Jordan Journal of Mathematics and Statistics (JJMS) 9(2), 2016, pp

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1 Jordan Journal of Mathematics and Statistics JJMS) 92), 2016, pp HOPF BIFURCATION ANALYSIS IN A SYSTEM FOR CANCER VIROTHERAPY WITH EFFECT OF THE IMMUNE SYSTEM A. ASHYANI 1), H.M. MOHAMMADINEJAD 2) AND O. RABIEIMOTLAGH 3) Abstract. We consider a system of differential equations which is motivated biologically and simulates a cancer virotherapy. The existence of equilibrium points and their local stability are studied using the characteristic equation. We investigate Hopf bifurcation around the interior equilibrium point. 1. Introduction Cancer is usually nown by an unnatural growth in cell numbers which is called tumor and sometimes causes to death. Hence, Finding an effective method for controlling the growth rate of tumor always considered as a very important issue in medical science. Many efforts have been done to promote therapeutic ways for cancer. Surgery is the oldest one but it also contains collateral effects so that it is preferable that tumor is treated or controlled without surgery. Cancer virotherapy is the new way which is used recently and it is effective in treatment and it uses virus to damage tumor cells. This method of therapy has been nown in the early twentieth century. However, until 1949, serious experiments about it can not be found [6, 8, 4]. In the first experiments, virus found in the nature were used, but it failed because the immune system response damped the infection. Thus, prevented the virus from destroying the cancer [6, 3, 9]. The next step was done 2000 Mathematics Subject Classification. 34A34, 34D20, 34D23. Key words and phrases. Stability, Asymptotic Stability, Hopf bifurcation. Copyright c Deanship of Research and Graduate Studies, Yarmou University, Irbid, Jordan. Received: Aug. 10, 2015 Accepted: March 9,

2 94 A. ASHYANI, H.M. MOHAMMADINEJAD AND O. RABIEIMOTLAGH by genetic researchers. They produced a new generation of viruses in the laboratory which were capable for cancer virotherapy. The most important fact in the cancer treatment is the fewer non-tumor cells destroyed, the better treatment it could be. The best choice to achieve this goal is cancer virotherapy. These days, such viruses are categorized as oncolytic viruses. There is a ind of iller cells in body which is nown as cytotoxic T lymphocytes CTL). It destroy foreign cells by nowing their special signals that have been sent to immune system. Recognizing cancer cells by immune system tumor-specific CTL starts to destroy them. Although, it may not damage all cancer cells, the cancer virotherapy after injection virus-specific CTL starts to now viruses and try to ill them. Therefore, both viruses and infected cells will be attaced by CTL. On the other hand, it helps to immune system for nowing more tumor cells which have not been recognized before. Then, tumor-specific CTL affects on both infected and non-infected cells to ill them [2, 5]. Experimental observations show that the interaction between the oncolytic virus and tumor might be too complicated. For this reason, it is almost impossible to figure out a complete theory, for finding all relations and analyzing them using the biological observations only. These convince researches to construct mathematical models simulating the virotherapy for studying unnown aspects of the phenomenon. Many of these effective models which are previously developed, based on system of differential equations [13, 12, 11, 7]. In 2001 Wodarz proposed the following system [13, 12]: 1.1) ẋ = rx1 x + y x + y ) dx βxy, ẏ = βxy + sy1 ) ay where yt) and xt) are the number of infected and non-infected cells, respectively. Parameters s and r are the growth rate of infected and non-infected cells in a logistic

3 HOPF BIFURCATION FOR CANCER VIROTHERAPY 95 model, respectively. Size of tumor of infected and non-infected tumor cells are considered to be limited by the carrying capacity. If the total number of infected and non-infected tumor cells becomes more than, the patient will die. Because of this fact we should have x + y <. Parameter a is the death rate of infected cells by virus, and d is the death rate of non-infected cells by the immune system. Finally, β is the rate of viral infection spread in tumor cells. In the same article, Wodarz presented the following system: 1.2) ẋ = rx1 x + y ) dx βxy P txz t, ẏ = βxy + sy1 x + y ) ay P vyz v P t yz t, Z v = C v yz v bz v, Z t = C t yz t x + y) bz t. This system contains four variables. Here, the number of virus-specific CTL and tumor-specific CTL are denoted by Z v and Z t, respectively. The rate of virus-specific CTL proliferate is C v yz v and the rate of illing the infected cells is P v yz v. Virusspecific CTL will die at a rate bz v in the absence of infection. Tumor-specific CTL propagates at a rate C t on infected and non-infected cells x + y). However, The warning signal, which is led to nowing tumor, has been recognized to be more useful in the presence of y virus. Finally, Tumor-specific CTL ills infected and non-infected cells at a rate P t Z t. Wodarz introduced the System 1.2), but he did not present further mathematical analysis or simulation for it. In 2014, we analyzed System 1.1) [1]. Although, we now immune system response is an important factor which effect on virus therapy. Thus, we want to improve System 1.1) by considering immune system response with separate variable, which helped us to provide results that is closer than biology happening. We consider both CTL responses in one variable because some structure

4 96 A. ASHYANI, H.M. MOHAMMADINEJAD AND O. RABIEIMOTLAGH of responses for virus-specific CTL and tumor-specific CTL are similar. In addition, this assumption does not change the system and mae its analysis easier by restricting the system into a third dimensional model. In this paper, we introduce the system for interaction between non-infected cells, infected cells and CTL response as 1.3) 1.4) 1.5) ẋ = rx1 x + y ) βxy p 1xz, ẏ = sy1 x + y ) + βxy p 2yz ay, ż = c 1 xz + c 2 yz bz. The non-infected cells death with response of the immune system is presented by the term p 1 xz. Infected tumor cells die with two process. They die with infection at a rate a and they die with response immune system which is given by the term p 2 yz. On the other hand, CTL for tumor grows is presented by the term c 1 xz and CTL for virus grows by the term c 2 yz. Moreover, in the absence of any infection, the cells will die at a rate b. All the parameters of the system are non negative with initial population conditions x0) 0, y0) 0, z0) 0. This paper is organized as follows. Section 2 contains the positivity and the boundedness of the model. In Section 3, we investigate the conditions for existence at positive equilibrium points and stability of important equilibrium points. We demonstrate the occurrence of Hopf bifurcation in Section 4.In Section 5, we present the numerical results. Finally, we draw some conclusion in Section Positively and boundedness of the solutions The positivity and boundedness of the system ) are necessary to mae the system has biologically meaningful. Therefore, we study them first.

