# American Institute of Mathematical Sciences

February  2017, 14(1): 1-15. doi: 10.3934/mbe.2017001

## 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  October 2016

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.

Citation: Emad Attia, Marek Bodnar, Urszula Foryś. Angiogenesis model with Erlang distributed delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 1-15. doi: 10.3934/mbe.2017001
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##### References:
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 $D_3$
Solutions of system (1.1) for parameters given by (3.1) and $\tau=10$, with time delay present only in the vessel formation term
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
Solutions of system (1.1) for parameters given by (3.1) and $\tau=5$, with time delay present in both terms
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
 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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