Theoretical conditions for the coexistence of viral strains with differences in phenotypic traits: A bifurcation analysis

Anel Nurtay; Matthew G. Hennessy; Josep Sardanyés; Lluís Alsedà; Santiago F. Elena

doi:10.1098/rsos.181179

Theoretical conditions for the coexistence of viral strains with differences in phenotypic traits: A bifurcation analysis

Anel Nurtay^*, Matthew G. Hennessy, Josep Sardanyés, Lluís Alsedà, Santiago F. Elena

^*Corresponding author for this work

Research output: Contribution to journal › Article (Academic Journal) › peer-review

9 Citations (Scopus)

Abstract

We investigate the dynamics of a wild-type viral strain which generates mutant strains differing in phenotypic properties for infectivity, virulence and mutation rates. We study, by means of a mathematical model and bifurcation analysis, conditions under which the wild-type and mutant viruses, which compete for the same host cells, can coexist. The coexistence conditions are formulated in terms of the basic reproductive numbers of the strains, a maximum value of the mutation rate and the virulence of the pathogens. The analysis reveals that parameter space can be divided into five regions, each with distinct dynamics, that are organized around degenerate Bogdanov-Takens and zero- Hopf bifurcations, the latter of which gives rise to a curve of transcritical bifurcations of periodic orbits. These results provide new insights into the conditions by which viral populations may contain multiple coexisting strains in a stable manner.

Original language	English
Article number	181179
Journal	Royal Society Open Science
Volume	6
Issue number	1
DOIs	https://doi.org/10.1098/rsos.181179
Publication status	Published - 2019

Bibliographical note

Funding Information:
The research leading to these results has received funding from 'la Caixa' Foundation. This work has been also partially funded by the 'Mar?a de Maeztu' Programme for Units of Excellence in R&D (MDM-2014-0445), as well as from projects MTM2014-52209-C2-1-P and MTM2017-86795-C3-1-P from the Spanish MINECO, and from the CERCA Programme of the Generalitat de Catalunya. M.H. has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk?odowska-Curie grant agreement no. 707658. Work in Val?ncia was supported by Spain's Agencia Estatal de Investigaci?n - FEDER grant no. BFU2015-65037-P to S.F.E. J.S. has been also funded by a 'Ram?n y Cajal' Fellowship (RYC-2017-22243).

Funding Information:
Data accessibility. MATCONT source files, output data and MATLAB files used to create the figures in the manuscript are available from the Dryad Digital Repository: https://doi.org/10.5061/dryad.56g7v08 [65]. Authors’ contributions. A.N. and M.H. carried out the analytical and numerical computations. A.N., M.H., L.A., J.S. and S.F.E. conceived the mathematical model. A.N. drafted the initial manuscript. All authors revised the article and gave final approval for publication. Competing interests. We declare we have no competing interests. Funding. The research leading to these results has received funding from ‘la Caixa’ Foundation. This work has been also partially funded by the ‘María de Maeztu’ Programme for Units of Excellence in R&D (MDM-2014-0445), as well as from projects MTM2014-52209-C2-1-P and MTM2017-86795-C3-1-P from the Spanish MINECO, and from the CERCA Programme of the Generalitat de Catalunya. M.H. has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 707658. Work in València was supported by Spain’s Agencia Estatal de Investigación - FEDER grant no. BFU2015-65037-P to S.F.E. J.S. has been also funded by a ‘Ramón y Cajal’ Fellowship (RYC-2017-22243). Acknowledgements. We thank Antoni Guillamon for interesting discussions about the zero-Hopf bifurcation. M.H., J.S. and L.A. acknowledge the hospitality of the Instituto de Biología Molecular y Celular de Plantas which led to fruitful meetings.

Publisher Copyright:
© 2019 The Authors.