5 HOPF BIFURCATION FOR CANCER VIROTHERAPY 97 Theorem 2.1. All the solutions of system ) which starting in the R 3 + remain positive. Also, they remain bounded in the region Ω R 3 + defined by Ω = {x, y, z) R 3 + x, x + y, x + y + p r + s) z c m }. Proof. Equation 1.3) can be written as dx/x = φ 1 x, y, z)dt, where φ 1 x, y, z) = r1 x+y ) βy p 1z. Then xt) = x0)e t 0 φ1x,y,z)dt 0 for all t 0 as x0) 0. Equation 1.4) gives dy/y = φ 2 x, y, z)dt, where φ 2 x, y, z) = s1 x+y ) + βx p 2z a. Then yt) = y0)e t 0 φ2x,y,z)dt 0 for all t 0 as y0) 0. Similarly, zt) 0 for all t 0. On the other hand, dx rx1 x+y ) rx1 x ). Hence, we consider dt du/dt = ru1 U ), with solving this differential equation we have Ut) = which gives lim sup t Then, lim sup t d dt Ut) =. We now dx/dt du/dt, so lim sup t xt). Simple calculations yield to x + y) = rx1 x + y rx1 x + y 1 e t 0 rt ), xt) lim sup Ut). t ) p 1xz + sy1 x + y ) p 2yz ay ) + sy1 x + y x + y ) δx + y)1 ) where δ = maxr, s). Hence, similarly lim supxt) + yt)) for all t 0. t Let W t) = xt) + yt) + p zt) where p = minp c 1, p 2 ) and c = maxc 1, c 2 ). Then, dw dt = dx dt + dy dt + p dz c + p c c 1xz + p c c 2yz p c dt = rx1 x + y bz rx1 x + y ) p 1xz + sy1 x + y ) p 2yz ay ) + sy1 x + y ) p c bz rx r x2 + sy s y2 p c bz = r r x r) 2 + s s y s) 2 p bz rx sy r + s) mw c

6 98 A. ASHYANI, H.M. MOHAMMADINEJAD AND O. RABIEIMOTLAGH where m = minr, s, b). Therefore, lim supw t)) r + s)/m. Thus, z is bounded t above and all the solutions of the system ) that started in R 3 + either approaches, enter or remain in the subset of Ω = {x, y, z) R 3 + x, x + y, x + y + p r + s) z c m }. Since the solutions of the system ) remain bounded in the positively invariant region Ω, the model is well posed in the epidemiologically and mathematically senses. 3. Existence of equilibrium points and stability analysis In this section, we study the stability of the equilibrium points Equilibrium points. System ) always has two equilibrium points E 0 = 0, 0, 0) and E 1 =, 0, 0). Other equilibrium points are given by E 2 = 0, s a), 0), E 3 = 0, b, bs + c 2s a) ), s c 2 c 2 p 2 E 4 = b, 0, c 1 b)r ar + βa s) ), E 5 = c 1 c 1 p 1 ββ + r s), rβ a) ββ + r s), 0), E 6 = ac 2p 1 b c 2 )p 2 r p 1 s) bp 2 β) c 1 c 2 )p 2 r p 1 s) + βc 1 p 2 c 2 p 1 ),, ac 1 p 1 b c 1 )p 2 r p 1 s) bp 1 β c 1 c 2 )p 2 r p 1 s) + βc 1 p 2 c 2 p 1 ) ac 2 r c 1 r + β)) + β c 2 r + c 1 s + br s + β)) ) := E = x, y, z ). c 1 c 2 )p 2 r p 1 s) + βc 1 p 2 c 2 p 1 ) Equilibrium point E 0 means that the immune system cells, non-infected and infected tumor cells do not exist. Hence, in biology it means that both non-infected and infected cells damaged in the absence of immune system. Equilibrium point E 1 means that only non-infected tumor cells remain and the tumor grows to its carrying capacity, which shows the inefficiency of virotherapy. This equilibrium has been nown as infection-free equilibrium.

7 HOPF BIFURCATION FOR CANCER VIROTHERAPY 99 The equilibrium points are biologically admissible if and only if all components are positive. If s > a, E 2 exists. Equilibrium point E 3 exists if c 2 s a) > bs. Furthermore, E 4 is meaningful if c > b. For existence of equilibrium point E 5 we have the following. Case1 If β + r s > 0, then the denominator of components of equilibrium point E 5 is positive. Thus, for positivity of components of E 5, the numerators should be positive. Therefore, ar + βa s) > 0 and β a > 0. Case2 If β + r s < 0, then the denominator of components of equilibrium point E 5 is negative. Thus, we want that numerators are negative. For this we have ar + βa s) < 0 and β a < 0. Equilibrium E 6 is called interior equilibrium point. It means that immune system cells, non-infected and infected cells exist. This is compatible with what exactly can be seen in reality. Hence, E 6 is the most interesting equilibrium point from the biological point of view. For existence of E 6 we have: Case1 If c 1 c 2 )p 2 r p 1 s)+βc 1 p 2 c 2 p 1 ) > 0, then the denominator of components of equilibrium point E 6 is positive. For existence of E 6, all components of E 6 should be positive. Therefore, all the numerators should be positive. Hence, we gain ac 2 p 1 b c 2 )p 2 r p 1 s) bp 2 β) > 0, ac 1 p 1 b c 1 )p 2 r p 1 s) bp 1 β > 0, ac 2 r c 1 r + β)) + β c 2 r + c 1 s + br s + β)) > 0. Case2 If c 1 c 2 )p 2 r p 1 s)+βc 1 p 2 c 2 p 1 ) < 0, then the denominator of components of equilibrium point E 6 is negative. Thus, all the numerators should be negative.

8 100 A. ASHYANI, H.M. MOHAMMADINEJAD AND O. RABIEIMOTLAGH Hence, we get ac 2 p 1 b c 2 )p 2 r p 1 s) bp 2 β) < 0, ac 1 p 1 b c 1 )p 2 r p 1 s) bp 1 β < 0, ac 2 r c 1 r + β)) + β c 2 r + c 1 s + br s + β)) < Stability analysis. We compute the variational matrix of the system ) at any point x, y, z). Real part of eigenvalues of this matrix at any point determine the local stability of equilibrium points. Variational matrix is given by J = J ij ) 3 3, where J 11 = r1 x + y)/) rx)/ p 1 z yβ, J 12 = rx)/ xβ, J 13 = p 1 x, J 21 = sy)/ + yβ, J 22 = s1 x + y)/) sy)/ a p 2 z + xβ, J 23 = p 2 y, J 31 = c 1 z, J 32 = c 2 z, J 33 = c 1 x + c 2 y b. At the equilibrium point E 0, the eigenvalues of J are r, b and s a. Since r is positive, equilibrium point E 0 is always unstable. As a result, tumor will grow and therapy is failed in the absence of immune response and virus. In the following theorem, we study the local stability of the infection-free equilibrium. Theorem 3.1. The infection-free equilibrium of the system ) is unstable for > a/β) or > b/c 1 ) and locally asymptotically stable for < a/β) and < b/c 1 ). Proof. The variational matrix of the system ) at equilibrium point E 1 is given by r r β p 1 JE 1 ) = 0 β a c 1 b

