Journal of Fractional Calculus and Applications, Vol.. July, No., pp. 6. ISSN: 9-5858. http://www.fcaj.webs.com/ ON FRACTIONAL ORDER CANCER MODEL E. AHMED, A.H. HASHIS, F.A. RIHAN Abstract. In this work a cancer model is given. It has the following realistic features: It is fractional order models which are more suitable to model complex systems. Cross reactivity of the immune system is taken into consideration. The stable memory state of the immune system is obtained. Finally it is known that immune system is multi-functional- multi-pathways. Hence two immune effectors are used.. Introduction The use of fractional-orders differential and integral operators in mathematical models has become increasingly widespread in recent years [, ]. These models have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, economic, viscoelasticity, biology, physics and engineering [4, 5]. Recently, a large amount of literatures developed concerning the application of fractional differential equations in nonlinear dynamics [6]. In this paper, we study the fractional-order model with two immune effectors interacting with the cancer cells. In Section, we present the fractional-order model and its equilibrium points and conditions that guarantee the local asymptotically stability of the steady states. In Section, we provide an a stable implicit approach for solving the underlying model of differential fractional order. Conclusions are presented in Section 4. Now we give the definition of fractional-order integration and fractional-order differentiation: Definition. The fractional integral of order β R + of the function f, t > is defined by (Caputo sense) I β f = t (t s) β f(s)ds () Γ(β) and the fractional derivative of order α (n, n) of f, t > is defined by D α f = I n α D n f, D = d, n =,,... () dt Mathematics Subject Classification. 4A, 4A, 4D. Key words and phrases. Cancer-immune system; Fractional-order differential equations; Steady sates; Stability. Submitted Jan.,. Accepted March 7,, Published July,.
E. AHMED, A. HASHIS, F. RIHAN JFCA-/ The following properties are some of the main ones of the fractional derivatives and integrals (see [4, 5, 6]). Remark. Let β, γ R + and α (, ). Then (i) If Ia β : L L and f(x) L, then Ia β Ia γ f(x) = Ia β+γ f(x); (ii) lim Ia β f(x) = Ia n f(x) uniformly on [a, b], n =,,,..., where I β n af(x) = x f(s)ds; (iii) lim β I β a f(x) = f(x) weakly; (iv) If f(x) is absolutely continuous on [a, b], then lim D α f(x) = df(x) α dx ; (v) If f(x) = k, where k is a constant, then D α k =.. Fractional order model The behavior of most biological systems has memory or after-effects (such as the delay due to the incubation time for vectors to become infectious [6]). The modeling of these systems by fractional ordinary differential equations has more advantages than classical integer-order modeling, in which such effects are neglected. Studying immune system (IS) cancer interactions is an important topic (see []). Trying to become as realistic as possible is a common goal. The problem is to try to obtain the known biological features without making the mathematics too complicated. Here we include the following features: Immune system is multi-functional- multipathways. Hence two immune effectors are used. Also Cross reactivity of the immune system is taken into consideration. The model is given by: D α T = at r T E r T, D α E = d E + T E T + k, < α, () D α = d + T T, + k where T T is the tumor cells, E E, are the immune effectors, and a, r, r, d, d, k, k are positive constants. The interaction terms in the second and third equations of model () satisfy the cross reactivity property of the immune system. The equilibrium points of the system () are: E = (,, ); E = ( d k /( d ), a/r, ); = ( d k /( d ),, a/r ) (4) To avoid the non-biological interior solution where both immune effectors coexist, we assume that (d k /( d )) << (d k /( d )) (5) The first equilibrium E is the nave, the second E is the memory and the third is endemic according to the value of the tumor size. Stability analysis shows that the nave state is unstable. However, the memory state is locally asymptotically stable if: d < d, and d <. (6) While the endemic state is locally asymptotically stable if d < d, and d <. (7)
JFCA-/ ON FRACTIONAL ORDER CANCER MODEL Hence we arrive to the following propositions: Proposition. If (6) is satisfied then the memory state E is locally asymptotically stable. The endemic state is also asymptotically stable if condition (7) is satisfied. Hence there is bifurcation at d =. Notice that this result is valid only for the fractional order case where stability conditions are more relaxed than integer order ones; See [].. Suggested numerical technique for model () Consider biological models in the form of a system of fractional order differential equations of the form D α X = F (t, X), with X = X() ( < α ) (8) where X = [x, x,..., x n ] T and F (t, X satisfies the Lipschitz condition F (t, X F (t, Y K X Y, K >, where Y is the solution of the perturbed system. Given mesh T = {t, t,..., t N } a discrete approximation to the fractional derivative can be obtained by a simple quadrature formula, using the Caputo fractional derivative () of order α, and using implicit Euler s method as follows (see [6]): D α x i (t n ) = Setting = Γ( α) Γ( α) t j= ( α)γ( α) dx i (s) (t n s) α ds ds n [ ] jh x j i xj i + O(h) (nh s) α ds (j )h h {[ ] n x j i xj i [(n + O(h) j + ) α (n j) α]} h α h j= = ( α)γ( α) n [ h α j= ( α)γ( α) n [ j= x j i xj i x j i xj i ] [(n j + ) α (n j) α] + ] [(n j + ) α (n j) α] O(h α ). (9) G(α, h) = ( α)γ( α) h α and ωα j = j α (j ) α, () then the first-order approximation method for the computation of Caputo s fractional derivative is then given by the expression n ( ) D α x i (t n ) = G(α, h) ωj α x n j+ i x n j i + O(h). () j= Numerical simulations of the fractional order model (), for the given parameters in the captions, are displayed in the Figures 4. It has been proved that the fractional order implicit difference approximation () is unconditionally stable.
