Pure Multiplicative Noises Induced Population Extinction in an Anti-tumor Model under Immune Surveillance

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1 Commun. Theor. Phys. Beijing, China pp c Chinese Physical Society and IOP Publishing Ltd Vol. 52, No. 3, September 15, 29 Pure Multiplicative Noises Induced Population Extinction in an Anti-tumor Model under Immune Surveillance WANG Can-Jun, 1,2, LI Di, 1 and MEI Dong-Cheng 2 1 Nonlinear Research Institute, Baoji University of Arts and Sciences, Baoji 7217, China 2 Department of Physics, Yunnan University, unming 6591, China Received October 6, 28; Revised February 16, 29 Abstract The dynamical characters of a theoretical anti-tumor model under immune surveillance subjected to a pure multiplicative noise are investigated. The effects of pure multiplicative noise on the stationary probability distribution SPD and the mean first passage time MFPT are analysed based on the approximate Fokker Planck equation of the system in detail. For the anti-tumor model, with the multiplicative noise intensity D increasing, the tumor population move towards to extinction and the extinction rate can be enhanced. Numerical simulations are carried out to check the approximate theoretical results. Reasonably good agreement is obtained. PACS numbers: 5.4.-a ey words: multiplicative noise, anti-tumor model 1 Introduction In past decades, nonlinear systems with noise disturbance have attracted extensive investigations and have been widely applied in the field of physics system, [111] laser system, [1215] biological system, [1628] ecological system, [2933] and so on. Many novel phenomena are found, such as, noise induced transition, [3,4,27] reentrance phenomena, [5,16] stochastic resonance, [21,3436] noise enhance stability, [3739] current reveal, [442] etc. Recently, noise biodynamics has become one of hot topics, and attracted attention of many scientists, especially, the the growth of tumor cells with noise perturbation. Recent studies suggested that these various responses of tumor cells to the treatments, once taken together, imply that there is an interesting and significant case for combining chemotherapy and immunotherapy in tumor treatments, and have mainly focused on the growth law of tumor cells via dynamics. It found many new phenomena, for example phase transition [21,28] and stochastic resonance. [23,36] In most of these areas the noise affects the dynamics through system variables, i.e., the noise is multiplicative in nature. The anti-tumor model under immune surveillance is as a basic model to describe the tumor growth process and has been investigated from different views. Reference [21] reports on a simple model of spatially extended antitumor system with a fluctuation in growth rate, which can undergo a nonequilibrium phase transition. Three states as excited, subexcited, and nonexcited states of a tumor are defined to describe its growth. The multiplicative noise is found to have opposite effects: The positive effect on a nonexcited tumor and the negative effect on an excited tumor. Reference [23] shows the pure multiplicative noise-induced stochastic resonance, which appears in an anti-tumor system modulated by a seasonal external field. For optimally selected values of the multiplicative noise intensity stochastic resonance is observed, which is manifested by the quasi symmetry of two potential minima. In the paper, we only consider the effect of a fluctuation of growth rate on anti-tumor model under immune surveillance. In other words, the system only subjects to a pure multiplicative noise. The dynamical characters including the steady probability distribution SPD and the mean first passage time MFPT are revealed from theory and numerical simulations. Lefever and Garay [43] studied the growth of the tumor under immune surveillance against cancer using the enzyme dynamics model. The model is, Normal Cells X λ 2X, γ X, k X + E 1 k E 2 E + P, P k3, 1 in which X, P, E, and E are cancer cells, dead cancer cells, immune cells, and the compounds of cancer cells and immune cells, respectively. The symbols, γ, λ, k 1, k 2, and k 3, are velocity coefficients. This model reveals that normal cells can transform into cancer cells, and then the cancer cells reproduce, decline, and die out ultimately. This model can be simplified to an equivalent single-variable nondimensional deterministic dynam- Supported by the Natural Science Foundation of China under Grant No , the Science Foundation of the Education Bureau of Shaanxi Province under Grant No. 9J331, and the Science Foundation of Baoji University of Science and Arts of China under Grant No. Zk725 cjwangbj@126.com

