Unidirectional wave propagation in a nonlocal dispersal endemic model with nonlinear incidence. (English) Zbl 07925055
Summary: This paper is concerned with existence and non-existence of traveling wave solutions in a nonlocal dispersal endemic model with nonlinear incidence. With the aid of upper-lower solutions method and Schauder’s fixed point theorem together with Lyapunov functional technique, we derive the existence of super-critical and critical traveling wave solutions connecting disease-free equilibrium to endemic equilibrium. In a combination with the theory of two-sided Laplace transform and local skilled analysis, we obtain the non-existence of sub-critical traveling wave solutions. Our results illustrate that: (i) the existence and non-existence of traveling waves are determined by the basic reproduction number and the wave speed; (ii) the critical wave speed is equal to the minimal wave speed; (iii) the traveling waves only propagate along one direction.
MSC:
| 35K57 | Reaction-diffusion equations |
| 37C65 | Monotone flows as dynamical systems |
| 92D30 | Epidemiology |
References:
| [1] | H. Baek, S. Kim and P. Kim, Permanence and stability of an Ivlev-type predator-prey system with impulsive control strategies, Math. Comput. Model., 2009, 50, 1385-1393. · Zbl 1185.34067 |
| [2] | I. Barbalat, Système d’equations differentielle d’oscillations nonlinearires, Rev. Roum. Math. Pures., 1959, 4, 267-270. · Zbl 0090.06601 |
| [3] | S. Bentout, S. Djilali, T. Kuniya and J. Wang, Mathematical analysis of a vaccination epidemic model with nonlocal diffusion, Math. Method. Appl. Sci., 2023, 46, 10970-10994. |
| [4] | V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deter-ministic epidemic model, Math. Biosci., 1978, 42, 43-61. · Zbl 0398.92026 |
| [5] | Y. Chen, J. Guo and F. Hamel, Traveling waves for a lattice dynamical system arising in a diffusive endemic model, Nonlinearity, 2017, 30, 2334-2359. · Zbl 1373.34022 |
| [6] | K. Cheng, S. Hsu and S. Lin, Some results on global stability of a predator-prey model, J. Math. Biol., 1981, 12, 115-126. · Zbl 0464.92021 |
| [7] | S. Djilali, Y. Chen and S. Bentout, Asymptotic analysis of SIR epidemic model with nonlocal diffusion and generalized nonlinear incidence functional, Math. Method. Appl. Sci., 2023, 46, 6279-6301. |
| [8] | S. Dunbar, Traveling waves in diffusive predator-prey equations: Periodic orbits and point-to-periodic heteroclinic orbits, SIAM J. Appl. Math., 1986, 46, 1057-1078. · Zbl 0617.92020 |
| [9] | Y. Enatsua, Y. Nakata and Y. Muroya, Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model, Nonlinear Anal.-Real, 2012, 13, 2120-2133. · Zbl 1257.34065 |
| [10] | S. Fu, J. Guo and C. Wu, Traveling wave solutions for discrete diffusive epi-demic model, J. Nonlinear Convex A., 2016, 17, 1793-1751. · Zbl 1353.39005 |
| [11] | R. Gardner, Existence of traveling wave solutions of predator-prey systems via the connection index, SIAM J. Appl. Math., 1984, 44, 56-79. · Zbl 0541.35044 |
| [12] | H. Hethcote, The mathematics of infectious disease, SIAM Rev., 2000, 42, 599-653. · Zbl 0993.92033 |
| [13] | Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Mod. Meth. Appl. S., 1995, 5, 935-966. · Zbl 0836.92023 |
| [14] | C. Hsu, C. Yang, T. Yang and T. Yang, Existence of traveling wave solutions for diffusive predator-prey type systems, J. Differ. Equations, 2012, 252, 3040-3075. · Zbl 1252.34036 |
| [15] | J. Huang, G. Lu and S. Ruan, Existence of traveling wave solutions in a diffu-sive predator-prey model, J. Math. Biol., 2003, 46, 132-152. · Zbl 1018.92026 |
| [16] | V. Ivlev, Experimental Ecology of the Feeding of Fishes, New Haven, CT: Yale University, 1961. |
| [17] | C. Kao, Y. Lou and W. Shen, Random dospersal vs. non-local dispersal, Dis-crete and Continuous Dynamical Systems, 2010, 26, 551-596. · Zbl 1187.35127 |
| [18] | R. Kooij and A. Zegeling, A predator prey model with Ivlev’s functional re-sponse, J. Math. Anal. Appl., 1996, 198, 473-489. · Zbl 0851.34030 |
| [19] | A. Korobeinikov and P. Maini, Nonlinear incidence and stability of infectious disease models, Math. Med. Biol., 2005, 22, 113-128. · Zbl 1076.92048 |
| [20] | W. Li and F. Yang, Traveling waves for a nonlocal dispersal SIR model with standard incidence, J. Integral Equ. Appl., 2014, 26, 243-273. · Zbl 1295.35170 |