4 E. AHMED, A. HASHIS, F. RIHAN JFCA-/.6.4 T,E,..8.6 E.8.6.4..4 T. 4 6 8.5 E.5.5 T.5.5 E.8 T,E,.5 T.6.4..5 4 6 8 E.5..4 T.6.8 Figure. Left banners show the numerical solutions of the fractional order model () when α =.75 (for top banner) and α =.95 (for bottom banner) and when the conditions given in (4), (5) are satisfied (a = r = r = ; d =., d =.7, k =., k =.7). Right (top and bottom banners): The relation between the Tumor cells and Effector cells E,. The system converges to a stable steady state (the memory state is locally asymptotically stable)..5 T E.8 T, E,.5.6.4..5 4 6 8 E.5 T.5 Figure. Left banner shows the numerical solution of () with integer order α = with the same parameters of in Figure. Right banner banner shows the relation between the Tumor cells and Effector cells E,. This shows that when α = the system oscillates. Proposition. We have seen that the presence of a fractional differential order in a differential equation can lead to a notable increase in the complexity of the observed behaviour, and the solution is continuously depends on all the previous states.
JFCA-/ ON FRACTIONAL ORDER CANCER MODEL 5.5.5 T,E,.5.5 T.5.5 E 4 6 8 E.5 T.4..5 T,E,.8.6 T.5.4.5. E 4 6 8 E.5.5 T.5 Figure. Left banners show the numerical solutions of the fractional order model () when α =.75 (for top banner) and α =.95 (for bottom banner) and when the conditions given in (6) are satisfied (a = r = r = ; d =.7, d =., k =.7, k =.). Right (top and bottom banners): The relation between the Tumor cells and Effector cells E,. The endemic state is locally asymptotically stable..8.8.6.6.4.4...5 E 4 T 6 8 E.5 T 4 x 8 Figure 4. shows the stability (left) of the solution of the fractional order system () when d =.9 = d < and un-stability (right) of the solution when d = d =. >. 4. Conclusion In this paper the following aspects have been taken into consideration: Fractional order model is used. Holling type III which models the immune system cross reactivity has been used to model cancer-immune system interaction. Two immune effectors have been used to model the fact that immune system is multi-functional multi-pathways. We obtained memory state whose stability depends on the value
6 E. AHMED, A. HASHIS, F. RIHAN JFCA-/ of one parameter namely the immune effector death rate. We provided unconditionally stable method for the resulting system. We note that the fractional order dynamical systems are more suitable to model the tumor-immune system interactions than their integer order counterpart. Acknowledgment The work is supported by the National Research Foundation (UAE), research project # FOS/IRG-/. References [] Ahmed E. and Elsaka H. A.A. (), On modeling two immune effectors two strain antigen interaction, Nonlinear biomedical physics, 4:6. [] Ahmed E., A.M.A. El-Sayed, H.A.A. Elsaka,(6), On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rossler, Chua and Chen systems, Phys. Lett. A. 58,. [] Itik M. and Banks S. (), Chaos in a three dimensional cancer model, Int.J. Bifur. Chaos.,, 7. [4] Podlubny, I. (999), Fractional Differential Equations, Academic Press. [5] Samko, S.G., Kilbas, A.A., Marichev, O.I. (99), Fractional Integrals and Derivatives, Gordon and Breach Sciences Publishers. [6] Rihan, F.A. (), Computational methods for delay parabolic and time fractional partial differential equations, Num. Meth. Partial Diff. Eqns., 6 (6) 556 57. El-Sayed Ahmed Department of Mathematics, Faculty of Science, Mansora University, Mansora, Egypt E-mail address: magd45@yahoo.com Adel H. Hashish Department of Physics, Faculty of Science, UAE University, Al-Ain, 755, UAE E-mail address: ahashish@uaeu.ac.ae Fathalla A. Rihan Department of Mathematical Sciences, Faculty of Science, UAE University, Al-Ain, 755, UAE E-mail address: frihan@uaeu.ac.ae