Time-periodic traveling wave solutions of a reaction–diffusion Zika epidemic model with seasonality
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Lin ZhaoAuthors Info & Claims
Abstract
In this paper, the full information about the existence and nonexistence of a time-periodic traveling wave solution of a reaction–diffusion Zika epidemic model with seasonality, which is non-monotonic, is investigated. More precisely, if the basic reproduction number, denoted by , is larger than one, there exists a minimal wave speed satisfying for each , the system admits a nontrivial time-periodic traveling wave solution with wave speed c, and for , there exist no nontrivial time-periodic traveling waves such that if , the system admits no nontrivial time-periodic traveling waves.
References
[1]
Ambrosio B, Ducrot A, and Ruan S Generalized traveling waves for time-dependent reaction–diffusion systems Math. Ann. 2021 381 1-27
[2]
Bacaëra N and Gomes M On the final size of epidemics with seasonality J. Math. Biol. 2009 71 1954-1966
[3]
Barnett NS and Dragomir SS Some Landau type inequalities for functions whose derivatives are of locally bounded variation Tamkang J. Math. 2006 37 301-308
[4]
Buonomo B, Chitnis N, and d’Onofrio A Seasonality in epidemic models: a literature review Ricerche mat. 2018 67 7-25
[5]
CDC. Ceters for Disease Control and Prevention: Zika virus. Accessed, July 24 (2019)
[6]
Chen J, Beier J, Cantrell R, Robert S, Cosner C, Fuller D, Guan Y, Zhang G, and Ruan S Modeling the importation and local transmission of vector-borne diseases in Florida: the case of Zika outbreak in 2016 J. Theor. Biol. 2018 455 342-356
[7]
Deng D, Wang J, and Zhang L Critical periodic traveling waves for a Kermack–McKendrick epidemic model with diffusion and seasonality J. Differ. Equ. 2022 322 365-395
[8]
Ding, C., Tao, N., Zhu, Y.: A mathematical model of Zika virus and its optimal control. In: 35th Chinese, pp. 2642–2645. IEEE (2016)
[9]
Ducrot A, Magal P, and Ruan S Travelling wave solutions in multigroup age-structured epidemic models Arch. Ration. Mech. Anal. 2010 195 311-331
[10]
Ducrot A and Magal P Travelling wave solutions for an infection-age structured model with external supplies Nonlinearity 2011 24 2891-2911
[11]
Eikenberry SE and Gumel AB Mathematical modeling of climate change and malaria transmission dynamics: a historical review J. Math. Biol. 2018 77 857-933
[12]
Fang J, Yu X, and Zhao X-Q Traveling waves and spreading speeds for time-space periodic monotone systems J. Funct. Anal. 2017 272 4222-4262
[13]
Foy B, Kobylinski K, Chilson Foy J, Blitvich B, da Rosa AT, Haddow A, Lanciotti R, and Tesh R Probable non-vector-borne transmission of Zika virus Emerging Infec. Dis. 2011 17 1-7
[14]
Grassly NC and Fraser C Seasonal infectious disease epidemiology Proc. R. Soc. B 2006 273 2541-2550
[15]
Gao Daozhou, Lou Yijun, He Daihai, Porco Travis C, Kuang Yang, Chowell Gerardo, and Ruan Shigui Prevention and control of Zika as a mosquito-borne and sexually transmitted disease: a mathematical modeling analysis Sci. Rep. 2016 6 1-10
[16]
Hethcote H The mathematics of infectious diseases SIAM Rev. 2000 42 599-653
[17]
Hethcote H and Levin S Levin SA, Hallam TG, and Gross L Periodicity in Epidemiological Models Applied Mathematical Ecology, Biomathematics 1989 Berlin Springer
[18]
Huang M, Wu S-L, and Zhao X-Q The principal eigenvalue for partially degenerate and periodic reaction–diffusion systems with time delay J. Differ. Equ. 2023 371 396-449
[19]
Khan, M.A., Shan, S.W., Ullah, S., Gmez-Aguilar, J.F.: A dynamical model of asymptomatic carrier Zika virus with optimal control strategies. Nonlinear Anal. Real World Appl. 50, 140–177 (2019)
[20]
Landau E Einige Ungleichungen fr zweimal differentzierban funktionen Proc. Lond. Math. Soc. 1913 13 43-49
[21]
Li J and Zou X Modeling spatial spread of infections diseases with a fixed latent period in a spatially continous domain Bull. Math. Biol. 2009 71 2048-2079
[22]
Liang X, Yi Y, and Zhao X-Q Spreading speeds and traveling waves for periodic evolution systems J. Differ. Equ. 2006 231 57-77
[23]
Liang X and Zhao X-Q Asymptotic speeds of spread and traveling waves for monotone semiflows with applications Commun. Pure Appl. Math. 2007 60 1-40
[24]
Lunardi A Analytic Semigroups and Optimal Regularity in Parabolic Problem 1995 Boston Birkhäuser
[25]
Macnamara F Zika virus: a report on three cases of human infection during an epidemic of jaundice in Nigeria Trans. R. Soc. Trop. Med. Hyg. 1954 48 139-145
[26]
Miyaoka T, Lenhart S, and Meyer J Optimal control of vaccination in a vector-borne reaction–diffusion model applied to Zika virus J. Math. Biol. 2019 79 1077-1104
