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On the Reproduction Number of a Gut Microbiota Model

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Abstract

A spatially structured linear model of the growth of intestinal bacteria is analysed from two generational viewpoints. Firstly, the basic reproduction number associated with the bacterial population, i.e. the expected number of daughter cells per bacterium, is given explicitly in terms of biological parameters. Secondly, an alternative quantity is introduced based on the number of bacteria produced within the intestine by one bacterium originally in the external media. The latter depends on the parameters in a simpler way and provides more biological insight than the standard reproduction number, allowing the design of experimental procedures. Both quantities coincide and are equal to one at the extinction threshold, below which the bacterial population becomes extinct. Optimal values of both reproduction numbers are derived assuming parameter trade-offs.

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References

  • Bacaër N, Guernaoui S (2006) The epidemic threshold of vector-borne diseases with seasonality. The case of cutaneous leishmaniasis in Chichaoua, Morocco. J Math Biol 53(3):421–436

    Article  MathSciNet  MATH  Google Scholar 

  • Barril C, Calsina A (2017) Stability analysis of an enteropathogen population growing within a heterogeneous group of animals. Discrec Contin Dyn Sys B 70(4):1231–1252

    Article  MathSciNet  MATH  Google Scholar 

  • Boldin B (2008) Persistence and spread of gastro-intestinal infections: the case of enterotoxigenic Escherichia coli in piglets. Bull Math Biol 70(7):2077–2101

    Article  MathSciNet  MATH  Google Scholar 

  • Calsina A, Palmada JM, Ripoll J (2011) Optimal latent period in a bacteriophage population model structured by infection-age. Math Models Methods Appl Sci 21(4):693–718

    Article  MathSciNet  MATH  Google Scholar 

  • Calsina A, Diekmann O, Farkas JZ (2016) Structured populations with distributed recruitment: from PDE to delay formulation. Math Models Methods Appl Sci 39:5175–5191

    Article  MathSciNet  MATH  Google Scholar 

  • Cortez MH (2013) When does pathogen evolution maximize the basic reproductive number in well-mixed host-pathogen systems? J Math Biol 67:1533–1583

    Article  MathSciNet  MATH  Google Scholar 

  • Cushing JM, Diekmann O (2016) The many guises of \(R_0\) (a didactic note). J Theor Biol 404:295–302

    Article  MathSciNet  MATH  Google Scholar 

  • Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations. J Math Biol 28(4):365–382

    Article  MathSciNet  MATH  Google Scholar 

  • Dublin LI, Lotka AJ (1925) On the true rate of natural increase. J Am Stat Assoc 20:305–339

    Google Scholar 

  • Engel K-J, Nagel R (2006) A short course on operator semigroups. Springer, Berlin

    MATH  Google Scholar 

  • Geritz SAH, Kisdi E, Meszena G, Metz JAJ (1998) Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evolut Ecol 12:35–57

    Article  Google Scholar 

  • Heesterbeek JAP (2002) A brief history of \(R_0\) and a recipe for its calculation. Acta Biotheor 50:189–204

    Article  Google Scholar 

  • Inaba H (2012) On a new perspective of the basic reproduction number in heterogeneous environments. J Math Biol 65(2):309–348

    Article  MathSciNet  MATH  Google Scholar 

  • Inaba H (2017) Age-structured population dynamics in demography and epidemiology. Springer, Berlin

    Book  MATH  Google Scholar 

  • Messenger SL, Molineux IJ, Bull JJ (1999) Virulence evolution in a virus obeys a trade-off. Proc R Soc B Biol Sci 266:397–404

    Article  Google Scholar 

  • Pugliese A (2002) On the evolutionary coexistence of parasite strains. Math Biosci 177:355–375

    Article  MathSciNet  MATH  Google Scholar 

  • Schaefer HH (1974) Banach lattices and positive operators. Springer, Berlin

    Book  MATH  Google Scholar 

  • Thieme H (2009) Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J Appl Math 70(1):188–211

    Article  MathSciNet  MATH  Google Scholar 

  • Van Den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48

    Article  MathSciNet  MATH  Google Scholar 

  • Webb GF (1985) Theory of nonlinear age-dependent population dynamics. Marcel Dekker, New York

    MATH  Google Scholar 

Download references

Acknowledgements

This work has been partially supported by the coordinated Projects MTM2014-52402-C3-2 and MTM2014-52402-C3-3 of the Spanish government. JR is also partially supported by the project MPCUdG2016/047 of the University of Girona and CB by the Spanish Ministry of Education Grant FPU13/04333.

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Barcelona, Spain

    Carles Barril & Àngel Calsina

  2. Departament d’Informàtica, Matemàtica Aplicada i Estadística, Universitat de Girona, Girona, Spain

    Jordi Ripoll

Authors
  1. Carles Barril
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  2. Àngel Calsina
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  3. Jordi Ripoll
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Correspondence to Jordi Ripoll.

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Barril, C., Calsina, À. & Ripoll, J. On the Reproduction Number of a Gut Microbiota Model. Bull Math Biol 79, 2727–2746 (2017). https://doi.org/10.1007/s11538-017-0352-8

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  • DOI: https://doi.org/10.1007/s11538-017-0352-8

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