<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In this paper, we adopt the model of [<xref ref-type="bibr" rid="j_amns.2019.2.00035_ref_012_w2aab3b7c39b1b6b1ab2b1c12Aa">12</xref>] by including fuzzy initial values to study the interaction of a monoclonal brain tumor and the macrophages for a condition of extinction of GB(Glioblastoma) by using Allee threshold. Numerical simulations will give detailed information on the behavior of the model at the end of the paper. We perform all the computations in this study with the help of the Maple software.</p></abstract>

In this paper, we adopt the model of [12] by including fuzzy initial values to study the interaction of a monoclonal brain tumor and the macrophages for a condition of extinction of GB(Glioblastoma) by using Allee threshold. Numerical simulations will give detailed information on the behavior of the model at the end of the paper. We perform all the computations in this study with the help of the Maple software.

An Allee Threshold Model for a Glioblastoma(GB)-Immune System(IS) Interaction with Fuzzy Initial Values

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution 4.0 International License.

In this paper, we adopt the model of [12] by including fuzzy initial values to study the interaction of a monoclonal brain tumor and the macrophages for a...

Time(t)	v	x	u	s	y	r
0.0	2.0000	1.6000	1.2000	0.3000	0.2000	0.10000
0.1	1.9800	1.6800	1.3762	0.3037	0.20950	0.11499
0.2	2.0005	1.7767	1.5461	0.30812	0.2194	0.13010
0.3	2.0589	1.8931	1.7179	0.31326	0.22972	0.14535
0.4	2.1543	2.0329	1.8995	0.31909	0.24046	0.16078
0.5	2.2876	2.2001	2.0980	0.3256	0.25164	0.17639
0.6	2.4605	2.3992	2.3203	0.33279	0.26324	0.19222
0.7	2.6760	2.6350	2.5731	0.34064	0.27528	0.20826
0.8	2.9375	2.9123	2.8626	0.34914	0.28775	0.22454
0.9	3.2487	3.2361	3.1948	0.35826	0.30065	0.24105
1.0	3.6133	3.6107	3.5750	0.36799	0.31397	0.25780

E	Allee constant	1.23
r₁	The growth rate of the macrophage	0.55
r₂	The growth rate of the macrophage	0.5
p	Division rate of the sensitive cells	0.148
K₁	Carriying capacity of the tumor cells	4.704
K₂	Carriying capacity of the macrophages	1.132
α₁	Logistic population rate of tumor cell population	α₁ɛ[0.5;0.95]
τ_i	Destroying rate caused from the interaction	0.01
d₁	Causes of drug treatment to the tumor cells	0.55
d₂	Causes of drug treatment to the macrophages	0.06
β₁	Logistic population rate of macrophages	β₁ɛ[0.05;0.25]

An Allee Threshold Model for a Glioblastoma(GB)-Immune System(IS) Interaction with Fuzzy Initial Values

Published Online: Oct 22, 2019

Page range: 499 - 508

Received: Mar 23, 2019

Accepted: Apr 22, 2019

DOI: https://doi.org/10.2478/amns.2019.2.00035

Keywords
Fuzzy number, Fuzzy derivative, Fuzzy differential equations (FDE), Fuzzy initial values

© 2020 Fatma Berna Benli et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Numerical values of Figure 3.

Values of the parameter of system (1).

An Allee Threshold Model for a Glioblastoma(GB)-Immune System(IS) Interaction with Fuzzy Initial Values

Published Online: Oct 22, 2019

Page range: 499 - 508

Received: Mar 23, 2019

Accepted: Apr 22, 2019

DOI: https://doi.org/10.2478/amns.2019.2.00035

KeywordsFuzzy number, Fuzzy derivative, Fuzzy differential equations (FDE), Fuzzy initial values

© 2020 Fatma Berna Benli et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Numerical values of Figure 3.

Values of the parameter of system (1).

Keywords
Fuzzy number, Fuzzy derivative, Fuzzy differential equations (FDE), Fuzzy initial values