Parametric dependence in model epidemics. I: contact-related parameters.

Area of Science:

• Nonlinear Dynamics
• Epidemiological Modeling
• Complex Systems Analysis

Background:

• Nonlinear dynamical systems exhibit bifurcations where small parameter changes cause qualitative behavioral shifts.
• Bifurcation diagrams visualize these dynamics, sometimes showing unique outcomes and other times coexisting sequences dependent on initial conditions.

Purpose of the Study:

• To investigate the emergence of unique parametric dependence (UPD) in an epidemiological model.
• To analyze how varying a second parameter simplifies multiple bifurcation sequences into a single, initial-condition-independent sequence.

Main Methods:

• Analysis of nonlinear dynamical systems and bifurcation theory.
• Modeling epidemiological dynamics with a focus on contact-related parameters.
• Examination of period-doubling, subharmonic resonance, attractor merging, and strange invariant sets.

Main Results:

• Demonstration of a phenomenon termed unique parametric dependence (UPD) in an epidemiological model.
• Identification of a second parameter that unifies disparate dynamical behaviors into a single sequence.
• Connection of UPD to established concepts like period-doubling and attractor merging.

Conclusions:

• The emergence of unique parametric dependence (UPD) offers a simplification in understanding complex epidemiological models.
• This finding integrates several theoretical threads in nonlinear dynamics developed since the 1980s.
• Contact-related parameters play a crucial role in simplifying model dynamics, with non-contact parameters to be explored separately.