9 HOPF BIFURCATION FOR CANCER VIROTHERAPY 101 Therefore, the eigenvalues are r, β a, and c 1 b. Because r is negative, stability of E 1 depends on the signs of β a and c 1 b. When > a/β) or > b/c 1 ), equilibrium point E 1 is unstable while it is asymptotically stable for < a/β) and < b/c 1 ). For local stability of interior equilibrium point we now that the characteristic equation of the variational matrix J at E 6 can be written as 3.1) DetJ λi 3 ) = λ 3 + a 2 λ 2 + a 1 λ + a 0 = 0 where a 0 = A 3 A 5 A 7 A 2 A 6 A 7 A 3 A 4 A 8 + A 1 A 6 A 8, 3.2) a 1 = A 2 A 4 A 1 A 5 + A 3 A 7 + A 6 A 8 ), a 2 = A 1 + A 5 ), where A 1 A 8 ) are introduced below. 3.3) A 1 = rac 2p 1 b c 2 )p 2 r p 1 s) bp 2 β) c 1 c 2 )p 2 r p 1 s) + c 2 p 1 + c 1 p 2 )β), A 2 = r + β)ac 2p 1 b c 2 )p 2 r p 1 s) bp 2 β), c 1 c 2 )p 2 r p 1 s) + c 2 p 1 + c 1 p 2 )β) A 3 = p 1 ac 2 p 1 + b c 2 )p 2 r p 1 s) + bp 2 β) c 1 + c 2 )p 2 r p 1 s) + c 2 p 1 c 1 p 2 )β, A 4 = s β) ac 1p 1 + b c 1 )p 2 r p 1 s) + bp 1 β), c 1 c 2 )p 2 r p 1 s) + c 2 p 1 + c 1 p 2 )β) A 5 = s ac 1p 1 + b c 1 )p 2 r p 1 s) + bp 1 β) c 1 c 2 )p 2 r p 1 s) + c 2 p 1 + c 1 p 2 )β), A 6 = p 2 ac 1 p 1 + b c 1 )p 2 r p 1 s) + bp 1 β) c 1 c 2 )p 2 r p 1 s) + c 2 p 1 + c 1 p 2 )β, A 7 = c 1ac 2 r c 1 r + β)) + β c 2 r + c 1 s + br s + β))), c 1 c 2 )p 2 r p 1 s) + c 2 p 1 + c 1 p 2 )β A 8 = c 2ac 2 r c 1 r + β)) + β c 2 r + c 1 s + br s + β))). c 1 c 2 )p 2 r p 1 s) + c 2 p 1 + c 1 p 2 )β

10 102 A. ASHYANI, H.M. MOHAMMADINEJAD AND O. RABIEIMOTLAGH Hence, in the following theorem we present some conditions for local stability of interior equilibrium point E 6 = x, y, z ). Theorem 3.2. Let a 0, a 1 and a 2 be defined in 3.2). The interior equilibrium point of the system ) is locally asymptotically stable if a 0 > 0 and a 2 a 1 a 0 > 0. Proof. According to Routh-Herwitz criterion of stability, the necessary and the sufficient conditions to all roots of characteristic equation 3.1) to be negative real parts are a 2 > 0, a 0 > 0, and a 2 a 1 a 0 > 0. It is clear that a 2 > 0. Thus, the interior equilibrium point E 6 is locally asymptotically stable if a 0 > 0 and a 2 a 1 a 0 > Hopf bifurcation In this section, we find the conditions for existence of Hopf bifurcation at the interior equilibrium point E 6. First we linearized equation ) in E 6. Let u 1 t) = xt) x, u 2 t) = yt) y, and u 3 t) = zt) z. It leads us to E 6 to be the origin. Then ) can be written as 4.1) u 1 = A 1 u 1 + A 2 u 2 + A 3 u 3 u 1 ru 1 + u 2 ) + p 1 u 3 + u 2 β), u 2 = A 4 u 1 + A 5 u 2 + A 6 u 3 u 2 su 1 + u 2 ) + p 2 u 3 u 1 β), u 3 = A 7 u 1 + A 8 u 2 + c 1 u 1 + c 2 u 2 )u 3 where A 1 A 8 ) are introduced in 3.3). The linearization of 4.1) at 0, 0) is u 1 = A 1 u 1 + A 2 u 2 + A 3 u 3, u 2 = A 4 u 1 + A 5 u 2 + A 6 u 3, u 3 = A 7 u 1 + A 8 u 2 with the characteristic equation λ 3 + a 2 λ 2 + a 1 λ + a 0 = 0, where a 2 = A 1 + A 5 ), a 1 = A 2 A 4 A 1 A 5 +A 3 A 7 +A 6 A 8 ), and a 0 = A 3 A 5 A 7 A 2 A 6 A 7 A 3 A 4 A 8 +A 1 A 6 A 8. If a 0 = a 1 a 2, we have a pair of imaginary eigenvalues λ 1,2 = ±i a 1. In this case, λ 3 = a 2 is the real eigenvalue. Thus let ɛ = a 0 a 1 a 2 and using the Implicit Function Theorem, we have λ 1,2 = ɛ ± i 2a 1 +a 2 2 a ) 1 + ɛ a1 a 2 2a 1 a 1 +a 2 2 ) ) + oɛ2 ) and

11 HOPF BIFURCATION FOR CANCER VIROTHERAPY 103 λ 3 = a 2 ɛ a 1 +a oɛ 2 ). Furthermore, if ɛ = 0, we may have Hopf bifurcation. On the other hand, solving a 2, a 1 and a 0 with respect to A 1, A 2 and A 3 respectively, we get A 1 = A 5 a 2, A 2 = A A 3 A 7 + A 6 A 8 + a 1 + A 5 a 2, and A 4 A 3 = A 5 2 A 6 A 7 A 4 A 5 A 6 A 8 + A 2 6 A 7 A 8 A 4 a 0 + A 6 A 7 a 1 + A 5 A 6 A 7 a 2 A 4 A 5 A 7 + A 6 A 2 7 A 2 4 A 8 + A 4 A 6 A 8 a 2 A 4 A 5 A 7 + A 6 A 7 2 A 4 2 A 8. In this case, we have 2-dimensional center manifold and afterwards by projecting this system on the center manifold we gain 2-dimensional system. First, we use the below transformation to get the standard system; where U 1 U 2 U 3 = P 1 u 1 u 2 u 3 P 1 P 2 P 3, and P = P 4 P 5 P P 1 = A 5A 6 A 7 A 8 A 4 A 6 A 2 8 A 4 A 8 a 1, P A 5 A 7 + A 4 A 8 ) 2 + A 2 4 = A 5A 6 A 2 7 A 4 A 6 A 7 A 8 A 4 A 7 a 1, 7a 1 A 5 A 7 + A 4 A 8 ) 2 + A 2 7a 1 P 3 = A 6A 8 + A 5 a 2 + a 2 2 a1 A 2, P 2 = 5A 7 A 4 A 5 A 8 + A 6 A 7 A 8 + A 7 a 1 ), A 5 A 7 A 4 A 8 + A 7 a 2 A 5 A 7 + A 4 A 8 ) 2 + A 2 7a 1 a1 A 4 A 5 A 7 A 6 A A 2 P 5 = 4A 8 ) A 6 A 7 A 4 a 2, P A 5 A 7 + A 4 A 8 ) 2 + A 2 6 =. 7a 1 A 5 A 7 A 4 A 8 + A 7 a 2 Therefore, system 4.1) transforms into the following standard form; U 1 = a 1 U 2 + G 1 U 1, U 2, U 3 ), 4.2) U 2 = a 1 U 1 + G 2 U 1, U 2, U 3 ), U 3 = a 2 U 3 + G 3 U 1, U 2, U 3 )