2 464 WANG Can-Jun, LI Di, and MEI Dong-Cheng Vol. 52 ics differential equation, [44] 1 dt = rx x βx2 1 + x 2, 2 where x is the population of tumor cells, r is their linear per capita birth rate, is the carrying capacity of the environment, and β is the immune coefficient, respectively. The potential corresponding to Eq. 2 is given by U x = 1 rx rx2 + βx β arctanx. 3 Under the condition x > the tumor cell population is all positive, Eq. 3 has two stable states and one unstable state. If we choose r = 1., = 1., and β = 2., the two stable states are x.6834 and x , and the unstable state is x u = 2. A bistable potential of U x is plotted in Fig. 1. Fig. 1 The bistable potential of Eq. 3. The parameter values are r = 1., = 1., and β = 2.. The stable states are x.6834 and x , and the unstable state is x u = 2.. It is known that all physical parameters are subjected to random perturbations that may have internal or external origin. So we should not consider only deterministic quantities. Here, we consider that the environmental fluctuations can influence birth rate in this model. Therefore, the growth rate r in Eq. 3 can be rewrite as r + Γt. Γt is the Gaussian white noises defined as Γt = and ΓtΓt = 2Dδtt, in which D is the noise intensity. The equivalent stochastic differential equation Langevin Equation of Eq. 2 can be generated as, 1 dt = rx x 1 + x 2 + x x Γt. 4 In the steady-state regime and under the constraint x >, the approximate Fokker Planck equation AFPE of the anti-tumor model under immune surveillance driven by a multiplicative Gaussian white noises Eq. 4 is derived [3] Px, t = t x where Ax = rx 1 x 2 AxPx, t + BxPx, t, 5 x2 1 + x 2 + Dx x 1 2x, 6 [ Bx = D x 1 x ] 2. 7 The stationary probability distribution SPD corresponding to Eq. 5 is obtained with Eqs. 6 7, P s x = N [ Bx exp Φx ], 8 D where N is a normalization constant, and Φx is the generalized potential function and its form follows Φx = 2β3 1 + x ln + arctanx x + r + Dln x x + β x arctanx + D ln x2. 9 In addition, the extrema of P st obey AxB x =, i.e. r1x/βx/1+x 2 Dr1x/12x/ =. By numerical calculation of Eq. 8, we analyze the effects of both the multiplicative noise intensity on the SPD from the view of the theory. In Fig. 2, we show SPD as a function of x for different D. It is found that a bimodal structure exists in Fig. 2. Only the height of the peak is changed with D increasing. On the other hand, with D increasing, this highest peak of the curves of SPD shifts from the right representing the large x to left representing the small x and tends zero, and a successive switch an asymmetric bimodal a symmetric bimodal an asymmetric bimodal occurs with an increase of noise intensities D, and a critical noise intensity D exists at which the symmetric bimodal appear of the switch process. Since x denotes the tumor cell population, it is clear that the tumor cell population gradually disappears with D increasing. In other words, the distribution of cell population, which was mainly peaked about zero for a large value of D signifying high extinction rates, moves toward zero with the increase of D. Fig. 2 The theoretical results of the steady state probability distribution functions P stx are plotted for the different multiplicative noise intensity D =.1,.2,.4, respectively. The other parameter values are the same as those in Fig. 1. In order to quantitatively investigate the stationary properties of the system, we introduce the moments of

3 No. 3 Pure Multiplicative Noises Induced Population Extinction in an Anti-tumor Model under Immune Surveillance 465 the variable x, and it given by, x n st = + The mean of the state variable x is x st = + x n P st x. 1 xp st x. 11 Making use of the expressions of Eq. 11, the effects of D and on x st can be analysed by the numerical calculation. The results of the numerical calculation of x st as a function of D are plotted on Fig. 3. Figure 3 shows that the larger D is, the x st smaller is. It also means that the tumor cell population decrease with D increasing. of numerical calculation. The results are shown in Fig. 4. Figure 4 shows that T monotonously decreases as D increases. It means that D can enhance the transition from the large tumor cell population state to the small tumor cell population state. Namely, for large D the tumor cell population are more obvious extinction. Fig. 4 The theoretical results of the mean first passage time MFPT are plotted as a function of the multiplicative noise strength D. The other parameter values are the same as those in Fig. 1. Fig. 3 The theoretical results of x st are plotted as a function of the multiplicative noise strength D. The other parameter values are the same as those in Fig. 1. For a stochastic dynamical system, we consider not only the steady state characters, which is described by SPD, but also transient properties, which is described by mean first passage time MFPT. The MFPT is the time from one state to the other state, and it is a random variable. Here the MFPT of the process xt to reach the large tumor cell populations state x + with initial condition xt = = x u the unstable state can be given by, [4546] Tx + x u = xu x + BxP st x x P st ydy. 12 When the intensities of noises terms D are small enough compared with the energy barrier height Φx = Φx + Φx u. We can apply the steepest-descent approximation to Eq. 12, and T can be simplified as following [47,48] 2π Tx + x u U x +U x u [ Φxu Φx + ] exp, 13 D here, the potential U x is given by Eq. 3. For small D, making use of Eqs. 13 and 9, the transient properties of this system can be analysed by means In order to check the validity of the approximate method employed in the derivation, it is necessary to perform numerical simulation. The method is given by Refs. [49 5]. The numerical simulations are performed by integrating the dynamical Eq. 4 with Γt = and ΓtΓt = 2Dδt t. Gaussian white noise is generated using the Box Muller algorithm and a pseudorandom number generator. The numerical data of time series are obtained using a simple forward Euler algorithm with a small time step t =.1: in which x t+ t = xt + fx t + gxq g xgxq 2 + O t 2, 14 Q = [4D t lna] 1/2 cos2πb, fx = rx 1 x 1 + x 2, gx = x x and a and b are all independent random numbers. The numerical results of the SPD P st x, the mean value and the MFPT are plotted in Figs. 5 7, respectively. The numerical simulations of the P st x are plotted in Fig. 5 with the different multiplicative noise intensity D. From Fig. 5, it is seen that the highest peak of the curves of P st x is shifted to small value of x as D increases. In other words, with D increasing, the peak at the large value of x gradually disappears and the height of the peak at the small value of x gradually becomes high. The height of the peak in P st x is slightly lower than that of the theoretical result shown in Fig. 2. The numerical simulations

