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Relating Eulerian and Lagrangian spatial models for vector-host disease dynamics through a fundamental matrix.

Esteban Vargas Bernal1, Omar Saucedo2, Joseph Hua Tien3

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This study links Eulerian and Lagrangian models for disease spread. Consistent models ensure host movement patterns match, impacting outbreak size and reproduction number predictions.

Keywords:
Eulerian approachFundamental matrixLagrangian approachMovementSpatial modelsVector-borne disease

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Area of Science:

  • Epidemiology
  • Mathematical Modeling
  • Disease Ecology

Background:

  • Modeling host movement is crucial for understanding vector-borne disease dynamics.
  • Eulerian and Lagrangian approaches offer distinct frameworks for spatial modeling.
  • Discrepancies between these models can affect disease transmission predictions.

Purpose of the Study:

  • To establish a mathematical relationship between Eulerian and Lagrangian host movement models in discrete spaces.
  • To define consistency between these two modeling frameworks based on residence times.
  • To analyze the impact of model consistency on disease outbreak size and basic reproduction number.

Main Methods:

  • Representing host movement using a graph Laplacian matrix (L) in the Eulerian approach.
  • Utilizing a mixing matrix (P) to capture time spent in foreign patches in the Lagrangian approach.
  • Deriving conditions for consistency between L and P matrices and examining disease metrics.

Main Results:

  • A sufficient condition for consistency between Eulerian (L) and Lagrangian (P) models was identified.
  • Disease quantities like final outbreak size and basic reproduction number were analyzed in both consistent and inconsistent scenarios.
  • In a two-patch model, similar disease outcomes were observed even with inconsistent models, highlighting potential discrepancies.

Conclusions:

  • The relationship between Eulerian and Lagrangian models is defined by matching residence times.
  • Model consistency is not always required for similar disease outcome predictions, particularly in simple systems.
  • Significant differences in final outbreak sizes can arise between consistent and inconsistent models, emphasizing the importance of the proposed relationship.