Stochastic stability and state shifts for a time-delayed cancer growth system subjected to correlated multiplicative and additive noises

Chaos, Solitons and Fractals 93 (2016) 1–13

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Stochastic stability and state shifts for a time-delayed cancer growth system subjected to correlated multiplicative and additive noisesR Kang Kang Wang a,b,∗, Ya Jun Wang a, Sheng Hong Li a,c, Jian Cheng Wu d a

School of Mathematics and Physics, Jiangsu University of Science and Technology, Zhenjiang 212003, China Department of Mathematics, Southeast University, Nanjing 210096, China College of Mathematics, Nanjing Normal University, Nanjing 210097, China d State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China b c

a r t i c l e

i n f o

Article history: Received 6 April 2016 Revised 17 September 2016 Accepted 24 September 2016

Keywords: Multiplicative noise Addictive noise Time delay Cancer growth model Mean treatment time System stability

a b s t r a c t In the present paper, we investigate the stationary probability distribution(SPD) and the mean treatment time of a time-delayed cancer growth system induced by cross-correlated intrinsic and extrinsic noises. Our main results show that the resonant-like phenomenon of the mean first-passage time (MFPT) appears in the tumor cell growth model due to the interaction of all kinds of noises and time delay. Due to the existence of the resonant-like peak value, by increasing the intensity of multiplicative noise and time delay, it is possible to restrain effectively the development of the cancer cells and enhance the stability of the system. During the process of controlling the diffusion of the tumor cells, it contributes to inhibiting the development of cancer by increasing the cross-correlated noise strength and weakening the additive noise intensity and time delay. Meanwhile, the proper multiplicative noise intensity is conducive to the process of inhibition. Conversely, in the process of exterminating cancer cells of a large density, it can exert positive effects on eliminating the tumor cells by increasing noises intensities and the value of time delay. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction During the past decades, it is always a hot issue of medical profession in the world to investigate the evolvement of cancer cells. Whereas, so far we still know very little about the mechanisms of its growth and destruction. Up to now, there are some important measures designed for the treatment, such as surgery, chemotherapy and radiotherapy, but all of those traditional treatment methods could not eradicate the cancer, and many patients experience recurrence and die of cancer eventually after all kinds of treatment methods. Recently, a method of adoptive cellular immunotherapy [1,2] has obtained increasing importance to investigate for cancer treatment, integrating with the traditional tools. In order to explore the mechanisms of the cure for the cancer growth, a lot of applied mathematical models reflecting the change rule of tumor growth are proposed [3–7]. Besides, environmental factors

R Project supported by the National Natural Science Foundation of China (Grant No 11072107, 61371114), the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures of Nanjing University of Aeronautics and astronautics, China (Grant No. 0113G01), China Postdoctoral Science Foundation Funded Project (Grant No: 2016M591737) and the Project Fund of Jiangsu University of Science and Technology, China (Grant No. 633051203). ∗ Corresponding author. E-mail address: [email protected] (K.K. Wang).

http://dx.doi.org/10.1016/j.chaos.2016.09.022 0960-0779/© 2016 Elsevier Ltd. All rights reserved.

often play critical roles in the process of cancer treatment, such as the temperature, radiations, chemical drugs, degree of vascularization of tissues, supply of oxygen and nutrients, and the immunological state of the host. Hence, it should be necessary to take into account the probable impacts of the all types of stochastic disturbances on the treatment process before launching the treatment experiment [8–13]. In the previous work, the effect of the noise in the cancer dynamics has been studied analytically and numerically. Stochastic characteristics in all types of cancer models have been investigated widely, such as the stationary probability distribution [14,15], mean first passage time [15–18], resonant activation [19–23], noise enhanced stability [24–29]. Lately, Zeng et.al [30] investigated the phenomenon of stochastic resonance in a tumor growth model under the presence of immune surveillance with consideration of time delay and cross-correlation between the multiplicative and additive noises. Afterwards, they [31] discussed the phenomenon of stochastic resonance in a vegetation ecological system with time delay, at which the vegetation dynamics is assumed to be disturbed by both intrinsic and extrinsic noises. In Ref [32], Zhong studied that stochastic resonance driven by pure multiplicative noise in a noise-resonant effects in cancer development induced by external fluctuations and periodic treatment. However, few works study the joint action of the correlated noise, due to the interaction of internal and the external noises and time delay which always exists in the developing process of cancer growth.