Keywords

Bifurcations
Epidemiology
Infection dynamics
Mathematical biology
Virus evolution

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@article{54ae5eb0d61f4120842a170616efc6b0,

title = "Theoretical conditions for the coexistence of viral strains with differences in phenotypic traits: A bifurcation analysis",

abstract = "We investigate the dynamics of a wild-type viral strain which generates mutant strains differing in phenotypic properties for infectivity, virulence and mutation rates. We study, by means of a mathematical model and bifurcation analysis, conditions under which the wild-type and mutant viruses, which compete for the same host cells, can coexist. The coexistence conditions are formulated in terms of the basic reproductive numbers of the strains, a maximum value of the mutation rate and the virulence of the pathogens. The analysis reveals that parameter space can be divided into five regions, each with distinct dynamics, that are organized around degenerate Bogdanov-Takens and zero- Hopf bifurcations, the latter of which gives rise to a curve of transcritical bifurcations of periodic orbits. These results provide new insights into the conditions by which viral populations may contain multiple coexisting strains in a stable manner.",

keywords = "Bifurcations, Epidemiology, Infection dynamics, Mathematical biology, Virus evolution",

author = "Anel Nurtay and Hennessy, \{Matthew G.\} and Josep Sardany{\'e}s and Llu{\'i}s Alsed{\`a} and Elena, \{Santiago F.\}",

note = "Funding Information: The research leading to these results has received funding from 'la Caixa' Foundation. This work has been also partially funded by the 'Mar?a de Maeztu' Programme for Units of Excellence in R\&D (MDM-2014-0445), as well as from projects MTM2014-52209-C2-1-P and MTM2017-86795-C3-1-P from the Spanish MINECO, and from the CERCA Programme of the Generalitat de Catalunya. M.H. has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk?odowska-Curie grant agreement no. 707658. Work in Val?ncia was supported by Spain's Agencia Estatal de Investigaci?n - FEDER grant no. BFU2015-65037-P to S.F.E. J.S. has been also funded by a 'Ram?n y Cajal' Fellowship (RYC-2017-22243). Funding Information: Data accessibility. MATCONT source files, output data and MATLAB files used to create the figures in the manuscript are available from the Dryad Digital Repository: https://doi.org/10.5061/dryad.56g7v08 [65]. Authors{\textquoteright} contributions. A.N. and M.H. carried out the analytical and numerical computations. A.N., M.H., L.A., J.S. and S.F.E. conceived the mathematical model. A.N. drafted the initial manuscript. All authors revised the article and gave final approval for publication. Competing interests. We declare we have no competing interests. Funding. The research leading to these results has received funding from {\textquoteleft}la Caixa{\textquoteright} Foundation. This work has been also partially funded by the {\textquoteleft}Mar{\'i}a de Maeztu{\textquoteright} Programme for Units of Excellence in R\&D (MDM-2014-0445), as well as from projects MTM2014-52209-C2-1-P and MTM2017-86795-C3-1-P from the Spanish MINECO, and from the CERCA Programme of the Generalitat de Catalunya. M.H. has received funding from the European Union{\textquoteright}s Horizon 2020 research and innovation programme under the Marie Sk{\l}odowska-Curie grant agreement no. 707658. Work in Val{\`e}ncia was supported by Spain{\textquoteright}s Agencia Estatal de Investigaci{\'o}n - FEDER grant no. BFU2015-65037-P to S.F.E. J.S. has been also funded by a {\textquoteleft}Ram{\'o}n y Cajal{\textquoteright} Fellowship (RYC-2017-22243). Acknowledgements. We thank Antoni Guillamon for interesting discussions about the zero-Hopf bifurcation. M.H., J.S. and L.A. acknowledge the hospitality of the Instituto de Biolog{\'i}a Molecular y Celular de Plantas which led to fruitful meetings. Publisher Copyright: {\textcopyright} 2019 The Authors.",

year = "2019",

doi = "10.1098/rsos.181179",

language = "English",

volume = "6",

journal = "Royal Society Open Science",

issn = "2054-5703",

publisher = "The Royal Society",

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TY - JOUR

T1 - Theoretical conditions for the coexistence of viral strains with differences in phenotypic traits

T2 - A bifurcation analysis

AU - Nurtay, Anel

AU - Hennessy, Matthew G.

AU - Sardanyés, Josep

AU - Alsedà, Lluís

AU - Elena, Santiago F.

