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Spreading speed and periodic traveling waves of a time periodic and diffusive SI epidemic model with demographic structure

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    * Corresponding author 

The first author was supported by the innovation fund project for colleges and universities of Gansu Province of China (2020A-062) and NSF of Gansu Province of China (21JR7RA549), the second author was supported by NSF (DMS-1204497), the third author was supported by NSF of China (12071193 and 11731005), and the forth author was supported by NSF of China (12171214, 11701242)

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  • We study the asymptotic spreading properties and periodic traveling wave solutions of a time periodic and diffusive SI epidemic model with demographic structure (follows the logistic growth). Since the comparison principle is not applicable to the full system, we analyze the asymptotic spreading phenomena for susceptible class and infectious class by comparing with respective relevant periodic equations with KPP-type. By applying fixed point theorem to a truncated problem on a finite interval, combining with limit idea, the existence of periodic traveling wave solutions are derived. The results show that the minimal wave speed exactly equals to the spreading speed of infectious class when susceptible class is abundant.

    Mathematics Subject Classification: Primary: 35K57, 35B40, 35C07; Secondary: 35B10, 92D30.


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  • [1] S. AltizerA. Dobson and Pa rviez Hosseini, et al., Seasonality and the dynamics of infectious disease, Ecol. Lett., 9 (2006), 467-484.  doi: 10.1111/j.1461-0248.2005.00879.x.
    [2] B. Ambrosio, A. Ducrot and S. Ruan, Generalized traveling waves for time-dependent reaction-diffusion systems, Math. Ann., 2020, 27 pp. doi: 10.1007/s00208-020-01998-3.
    [3] D. G. Aronson, The asymptotic speed of a propagation of a simple epidemic, in Nonlinear Diffusion, Research Notes in Mathematics, Pitman, London, 1977.
    [4] W. Beauvais, I. Musallam and J. Guitian, Vaccination control programs for multiple livestock host species: an age-stratified, seasonal transmission model for brucellosis control in endemic settings, Parasites & Vectors, 9 (2016), 10 pp. doi: 10.1186/s13071-016-1327-6.
    [5] V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.
    [6] O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differ. Equ., 33 (1979), 58-73.  doi: 10.1016/0022-0396(79)90080-9.
    [7] W. J. BoG. Lin and S. Ruan, Traveling wave solutions for time periodic reaction-diffusion systems, Discrete Contin. Dyn. Syst., 38 (2018), 4329-4351.  doi: 10.3934/dcds.2018189.
    [8] A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differ. Equ., 260 (2016), 8316-8357.  doi: 10.1016/j.jde.2016.02.023.
    [9] A. Ducrot and P. Magal, Traveling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 459-482.  doi: 10.1017/S0308210507000455.
    [10] A. Ducrot, T. Giletti and H. Matano, Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type, Calc. Var. Partial Differ. Equ., 58 (2019), 34 pp. doi: 10.1007/s00526-019-1576-2.
    [11] P. Hess, Periodic-parabolic boundary value problems and positivity, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991.
    [12] Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Model Methods Appl. Sci., 5 (1995), 935-966.  doi: 10.1142/S0218202595000504.
    [13] W. Huang, A geometric approach in the study of traveling waves for some classes of nonmonotone reaction-diffusion systems, J. Differ. Equ., 260 (2016), 2190-2224.  doi: 10.1016/j.jde.2015.09.060.
    [14] D. G. Kendall, Mathematical models of the spread of infection, In Mathematics and computer science in biology and medicine, pp. 213-225. London, UK: Medical Research Council, 1965.
    [15] X. LiangY. Yi and X. Q. Zhao, Spreading speeds and traveling waves for perioidc evolution systems, J. Differ. Equ., 231 (2006), 57-77.  doi: 10.1016/j.jde.2006.04.010.
    [16] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
    [17] A. Lunardi, Analytic Semigroups And Optimal Regularity In Parabolic Problems, Birkhäuser/Springer Basel AG, Basel, 1995.
    [18] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, mathematics for life science and medicine, Biol. Med. Phys. Biomed. Eng., Springer, Berlin, 2007, 97-122
    [19] M. N. SeleemS. M. Boyle and N. Sriranganathan, Brucellosis: a re-emerging zoonosis, Vet. Microbiol., 140 (2010), 392-398.  doi: 10.1016/j.vetmic.2009.06.021.
    [20] S. M. Wang, Z. Feng, Z. C. Wang and L. Zhang, Periodic traveling wave of a time periodic and diffusive epidemic model with nonlocal delayed transmission, Nonlinear Anal. Real World Appl., 55 (2020), 103117, 27 pp. doi: 10.1016/j.nonrwa.2020.103117.
    [21] X. S. WangH. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst., 32 (2012), 3303-3324.  doi: 10.3934/dcds.2012.32.3303.
    [22] Z. C. Wang and J. Wu, Traveling waves of a diffusive Kermack-McKendrick epidemic model with nonlocal delayed transmission, Proc. R. Soc. Lond. Ser. A, 466 (2009), 237-261.  doi: 10.1098/rspa.2009.0377.
    [23] Z. C. WangL. Zhang and X. Q. Zhao, Time periodic traveling waves for a periodic and diffusive SIR epidemic model, J. Dynam. Differ. Equ., 30 (2018), 379-403.  doi: 10.1007/s10884-016-9546-2.
    [24] H. F. WeinbergerK. Kawasaki and N. Shigesada, Spreading speeds for a partially cooperative 2-species reaction-diffusion model, Discrete Contin. Dyn. Syst., 23 (2009), 1087-1098.  doi: 10.3934/dcds.2009.23.1087.
    [25] Z. Xu, Traveling waves for a diffusive SEIR epidemic model, Commun. Pure Appl. Anal., 15 (2016), 871-892.  doi: 10.3934/cpaa.2016.15.871.
    [26] L. ZhangZ. C. Wang and X. Q. Zhao, Propagation dynamics of a time periodic and delayed reaction-diffusion model without quasi-monotonicity, Trans. Amer. Math. Soc., 372 (2019), 1751-1782.  doi: 10.1090/tran/7709.
    [27] L. ZhangZ. C. Wang and X. Q. Zhao, Time periodic traveling wave solutions for a Kermack-McKendrick epidemic model with diffusion and seasonality, J. Evol. Equ., 20 (2020), 1029-1059.  doi: 10.1007/s00028-019-00544-2.
    [28] L. Zhang and S. M. Wang, Critical periodic traveling waves for a periodic and diffusive epidemic model, Appl. Anal., 100 (2021), 2108-2121.  doi: 10.1080/00036811.2019.1677894.
    [29] T. ZhangW. Wang and K. Wang, Minimal wave speed for a class of non-cooperative diffusion-reaction system, J. Differ. Equ., 260 (2016), 2763-2791.  doi: 10.1016/j.jde.2015.10.017.
    [30] X. Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dynam. Differ. Equ., 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.
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