Angiogenesis model with Erlang distributed delays

Emad Attia; Marek Bodnar; Urszula Foryś; Emad Attia; Marek Bodnar; Urszula Foryś

doi:10.3934/mbe.2017001

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 1: 1-15. doi: 10.3934/mbe.2017001

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Angiogenesis model with Erlang distributed delays

1.
Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt
2.
Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw Banacha 2, 02-097 Warsaw, Poland

Received: 23 November 2015 Accepted: 06 April 2016 Published: 01 February 2017
MSC : Primary: 34K11, 34K13, 34K18, 37N25; Secondary: 92B05

We consider the model of angiogenesis process proposed by Bodnar and Foryś (2009) with time delays included into the vessels formation and tumour growth processes. Originally, discrete delays were considered, while in the present paper we focus on distributed delays and discuss specific results for the Erlang distributions. Analytical results concerning stability of positive steady states are illustrated by numerical results in which we also compare these results with those for discrete delays.
- Delay differential equations,
- stability analysis,
- Hopf bifurcation,
- angiogenesis,
- tumour growth
Citation: Emad Attia, Marek Bodnar, Urszula Foryś. Angiogenesis model with Erlang distributed delays[J]. Mathematical Biosciences and Engineering, 2017, 14(1): 1-15. doi: 10.3934/mbe.2017001

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Abstract

We consider the model of angiogenesis process proposed by Bodnar and Foryś (2009) with time delays included into the vessels formation and tumour growth processes. Originally, discrete delays were considered, while in the present paper we focus on distributed delays and discuss specific results for the Erlang distributions. Analytical results concerning stability of positive steady states are illustrated by numerical results in which we also compare these results with those for discrete delays.

References

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