| [21] | Y. Li, W. Li and F. Yang, Traveling waves for a nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput., 2014, 247, 723-740. · Zbl 1338.92131 |
| [22] | G. Lin, Invasion traveling wave solutions of a predator-prey system, Nonlinear Anal.-Theor., 2014, 96, 47-58. · Zbl 1284.35109 |
| [23] | R. May, Limit cycles in predator-prey communities, Science, 1972, 177, 900-902. |
| [24] | C. Parkinson and W. Wang, Analysis of a reaction-diffusion SIR epidemic models with noncompliant behavior, SIAM J. Appl. Math., 2023, 83, 1969-2002. · Zbl 1533.35032 |
| [25] | M. Rosenzweig, Paradox of enrichment: Destabilization of exploitation ecosys-tems in ecological time, Science, 1971, 171, 385-387. |
| [26] | J. Sugie, Two-parameter bifurcation in a predator-prey system of Ivlev type, J. Math. Anal. Appl., 1998, 217, 349-371. · Zbl 0894.34025 |
| [27] | C. Tadmon, B. Tsanou and A. F. Feukouo, Avian-human influenza epidemic model with diffusion, nonlocal delay and spatial homogeneous environment, Nonlinear Anal.-Real, 2022, 67, 103615. · Zbl 1492.35380 |
| [28] | H. Wang, K. Wang and Y. J. Kim, Spatial segregation in reaction-diffusion epimidemic models, SIAM J. Appl. Math., 2022, 82, 1680-1709. · Zbl 1501.35245 |
| [29] | H. Wang and X. Wang, Traveling wave phenomena in a Kermack-Mckendrick SIR model, J. Dyn. Differ. Equ., 2016, 28, 143-166. · Zbl 1341.92078 |
| [30] | Z. Wang and J. Wu, Traveling waves of a diffusive Kermack-Mckendrick epi-demic model with non-local delayed transmission, P. Roy. Soc. Lond. A, 2010, 466, 237-261. · Zbl 1195.35291 |
| [31] | J. Wei, J. Zhou, Z. Zhen and L. Tian, Super-critical and critical traveling waves in a three-component delayed disease system with mixed diffusion, J. Comput. Appl. Math., 2020, 367, 112451. · Zbl 1428.35631 |
| [32] | D. Widder, The Laplace Transform, NJ: Princeton University Press, Princeton, 1941. · JFM 67.0384.01 |
| [33] | J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Differ. Equ., 2001, 13, 683-692. |
| [34] | Z. Xiang and X. Song, The dynamical behaviors of a food chain model with impulsive effect and Ivlev functional response, Chaos Soliton. Fract., 2009, 39, 2282-2293. · Zbl 1197.34012 |
| [35] | Z. Xu, Wave propagation in an infectious disease model, J. Math. Anal. Appl., 2017, 449, 853-871. · Zbl 1380.35047 |
| [36] | F. Yang, Y. Li, W. Li and Z. Wang, Traveling waves in a nonlocal disper-sal Kermack-Mckendrick epidemic model, Discrete and Continuous Dynamical Systems B, 2013, 18, 1969-1993. · Zbl 1277.35107 |
| [37] | F. Yang and W. Li, Traveling waves in a nonlocal dispersal SIR model with critical wave speed, J. Math. Anal. Appl., 2018, 458, 1131-1146. · Zbl 1378.92078 |
| [38] | F. Yang, W. Li and Z. Wang, Traveling waves in a nonlocal dispersal SIR epidemic model, Nonlinear Anal.-Real, 2015, 23, 129-147. · Zbl 1354.92096 |
| [39] | Z. Zhen, J. Wei, L. Tian, J. Zhou and W. Chen, Wave propagation in a diffusive SIR epidemic model with spatio-temporal delay, Math. Method. Appl. Sci., 2018, 41, 7074-7098. · Zbl 1402.35152 |
| [40] | Z. Zhen, J. Wei, J. Zhou and L. Tian, Wave propagation in a nonlocal diffusion epidemic model with nonlocal delayed effects, Appl. Math. Comput., 2018, 339, 15-37. · Zbl 1428.35176 |
| [41] | J. Zhou, L. Song and J. Wei, Mixed types of waves in a discrete diffusive epidemic model with nonlinear incidence and time delay, J. Differ. Equations, 2020, 268, 4491-4524. · Zbl 1440.34089 |
| [42] | J. Zhou, J. Xu, J. Wei and H. Xu, Existence and non-existence of traveling wave solutions for a nonlocal dispersal SIR epidemic model with nonlinear incidence rate, Nonlinear Anal.-Real, 2018, 41, 204-231. · Zbl 1383.35232 |
| [43] | J. Zhou and Y. Yang, Traveling waves for a nonlocal dispersal SIR model with general noneral nonlinear incidence rate and spatio-temporal delay, Discrete and Continuous Dynamical Systems B, 2017, 22, 1719-1741. · Zbl 1360.92124 |
| [44] | K. Zhou, M. Han and Q. Wang, Traveling wave solutions for delayed diffusive SIR epidemic model with nonlinear incidence rate and external supplies, Math. Method. Appl. Sci., 2017, 40, 2772-2783. · Zbl 1362.92088 |
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