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A Reaction–Diffusion Model of Vector-Borne Disease with Periodic Delays

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Abstract

A vector-borne disease is caused by a range of pathogens and transmitted to hosts through vectors. To investigate the multiple effects of the spatial heterogeneity, the temperature sensitivity of extrinsic incubation period and intrinsic incubation period, and the seasonality on disease transmission, we propose a nonlocal reaction–diffusion model of vector-borne disease with periodic delays. We introduce the basic reproduction number \(\mathfrak {R}_0\) for this model and then establish a threshold-type result on its global dynamics in terms of \(\mathfrak {R}_0\). In the case where all the coefficients are constants, we also prove the global attractivity of the positive constant steady state when \(\mathfrak {R}_0>1\). Numerically, we study the malaria transmission in Maputo Province, Mozambique.

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References

  • Atkinson, M.P., Su, Z., Alphey, N., Alphey, L.S., Coleman, P.G., Wein, L.M.: Analyzing the control of mosquito-borne diseases by a dominant lethal genetic system. Proc. Natl. Acad. Sci. USA 104, 9540–9545 (2007)

    Article  Google Scholar 

  • Bacaër, N., Guernaoui, S.: The epidemic threshold of vector-borne diseases with seasonality. J. Math. Biol. 53, 421–436 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Bai, Z., Peng, R., Zhao, X.-Q.: A reaction-diffusion malaria model with seasonality and incubation period. J. Math. Biol. 77, 201–228 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • Cai, L., Li, X., Fang, B., Ruan, S.: Global properties of vector-host disease models with time delays. J. Math. Biol. 74, 1397–1423 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Chan, M., Johansson, M.A.: The incubation periods of Dengue Viruses. PLoS ONE (2012). https://doi.org/10.1371/journal.pone.0050972

    Google Scholar 

  • Chitnis, N., Hyman, J.M., Cushing, J.M.: Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull. Math. Biol. 70, 1272–1296 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Craig, M.H., Snow, R.W., le Sueur, D.: A climate-based distribution model of malaria transmission in sub-Saharan Africa. Parasitol Today 15, 105–111 (1999)

    Article  Google Scholar 

  • Daners, D., Medina, P.: Abstract evolution equations, periodic problems and applications. In: Pitman Research Notes in Mathematics Series, vol. 279. Longman, Harlow, UK (1992)

  • Esteva, L., Vargas, C.: Analysis of a dengue disease transmission model. Math. Biosci. 150, 131–151 (1998)

    Article  MATH  Google Scholar 

  • Ewing, D.A., Cobbold, C.A., Purse, B.V., Nunn, M.A., White, S.M.: Modelling the effect of temperature on the seasonal population dynamics of temperate mosquitoes. J. Theor. Biol. 400, 65–79 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Fan, G., Liu, J., van den Driessche, P., Wu, J., Zhu, H.: The impact of maturation delay of mosquitoes on the transmission of West Nile virus. Math. Biosci. 228, 119–126 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Hattaf, K., Yousfi, N.: Global stability for reaction–diffusion equations in biology. Comput. Math. Appl. 66, 1488–1497 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Heffernan, J.M., Smith, R.J., Wahl, L.M.: Perspectives on the basic reproductive ratio. J. R. Soc. Interface 2, 281–293 (2005)

    Article  Google Scholar 

  • Jin, Y., Zhao, X.-Q.: Spatial dynamics of a nonlocal periodic reaction–diffusion model with stage structure. SIAM J. Math. Anal. 40, 2496–2516 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Jones, P., Harpham, C., Kilsby, C., Glenis, V., Burton, A.: Projections of future daily climate for the UK from the weather generator. Technical Report (2009)

  • Lashari, A., Aly, S., Hattaf, K., Zaman, G., Jung, I., Li, X.: Presentation of malaria epidemics using multiple optimal controls. J. Appl. Math. 2012, 17 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Lashari, A., Hattaf, K., Zaman, G., Li, X.: Backward bifurcation and optimal control of a vector borne disease. Appl. Math. Inf. Sci. 7, 301–309 (2013)

    Article  MathSciNet  Google Scholar 

  • Liang, X., Zhang, L., Zhao, X.-Q.: Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for Lyme disease). J. Dyn. Differ. Equ. (2017). https://doi.org/10.1007/s10884-017-9601-7

