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Stability analysis for models of diseases without immunity

H W Hethcote, H W Stech, P van den Driessche

    Journal of Mathematical Biology
    |January 1, 1981
    PubMed
    Summary
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    This study models infectious disease spread in a closed population using distributed delays. Results show that disease dynamics either lead to extinction or an endemic equilibrium, regardless of delay.

    Area of Science:

    • Epidemiology
    • Mathematical Biology
    • Dynamical Systems

    Background:

    • Understanding disease transmission dynamics is crucial for public health interventions.
    • Constant parameter models offer a simplified view of infectious disease spread.
    • Incorporating delays can provide a more realistic representation of disease progression.

    Purpose of the Study:

    • To develop and analyze a cyclic epidemiological model with distributed delays.
    • To investigate the impact of distributed delays on disease extinction and endemic states.
    • To determine the stability of disease-free and endemic equilibria in the presence of delays.

    Main Methods:

    • Formulation of a compartmental epidemiological model (Susceptible-Exposed-Infectious).
    • Introduction of distributed delays to represent disease progression.

    Related Experiment Videos

  • Mathematical analysis using coupled Volterra integral equations.
  • Stability analysis of model equilibria.

    • Main Results:

    • The model incorporates susceptible, exposed, and infectious classes within a closed population.
    • Distributed delays were introduced, leading to a formulation with two coupled Volterra integral equations.
    • Analysis confirmed that delays do not alter the fundamental nature of disease thresholds or asymptotic stability.
    • The disease trajectory invariably leads to either extinction or an endemic steady state.

    Conclusions:

    • The developed epidemiological model with distributed delays provides insights into disease dynamics.
    • The presence of distributed delays does not qualitatively change the long-term outcomes of disease spread.
    • The model predicts that infectious diseases will either disappear or persist at a constant level in the population.