N1 - Funding Information: The research leading to these results has received funding from 'la Caixa' Foundation. This work has been also partially funded by the 'Mar?a de Maeztu' Programme for Units of Excellence in R&D (MDM-2014-0445), as well as from projects MTM2014-52209-C2-1-P and MTM2017-86795-C3-1-P from the Spanish MINECO, and from the CERCA Programme of the Generalitat de Catalunya. M.H. has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk?odowska-Curie grant agreement no. 707658. Work in Val?ncia was supported by Spain's Agencia Estatal de Investigaci?n - FEDER grant no. BFU2015-65037-P to S.F.E. J.S. has been also funded by a 'Ram?n y Cajal' Fellowship (RYC-2017-22243). Funding Information: Data accessibility. MATCONT source files, output data and MATLAB files used to create the figures in the manuscript are available from the Dryad Digital Repository: https://doi.org/10.5061/dryad.56g7v08 [65]. Authors’ contributions. A.N. and M.H. carried out the analytical and numerical computations. A.N., M.H., L.A., J.S. and S.F.E. conceived the mathematical model. A.N. drafted the initial manuscript. All authors revised the article and gave final approval for publication. Competing interests. We declare we have no competing interests. Funding. The research leading to these results has received funding from ‘la Caixa’ Foundation. This work has been also partially funded by the ‘María de Maeztu’ Programme for Units of Excellence in R&D (MDM-2014-0445), as well as from projects MTM2014-52209-C2-1-P and MTM2017-86795-C3-1-P from the Spanish MINECO, and from the CERCA Programme of the Generalitat de Catalunya. M.H. has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 707658. Work in València was supported by Spain’s Agencia Estatal de Investigación - FEDER grant no. BFU2015-65037-P to S.F.E. J.S. has been also funded by a ‘Ramón y Cajal’ Fellowship (RYC-2017-22243). Acknowledgements. We thank Antoni Guillamon for interesting discussions about the zero-Hopf bifurcation. M.H., J.S. and L.A. acknowledge the hospitality of the Instituto de Biología Molecular y Celular de Plantas which led to fruitful meetings. Publisher Copyright: © 2019 The Authors.

PY - 2019

Y1 - 2019

N2 - We investigate the dynamics of a wild-type viral strain which generates mutant strains differing in phenotypic properties for infectivity, virulence and mutation rates. We study, by means of a mathematical model and bifurcation analysis, conditions under which the wild-type and mutant viruses, which compete for the same host cells, can coexist. The coexistence conditions are formulated in terms of the basic reproductive numbers of the strains, a maximum value of the mutation rate and the virulence of the pathogens. The analysis reveals that parameter space can be divided into five regions, each with distinct dynamics, that are organized around degenerate Bogdanov-Takens and zero- Hopf bifurcations, the latter of which gives rise to a curve of transcritical bifurcations of periodic orbits. These results provide new insights into the conditions by which viral populations may contain multiple coexisting strains in a stable manner.

AB - We investigate the dynamics of a wild-type viral strain which generates mutant strains differing in phenotypic properties for infectivity, virulence and mutation rates. We study, by means of a mathematical model and bifurcation analysis, conditions under which the wild-type and mutant viruses, which compete for the same host cells, can coexist. The coexistence conditions are formulated in terms of the basic reproductive numbers of the strains, a maximum value of the mutation rate and the virulence of the pathogens. The analysis reveals that parameter space can be divided into five regions, each with distinct dynamics, that are organized around degenerate Bogdanov-Takens and zero- Hopf bifurcations, the latter of which gives rise to a curve of transcritical bifurcations of periodic orbits. These results provide new insights into the conditions by which viral populations may contain multiple coexisting strains in a stable manner.

KW - Bifurcations

KW - Epidemiology

KW - Infection dynamics

KW - Mathematical biology

KW - Virus evolution

UR - https://www.scopus.com/pages/publications/85062293025

U2 - 10.1098/rsos.181179

DO - 10.1098/rsos.181179

M3 - Article (Academic Journal)

C2 - 30800366

AN - SCOPUS:85062293025

SN - 2054-5703

VL - 6

JO - Royal Society Open Science

JF - Royal Society Open Science

IS - 1

M1 - 181179

ER -

Theoretical conditions for the coexistence of viral strains with differences in phenotypic traits: A bifurcation analysis

Abstract

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Keywords

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