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Modeling diseases with latency and relapse.

P van den Driessche1, Lin Wang, Xingfu Zou

  • 1Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada, V8W 3P4. pvdd@math.uvic.ca

Mathematical Biosciences and Engineering : MBE
|July 31, 2007
PubMed
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This study introduces a mathematical model for diseases with latent periods and relapses, like tuberculosis. The basic reproduction number (R(0)) predicts disease extinction or endemic states, crucial for understanding disease dynamics.

Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Disease Modeling

Background:

  • Many infectious diseases, such as tuberculosis and herpes, feature a latent (exposed) period and the possibility of relapse.
  • Understanding disease dynamics is critical for effective public health interventions and disease management.

Purpose of the Study:

  • To develop a general mathematical model for diseases characterized by an exposed period and relapse.
  • To identify and analyze the threshold properties of the basic reproduction number (R(0)) for disease persistence.

Main Methods:

  • Formulation of a general mathematical model incorporating an exposed class and relapse.
  • Analysis of the basic reproduction number (R(0)) and its relationship to disease-free and endemic equilibria.
  • Utilizing delay differential equations for specific probability distributions (step-function).

Related Experiment Videos

  • Numerical simulations to validate model predictions, particularly for bovine tuberculosis.

    • Main Results:

    • The disease-free equilibrium is globally asymptotically stable when R(0) < 1.
    • R(0) = 1 serves as a sharp threshold for disease extinction versus endemicity under exponential distribution assumptions.
    • For a step-function probability, the endemic equilibrium is locally asymptotically stable if R(0) > 1.
    • Numerical simulations confirm that infected populations tend towards the endemic state when R(0) > 1.

    Conclusions:

    • The proposed mathematical model effectively captures the dynamics of diseases with latent periods and relapse.
    • The basic reproduction number (R(0)) is a key determinant of disease persistence, distinguishing between extinction and endemicity.
    • The model provides a framework for predicting disease spread and stability, with implications for diseases like bovine tuberculosis.