12 104 A. ASHYANI, H.M. MOHAMMADINEJAD AND O. RABIEIMOTLAGH where G 1 U 1, U 2, U 3 ) = 1 P 1 P 3 )P 5 + P 2 P 6 P 4 ) ) P 2 P 6 P 3 P 5 )U 1 + U 3 ) c 1 P 1 U 1 + P 2 U 2 + P 3 U 3 ) + c 2 P 4 U 1 + P 5 U 2 + P 6 U 3 )) ) + P 2 P 4 U 1 + P 5 U 2 + P 6 U 3 )p 2 U 1 + U 3 ) + sp 1 + P 4 )U 1 + P 2 + P 5 )U 2 + P 3 + P 6 )U 3 ) P 1 U 1 + P 2 U 2 + P 3 U 3 )β) P 5 P 1 U 1 + P 2 U 2 + P 3 U 3 )p 1 U 1 + U 3 ) + rp 1 + P 4 )U 1 ) + P 2 + P 5 )U 2 + P 3 + P 6 )U 3 ) + P 4 U 1 + P 5 U 2 + P 6 U 3 )β), G 2 U 1, U 2, U 3 ) = 1 P 1 P 3 )P 5 + P 2 P 6 P 4 ) ) P 3 P 4 P 1 P 6 )U 1 + U 3 ) c 1 P 1 U 1 + P 2 U 2 + P 3 U 3 ) + c 2 P 4 U 1 + P 5 U 2 + P 6 U 3 )) ) + P 1 P 3 )P 4 U 1 + P 5 U 2 + P 6 U 3 ) p 2 U 1 + U 3 ) sp 1 + P 4 )U 1 + P 2 + P 5 )U 2 + P 3 + P 6 )U 3 ) + P 1 U 1 + P 2 U 2 + P 3 U 3 )β) + P 4 P 6 )P 1 U 1 + P 2 U 2 + P 3 U 3 ) p 1 U 1 + U 3 ) + rp 1 + P 4 )U 1 ) + P 2 + P 5 )U 2 + P 3 + P 6 )U 3 ) + P 4 U 1 + P 5 U 2 + P 6 U 3 )β), G 3 U 1, U 2, U 3 ) = 1 P 1 P 3 )P 5 + P 2 P 6 P 4 ) ) P 1 P 5 P 2 P 4 )U 1 + U 3 ) c 1 P 1 U 1 + P 2 U 2 + P 3 U 3 ) + c 2 P 4 U 1 + P 5 U 2 + P 6 U 3 )) ) + P 5 P 1 U 1 + P 2 U 2 + P 3 U 3 )p 1 U 1 + U 3 ) + rp 1 + P 4 )U 1 + P 2 + P 5 )U 2 + P 3 + P 6 )U 3 ) + P 4 U 1 + P 5 U 2 + P 6 U 3 )β) P 2 P 4 U 1 + P 5 U 2 + P 6 U 3 )p 2 U 1 + U 3 ) + sp 1 + P 4 )U 1 ) + P 2 + P 5 )U 2 + P 3 + P 6 )U 3 ) P 1 U 1 + P 2 U 2 + P 3 U 3 )β).

13 HOPF BIFURCATION FOR CANCER VIROTHERAPY 105 By the existence theorem in the center manifold theory [10], there exists a center manifold which can be expressed locally as W c 0) = {U 1, U 2, U 3 ) R 3 U 3 = hu 1, U 2 ), U 3 < δ, h0, 0) = 0, Dh0, 0) = 0} for δ sufficiently small. We now compute W c 0). Assume that hu 1, U 2 ) have the following form: 4.3) U 3 = hu 1, U 2 ) = b 1 U b 2 U b 3 U 1 U 2 + ou 1, U 2 ). Based on the invariance of W c 0) under the dynamics of 4.2), the center manifold must satisfy 4.4) N U 1, U 2 ) = a 2 hu 1, U 2 ) + G 3 U 1, U 2, hu 1, U 2 )) h U 1, U 2 ) a 1 U 2 + G 1 U 1, U 2, hu 1, U 2 )) ) U 1 h U 2 U 1, U 2 ) a1 U 1 + G 2 U 1, U 2, hu 1, U 2 )) ).

