Application of Non-singular Kernel in a Tumor Model with Strong Allee Effect

Vol.:(0123456789) Differential Equations and Dynamical Systems https://doi.org/10.1007/s12591-022-00622-x 1 3 ORIGINAL RESEARCH Application ofNon‑singular Kernel inaTumor Model withStrong Allee Effect SubhasKhajanchi1· MrinmoySardar2· JuanJ.Nieto3 Dedicated to the memory of Professor H. I. Freedman. Accepted: 7 December 2022 © The Author(s) 2022 Abstract We obtain the analytical solutions in implicit form of a tumor cell population differential equation with strong Allee effect. We consider the ordinary case and then a fractional version. Some particular cases are plotted. Keywords Tumor growth model· Allee effect· Caputo-Fabrizio fractional derivative· Non-singular kernel Introduction The mathematical modeling and analysis of tumor growth is a crucial point to understand different aspects of cancer and to discover potential treatments [1–4]. Prof. H.I. Freedman has made important contributions in tumor-immune competitive systems and their application to chemotherapy [5, 6]. In this paper, we investigate a tumor model with strong Allee effect in the following form [7] (1) dT dt =𝛼T ( 1− T k )( T−c ), * Juan J. Nieto [email protected] Subhas Khajanchi subhaskhajanc[email protected] Mrinmoy Sardar sardar.mr[email protected] 1 Department ofMathematics, Presidency University, 86/1 College Street, Kolkata700073, India 2 Department ofMathematics, Jadavpur University, Kolkata700032, India 3 CITMAga, Universidade de Santiago de Compostela, 15782SantiagodeCompostela, Spain Differential Equations and Dynamical Systems 1 3 with initial size of the tumor population is T(0)=T0≥0 . We assume that tumor cells grow logistically as tumor growth slows down due to insufficient nutrients. Here, 𝛼 represents the intrinsic growth rate and k is the maximum carrying capacity of tumor cells [8]. It can be observed that 0, k and c are constant solutions. The parameter c≥0 represents the strong Allee threshold. The growth of population subject to Allee effects is reduced at low density and it is related to the existence of a threshold size for the visibility of that population and they may altered the long-term persistence of the population. Note that for T between 0 and k, T is increasing if T>c , and T is decreasing if T<c . There is a long history of fractional order derivative and it plays a vital role in nonlinear mathematical model. There are some limitations in integer derivatives of nonlinear models. Therefore the application of fractional order derivative is very important in the mathematical model. In fractional order differential equation, there are mainly two types of derivatives known as Caputo-Fabrizio derivative and Riemann-Liouville derivative. In our study, we used Caputo-Fabrizio fractional order derivative. The following fractional derivative of the model (1) has been considered as with 𝜆∈(0, 1) and D𝜆 the Caputo fractional derivative of a C1 function and T is defined as In this paper, we investigate the fractional version of the model (1) where 𝔻 is the Caputo-Fabrizio fractional derivative [9–11] and in the line of [12]. The paper is organized in the following way. We investigate the Caputo-Fabrizio fractional order derivative and its corresponding fractional integral in the Sect.2. We implicitly solved the Caputo-Fabrizio fractional derivative with order 𝜆∈(0, 1) of the proposed model (1) in the next Section. In the same Section, we plot the fractional Caputo-Fabrizio logistic equation with strong Allee effect (7) for different values of 𝜆 by using MATHEMATICA. The paper ends with a brief conclusion. Fractional Calculus withNon‑singular Kernel Let us assume that 𝜆∈(0, 1) . The classical Riemann-Lioville fractional integral is given by We have [13], and (2) D 𝜆T(t)=𝛼T ( 1−T k)( T−c ), D 𝜆T(t)= 1 Γ(1− 𝜆 )∫t 0 (t−x)−𝜆T�(x)dx . (3) 𝔻 𝜆T(t)=𝛼T ( 1−T k)( T−c ), J 𝜆T(t)= 1 Γ( 𝜆 ) ∫ t 0 (t−x)𝜆−1T(x)dx . D𝜆J𝜆T(t)=T(t), J𝜆 D 𝜆 T(t)=T(t)+K 1, Differential Equations and Dynamical Systems 1 3 where K1 is an arbitrary constant. For real smooth function g, the Caputo-Fabrizio fractional derivative is given by It is corresponding to the Caputo fractional derivative replacing the constant 1 Γ(1− 𝜆 ) by 1 1−𝜆 and the singular kernel by another kernel Thus, the corresponding fractional integral [14, 15] of the function h is given by Again, we have K1 is an arbitrary constant. Due to [15], we point out that This is quite different to the first order integral and derivative, the Caputo fractional derivative and the Riemann-Liouville fractional derivative and integral. Solution fortheFractional Differential Equation Assume that T is a solution of (3). Integration leads to where Therefore by (6), we get where T(0)=T0 is the initial value. Taking derivative both sides of the above equation leads to (4) 𝔻 𝜆g(t)= 1 1− 𝜆 ∫t 0 e−𝜆 1−𝜆(t−x)g�(x)dx . m(t,x)=(t−x)−𝜆 n(t,x)=e − 𝜆 1−𝜆(t−x) . (5) 𝕀 𝜆h(t)=(1−𝜆)[h(t)−h(0)] + 𝜆∫ t 0 h(x)dx . (6) 𝕀𝜆 𝔻 𝜆g(t)=g(t)+K1, 𝔻 𝜆𝕀𝜆 g(t)=g(t)−g(0)e − 𝜆 1−𝜆t . 