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A reaction–diffusion malaria model with seasonality and incubation period

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Abstract

In this paper, we propose a time-periodic reaction–diffusion model which incorporates seasonality, spatial heterogeneity and the extrinsic incubation period (EIP) of the parasite. The basic reproduction number \(\mathcal {R}_0\) is derived, and it is shown that the disease-free periodic solution is globally attractive if \(\mathcal {R}_0<1\), while there is an endemic periodic solution and the disease is uniformly persistent if \(\mathcal {R}_0>1\). Numerical simulations indicate that prolonging the EIP may be helpful in the disease control, while spatial heterogeneity of the disease transmission coefficient may increase the disease burden.

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References

  • Abboubakar H, Buonomo B, Chitnis N (2016) Modelling the effects of malaria infection on mosquito biting behaviour and attractiveness of humans. Ricerche Mat 65:329–346

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Buonomo B, Vargas-De-León C (2013) Stability and bifurcation analysis of a vector-bias model of malaria transmission. Math Biosci 242:59–67

    Article  MathSciNet  MATH  Google Scholar 

  • Chamchod F, Britton NF (2011) Analysis of a vector-bias model on malaria transmission. Bull Math Biol 73:639–657

    Article  MathSciNet  MATH  Google Scholar 

  • Cosner C, Beier JC, Cantrell RS, Impoinvil D, Kapitanski L, Potts MD, Troyo A, Ruan S (2009) The effects of human movement on the persistence of vector-borne diseases. J Theor Biol 258:550–560

    Article  MathSciNet  Google Scholar 

  • Daners D, Medina PK (1992) Abstract evolution equations, periodic problems and applications, Pitman research notes in mathematics series, vol 279. Longman, Harlow

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Ewing DA, Cobbold CA, Purse BV, Nunn MA, White SM (2016) Modelling the effect of temperature on the seasonal population dynamics of temperate mosquitoes. J Theor Biol 400:65–79

    Article  MathSciNet  MATH  Google Scholar 

  • Forouzannia F, Gumel AB (2014) Mathematical analysis of an age-structured model for malaria transmission dynamics. Math Biosci 247:80–94

    Article  MathSciNet  MATH  Google Scholar 

  • Friedman A (1964) Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  • Grassly NC, Fraser C (2006) Seasonal infectious disease epidemiology. Proc R Soc B 273:2541–2550

    Article  Google Scholar 

  • Gutierrez JB, Galinski MR, Cantrell S, Voit EO (2015) From within host dynamics to the epidemiology of infectious disease scientific overview and challenges. Math Biosci 270:143–155

    Article  MathSciNet  MATH  Google Scholar 

  • Hay SI, Were EC, Renshaw M, Noor AM, Ochola SA, Olusanmi I, Alipui N, Snow RW (2003) Forecasting, warning, and detection of malaria epidemics: a case study. Lancet 361:1705–1706

    Article  Google Scholar 

  • Hosack GR, Rossignol PA, van den Driessche P (2008) The control of vector-borne disease epidemics. J Theor Biol 255:16–25

    Article  MathSciNet  Google Scholar 

  • Kingsolver JG (1987) Mosquito host choice and the epidemiology of malaria. Am Nat 130:811–827

    Article  Google Scholar 

  • Lacroix R, Mukabana WR, Gouagna LC, Koella JC (2005) Malaria infection increases attractiveness of humans to mosquitoes. PLoS Biol 3:1590–1593

    Article  Google Scholar 

  • Liang X, Zhao X-Q (2007) Asymptotic speeds of spread and traveling waves formonotone semiflows with applications. Commun Pure Appl Math 60:1–40

    Article  MATH  Google Scholar 

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

    Google Scholar 

  • Lou Y, Zhao X-Q (2010) A climate-based malaria transmission model with structured vector population. SIAM J Appl Math 70:2023–2044

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Macdonald G (1957) The epidemiology and control of malaria. Oxford University Press, London

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Metz JAJ, Diekmann O (1986) The dynamics of physiologically structured populations. Springer, New York

    Book  MATH  Google Scholar 

  • Niger AM, Gumel AB (2008) Mathematical analysis of the role of repeated exposure on malaria transmission dynamics. Differ Equ Dyn Syst 16:251–287

    Article  MathSciNet  MATH  Google Scholar 

  • Okuneye K, Gumel AB (2017) Analysis of a temperature- and rainfall-dependent model for malaria transmission dynamics. Math Biosci 287:72–92

    Article  MathSciNet  MATH  Google Scholar 

  • Ross R (1911) The prevention of malaria, 2nd edn. Murray, London

    Google Scholar 

  • Smith HL (1995) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, mathematical surveys and monographs, vol 41. American Mathematical Society, Providence

    Google Scholar 

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

    Article  Google Scholar 

  • Tatem AJ, Hay SI, Rogers DJ (2006) Global traffic and disease vector dispersal. Proc Natl Acad Sci USA 103:6242–6247

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Vargas-De-León C (2012) Global analysis of a delayed vector-bias model for malaria transmission with incubation period in mosquitoes. Math Biosci Eng 9:165–174

    Article  MathSciNet  MATH  Google Scholar 

  • Wang X, Zhao X-Q (2017) A periodic vector-bias malaria model with incubation period. SIAM J Appl Math 77:181–201

    Article  MathSciNet  MATH  Google Scholar 

  • Wu J (1996) Theory and applications of partial functional differential equations. Springer, New York

    Book  MATH  Google Scholar 

  • Xiao Y, Zou X (2014) Transmission dynamics for vector-borne diseases in a patchy environment. J Math Biol 69:113–146

    Article  MathSciNet  MATH  Google Scholar 

  • Xu Z, Zhao X-Q (2012) A vector-bias malaria model with incubation period and diffusion. Discrete Contin Dyn Syst Ser B 17:2615–2634

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao X-Q (2017b) Dynamical systems in population biology, 2nd edn. Springer, New York

    Book  MATH  Google Scholar 

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Acknowledgements

We are grateful to two anonymous referees for careful reading and valuable comments which led to improvements of our original manuscript. We also sincerely thank Lei Zhang for his helpful discussions on the numerical computation of \(\mathcal {R}_0\).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, China

    Zhenguo Bai

  2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, China

    Rui Peng

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

    Xiao-Qiang Zhao

Authors
  1. Zhenguo Bai
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  2. Rui Peng
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  3. Xiao-Qiang Zhao
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Corresponding author

Correspondence to Zhenguo Bai.

Additional information

Bai’s research was supported by NSF of China (11401453); Peng’s research was supported by NSF of China (Nos. 11671175, 11271167, 11571200), the Priority Academic Program Development of Jiangsu Higher Education Institutions, Top-notch Academic Programs Project of Jiangsu Higher Education Institutions (No. PPZY2015A013) and Qing Lan Project of Jiangsu Province; and Zhao’s research was supported in part by the NSERC of Canada.

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Bai, Z., Peng, R. & Zhao, XQ. A reaction–diffusion malaria model with seasonality and incubation period. J. Math. Biol. 77, 201–228 (2018). https://doi.org/10.1007/s00285-017-1193-7

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  • DOI: https://doi.org/10.1007/s00285-017-1193-7

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