14 106 A. ASHYANI, H.M. MOHAMMADINEJAD AND O. RABIEIMOTLAGH Substituting 4.3) into 4.4) and equating terms of lie powers to zero, we obtain b 1 = 1 2a 2 2b 3 a1 + 2 A 8 c 1 + A 7 c 2 )A 2 5A 2 7 2A 4 A 5 A 7 A 8 + A 2 4A A 2 7a 1 ) A 5 A 6 A 7 A 4 A 6 A 8 + a 1 )) 2 + 1/)A 8 A 4 A 5 A 7 A 6 A A 2 4A 8 ) A 5 A 6 A 7 + A 4 A 6 A 8 + a + 1))A 2 5A 2 7p 1 + A 2 4A 2 8p 1 A 5 A 7 2A 4 A 8 p 1 + A 6 A 7 r A 8 r + A 7 β)) + A 2 7p 1 a 1 + A 4 A 7 r A 8 r + A 7 β)a 6 A 8 + a 1 )) 1/)A 7 A 5 A 6 A 7 A 4 A 6 A 8 + a 1 ))A 2 5A 7 A 4 A 5 A 8 + A 7 A 6 A 8 + a 1 )) A 2 5A 2 7p 2 + A 2 4A 2 8p 2 A 5 A 7 2A 4 A 8 p 2 + A 6 A 7 s A 8 s + A 8 β)) + A 2 7p 2 a 1 + A 4 A 7 s A 8 s + A 8 β)a 6 A 8 + a 1 )))A 5 A 7 A 4 A 8 + A 7 a 2 ) )/ A 4 A 5 A 7 A 6 A A 2 4A 8 )A 2 5A 2 7 2A 4 A 5 A 7 A 8 + A 2 4A 2 8 ) + A 2 7a 1 ) 2 a 1 + a 2 2)), b 2 = 1/2 a 1 )) b 3 a 2 + a1 A 4 A 8 + A 7 A 5 + a 2 ))A 5 A 7 A 4 A 8 ) 2 + A 2 7a 1 ) A 2 5A 7 c 1 + A 2 4A 8 c 2 + A 6 A 7 A 8 c 1 A 7 c 2 ) A 4 A 5 A 8 c 1 + A 7 c 2 ) + A 7 c 1 a 1 ) A 5 A 6 A 7 A 4 A 6 A 8 + a 1 )) A 7 A 5 A 6 A 7 A 4 A 6 A 8 + a 1 ))A 2 5A 7 A 4 A 5 A 8 + A 7 A 6 A 8 + a 1 )) A 2 4A 8 A 4 A 5 A 7 + A 8 ) + A 7 A 2 5 A 6 A 7 + A 6 A 8 + a 1 ))s + A 2 5A 7 A 4 A 5 A 8 + A 7 A 6 A 8 + a 1 ))β))/ A 8 A 7 A 4 A 5 + A 6 A 7 ) + A 2 4A 8 ) A 5 A 6 A 7 + A 4 A 6 A 8 + a 1 ))A 2 4A 8 r + β) A 4 A 5 A 7 + A 8 )r + A 7 β) + A 7 A 2 5 A 6 A 7 + A 6 A 8 + a 1 )r A 6 A 7 β)))/ A 4 A 5 A 7 + A 6 A 2 7 A 2 4A 8 )A 2 5A 7 A 4 A 5 A 8 + A 7 A 6 A 8 + a 1 ))A 2 5A 2 7p 1 + A 2 4A 2 8p 1 + A 2 7a 1 p 1 + A 4 A 6 A 8 + a 1 )A 7 r A 8 r + A 7 β) A 5 A 7 2A 4 A 8 p 1 + A 6 A 7 r A 8 r + A 7 β))))/ + A 4 A 5 A 7 + A 6 A 2 7 A 2 4A 8 )A 2 5A 7 A 4 A 5 A 8 + A 7 A 6 A 8

15 HOPF BIFURCATION FOR CANCER VIROTHERAPY a 1 ))A 2 5A 2 7p 2 + A 2 4A 2 8p 2 + A 2 7a 1 p 2 + A 4 A 6 A 8 + a 1 )A 7 s A 8 s + A 8 β) A 5 A 7 2A 4 A 8 p 2 + A 6 A 7 s A 8 s + A 8 β))))/) )/ A 7 A 4 A 5 + A 6 A 7 ) + A 2 4A 8 )A 5 A 7 A 4 A 8 ) 2 + A 2 7a 1 ) 2 a 1 + a 2 2)) + 1/a 2 a1 2b3 a1 + 2 A 4 A 8 + A 7 A 5 + a 2 )) A 8 c 1 + A 7 c 2 )A 5 A 7 A 4 A 8 ) 2 + A 2 7a 1 )A 5 A 6 A 7 A 4 A 6 A 8 + a 1 )) 2 + A 8 A 7 A 4 A 5 + A 6 A 7 ) + A 2 4A 8 ) A 5 A 6 A 7 + A 4 A 6 A 8 + a 1 )) A 2 5A 2 7p 1 + A 2 4A 2 8p 1 + A 2 7a 1 p 1 + A 4 A 6 A 8 + a 1 )A 7 r A 8 r + A 7 β) A 5 A 7 2A 4 A 8 p 1 + A 6 A 7 r A 8 r + A 7 β))))/ 1/A7A 5 A 6 A 7 A 4 A 6 A 8 + a 1 ))A 2 5A 7 A 4 A 5 A 8 + A 7 A 6 A 8 + a 1 ))A 2 5A 2 7p 2 + A 2 4A 2 8p 2 + A 2 7a 1 p 2 + A 4 A 6 A 8 + a 1 )A 7 s A 8 s + A 8 β) A 5 A 7 2A 4 A 8 p 2 + A 6 A 7 s A 8 s / + A 8 β))))) A 7 A 4 A 5 + A 6 A 7 ) + A 2 4A 8 )A 5 A 7 A 4 A 8 ) 2 + A 2 7a 1 ) 2 a 1 + a 2 2)) )), b 3 = a1 A 5 A 7 A 4 A 8 + A 7 a 2 ) 2/)r s + β)a 1 A 3 4A 5 A A 2 7A A 6 A 8 + a 1 )A A 6 A 7 + A 8 ) + a 1 ) A 4 A 5 A 7 2A 6 A A 2 5A 7 + 2A 8 ) + A 7 + 2A 8 )a 1 ) + A 2 4A 8 A 2 52A 7 + A 8 ) + A 7 A 6 A 8 + a 1 ))) 2 A 4 A 5 A 7 A 6 A A 2 4A 8 A 8 c 1 + A 7 c 2 )A 2 5A 2 7 2A 4 A 5 A 7 A 8 + A 2 4A A 2 7a 1 )A 5 A 6 A 7 A 4 A 6 A 8 + a 1 )) 2 + A 8 A 4A 5 A 7 A 6 A A 2 4A 8 ) A 5 A 6 A 7 + A 4 A 6 A 8 + a 1 ))A 2 5A 2 7p 1 + A 2 4A 2 8p 1 A 5 A 7 2A 4 A 8 p 1 + A 6 A 7 r A 8 r + A 7 β)) + A 2 7p 1 a 1 + A 4 A 7 r A 8 r + A 7 β)a 6 A 8 + a 1 )) A 7 A 5A 6 A 7 A 4 A 6 A 8 + a 1 ))A 2 5A 7 A 4 A 5 A 8 + A 7 A 6 A 8 + a 1 ))A 2 5A 2 7p 2 + A 2 4A 2 8p 2 A 5 A 7 2A 4 A 8 p 2 + A 6 A 7 s A 8 s + A 8 β)) + A 2 7p 2 a 1 + A 4 A 7 s A 8 s + A 8 β)a 6 A 8 + a 1 ))) 1 A 4 A 5 A 7 A 6 A A 2 4A 8 A 2 5A 2 7 2A 4 A 5 A 7 A 8 + A 2 4A A 2 7a 1 )A 2 5A 7 c 1 + A 2 4A 8 c 2 + A 6 A 7 A 8 c 1 A 7 c 2 )