4 466 WANG Can-Jun, LI Di, and MEI Dong-Cheng Vol. 52 of the x st as a function of D are plotted in Fig. 6. From Fig. 6, it is seen that the x st monotonically decreases as D increases. The numerical simulations of the MFPT as a function of D are plotted in Fig. 7. From Fig. 7, it is seen that the MFPT monotonically also decreases as D increases. It is clear that the analytic results shown in Figs. 2 4 are consistent with the numerical computations. The dynamical characters of a theoretical anti-tumor model under immune surveillance subjected to a pure multiplicative noise are investigated from theory and numerical simulation. The effects of the noise intensity D on the SPD, x st and MFPT are analysed based the Fokker Planck equation in detail. The results are as follow. Fig. 5 The numerical simulations of the P stx are plotted as a function of x for different noise intensity D =.1,.2,.4, respectively. The other parameter values are the same as those in Fig. 1. Fig. 6 The numerical simulations of x st are plotted as a function of noise intensity D. The other parameter values are the same as those in Fig. 1. Fig. 7 The numerical simulations of the MFPT are plotted as a function of noise intensity D. The other parameter values are the same as those in Fig. 1. i There exists a bimodal structure in the curves of P st x. The highest peak of P st x shifts from the large value x to the small value x with D increasing. As D increases, the mean value of the state variable x x st monotonously decreases. It means that the tumor population move towards to extinction as the increase of D. ii The MFPT monotonously decreases with D increasing. For the anti-tumor model under immune surveillance, x denotes the tumor cell population, this result shows that the transition rate from large tumor cell population to the small tumor cell population is speeded up with the environmental fluctuations enhancing. That is to say that the extinction rate is enhanced as D increases. The analytic results are consistent with the numerical computations. References [1] F. Moss and P.V.E. McClintock, Noise in Nonlinear Dynamical Systems, Cambridge Univ. Press, Cambridge [2] A. Fulinski and T. Telejko, Phys. Lett. A [3] D.J. Wu, L. Cao, and S.Z. e, Phys. Rev. E [4] L. Cao, D.J. Wu, and S.Z. e, Phys. Rev. E [5] Y. Jia and J.R. Li, Phys. Rev. E ; Phys. Rev. E ; Phys. Rev. Lett [6] C.W. Xie and D.C. Mei, Chin. Phys. Lett [7] D.C. Mei, G.Z. Xie, L. Cao, and D.J. Wu, Phys. Rev. E [8] H.S. Wio and S. Bouzat, Braz. J. Phys [9] C.J. Wang, S.B. Chen, and D.C. Mei, Chin. Phys [1] P. Zhu, Eur. Phys. J. B [11] B.Q. Ai, H. Zheng, and L.G. Liu, Eur. Phys. J. B [12] S.Q. Zhu, Phys. Rev. A [13] L. Cao and D.J. Wu, Phys. Lett. A [14] S.B. Chen and D.C. Mei, Chin. Phys [15] D. Wu, X.Q. Luo, and S.Q. Zhu, Commun. Theor. Phys

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