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Moreover, few works analyze theoretically the characteristics of the steady states and transient properties of cancer growth system subjected to the correlated noises and time delay. In fact, the combination of noises and time delay is ubiquitous in nature and often changes fundamental dynamics of the system. Zeng et al [34,35] studied Schöogl model with time-delayed feedback to investigate the switching behavior of a bistable chemical reaction system in the presence of cross-correlated multiplicative and additive noise sources. They analyze the impact of noise and time delay in the monomer-dimer (MD) surface reaction model by using theoretical analysis. Meanwhile, they [36,37] investigated the noise-and delay- induced regime shifts in an ecological system of vegetation, the transport of an inertial Brownian motor moving in an asymmetric periodic potential, where it is driven by a time periodic and a constant biasing driving force. Also, Zeng et.al [38] discussed the dynamical properties of an anti-tumor cell growth system in the presence of delay and correlated noises. Valenti et.al [42,43] analyzed the dynamics of the FitzHugh–Nagumo (FHN) model in the presence of colored noise and a periodic signal and a spatially extended system of two competing species in the presence of two noise sources. They [44–47] also studied variable randomness of the stochastic process by varying the memory and investigated a mathematical model describing the growth of tumor in the presence of immune response of a host organism. In recent years, some study on the probability density function of the residence times in metastable systems, and on the role of additive and multiplicative noise sources in biological and magnetic systems are made in [48– 50]. In the present paper, based on the stochastic cancer development model, the characteristics of stationary probability distribution for the system and the mean treatment time caused by the impact of the additive noise, multiplicative noise, the crosscorrelated noise between them and the time delay are explored. The paper aims to contributing to a comprehensive and systematic understanding of the behavior of the tumor cells caused by the influences of the additive and multiplicative noises with the time delay. In Section 2, we introduce the stochastic cancer growth system. In Sections 3 and 4, the steady state properties for the generalized potential function and the SPDF of the cancer growth system are discussed in detail. In Section 5, the expressions for the mean treatment time between two stable states of the system are derived, and the effects of the noises and time delay on the MFPT of the stochastic system are analyzed numerically. A detailed conclusion and some description are given in the final section. 2. A stochastic cancer growth model including noise terms and time delay In the light of the derivation of Lefever and Garay [8,9], we can put forward the deterministic dynamical equation of the cancer growth model as follows:

dx(t ) x = ( 1 − θ x )x − β , dt 1+x

(1)

where x stands for the number of tumor cells, θ and β represent the influence coefficients which control the varying of number of tumor cells. θ > 0, β > 0. In fact, the status of tumor cells are always disturbed by all types of internal or external factors. Hence, we can rewrite the coefficient β as β + ξ (t), where ξ (t) indicates the fluctuation of many external factors, such as temperature, drugs, radiotherapy on the stochastic system. Meanwhile, taking into account some factors such as interaction between organs and the nervous system fluctuation, hormonal change and time delay that the drugs and radiotherapy need to militate in the human body, we have reason to introduce an additive noise η(t) and a time delay term τ into the tumor cell model in order to describe

Fig. 1. The bistable potential V(x). The parameter values of the potential V(x) are θ = 0.1, β = 2.6. The two stable states are xs1 = 0, and xs2 ≈ 6.56155, and one unstable state is xu ≈ 2.43847.

the effects of all external factors on the development of cancer cells:

dx(t ) x(t ) = x(t − τ )(1 − θ x(t − τ )) − [β + ξ (t )] + η (t ), dt 1 + x(t ) (2) in which ξ (t) and η(t) denote Gaussian white noises, whose statistical properties are defined as follows:

ξ (t ) = η (t ) = 0,   ξ (t )ξ (t  ) = 2Q δ (t − t  ),   η (t )η (t  ) = 2Mδ (t − t  ),      ξ (t )η (t  ) = η (t )ξ (t  ) = 2λ QMδ (t − t  ),

(3)

where both of Q and Mare the intensities of the multiplicative noise and the additive noise respectively. On the other hand, because all kinds of external fluctuations from drugs, radiotherapy and so on can interact with the intrinsic impact from human organs and the nervous system, a cross-correlated noise can be produced naturally by the external noise and the internal one. Here, we denote by λ the strength of the cross-correlated noise. − 1 < λ < 0 represents the negative correlation strength between the two noises; 0 < λ < 1denotes the positive one between the two noises. The deterministic potential function corresponding to Eq. (1) can be written as follows:

V (x ) =

θ 3

x3 −

1 2 x + β x − β ln(1 + x ). 2

(4)

Its figure in the interval [0,10] is plotted in Fig 1. Without counting the impact of noise terms and time delay, the fixed points of Eq. (1) are totally dependent on θ and β : (1) For θ > 1 and  0 < β < 1, a stable point xs = 0 and an unstable point 1−θ +

(1+θ )2 −4βθ

xu = ; 2θ (2) For 0 < θ < 1 and 0 < β < (1 + θ )2 /4θ , two stable point xs1 = 0,  xs2 = 

1−θ −

1−θ +

(1+θ )2 −4βθ and 2θ

(1+θ )2 −4βθ ; 2θ

an

unstable

point

xu =

(3) For 0 < θ < 1 and β = (1 + θ )2 /4θ , a stable point xs = 0 and an unstable point xu = (1 − θ )/2θ . In the following section, we will

K.K. Wang et al. / Chaos, Solitons and Fractals 93 (2016) 1–13

3

consider only the parameters in a region of 0 < θ < 1, 0 < β < (1 + θ )2 /4θ and the model has two stable states. By using the small delay approximation [31,39], we can get the consistent Markovian approximation. Therefore, the logistic Eq. (2) can be rewritten as

dx(t ) = fe f f (x ) + ge f f (x )ξ (t ) + η (t ). dt

(5)

In which the effective coefficients feff (x) and geff (x) are

f e f f (x ) =

ge f f ( x ) =



+∞

−∞



+∞

−∞



xτ ( 1 − θ xτ ) − β



x P1 (xτ , t − τ |x, t )dxτ , 1+x

−x P1 (xτ , t − τ |x, t )dxτ , 1+x

(6)

where P1, 2 (xτ , t − τ |x, t) stand for the conditional probability density of the stochastic process x(t)



P1 (xτ , t − τ |x, t ) =

P2 (xτ , t − τ |x, t ) =



[xτ − x − f ( x )τ ] 1 exp − 2π G ( x )τ 2G ( x )τ

2



[ x τ − x − g( x ) τ ] 1 exp − 2π G ( x )τ 2G ( x )τ

2



Fig. 2. Three-dimensional curve of the generalized potential U(x) versus x and Q. The values of the system parameters are λ = 0.5, M = 0.1, τ = 0.3, θ = 0.1, β = 2.6.

,

 (7)

x x f (x ) = (1 − θ x )x − β 1+ g(x ) = − 1+ G(x ) = Q g2 (x ) + x, x,  2λ QMg(x ) + M. Substituting Eq. (7) into (6), we obtain

where



f e f f ( x ) = ( 1 − θ x )x − β



ge f f ( x ) = −

x ( 1 + ( 2θ x − 1 )τ ), 1+x

x ( 1 + ( 2θ x − 1 )τ ), 1+x

(8)

Hence, according to Ref [40,41], we can derive the corresponding delay Fokker–Planck equation of the stochastic process x(t) under the influence of correlated noises terms based on Eq. (5) with Eq. (3) as follows

∂ P (x, t ) ∂ ∂2 2 = − μ(x )P (x, t ) + σ (x )P (x, t ). ∂t ∂x ∂ x2

(9)

In which P(x, t) is the probability distribution function. The drift coefficient μ(x) and the diffusion coefficient σ 2 (x) are defined as follows:



 μ(x ) = f (x ) + Qg(x )g (x ) + λ QMg (x ) (1 + (2θ x − 1 )τ ) (10)

 σ 2 (x ) = (Q g2 (x ) + 2λ QMg(x ) + M )(1 + (2θ x − 1 )τ )2

(11)

In accordance with Eqs. (9)–(11), we can obtain the steady probability distribution of the Fokker–Planck equation of the cancer growth system as follows:

Pst (x ) = lim P (x, t ) = t→∞

=

 

N B (x ) N B (x )



exp x

μ ( x )  dx σ 2 ( x )

exp [−U (x )],



(12)

where N is the normalization constant, and U(x) is the modified generalized potential function, whose explicit expression is given in the Appendix.

