Angiogenesis model with Erlang distributed delays

Emad Attia; Marek Bodnar; Urszula Foryś

doi:10.3934/mbe.2017001

Article Contents

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: November 23, 2015

Accepted: April 06, 2016

Published: September 2017

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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.

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Mathematics Subject Classification: Primary: 34K11, 34K13, 34K18, 37N25; Secondary: 92B05.

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Figure 1. Critical average delay, that is $m/a_\text{cr}$ for various values of $m$ in the dependance on $\delta$ in the case when only the process of tumour growth is delayed; left -graphs for the steady state $D_1$, right -graphs for the steady state $D_3$

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Figure 2. Solutions of system (1.1) for parameters given by (3.1) and $\tau=10$, with time delay present only in the vessel formation term

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Figure 3. Solutions of system (1.1) for parameters given by (3.1) and $\tau=10$, with time delay present only in the tumour growth term. Here, for Erlang distribution, the steady state is stable, and solutions for $m=1$ and $m=5$ are almost identical

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Figure 4. Solutions of system (1.1) for parameters given by (3.1) and $\tau=5$, with time delay present in both terms

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Table 1. Critical values of $\tau$ at which the positive steady state loses stability

	steady state $D_1$					steady state $D_3$
$\delta$	0.332	0.346	0.36	0.368	0.378	0.3	0.332	0.346	0.36	0.368
discrete	66.7	33.4	29.3	43.6	182	4.49	5.89	7.53	13.0	94.0
$m=1$	steady state does not lose stability
$m=2$	176	54.7	69.1	106.1	460	5.58	9.36	14.4	32.2	284
$m=5$	89.9	29.6	37.4	56.6	234	4.03	5.97	8.34	16.6	135

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