16 108 A. ASHYANI, H.M. MOHAMMADINEJAD AND O. RABIEIMOTLAGH A 4 A 5 A 8 c 1 + A 7 c 2 ) + A 7 c 1 a 1 )A 5 A 6 A 7 A 4 A 6 A 8 + a 1 )) A 4A 5 A 7 A 6 A A 2 4A 8 A 2 5A 7 + A 4 A 5 A 8 A 7 A 6 A 8 + a 1 ))A 2 5A 2 7p 1 + A 2 4A 2 8p 1 A 5 A 7 2A 4 A 8 p 1 + A 6 A 7 r A 8 r + A 7 β)) + A 2 7p 1 a 1 + A 4 A 7 r A 8 r + A 7 β)a 6 A 8 + a 1 )) + A 4A 5 A 7 A 6 A A 2 4A 8 A 2 5A 7 + A 4 A 5 A 8 A 7 A 6 A 8 + a 1 ))A 2 5A 2 7p 2 + A 2 4A 2 8p 2 A 5 A 7 2A 4 A 8 p 2 + A 6 A 7 s A 8 s + A 8 β)) + A 2 7p 2 a 1 + A 4 A 7 s A 8 s + A 8 β)a 6 A 8 + a 1 )) A 8 A 4 A 5 A 7 A 6 A A 2 4A 8 ) A 5 A 6 A 7 + A 4 A 6 A 8 + a 1 ))A 2 4A 8 r + β) A 4 A 5 A 8 r + A 7 r + β)) + A 7 A 2 5r A 6 A 7 r + A 6 A 8 r A 6 A 7 β + ra 1 )) A 7 /)A 5 A 6 A 7 A 4 A 6 A 8 + a 1 ))A 2 5A 7 A 4 A 5 A 8 + A 7 A 6 A 8 + a 1 )) A 2 4A 8 s + A 4 A 5 A 7 s + A 8 s β)) + A 7 A 6 A 7 s A 6 A 8 s + A 6 A 8 β + A 2 5 s + β) sa 1 + βa 1 )))a 2 ) )/ A 2 5A 2 7 2A 4 A 5 A 7 A 8 + A 2 4A A 2 7a 1 ) 2 a 1 + a 2 2)4a 1 + a 2 2)) Finally, by substituting 4.3) into 4.2) we obtain the vector field reduced to the center manifold by U 1 = a 1 U 2 + η 1 U 1 U 2 + η 2 U η 3 U 2 2, 4.5) U 2 = a 1 U 1 + θ 1 U 1 U 2 + θ 2 U θ 3 U 2 2 where a1 1 η 1 = A 4 A 8 + A 7 A 5 + a 2 )) A 4 A 8 + A 7 A 5 + a 2 ) A 5A 7 A 4 A 8 ) 2 + A 7 c 1 a 1 ) + A 2 7a 1 )A 2 5A 7 c 1 + A 2 4A 8 c 2 + A 6 A 7 A 8 c 1 A 7 c 2 ) A 4 A 5 A 8 c 1 + A 7 c 2 )A 6 A 5 A 7 A 4 A 8 ) 2 + A 2 7a 1 ) + A 7 A 5 A 6 A 7 A 4 A 6 A 8 + a 1 ))a 2 + A 4 A 5 A 7 + A 6 A 2 7 A 2 4A 8 )a 2 2) + 1 A7 A 5 A 6 A 7 A 4 A 6 A 8 + a 1 ))A 2 5A 7

17 HOPF BIFURCATION FOR CANCER VIROTHERAPY 109 A 4 A 5 A 8 + A 7 A 6 A 8 + a 1 )) A 2 4A 8 A 4 A 5 A 7 + A 8 ) + A 7 A 2 5 A 6 A 7 + A 6 A 8 + a 1 ))s + A 2 5A 7 A 4 A 5 A 8 + A 7 A 6 A 8 + a 1 ))β) + A 8 A 7 A 4 A 5 + A 6 A 7 ) + A 2 4A 8 ) A 5 A 6 A 7 + A 4 A 6 A 8 + a 1 ))A 2 4A 8 r + β) A 4 A 5 A 7 + A 8 )r + A 7 β) + A 7 A 2 5 A 6 A 7 + A 6 A 8 + a 1 )r A 6 A 7 β)) + A 4 A 5 A 7 + A 6 A 2 7 A 2 4A 8 )A 2 5A 7 A 4 A 5 A 8 + A 7 A 6 A 8 + a 1 ))A 2 5A 2 7p 1 + A 2 4A 2 8p 1 + A 2 7a 1 p 1 + A 4 A 6 A 8 + a 1 )A 7 r A 8 r + A 7 β) A 5 A 7 2A 4 A 8 p 1 + A 6 A 7 r A 8 r + A 7 β))) A 4 A 5 A 7 + A 6 A 2 7 A 2 4A 8 )A 2 5A 7 A 4 A 5 A 8 + A 7 A 6 A 8 + a 1 ))A 2 5A 2 7p 2 + A 2 4A 2 8p 2 + A 2 7a 1 p 2 + A 4 A 6 A 8 + a 1 )A 7 s A 8 s + A 8 β) A 5 A 7 2A 4 A 8 p 2 + A 6 A 7 s A 8 s + A 8 β))) )))/ 2 A 7 A 4 A 5 + A 6 A 7 ) + A 2 4A 8 )A 5 A 7 A 4 A 8 ) 2 + A 2 7a 1 ) 2 a 1 + a 2 2)), η 2 = 1 A 4 A 8 + A 7 A 5 + a 2 )) A8 c 1 A 7 c 2 )A 5 A 7 A 4 A 8 A 7 A 5 + a 2 ) A 4 A 8 ) 2 + A 2 7a 1 ) A 5 A 6 A 7 + A 4 A 6 A 8 + a 1 ))A 6 A 5 A 7 A 4 A 8 ) 2 + A 2 7a 1 ) + A 7 A 5 A 6 A 7 A 4 A 6 A 8 + a 1 ))a 2 + A 4 A 5 A 7 + A 6 A 2 7 A 2 4A 8 )a 2 2) ) 1 A8 A 7 A 4 A 5 + A 6 A 7 ) + A 2 4A 8 ) A 5 A 6 A 7 + A 4 A 6 A 8 + a 1 ))A 2 5A 2 7p 1 + A 2 4A 2 8p 1 + A 2 7a 1 p 1 + A 4 A 6 A 8 + a 1 )A 7 r A 8 r + A 7 β) A 5 A 7 2A 4 A 8 p 1 + A 6 A 7 r A 8 r + A 7 β))) + A 7 A 5 A 6 A 7 A 4 A 6 A 8 + a 1 ))A 2 5A 7 A 4 A 5 A 8 + A 7 A 6 A 8 + a 1 ))A 2 5A 2 7p 2 + A 2 4A 2 8p 2 + A 2 7a 1 p 2 + A 4 A 6 A 8 + a 1 )A 7 s A 8 s + A 8 β) A 5 A 7 2A 4 A 8 p 2 + A 6 A 7 s A 8 s + A 8 β))) )))/ A 7 A 4 A 5 A 6 A 7 ) + A 2 4A 8 )A 5 A 7 A 4 A 8 ) 2 + A 2 7a 1 ) 2 a 1 + a 2 2)), η 3 = a 1 A 2 5A 7 A 4 A 5 A 8 + A 7 A 6 A 8 + a 1 ))A 2 4A 8 A 4 A 5 A 7 + A 8 ) )/ + A 7 A 2 5 A 6 A 7 + A 6 A 8 + a 1 )) A 4 A 8 + A 7 A 5 + a 2 ))r s + β) A 5 A 7 A 4 A 8 ) 2 + A 2 7a 1 ) 2 a 1 + a 2 2)),

