.; Li, W., Traveling waves of a nonlocal dispersal SEIR model with standard incidence, Nonlinear Anal., Real World Appl., 49, 196-216 (2019) · Zbl 1429.92135 · doi:10.1016/j.nonrwa.2019.03.003 |
| [11] | Ma, S., Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differ. Equ., 237, 259-277 (2007) · Zbl 1114.34061 · doi:10.1016/j.jde.2007.03.014 |
| [12] | Li, W.; Lin, G.; Ruan, S., Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19, 1253-1273 (2006) · Zbl 1103.35049 · doi:10.1088/0951-7715/19/6/003 |
| [13] | Faria, T.; Wu, H., Traveling waves for delayed reaction-diffusion equations with global response, Proc. Math. Phys. Eng. Sci., 462, 2065, 229-261 (2006) · Zbl 1149.35368 |
| [14] | Murray, J.; Stanley, E.; Brown, D., On the spatial spread of rabies among foxes, Proc. R. Soc. Lond., 229, 1255, 111-150 (1986) |
| [15] | Hosono, Y.; Ilyas, B., Existence of traveling waves with any positive speed for a diffusive epidemic model, Nonlinear World, 1, 3, 277-290 (1994) · Zbl 0809.34061 |
| [16] | Chen, W.; Tuerxun, N.; Teng, Z., The global dynamics in a wild-type and drug-resistant HIV infection model with saturated incidence, Adv. Differ. Equ., 2020, 1 (2020) · Zbl 1487.92037 · doi:10.1186/s13662-020-2497-2 |
| [17] | Tian, B.; Yuan, R., Traveling waves for a diffusive SEIR epidemic model with non-local reaction and with standard incidences, Nonlinear Anal., Real World Appl., 37, 162-181 (2017) · Zbl 1373.35074 · doi:10.1016/j.nonrwa.2017.02.007 |
| [18] | Yang, F.; Li, W.; Wang, Z., Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model, Discrete Contin. Dyn. Syst., 18, 1969-1993 (2013) · Zbl 1277.35107 |
| [19] | Chen, H.; Yuan, R., Traveling waves of a nonlocal dispersal Kermack-McKendrick epidemic model with delayed transmission, J. Evol. Equ., 17, 979-1002 (2017) · Zbl 1381.92090 · doi:10.1007/s00028-016-0362-2 |
| [20] | Zhang, G.; Li, W.; Wang, Z., Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differ. Equ., 252, 5096-5124 (2012) · Zbl 1250.35061 · doi:10.1016/j.jde.2012.01.014 |
| [21] | Capasso, V.; Serio, G., A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42, 1-2, 43-61 (1978) · Zbl 0398.92026 · doi:10.1016/0025-5564(78)90006-8 |
| [22] | Brown, G.; Hasibuan, R., Conidial discharge and transmission efficiency of Neozygites floridana, an entomopathogenic fungus infecting two-spotted spider mites under laboratory conditions, J. Invertebr. Pathol., 65, 1, 10-16 (1995) · doi:10.1006/jipa.1995.1002 |
| [23] | Zhou, J.; Xu, J.; Wei, J.; Xu, H., Existence and non-existence of traveling wave solutions for a nonlocal dispersal SIR epidemic model with nonlinear incidence rate, Nonlinear Anal., Real World Appl., 41, 204-231 (2018) · Zbl 1383.35232 · doi:10.1016/j.nonrwa.2017.10.016 |
| [24] | Wu, J.; Zou, X., Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Differ. Equ., 13, 651-687 (2001) · Zbl 0996.34053 · doi:10.1023/A:1016690424892 |
| [25] | Zhu, C.; Li, W.; Yang, F., Traveling waves in a nonlocal dispersal SIRH model with relapse, Comput. Math. Appl., 73, 1707-1723 (2017) · Zbl 1370.92172 · doi:10.1016/j.camwa.2017.02.014 |
| [26] | Li, Y.; Li, W.; Yang, F., Traveling waves for a nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput., 247, 723-740 (2014) · Zbl 1338.92131 · doi:10.1016/j.amc.2014.09.072 |
| [27] | Naresh, R.; Tripathi, A.; Tchuenchec, J.; Sharma, D., Stability analysis of a time delayed SIR epidemic model with nonlinear incidence rate, Comput. Math. Appl., 58, 2, 348-359 (2009) · Zbl 1189.34098 · doi:10.1016/j.camwa.2009.03.110 |
