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Case fatality models for epidemics in growing populations.

Karl Peter Hadeler1, Klaus Dietz2, Muntaser Safan3

  • 1Biomathematics, University of Tübingen, 72076 Tübingen, Germany.

Mathematical Biosciences
|September 27, 2016
PubMed
Summary

This study re-examines the SIR model for growing populations, introducing case fatality to better understand epidemiological dynamics and thresholds. It highlights differences in basic reproduction numbers between case fatality and differential mortality models.

Keywords:
Asymptotically homogeneous systemBasic reproduction numberCase fatalityEpidemic modelGrowing populationStability

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Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Population Dynamics

Background:

  • The SIR (Susceptible-Infected-Recovered) model is a cornerstone in epidemiological modeling.
  • Thieme's (1992) asymptotically homogeneous SIR model considers growing populations and general incidence functions.
  • Existing models often use differential mortality, which can obscure certain epidemiological insights.

Purpose of the Study:

  • To reconsider Thieme's SIR model with alternative parameterizations for clearer epidemiological insights.
  • To investigate the impact of using case fatality instead of differential mortality.
  • To analyze persistent distributions, growth exponents, and stability of infected states.

Main Methods:

  • Re-parameterization of the asymptotically homogeneous SIR model.
  • Analysis of epidemiological relations and thresholds, focusing on case fatality.
  • Computation and discussion of persistent distributions and growth exponents.
  • Interpretation of asymptotically exponentially growing states using stability theory.

Main Results:

  • Case fatality parameterization offers distinct insights compared to differential mortality.
  • The basic reproduction number is shown to depend on differential mortality but not on case fatality.
  • Persistent distributions and growth exponents of infected solutions were computed and analyzed.
  • Limiting cases without recovery reveal the existence of two infected solutions.

Conclusions:

  • The choice of parameterization (case fatality vs. differential mortality) significantly impacts model interpretation and derived epidemiological thresholds.
  • Understanding these parameterizations is crucial for accurate disease modeling in growing populations.
  • The study provides a deeper analysis of disease dynamics, stability, and long-term behavior in SIR models.