Abstract
This paper is concerned with the conditions of existence and nonexistence of traveling wave solutions (TWS) for a class of discrete diffusive epidemic model. We find that the existence of TWS is determined by the so-called basic reproduction number and the critical wave speed: When the basic reproduction number \(\mathfrak {R}_0>1\), there exists a critical wave speed \(c^*>0\), such that for each \(c \ge c^*\) the system admits a nontrivial TWS and for \(c<c^*\) there exists no nontrivial TWS for the system. In addition, the boundary asymptotic behavior of TWS is obtained by constructing a suitable Lyapunov functional and employing Lebesgue dominated convergence theorem. Finally, we apply our results to two discrete diffusive epidemic models to verify the existence and nonexistence of TWS.
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References
Anderson, R.M., May, R.M.: Infectious Diseases in Humans: Dynamics and Control. Oxford University Press, Oxford (1991)
Bai, Z., Zhang, S.: Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay. Commun. Nonlinear Sci. Numer. Simulat. 22, 1370–1381 (2015)
Bates, P.W., Chmaj, A.: A discrete convolution model for phase transitions. Arch. Ration. Mech. Anal. 150, 281–305 (1999)
Briggs, C.J., Godfray, H.C.J.: The dynamics of insect-pathogen interactions in stage-structured populations. Am. Nat. 145, 855–887 (1995)
Brucal-Hallare, M., Vleck, E.V.: Traveling wavefronts in an antidiffusion lattice Nagumo model. SIAM J. Appl. Dyn. Syst. 10, 921–959 (2011)
Capasso, V., Serio, G.: A generalization of the Kermack-Mackendric deterministic model. Math. Biosci. 42, 43–61 (1978)
Chang, K.-C.: Methods in Nonlinear Analysis. Springer Monographs in Mathematics. Springer, Berlin (2005)
Chen, X., Guo, J.-S.: Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics. Math. Ann. 326, 123–146 (2003)
Chen, Y.-Y., Guo, J.-S., Hamel, F.: Traveling waves for a lattice dynamical system arising in a diffusive endemic model. Nonlinearity 30, 2334–2359 (2017)
Cui, J., Sun, Y., Zhu, H.: The impact of media on the control of infectious diseases. J. Dyn. Differ. Equ. 20, 31–53 (2008)
Ducrot, A., Magal, P.: Travelling wave solutions for an infection-age structured epidemic model with external supplies. Nonlinearity 24, 2891–2911 (2011)
Erneux, T., Nicolis, G.: Propagating waves in discrete bistable reaction diffusion systems. Physica D 67, 237–244 (1993)
Fang, J., Wei, J., Zhao, X.-Q.: Spreading speeds and travelling waves for non-monotone time-delayed lattice equations. Proc. R. Soc. A-Math. Phys. Eng. Sci. 466, 1919–1934 (2010)
Fu, S.-C., Guo, J.-S., Wu, C.-C.: Traveling wave solutions for a discrete diffusive epidemic model. J. Nonlinear Convex Anal. 17, 1739–1751 (2016)
Fu, S.-C.: Traveling waves for a diffusive SIR model with delay. J. Math. Anal. Appl. 435, 20–37 (2016)
Guo, J.-S., Wu, C.-H.: Traveling wave front for a two-component lattice dynamical system arising in competition models. J. Differ. Equ. 252, 4357–4391 (2012)
Han, X., Kloeden, P.E.: Lattice dynamical systems in the biological sciences. In: Yin, G., Zhang, Q. (eds.) Modeling, Stochastic Control, Optimization, and Applications. Springer, Cham (2019)
He, J., Tsai, J.-C.: Traveling waves in the Kermark–McKendrick epidemic model with latent period. Z. Angew. Math. Phys. 70, 2722 (2019)
Heesterbeek, J.A.P., Metz, J.A.J.: The saturating contact rate in marriage and epidemic models. J. Math. Biol. 31, 529–539 (1993)
Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42, 599–653 (2000)
Hosono, Y., Ilyas, B.: Traveling waves for a simple diffusive epidemic model. Math. Models Methods Appl. Sci. 5, 935–966 (1995)
Kapral, R.: Discrete models for chemically reacting systems. J. Math. Chem. 6, 113–163 (1991)
Kermack, W., McKendrick, A.: A contribution to mathematical theory of epidemics. Proc. R. Soc. A-Math. Phys. Eng. Sci. 115, 700–721 (1927)
Korobeinikov, A., Maini, P.K.: Nonlinear incidence and stability of infectious disease models. Math. Med. Biol. 22, 113–128 (2005)
Korobeinikov, A.: Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol. 68, 615–626 (2006)
Lam, K.-Y., Wang, X., Zhang, T.: Traveling waves for a class of diffusive disease-transmission models with network structures. SIAM J. Math. Anal. 50, 5719–5748 (2018)
Li, W.-T., Xu, W.-B., Zhang, L.: Traveling waves and entire solutions for an epidemic model with asymmetric dispersal. Discret. Contin. Dyn. Syst. 37, 2483–2512 (2017)
Li, Y., Li, W.-T., Lin, G.: Traveling waves of a delayed diffusive SIR epidemic model. Commun. Pur. Appl. Anal. 14, 1001–1022 (2015)
Li, Y., Li, W.-T., Yang, F.-Y.: Traveling waves for a nonlocal dispersal SIR model with delay and external supplies. Appl. Math. Comput. 247, 723–740 (2014)