Fig. 3. Three-dimensional curve of the generalized potential U(x) versus x and M. The values of the system parameters are λ = 0.5, Q = 0.3, τ = 0.3, θ = 0.1, β = 2.6.

3. Effect of different noise terms on the generalized potential function We present a pictorial description of the three-dimensional curves of the generalized potential function U(x) which is excited by the correlated noises and time delay, respectively in Figs. 2–4. U(x) is an asymmetric bistable potential [33,36]. In the aspect of physics, the minimum of U(x) with regard to xs1 (or xs2 ) is also called the potential well, and the maximum of U(x) with regard to xu is called the potential barrier [36]. For convenience we call the potential well with regard to xs1 (or xs2 ) the left (or right) well. The multiplicative and additive noise terms can drive the particle to escape from a local minimum and reach other well. Here, we denote the depth of left well by d1 = U(xu ) − U(xs1 ), and the depth of right well by d2 = U(xu ) − U(xs2 ). In Fig. 2, as we see, by increasing the multiplicative noise intensity Q, the depths of two well decrease at the same time, but finally the depth of left well d1 exceeds that of the right well d2 . It implies that the system state is more easily attracted by the left well xs1 . Analogously, by increasing the additive noise intensity M(see Fig. 3), the two well depths also decrease altogether, and keep almost the same. It shows that the particle of the system can be attracted by either of two potential well. From Fig. 4, it is easy to see that the depth of the right well d2 is de-

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T ( xs2 → xs1 ) =



2π exp[U (xs2 ) − U (xu )] |V  (xu )V  (xs2 )|

(18)

in which U(x) and V(x) have been defined by Eq. (A1) of the Appendix and Eq. (4), respectively. 4. Steady state properties for a stochastic cancer growth system influenced by noises and time delay

Fig. 4. Three-dimensional curve of the generalized potential U(x) versus x and λ. The values of the system parameters are Q = 0.3, M = 0.1, τ = 0.3, θ = 0.1, β = 2.6.

creased, and that of the left well d1 is increased by increasing the strength λ, which implies the strength of the correlation noise can contribute to the extinction of tumor cells. With regard to the stochastic cancer growth system, we need to take into account the transient properties beyond its steady states by investigating the mean first-passage time (MFPT), which is the time passing through a potential barrier from one stable state to the other one. In fact, we can begin with Kolomogorov’s backward equation which is equivalent with (9). This finally leads to the MFPT equation for T(x).

μ (x )

∂ ∂2 T (x ) + σ 2 (x ) 2 T (x ) = −1, ∂x ∂x

(13)

with boundary conditions



dT (x )  dx 

= 0, T (x )|x=xs2 = 0,

(14)

x = x s1

which corresponds to a reflecting boundary at x = xs2 and an absorbing boundary at x = xs1 . By solving Eq. (13) with boundary conditions (14), the MFPT of the process x(t) to reach the extinction state xs1 with initial condition x(t = 0) = xs2 can be given by

T ( xs2 → xs1 ) =



x s2 x s1

σ

2

dx (x )Pst (x )

Similarly, we obtain

T ( xs1 → xs2 ) =



x s2 x s1

dx σ 2 (x )Pst (x )



+∞ x



x 0

Pst (y )dy.

Pst (y )dy,

(15)

(16)

We denote by T(xs1 → xs2 ) the mean development time for tumor cells going from the extinction state xs1 = 0 to the surviving steady state xs2 , and on the other hand, denote by T(xs2 → xs1 ) the mean extinction time for tumor cells coming back from the surviving steady state to the extinction steady state. We call the mean development time and the mean extinction time by a joint name the mean treatment time for the tumor cells. Because Eqs. (15) and (16) are too complicated to calculate, we consider the approximate expressions of the mean first passage time. When the noise intensity is small enough compared with the energy barriers

U (x ) = U (xu ) − U (xs1 ) and U (x ) = U (xu ) − U (xs2 ), by applying the steepest-descend approximation, we can obtain the modified MFPT as follows:

T ( xs1 → xs2 ) =



2π

|V  (x

u

)V  (xs1 )|

exp[U (xs1 ) − U (xu )]

(17)

