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A Kermack-McKendrick type epidemic model with double threshold phenomenon (and a possible application to Covid-19)

Joan Ponce1, Horst R Thieme2

  • 1School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287-1804, USA. jponce15@asu.edu.

Journal of Mathematical Biology
|February 13, 2026
PubMed

Insights

This study models how overcoming natural disease resistance influences epidemic spread, revealing an Allee effect that explains varying outbreak severity in similar geographic areas.

Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Infectious Disease Dynamics

Background:

  • K.L. Cooke (1967) proposed infected individuals become infective upon overcoming natural resistance.
  • Existing epidemic models often simplify host resistance mechanisms.

Purpose of the Study:

  • To incorporate host resistance overcoming into a Kermack-McKendrick epidemic model.
  • To investigate the impact of uniform versus distributed host resistance on epidemic dynamics.
  • To explain variations in epidemic severity using a deterministic model.

Main Methods:

  • Developed a Kermack-McKendrick type epidemic model incorporating post-latency age.
  • Analyzed scenarios with homogeneous and heterogeneous host resistance.
  • Examined the relationship between basic reproductive number and final epidemic size.
  • Investigated the generation of an Allee effect in the force of infection.

Main Results:

  • The model demonstrates threshold behavior for epidemic size concerning the basic reproductive number.
  • An Allee effect was observed for the final cumulative force of infection.
  • This effect arises from the interplay between primary infection force and overcoming resistance.
  • Model results provide a deterministic explanation for differing epidemic severities in similar regions.

Conclusions:

  • Host resistance overcoming is a critical factor in epidemic modeling.
  • The generated Allee effect offers a novel explanation for epidemic severity disparities.
  • This modeling approach enhances understanding of infectious disease spread and control strategies.
Keywords:
Cumulative doseDensity-dependent incidenceFinal size curveHost heterogeneityMinimal solutionsState-dependent delay

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