18 110 A. ASHYANI, H.M. MOHAMMADINEJAD AND O. RABIEIMOTLAGH θ 1 = A 4 A 8 + A 7 A 5 + a 2 )) A A 7 A 4 A 5 + A 6 A 7 ) + A 2 4A 8 )A 5 A 7 A 4 A 8 ) 2 + A 2 7a 1 ) 2 a 1 + a 2 5 A 7 2) A 4 A 8 ) 2 + A 2 7a 1 )A 2 5A 7 c 1 + A 2 4A 8 c 2 + A 6 A 7 A 8 c 1 A 7 c 2 ) A 4 A 5 A 8 c A 7 c 2 ) + A 7 c 1 a 1 ) A 5 A 6 A 7 + A 4 A 6 A 8 + a 1 ))a 2 + A 4 A 8 + A 7 A 5 + a 2 )) A 7 A 5 A 6 A 7 A 4 A 6 A 8 + a 1 )) A 4 A 7 A 8 A 5 a 1 + 2A A 6 A 8 + a 1 )a 2 + 2A 5 a 2 2) + A 2 4A 2 8a 1 + a 2 A 5 + a 2 )) + A 2 7A a 1 )a 2 A 5 + a 2 ) + A 6 A 8 a 1 + A 5 a 2 ))) A 2 4A 8 A 4 A 5 A 7 + A 8 ) + A 7 A 2 5 A 6 A 7 + A 6 A 8 + a 1 ))s + A 2 5A 7 A 4 A 5 A 8 + A 7 A 6 A 8 + a 1 ))β)) A 8 A 7 A 4 A 5 + A 6 A 7 ) + A 2 4A 8 ) A 5 A 6 A 7 + A 4 A 6 A 8 + a 1 ))A 4 A 8 a 2 + A 7 a 1 A 5 a 2 ))A 2 4A 8 r + β) A 4 A 5 A 7 + A 8 )r + A 7 β) + A 7 A 2 5 A 6 A 7 + A 6 A 8 + a 1 )r A 6 A 7 β))) + A 7 A 4 A 5 + A 6 A 7 ) + A 2 4A 8 )A 2 5A 7 A 4 A 5 A 8 + A 7 A 6 A 8 + a 1 ))A 4 A 8 a 2 + A 7 a 1 A 5 a 2 ))A 2 5A 2 7p 1 + A 2 4A 2 8p 1 + A 2 7a 1 p 1 + A 4 A 6 A 8 + a 1 )A 7 r A 8 r + A 7 β) A 5 A 7 2A 4 A 8 p 1 + A 6 A 7 r A 8 r + A 7 β)))) + A 7 A 4 A 5 + A 6 A 7 ) + A 2 4A 8 ) A 4 A 7 A 8 A 5 a 1 + 2A A 6 A 8 + a 1 )a 2 + 2A 5 a 2 2) + A 2 4A 2 8a 1 + a 2 A 5 + a 2 )) + A 2 7A a 1 )a 2 A 5 + a 2 ) + A 6 A 8 a 1 + A 5 a 2 )))A 2 5A 2 7p 2 + A 2 4A 2 8p 2 + A 2 7a 1 p 2 + A 4 A 6 A 8 + a 1 )A 7 s A 8 s + A 8 β) A 5 A 7 2A 4 A 8 p 2 + A 6 A 7 s )) A 8 s + A 8 β))), θ 2 = A 4 A 8 + A 7 A 5 + a 2 )) A 8 c 1 A 7 c 2 )A 5 A 7 A 4 A 8 ) 2 + A 2 7a 1 )A 5 A 6 A 7 A 4 A 6 A 8 + a 1 )) 2 1 a 2 + A 8 A 7 A 4 A 5 + A 6 A 7 ) A 4 A 8 + A 7 A 5 + a 2 )) + A 2 4A 8 ) A 5 A 6 A 7 + A 4 A 6 A 8 + a 1 ))A 4 A 8 a 2 + A 7 a 1 A 5 a 2 ))A 2 5A 2 7p 1 + A 2 4A 2 8p 1 + A 2 7a 1 p 1 + A 4 A 6 A 8 + a 1 )A 7 r A 8 r + A 7 β) A 5 A 7 2A 4 A 8 p 1 + A 6 A 7 r A 8 r + A 7 β))) + A 7 A 5 A 6 A 7 A 4 A 6 A 8 + a 1 )) A 4 A 7 A 8 A 5 a 1 + 2A A 6 A 8 + a 1 )a 2 + 2A 5 a 2 2) + A 2 4A 2 8a 1 + a 2 A 5 + a 2 )) + A 2 7A a 1 )a 2

19 HOPF BIFURCATION FOR CANCER VIROTHERAPY 111 θ 3 = A 5 + a 2 ) + A 6 A 8 a 1 + A 5 a 2 )))A 2 5A 2 7p 2 + A 2 4A 2 8p 2 + A 2 7a 1 p 2 + A 4 A 6 A 8 ))/ + a 1 )A 7 s A 8 s + A 8 β) A 5 A 7 2A 4 A 8 p 2 + A 6 A 7 s A 8 s + A 8 β))) A 7 A 4 A 5 + A 6 A 7 ) + A 2 4A 8 ) a 1 A 5 A 7 A 4 A 8 ) 2 + A 2 7a 1 ) 2 a 1 + a 2 2)), a 1 A 2 A 5 A 7 A 4 A 8 ) 2 + A 2 7a 1 ) 2 a 1 + a 5A A 4 A 5 A 8 A 7 A 6 A 8 2)) + a 1 ))A 4 A 8 a 2 + A 7 a 1 A 5 a 2 ))A 2 4A 8 r + β) A 4 A 5 A 7 + A 8 )r + A 7 β) + A 7 A 2 5 A 6 A 7 + A 6 A 8 + a 1 )r A 6 A 7 β)) A 4 A 7 A 8 A 5 a 1 + 2A A 6 A 8 + a 1 )a 2 + 2A 5 a 2 2) + A 2 4A 2 8a 1 + a 2 A 5 + a 2 )) + A 2 7A a 1 )a 2 A 5 + a 2 ) + A 6 A 8 a 1 + A 5 a 2 )))A 2 4A 8 s + A 7 A 2 5 A 6 A 7 + A 6 A 8 + a 1 )s A 7 A A 6 A 8 ) + a 1 )β + A 4 A 5 A 7 + A 8 )s + A 8 β)). At the bifurcation point, System 4.5) becomes where U 1 = 0 a 1 U 1 + f 1 U 1, U 2 ) U 2 a1 0 f 2 U 1, U 2 ) U 2 f 1 U 1, U 2 ) = η 1 U 1 U 2 + η 2 U η 3 U 2 2 and f 2 U 1, U 2 ) = θ 1 U 1 U 2 + θ 2 U θ 3 U 2 2. The coefficient a0) a is given by a = 1 16 f 1 U1 U 1 U 1 + f 1 U 1 U 2 U 2 + f 2 U 1 U 1 U 2 + f 2 U 2 U 2 U 2 ) a 1 f 1 U1 U 2 f 1 U 1 U 1 + f 1 U 2 U 2 ) f 2 U 1 U 2 f 2 U 1 U 1 + f 2 U 2 U 2 ) f 1 U 1 U 1 f 2 U 1 U 1 + f 1 U 2 U 2 f 2 U 2 U 2 ). Therefore, a = 1 8 a 1 η1 η 2 + η 3 ) θ 1 θ 2 + θ 3 ) 2η 2 θ 2 + 2η 3 θ 3 ) 0. Hence, from Hopf bifurcation theory in [10] we get Hopf bifurcation. Also, the periodic orbit is asymptotically stable for a < 0 and unstable for a > 0.