    Google Scholar 

  • Lou, Y., Zhao, X.-Q.: A reaction-diffusion malaria model with incubation period in the vector population. J. Math. Boil. 62, 543–568 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Lou, Y., Zhao, X.-Q.: A theoretical approach to understanding population dynamics with seasonal developmental durations. J. Nonlinear Sci. 27, 573–603 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Macdonald, G.: The Epidemiology and Control of Malaria. Oxford University Press, London (1957)

    Google Scholar 

  • Magal, P., Zhao, X.-Q.: Global attractors and steady states for uniformly persistent dynamical systems. SIAM J. Math. Anal. 37, 251–275 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Martin, R.H., Smith, H.L.: Abstract functional differential equations and reaction–diffusion systems. Trans. Am. Math. Soc. 321, 1–44 (1990)

    MathSciNet  MATH  Google Scholar 

  • Mirski, T., Bartoszcze, M., Bielawska-Drozd, A.: Impact of climate change on infectious diseases. Pol. J. Environ. Stud. 21, 525–532 (2012)

    Google Scholar 

  • Nisbet, R.M., Gurney, W.S.: The systematic formulation of population models for insects with dynamically varying instar duration. Theor. Popul. Biol. 23, 114–135 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  • Omori, R., Adams, B.: Disrupting seasonality to control disease outbreaks: the case of koi herpes virus. J. Theor. Biol. 271, 159–165 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Paaijmans, K.P., Read, A.F., Thomas, M.B.: Understanding the link between malaria risk and climate. PNAS 106, 13844–13849 (2009)

    Article  Google Scholar 

  • Ross, R.: The Prevention of Malaria, 2nd edn. John Murray, London (1911)

    Google Scholar 

  • Ruan, S., Xiao, D., Beier, J.C.: On the delayed Ross–Macdonald model for malaria transmission. Bull. Math. Biol. 70, 1098–1114 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Smith, D.L., Dushoff, J., McKenzie, F.E.: The risk of a mosquito-borne infection in a heterogeneous environment. PLoS Biol. 2, 1957–1964 (2004)

    Article  Google Scholar 

  • Thieme, H.R.: Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J. Appl. Math. 70, 188–211 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Thieme, H.R., Zhao, X.-Q.: A non-local delayed and diffusive predator-prey model. Nonlinear Anal. RWA 2, 145–160 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, X., Zhao, X.-Q.: A malaria transmission model with temperature-dependent incubation period. Bull. Math. Biol. 79, 1155–1182 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, W., Zhao, X.-Q.: Basic reproduction numbers for reaction–diffusion epidemic models. SIAM J. Appl. Dyn. Syst. 11, 1652–1673 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Xu, D., Zhao, X.Q.: Dynamics in a periodic competitive model with stage structure. J. Math. Anal. Appl. 311, 417–438 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, L., Wang, Z., Zhao, X.-Q.: Threshold dynamics of a time periodic reaction–diffusion epidemic model with latent period. J. Differ. Equ. 258, 3011–3036 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao, X.-Q.: Basic reproduction ratios for periodic compartmental models with time delay. J. Dyn. Differ. Equ. 29, 67–82 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao, X.-Q.: Dynamical Systems in Population Biology, 2nd edn. Springer, New York (2017)

    Book  MATH  Google Scholar 

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Acknowledgements

We sincerely thank Dr. Zhenguo Bai and Lei Zhang for helpful discussions on mathematical modeling and numerical computation of \(\mathfrak {R}_0\). We are also grateful to two referees for their careful reading and valuable suggestions which led to an improvement of our original manuscript.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Memorial University of Newfoundland, Saint John’s, NL, A1C 5S7, Canada

    Ruiwen Wu & Xiao-Qiang Zhao

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  1. Ruiwen Wu
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  2. Xiao-Qiang Zhao
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Correspondence to Ruiwen Wu.

Additional information

Communicated by Sue Ann Campbell.

Research supported in part by the NSERC of Canada.

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Wu, R., Zhao, XQ. A Reaction–Diffusion Model of Vector-Borne Disease with Periodic Delays. J Nonlinear Sci 29, 29–64 (2019). https://doi.org/10.1007/s00332-018-9475-9

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  • DOI: https://doi.org/10.1007/s00332-018-9475-9

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