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Transition to multitype mixing in d-dimensional spreading dynamics.

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Infectious disease spread depends on geography and population mixing. This study models disease dynamics using multitype branching processes, revealing a transition to multitype exponential growth in larger populations.

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Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Network Science

Background:

  • Infectious disease spread is influenced by geographic factors and population interactions.
  • Local populations exhibit homogeneous mixing, while human mobility connects distant populations.

Purpose of the Study:

  • To model the spatial dynamics of infectious disease spreading.
  • To analyze the impact of intra- and intertype mixing patterns on disease transmission.
  • To investigate the transition in growth patterns based on population size.

Main Methods:

  • Modeling spatial location as distinct types.
  • Utilizing multitype branching process theory to calculate infection numbers over time.
  • Employing eigenvalue problems for tridiagonal matrices in one dimension.
  • Extending analysis to d dimensions using graph Cartesian products.
  • Conducting numerical simulations to observe growth dynamics.

Main Results:

  • The analysis in one dimension simplifies to an eigenvalue problem of a tridiagonal Teoplitz matrix.
  • In d dimensions, eigenvalues and eigenvectors are constructed using graph Cartesian products.
  • Numerical simulations show a transition from linear to multitype mixing exponential growth as population size increases.

Conclusions:

  • The multitype mixing approximation is crucial for understanding disease spread in large populations.
  • This model is particularly relevant for countries with extensive networks of large cities (over 100,000 inhabitants).
  • The findings suggest multitype mixing is the dominant scenario for disease dynamics in many real-world settings.