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Some epidemiological models with nonlinear incidence.

H W Hethcote1, P van den Driessche

  • 1Department of Mathematics, University of Iowa, Iowa City 52242.

Journal of Mathematical Biology
|January 1, 1991
PubMed
Summary
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Nonlinear epidemiological models exhibit complex dynamics, including multiple equilibria and periodic solutions, differing significantly from standard bilinear models. These findings are crucial for understanding disease spread with temporary immunity and delayed removal.

Area of Science:

  • Mathematical Epidemiology
  • Dynamical Systems Theory
  • Disease Modeling

Background:

  • Traditional epidemiological models often use bilinear incidence rates, which may oversimplify complex disease dynamics.
  • Nonlinear incidence rates can lead to richer and more varied mathematical behaviors in disease transmission models.
  • Understanding disease dynamics is critical for public health interventions and policy-making.

Purpose of the Study:

  • To investigate the dynamic behaviors of epidemiological models incorporating nonlinear incidence rates.
  • To analyze models with vital dynamics, temporary immunity, and delayed removal.
  • To identify conditions leading to multiple equilibria and periodic solutions (Hopf bifurcation).

Main Methods:

  • Analysis of epidemiological models with nonlinear incidence.

Related Experiment Videos

  • Investigation of equilibria and their stability.
  • Application of Hopf bifurcation theory to identify periodic solutions.

    • Main Results:

    • Nonlinear incidence rates lead to significantly different dynamics compared to bilinear rates.
    • Models with temporary immunity and vital dynamics can exhibit multiple equilibria.
    • Hopf bifurcation can generate periodic solutions from endemic equilibria in these nonlinear models.
    • Analogous results were found for a second model featuring a delay in the removed class.

    Conclusions:

    • Nonlinear incidence rates are essential for capturing complex behaviors in epidemiological models.
    • The presence of multiple equilibria and periodic solutions highlights the potential for intricate disease dynamics.
    • These findings provide a more nuanced understanding of disease transmission patterns under specific epidemiological conditions.