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Evolutionary dynamics in an SI epidemic model with phenotype-structured susceptible compartment

Journal of Mathematical Biology volume 83, Article number: 72 (2021) Cite this article

Abstract

We present an SI epidemic model whereby a continuous structuring variable captures variability in proliferative potential and resistance to infection among susceptible individuals. The occurrence of heritable, spontaneous changes in these phenotypic characteristics and the presence of a fitness trade-off between resistance to infection and proliferative potential are explicitly incorporated into the model. The model comprises an ordinary differential equation for the number of infected individuals that is coupled with a partial integrodifferential equation for the population density function of susceptible individuals through an integral term. The expression for the basic reproduction number \(\mathcal {R}_0\) is derived, the disease-free equilibrium and endemic equilibrium of the model are characterised and a threshold theorem involving \(\mathcal {R}_0\) is proved. Analytical results are integrated with the results of numerical simulations of a calibrated version of the model based on the results of artificial selection experiments in a host-parasite system. The results of our mathematical study disentangle the impact of different evolutionary parameters on the spread of infectious diseases and the consequent phenotypic adaption of susceptible individuals. In particular, these results provide a theoretical basis for the observation that infectious diseases exerting stronger selective pressures on susceptible individuals and being characterised by higher infection rates are more likely to spread. Moreover, our results indicate that heritable, spontaneous phenotypic changes in proliferative potential and resistance to infection can either promote or prevent the spread of infectious diseases depending on the strength of selection acting on susceptible individuals prior to infection. Finally, we demonstrate that, when an endemic equilibrium is established, higher levels of resistance to infection and lower degrees of phenotypic heterogeneity among susceptible individuals are to be expected in the presence of infections which are characterised by lower rates of death and exert stronger selective pressures.

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Acknowledgements

T.L. gratefully acknowledges the hospitality provided by the Department of Mathematics of the Università di Trento during his research stays and support from the MIUR grant “Dipartimenti di Eccellenza 2018-2022” Project nr E11G18000350001.

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Authors and Affiliations

  1. Department of Mathematical Sciences “G. L. Lagrange”, Dipartimento di Eccellenza 2018-2022, Politecnico di Torino, 10129, Turin, Italy

    Tommaso Lorenzi

  2. Department of Mathematics, Università di Trento, Via Sommarive 14, 38123, Povo, TN, Italy

    Andrea Pugliese, Mattia Sensi & Agnese Zardini

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  1. Tommaso Lorenzi
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  2. Andrea Pugliese
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Correspondence to Tommaso Lorenzi.

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Lorenzi, T., Pugliese, A., Sensi, M. et al. Evolutionary dynamics in an SI epidemic model with phenotype-structured susceptible compartment. J. Math. Biol. 83, 72 (2021). https://doi.org/10.1007/s00285-021-01703-1

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  • DOI: https://doi.org/10.1007/s00285-021-01703-1

Keywords

  • SI models
  • Mathematical epidemiology
  • Phenotypic heterogeneity
  • Spontaneous phenotypic changes
  • Partial integrodifferential equations

Mathematics Subject Classification

  • 35Q92
  • 35B40
  • 35R09
  • 92D30
  • 35J60