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Integral equation models for endemic infectious diseases

H W Hethcote, D W Tudor

    Journal of Mathematical Biology
    |March 1, 1980
    PubMed
    Summary
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    This study models endemic infectious diseases with permanent immunity using nonlinear integral equations. Findings show that distributed delays do not alter disease thresholds or long-term population behavior.

    Area of Science:

    • Mathematical epidemiology
    • Infectious disease modeling
    • Nonlinear dynamics

    Background:

    • Endemic infectious diseases conferring permanent immunity pose significant public health challenges.
    • Mathematical models are crucial for understanding disease dynamics and control strategies.
    • Previous models often simplified the duration of the infectious period.

    Purpose of the Study:

    • To develop and analyze a mathematical model for endemic infectious diseases with permanent immunity.
    • To incorporate vital dynamics, immunization, and a distributed infectious period into the model.
    • To determine the well-posedness, threshold criteria, and asymptotic behavior of the proposed model.

    Main Methods:

    • Utilized a system of nonlinear Volterra integral equations of convolution type.

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  • Incorporated constant parameters for vital dynamics (births and deaths), immunization, and distributed infectious periods.
  • Applied mathematical analysis to determine model well-posedness, threshold criteria, and asymptotic behavior.

    • Main Results:

    • The developed models were demonstrated to be well-posed.
    • Threshold criteria governing disease persistence were determined.
    • Asymptotic behavior analysis revealed key long-term dynamics.

    Conclusions:

    • Distributed delays within the infectious period do not affect the fundamental threshold criteria for disease persistence.
    • The asymptotic behaviors of the models remain unchanged despite the inclusion of distributed delays.
    • The study provides a robust mathematical framework for analyzing infectious diseases with permanent immunity.