20 112 A. ASHYANI, H.M. MOHAMMADINEJAD AND O. RABIEIMOTLAGH 5. Numerical simulation In this section, we carry out some numerical simulations to verify the previous results. Suppose that r = 1.1, β = 1, s = 1, = 0.7, a = , 5.1) b = 0.1, p 1 = 1, p 2 = 2, c 1 = 2, c 2 = 0.1. In this case, E 6 = , , ) and System ) is given as ẋ = 1.1x1 x + y ) xy xz, 0.7 ẏ = y1 x + y ) + xy 2yz y, 0.7 ż = 2xz + 0.1yz 0.1z. Furthermore, restricted system to the center manifold is approximately given by U 1 = U U 1 U U U 2 2, U 2 = U U 1 U U U 2 2. Thus a 0 = a 1 a , λ 1,2 ±i and λ 3 = In this case a = Therefore, we have Hopf bifurcation that bifurcating periodic orbit is stable. Thus, E 6 is unstable. These results are presented in Figures 1 and 2).

21 HOPF BIFURCATION FOR CANCER VIROTHERAPY 113 Figure 1. Components of the solution U 1 t) and U 2 t) with parameters given in 5.1). Figure 2. The bifurcating periodic solution. 6. Conclusion In this paper, we consider a virus therapy for cancer and analyze stability of equilibrium points of this system. Equilibrium E 0 is unstable. It means that with this therapy, tumor does not completely destroy. Thus, we loo for on other equilibrium points. Infection-free equilibrium point E 1 exists and may be stable. Stability of E 1 is not useful because it means that non-infected tumor cells exist then therapy fails. We note that the interior equilibrium point E 6 is the most interesting equilibrium point from the biological point of view since its existence means that both of the

22 114 A. ASHYANI, H.M. MOHAMMADINEJAD AND O. RABIEIMOTLAGH non-infected and infected tumor cells and CTL exist. Its stability means that the tumor growth is controlled in a way that it can not reach to the carrying capacity. Hence, under the conditions of the parameters in theorem 3.2, it means that with this therapy we could control the size of the tumor which is x + y, but tumor exists and not completely wasted. In addition, we noted that when a 0 = a 1 a 2, we have Hopf bifurcation in E 6. Moreover, theorem 3.2 means this equilibrium point is unstable. Hence, existence of Hopf bifurcation is not useful because Hopf cycle means that the size of tumor cells decreases. Exactly when we suppose that therapy is useful, tumor size increases, and this behavior repeated See Figure 2). Although, this behavior means when we suppose that tumor treated, after some times we see that tumor grows. Thus, patient will die. Therefore, we attempted that value of parameters must be controlled in the way that we have not Hopf cycle. References [1] A. Ashyani, H. M. Mohammadinejad, O. RabieiMotlagh Stability analysis of mathematical model of virus therapy for cancer.to be appear in Iranian Journal of Mathematical Sciences and Informatics 2016). [2] E. J. Fuchs, P. Matzinger. Is cancer dangerous to the immune system? Semin Immuno. 18, 1996), , [3] R. J. Huebner, W. P. Rowe, W. E. Schatten, R. R. Smith, L. B. Thomas. Studies on the use of viruses in the treatment of carcinoma of the cervix. Cancer. 9, 1956), [4] C. M. Kunin. Cellular susceptibility to enteroviruses. Bacteriological Reviews. 28, No. 4, 1964), [5] P. Matzinger, An innate sense of danger. Semin Immunol. 10, 1998), , [6] A.E. Moore. The destructive effect of the virus of Russian far east encephalitis on the transplantable mouse sarcoma 180. Cancer. 2, 1949), [7] A. S. Novozhilov, F. S. Berezovsaya, E. V. Koonin, G. P. Karev. Mathematical modeling of tumor therapy with oncolytic viruses: regimes with complete tumor elimination within the framewor of deterministic models. Biology Direct. 1, 2006), 6 24.

23 HOPF BIFURCATION FOR CANCER VIROTHERAPY 115 [8] L. Pelner, GA. Fowler, HC. Nauts. Effects of concurrent infections and their toxins on the course of leuemia. Acta. Med. Scand. Suppl. 338, 1958), [9] A. R. Pond, E. E. Manuelidis. Oncolytic effect of poliomyelitis virus on human epidermoid carcinoma Hela tumor) heterologously transplanted to guinea pigs. The American J. of Pathology. 45, No. 2, 1964), [10] S. Wiggins: Introduction to applied nonlinear dynamical systems and chaos, Berlin, Springer, [11] D. Wodarz, N. L. Komarova. Computational biology of cancer: lecture notes and mathematical modeling. World Scientific Publishing Company, Singapore, [12] D. Wodarz. Computational approaches to study oncolytic virus therapy: insights and challenges. Gene. Ther. Mol. Biol. 8, 2004), [13] D. Wodarz. Viruses as antitumor weapons: defining conditions for tumor remission. Cancer Research. 61, 2001), ) Department of Mathematics, University of Birjand, Birjand, Iran. address: a.ashyani@birjand.ac.ir 2) Department of Mathematics, University of Birjand, Birjand, Iran. address: hmohammadin@birjand.ac.ir 3) Department of Mathematics, University of Birjand, Birjand, Iran. address: orabieimotlagh@birjand.ac.ir