Liu, W.M., Levin, S.A., Iwasa, X.: Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models. J. Math. Biol. 23, 187–204 (1986)
Muroya, Y., Kuniya, T., Enatsu, Y.: Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection. Discret. Contin. Dyn. Syst. Ser. B 20, 3057–3091 (2015)
Rudin, W.: Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics, 3rd edn. McGraw-Hill, New York (1976)
Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill, New York (1991)
San, X.F., Wang, Z.-C.: Traveling waves for a two-group epidemic model with latent period in a patchy environment. J. Math. Anal. Appl. 475, 1502–1531 (2019)
Shu, H., Pan, X., Wang, X.-S., Wu, J.: Traveling waves in epidemic models: non-monotone diffusive systems with non-monotone incidence rates. J. Dyn. Differ. Equ. 31, 883–901 (2019)
Thieme, H.R.: Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators. J. Differ. Equ. 250, 3772–3801 (2011)
Tian, B., Yuan, R.: Traveling waves for a diffusive SEIR epidemic model with standard incidences. Sci. China Math. 60, 813–832 (2017)
Wang, W., Ma, W.: Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discret. Contin. Dyn. Syst. Ser. B 23, 3213–3235 (2018)
Weng, P., Huang, H., Wu, J.: Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction. IMA J. Appl. Math. 68, 409–439 (2003)
Widder, D.V.: The Laplace Transform. Princeton Mathematical Series 6. Princeton University Press, Princeton (1941)
Wu, C.-C.: Existence of traveling waves with the critical speed for a discrete diffusive epidemic model. J. Differ. Equ. 262, 272–282 (2017)
Wu, S., Weng, P., Ruan, S.: Spatial dynamics of a lattice population model with two age classes and maturation delay. Eur. J. Appl. Math. 26, 61–91 (2015)
Xiao, D., Ruan, S.: Global analysis of an epidemic model with a nonlinear incidence rate. Math. Biosci. 208, 419–429 (2007)
Xiao, D., Zhou, Y.: Qualitative analysis of an epidemic model. Can. Appl. Math. Q 14, 469–492 (2006)
Xu, R., Ma, Z.: Global stability of a SIR epidemic model with nonlinear incidence rate and time delay. Nonlinear Anal. Real World Appl. 10, 3175–3189 (2009)
Xu, Z., Guo, T.: Traveling waves in a diffusive epidemic model with criss-cross mechanism. Math. Meth. Appl. Sci. 42, 2892–2908 (2019)
Yang, F.-Y., Li, Y., Li, W.-T., Wang, Z.-C.: Traveling waves in a nonlocal dispersal Kermack–McKendrick epidemic model equation with monostable convolution type nonlinearity. Discret. Contin. Dyn. Syst. Ser. B 18, 1969–1993 (2013)
Yang, Z., Zhang, G.: Stability of non-monotone traveling waves for a discrete diffusion equation with monostable convolution type nonlinearity. Sci. China Math. 61, 1789–1806 (2018)
Zhang, Q., Wu, S.-L.: Wave propagation of a discrete SIR epidemic model with a saturated incidence rate. Int. J. Biomath. 12, 1950029 (2019)
Zhang, S., Xu, R.: Travelling waves and global attractivity of an SIRS disease model with spatial diffusion and temporary immunity. Appl. Math. Comput. 224, 635–651 (2013)
Zhang, Y., Li, Y., Zhang, Q., Li, A.: Behavior of a stochastic SIR epidemic model with saturated incidence and vaccination rules. Physica A 501, 178–187 (2018)
Zhang, S.-P., Yang, Y.-R., Zhou, Y.-H.: Traveling waves in a delayed SIR model with nonlocal dispersal and nonlinear incidence. J. Math. Phys. 59, 011513 (2018)
Zhao, L., Wang, Z.-C., Ruan, S.: Traveling wave solutions in a two-group epidemic model with latent period. Nonlinearity 30, 1287–1325 (2017)
Zhou, J., Song, L., Wei, J.: Mixed types of waves in a discrete diffusive epidemic model with nonlinear incidence and time delay. J. Differ. Equ. 268, 4491–4524 (2020)
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)
Zhou, J., Yang, Y., Hsu, C.-H.: Traveling waves for a nonlocal dispersal vaccination model with general incidence. Discret. Contin. Dyn. Syst. Ser. B 25, 1469–1495 (2020)
Acknowledgements
The authors would like to thank the editor and anonymous reviewers for their valuable comments and suggestions which led to a significant improvement of this work. R Zhang and S Liu were supported by Natural Science Foundation of China (11871179; 11771374), J. Wang was supported by National Natural Science Foundation of China (nos. 12071115, 11871179), Natural Science Foundation of Heilongjiang Province (nos. LC2018002, LH2019A021) and Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems.
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Zhang, R., Wang, J. & Liu, S. Traveling Wave Solutions for a Class of Discrete Diffusive SIR Epidemic Model. J Nonlinear Sci 31, 10 (2021). https://doi.org/10.1007/s00332-020-09656-3
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DOI: https://doi.org/10.1007/s00332-020-09656-3
Keywords
- Lattice dynamical system
- Schauder’s fixed point theorem
- Traveling wave solutions
- Diffusive epidemic model
- Lyapunov functional