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Angiogenesis model with Erlang distributed delays.

Emad Attia1, Marek Bodnar, Urszula Forys

  • 1Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt.

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Summary
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This study analyzes the angiogenesis model with distributed time delays, focusing on Erlang distributions. Results show how these delays impact vessel formation and tumor growth stability compared to discrete delays.

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Area of Science:

  • Mathematical Biology
  • Tumor Microenvironment Dynamics
  • Angiogenesis Modeling

Background:

  • The Bodnar and Foryś (2009) model describes angiogenesis, the process of new blood vessel formation.
  • Previous models incorporated discrete time delays in vessel formation and tumor growth.
  • Understanding the impact of time delays is crucial for cancer research and treatment strategies.

Purpose of the Study:

  • To extend the Bodnar and Foryś angiogenesis model by incorporating distributed time delays.
  • To analyze the stability of positive steady states in the presence of distributed delays, specifically using Erlang distributions.
  • To compare the effects of distributed delays with previously studied discrete delays.

Main Methods:

  • Mathematical modeling of the angiogenesis process.
  • Analysis of time-delayed differential equations.
  • Application of stability theory to determine steady-state behavior.
  • Numerical simulations to illustrate analytical findings and compare delay types.

Main Results:

  • Analytical conditions for the stability of positive steady states were derived for distributed delays.
  • Specific results were obtained for the Erlang distribution, a type of distributed delay.
  • Numerical simulations confirmed the analytical stability results.
  • Comparisons demonstrated differences in stability behavior between distributed and discrete time delays.

Conclusions:

  • Distributed time delays, particularly those modeled by Erlang distributions, significantly influence the stability of angiogenesis models.
  • The choice of delay distribution (distributed vs. discrete) impacts predictions of tumor growth and vessel formation.
  • This extended model provides a more nuanced understanding of angiogenesis dynamics relevant to cancer progression.