𝕀𝜆 𝔻 𝜆T(t)= 𝕀 𝜆P(t), P (t)=𝛼T(t) ( 1−T(t) k)( T(t)−c ). T (t)−T0=(1−𝜆) [ 𝛼T(t) ( 1−T(t) k )( T(t)−c ) −𝛼T0 ( 1− T0 k )( T0−c )] +𝜆∫t 0 𝛼T(s) ( 1−T(s) k )( T(s)−c ) ds, Differential Equations and Dynamical Systems 1 3 We notice that if 𝜆=1 then we recover the Allee model (1). After some algebraic manipulation, for 0<𝜆<1 equation (7) becomes with Solving above equation by integration, we get where Putting t=0 in the above equation, we have where Hence, we obtained the solution in an implicit form as (7) T �(t) =(1−𝜆) [ 𝛼T(t) ( 1−T(t) k ) T�(t)+𝛼(T(t)−c) ( T�(t)−2T(t)T � (t) k )] +𝜆𝛼T(t) ( 1−T(t) k)( T(t)−c ) . T �(t)= 𝛼𝜆 3(𝜆−1) ⋅ T 3 (t)−a1T 2 (t)+a2T(t ) T2(t)−2 3a1T(t)+a3 ⇒l=T�(t) T2(t)−2 3a1T(t)+a3 T3 ( t )− a 1 T2 ( t )+ a 2 T ( t ) , a 1 =(k+c), a 2=kc, a 3=1+𝛼c(1−𝜆) 3𝛼(1−𝜆) k , l=𝛼𝜆 3( 𝜆 −1) . lt +c1=1 3ln |T2(t)|+ (a 3 a2 −1 3 ) ln |T(t)|+ ( 1 6− a 3 2a2 ) ln |T3(t)| +(a1a3 2a2 − a1 6)1 c−k(ln |T(t)−c|−ln |T(t)−k|),[c1is an integral constant. ] ⇒ exp(lt +c1) =(T2(t)) 1 3(T(t)) ( a3 a2 −1 3 ) (T3(t)) ( 1 6−a3 2a2 )( T(t)−c T(t)−k) 3a1a3−a1a2 6a2(c−k) , T 2(t)=T 3 (t)−a1T 2 (t)+a2T(t) , T3( t )= T2 ( t )− a 1 T ( t )+ a 2. exp (c1)=(T2(0)) 1 3T(a3 a2 −1 3) 0(T3(0)) ( 1 6−a3 2a2 )( T0−c T 0 −k ) 3a 1 a 3 −a 1 a 2 6a2(c−k) , T 2(0)=T 3 0−a1T 2 0+a2T0 , T 3 (0)=T2 0 −a 1 T 0 +a 2 . Differential Equations and Dynamical Systems 1 3 We plot the solutions of the proposed model (7) by choosing a suitable set of parameter values and for different values of 𝜆 . To numerically plot the solutions of (7), we use the initial size of the tumor cell population is T0=0.5 . To compare the solution of the classical (𝜆=1) logistic differential equation with Allee effect (1), and the Caputo-Fabrizio fractional logistic equation with Allee effect (7) for 𝜆=1∕2 and 𝜆=1∕4 . See the Fig.1. Rest of the parameter values are chosen as 𝛼 = 1, k = 1 and c = 0.25. The solution of the classical logistic equation with strong Allee effect is above the solution of the fractional order Caputo-Fabrizio logistic equation with strong Allee effect (7) and above all the solution of the Caputo-fractional logistic differential equation. Conclusions In this paper, we proposed a mathematical model of tumor cell population with strong Allee effect, which is constructed by Sardar etal. [7] . Then we introduce fractional order derivative in our model. To solve the fractional logistic differential equation with Allee effect, we use fractional calculus with non-singular kernel. We calculate the analytical solutions in implicit form of the tumor cell population with strong Allee effect. Finally, we plot our solutions for different values of 𝜆 . Acknowledgements The research of Juan J. Nieto has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain under grant PID2020-113275GB-I00, and by Xunta de Galicia, grant ED431C 2019/02. Subhas Khajanchi acknowledges the financial support from the Department of Science and Technology (DST), Govt. of India, under the Scheme “Fund for Improvement of S &T Infrastructure (FIST)” [File No. SR/FST/MS-I/2019/41]. (8) e lt ⋅(T2(0)) 1 3T ( a3 a2 −1 3 ) 0(T3(0)) ( 1 6−a3 2a2 )( T0−c T0−k ) 3a 1 a 3 −a 1 a 2 6a2(c−k) =(T2(t)) 1 3(T(t)) ( a3 a2 −1 3 ) ⋅ (T3(t))(1 6−a3 2a2)(T(t)−c T(t)−k) 3a1a3−a1a2 6a2(c−k) , ⇒exp(lt)= ( T2(t) T 2 (0) ) 1 3 ( T(t) T 0) (a3 a2 −1 3) ( T3(t) T 3 (0) ) (1 6−a3 2a2) ( (T(t)−c)(T0−k) (T(t)−k)(T 0 −c) ) 3a1a3−a1a2 6a2(c−k) . Fig. 1 Solution of the ordinary logistic differential equation with strong Allee effect for the initial value T0 = 0.5. Classical logistic equation with strong Allee effect (blue colour), fractional Caputo-Fabrizio logistic equation with strong Allee effect (7) for 𝜆=1∕2 (black) and for 𝜆=1∕4 (green) Differential Equations and Dynamical Systems 1 3 Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Declarations Conflict of interest The authors declare that they have no conflict of interest. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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