[27]
Murray JD Mathematical Biology 1989 Berlin Springer
[28]
Nishiura H, Kinoshita R, Mizumoto K, Yasuda Y, and Nah K Transmission potential of Zika virus infection in the south pacific Int. J. Infect. Dis. 2016 45 95-97
[29]
Rass L and Radcliffe J Spatial Deterministic Epidemics, Mathematical Surveys and Monographs 102 2003 Providence American Mathematical Society
[30]
Ruan S Spatial-Temporal Dynamics in Nonlocal Epidemiological Models 2007 Berlin Springer 99-122
[31]
Ruan, S., Wu, J.: Modeling spatial spread of communicable diseases involving animal hosts. In: Spatial Ecology, pp. 293–316. Chapman & Hall/CRC, Boca Raton (2009)
[32]
Simpson D Zika virus infection in man Trans. R. Soc. Trop. Med. Hygiene 1964 58 335-337
[33]
Soper HE The interpretation of periodicity in disease prevalence J. R. Stat. Soc. 1929 92 34-73
[34]
Suparit P, Wiratsudakul A, and Modchang C A mathematical model for Zika virus transmission dynamics with a time-dependent mosquito biting rate Theor. Biol. Med. Model. 2018 15 1-11
[35]
Wang L and Wu P Threshold dynamics of a Zika model with environmental and sexual transmissions and spatial heterogeneity Z. Angew. Math. Phys. 2022 73 171
[36]
Wang S-M, Feng Z, Wang Z-C, and Zhang L Periodic traveling wave of a time periodic and diffusive epidemic model with nonlocal delayed transmission Nonlinear Anal. Real World Appl. 2020 55 103117
[37]
Wang W and Zhao X-Q Threshold dynamics for compartmental epidemic models in periodic environments J. Dyn. Differ. Equ. 2008 20 699-717
[38]
Wang Z-C and Wu J Traveling waves of a diffusive Kermack–McKendrick epidemic model with nonlocal delayed transmission Proc. R. Soc. A 2010 466 237-261
[39]
Wang Z-C, Wu J, and Liu R Traveling waves of the spread of avian influenza Proc. Am. Math. Soc. 2012 140 3931-3946
[40]
Wang Z-C, Zhang L, and Zhao X-Q Time periodic traveling waves for a periodic and diffusive SIR epidemic model J. Dyn. Differ. Equ. 2018 30 379-403
[41]
Weinberger HF Long-time behavior of a class of biological model SIAM J. Math. Anal. 1982 13 353-396
[42]
World Health Organization (WHO).: WHO statement on the first meeting of the International Health Regulations: Emergency Committee on Zika virus and observed increase in neurological disorders and neonatal malformations, February 1, 2016 (2005). http://www.who.int/mediacentre/news/statements/2016/1st-emergency-committee-zika/en/. Accessed 26 Feb 2016
[43]
Wu S-L, Zhao H, Zhang X, and Hsu C-H Spatial dynamics for a time-periodic epidemic model in discrete media J. Differ. Equ. 2023 374 699-736
[44]
Wu W and Teng Z Periodic wave propagation in a diffusive SIR epidemic model with nonlinear incidence and periodic environment J. Math. Phys. 2022 63 12
[45]
Xu D and Zhao X-Q Dynamics in a periodic competitive model with stage structure J. Math. Anal. Appl. 2005 311 417-438
[46]
Yang L and Li Y Periodic traveling waves in a time periodic SEIR model with nonlocal dispersal and delay Discrete Contin. Dyn. Syst. Ser. B 2023 28 9 5087-5104
[47]
Yang X and Lin G Spreading speeds and traveling waves for a time periodic DS-I-A epidemic model Nonlinear Anal. Real World Appl. 2022 66 1-27
[48]
Zhang L, Wang Z-C, and Zhao X-Q Time periodic traveling wave solutions for a Kermack–McKendrick epidemic model with diffusion and seasonality J. Evol. Equ. 2020 20 1029-1059
[49]
Zhang R and Zhao H Traveling wave solutions for Zika transmission model with nonlocal diffusion J. Math. Anal. Appl. 2022 513 1-29
[50]
Zhao X-Q Basic reproduction ratios for periodic compartmental models with time delay J. Dyn. Differ. Equ. 2017 29 67-82
[51]
Zhao L, Wang Z-C, and Ruan S Traveling wave solutions in a two-group SIR epidemic model with constant recruitment J. Math. Biol. 2018 77 1871-1915
[52]
Zhao L, Wang Z-C, and Ruan S Traveling wave solutions of a two-group epidemic model with latent period Nonlinearity 2017 30 1287-1325
[53]
Zhao L, Wang Z-C, and Zhang L Propagation dynamics for a time-periodic reaction–diffusion SI epidemic model with periodic recruitment Z. Angew. Math. Phys. 2021 72 1-20
[54]
Zhao L Spreading speed and travelling wave solutions of a reaction–diffusion Zika model with constant recruitment Nonlinear Anal. Real World Appl. 2023 74 103942
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Publication History
Published: 10 February 2024
Accepted: 20 December 2023
Revision received: 29 October 2023
Received: 09 July 2023
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- National Natural Science Foundation of China
- Natural Science Foundation of Gansu
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