doi: 10.3934/cpaa.2021145
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Spreading speed and periodic traveling waves of a time periodic and diffusive SI epidemic model with demographic structure

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

2. 

School of Information Engineering, Lanzhou University of Finance and Economics, Lanzhou, Gansu 730020, China

3. 

School of Mathematical and Statistical Sciences, University of Texas, Edinburg, Texas 78539, USA

* Corresponding author

Received  January 2021 Revised  June 2021 Early access August 2021

Fund Project: 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)

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.

Citation: Shuang-Ming Wang, Zhaosheng Feng, Zhi-Cheng Wang, Liang Zhang. Spreading speed and periodic traveling waves of a time periodic and diffusive SI epidemic model with demographic structure. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021145
References:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

S. AltizerA. Dobson and Pa rviez Hosseini, Seasonality and the dynamics of infectious disease, Ecol. Lett., 9 (2006), 467-484.  doi: 10.1111/j.1461-0248.2005.00879.x.  Google Scholar

[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.  Google Scholar

[3]

D. G. Aronson, The asymptotic speed of a propagation of a simple epidemic, in Nonlinear Diffusion, Research Notes in Mathematics, Pitman, London, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[16]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. Google Scholar

[17]

A. Lunardi, Analytic Semigroups And Optimal Regularity In Parabolic Problems, Birkhäuser/Springer Basel AG, Basel, 1995.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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