| [28] | Liu, W.; Levin, S.; Isawa, Y., Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models, Math. Biosci., 23, 187-204 (1986) · Zbl 0582.92023 |
| [29] | Ma, S., Traveling wave fronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differ. Equ., 171, 294-314 (2001) · Zbl 0988.34053 · doi:10.1006/jdeq.2000.3846 |
| [30] | Takeuchi, Y.; Ma, W.; Beretta, E., Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal., 42, 931-947 (2000) · Zbl 0967.34070 · doi:10.1016/S0362-546X(99)00138-8 |
| [31] | Bai, Z.; Wu, S., Traveling waves in a delayed SIR epidemic model with nonlinear incidence, Appl. Math. Comput., 263, 221-232 (2015) · Zbl 1410.35016 · doi:10.1016/j.amc.2015.04.048 |
| [32] | Gan, Q.; Xu, R.; Yang, P., Traveling waves of a delayed SIRS epidemic model with spatial diffusion, Nonlinear Anal., Real World Appl., 12, 52-68 (2011) · Zbl 1202.35046 · doi:10.1016/j.nonrwa.2010.05.035 |
| [33] | Zhen, Z.; Wei, J.; Zhou, J.; Tian, L., Wave propagation in a nonlocal diffusion epidemic model with nonlocal delayed effects, Appl. Math. Comput., 339, 15-37 (2018) · Zbl 1428.35176 · doi:10.1016/j.amc.2018.07.007 |
| [34] | Zhang, S.; Yang, Y.; Zhou, Y., Traveling waves in a delayed SIR model with nonlocal dispersal and nonlinear incidence, J. Math. Phys., 59, 1 (2018) · Zbl 1392.92112 · doi:10.1063/1.5021761 |
| [35] | Smith, J.; Fishrein, D.; Rupprecht, C., Unexplained rabies in three immigrants in the United States. A virologic investigation, N. Engl. J. Med., 324, 205-211 (1991) · doi:10.1056/NEJM199101243240401 |
| [36] | Charlton, K.; Nadin-Davis, S.; Casey, G.; Wandele, A., The long incubation period in rabies: delayed progression of infection in muscle at the site of exposure, Acta Neuropathol., 94, 73-77 (1997) · doi:10.1007/s004010050674 |
| [37] | Zhang, L.; Wang, Z.; Zhao, X., Time periodic traveling wave solutions for a Kermack-McKendrick epidemic model with diffusion and seasonality, J. Evol. Equ. (2019) · Zbl 1447.35104 · doi:10.1007/s00028-019-00544-2 |
| [38] | Xu, Z., Traveling waves in a Kermack-McKendrick epidemic model with diffusion and latent period, Nonlinear Anal., Theory Methods Appl., 111, 66-81 (2014) · Zbl 1338.92144 · doi:10.1016/j.na.2014.08.012 |
| [39] | Ruan, S.; Wang, W., Dynamical behavior of an epidemic model with nonlinear incidence rate, J. Differ. Equ., 188, 135-163 (2003) · Zbl 1028.34046 · doi:10.1016/S0022-0396(02)00089-X |
| [40] | Smith, L.; Zhao, X., Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31, 514-534 (2000) · Zbl 0941.35125 · doi:10.1137/S0036141098346785 |
| [41] | Liang, X.; Zhao, X., Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Commun. Pure Appl. Math., 60, 1-40 (2007) · Zbl 1106.76008 · doi:10.1002/cpa.20154 |
| [42] | Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, 17-22 (1983), New York: Springer, New York · Zbl 0516.47023 |
| [43] | Wu, J., Theory and Applications of Partial Functional Differential Equations, 20-30 (1996), New York: Springer, New York · Zbl 0870.35116 |
| [44] | Smith, H., Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems (1995) · Zbl 0821.34003 |
| [45] | Zhao, X., Dynamical Systems in Population Biology, 44-48 (2003), New York: Springer, New York · Zbl 1023.37047 |
| [46] | Li, W.; Yang, F., Traveling waves for a nonlocal dispersal SIR model with standard incidence, J. Integral Equ. Appl., 26, 243-273 (2014) · Zbl 1295.35170 · doi:10.1216/JIE-2014-26-2-243 |
| [47] | Yang, F.; Li, W., Traveling waves in a nonlocal dispersal SIR model with critical wave speed, J. Math. Anal. Appl., 458, 1131-1146 (2018) · Zbl 1378.92078 · doi:10.1016/j.jmaa.2017.10.016 |
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.
