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Modeling relapse in infectious diseases.

P van den Driessche1, Xingfu Zou

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

Mathematical Biosciences
|November 23, 2006
PubMed
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This study models infectious disease relapses, like herpes, using an integro-differential equation. The basic reproduction number (R(0)) was established, showing no sustained oscillations in herpes relapse models.

Area of Science:

  • Mathematical modeling of infectious diseases
  • Epidemiology
  • Dynamical systems theory

Background:

  • Infectious diseases can exhibit relapse phenomena, complicating disease dynamics and control.
  • Herpes simplex virus is a common example of a disease with recurrent outbreaks.
  • Mathematical models are crucial for understanding disease transmission and relapse patterns.

Purpose of the Study:

  • To propose a novel integro-differential equation model for general infectious disease relapse.
  • To analyze the stability and threshold properties of the model using the basic reproduction number (R(0)).
  • To investigate the specific case of constant relapse periods and explore numerical simulations for herpes.

Main Methods:

  • Formulation of an integro-differential equation to capture relapse dynamics.

Related Experiment Videos

  • Identification and analysis of the basic reproduction number (R(0)).
  • Application of linear stability analysis, Lyapunov-Razumikhin technique, and monotone dynamical systems theory.
  • Numerical simulations using parameters relevant to herpes infections.

    • Main Results:

    • The basic reproduction number (R(0)) was identified and its threshold property established.
    • Theoretical analysis confirmed global results for models with constant relapse periods.
    • Numerical simulations for herpes did not reveal any evidence of sustained oscillatory solutions.
    • The model provides a framework for understanding relapse in various infectious diseases.

    Conclusions:

    • The proposed integro-differential equation model effectively captures general infectious disease relapse phenomena.
    • The basic reproduction number (R(0)) serves as a critical threshold for disease persistence.
    • The analysis, particularly for herpes, suggests that relapse dynamics may not inherently lead to sustained oscillations.
    • This modeling approach offers valuable insights for predicting and managing relapsing infectious diseases.