In line with Eq. (12) of the SPDF, the impacts of the multiplicative and additive noises, the correlation strength between multiplicative and additive noises on the stationary probability distribution function of the stochastic tumor cells growth system can be discussed by the numerical calculations. We provide a series of numerical simulations for the steady probability distribution and MFPT of the systemfor the different parameters. The SPDF Eq. (12) with Eqs. (2,8–10) and the MFPT Eqs. (17,18) with Eqs. (2,8– 10) are simulated through the Euler arithmetic with a small time step t = 0.01. For each value of noise intensities and time delay, the statistics for the SNR has been taken from 10 0 0 0 simulation runs. The initial condition in the simulations is chosen as x(t − τ ) = xs1 when t ≤ τ . The comparison of the SPDF and MFPT between the analytical calculations and numerical simulations for different noise intensities and time delays are presented in Figs. 5, 9 and 10. Fig. 5 shows the steady state probability distribution function Pst (x) as a function of the number of tumor cells x for different values of the intensity of the multiplicative noise Q. As one see, in the case of λ = 0.5, with the increase of Q, the height of the probability peak at the surviving state xs2 of tumor cells is decreased greatly. At the same time, the probability of the system at the steady state of extinction state xs1 is increased substantially due to the increase of Q. Comparing Fig. 5(b) with Fig. 5(a), we can easily discover in Fig. 5(b) that the increase of Q can also decrease the probability density of the system at the steady state xs2 to some extent, but it can hardly change the probability of the system at the extinction state xs1 . With increasing Q, the probability at the unstable state xu is increased too. On the whole, we can see that no matter the cross-correlated noise strength λ is positive or negative, it can always play an effective and positive role in maintaining the diffusion of the cancer cells to increase the intensity Q. In particular, in the case of λ > 0, the increase of Q can accelerate the extinction of the tumor cells greatly. Fig. 6 shows the steady probability distribution function Pst (x) as a function of the number of tumor cells x for different values of time delay τ and two values of the parameterλ. The two panels (a) and (b) of Fig. 6 show almost identical physical characteristics. That is to say, as the time delay τ increases, the probability peak at the surviving steady state xs2 drops sharply and tends to disappear. In the meantime, the probability peak at the extinction state xs1 increases significantly. In the end, the curve of the probability distribution function almost degenerates into a monotone decreasing function on x. It shows that time delay τ always plays a positive role in inhibiting and destroying cancer cells in any case. The influence of the additive noise intensity M on the SPDF Pst (x) is displayed in Fig. 7. From Fig. 7(a) one can see that with the increase of M, the peak at the surviving steady state xs2 will decline substantially, but it is interesting that the probability at the extinction state xs1 will simultaneously fall rapidly. In Fig. 7(b), the intensity M can produce the similar effect on SPDF. The difference lies in the fact that in the case of λ = −0.5 the effect of M on the surviving steady state xs2 is not very significant compared to that shown in Fig. 7(a). In conclusion, no matter how two noises associate with each other, the intensity of M plays a crucial role in refraining the diffusion of cancer cells.

K.K. Wang et al. / Chaos, Solitons and Fractals 93 (2016) 1–13

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Fig. 5. The Pst (x) as a function of the density of tumor cells x for different values of Q(lines) and numerical simulations (symbols), the parameters take M = 0.1, τ = 0.3, θ = 0.1, β = 2.6. (a) λ = 0.5 (b) λ = −0.5.

Fig. 6. The Pst (x) as a function of the density of tumor cells x for different values of τ . The values of the system parameters are Q = 0.3, M = 0.1, θ = 0.1, β = 2.6. (a) λ = 0.5 (b) λ = −0.5.

Fig. 7. The Pst (x) as a function of the density of tumor cells x for different values of M. The values of the system parameters are Q = 0.3, τ = 0.3, θ = 0.1, β = 2.6. (a) λ = 0.5 (b) λ = −0.5.

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K.K. Wang et al. / Chaos, Solitons and Fractals 93 (2016) 1–13

some three-dimensional MFPT images in Figs. 9–10, 12–15. When the patients are in the early stage of the catching cancer, the doctors hope to extend the spread time of cancer cells in the body as far as possible. Therefore, we wish to prolong the development time of the cancer system further. As one can see, for the case of λ < 0, the mean treatment time is always a monotonic decreasing function on the noise intensity Q. In the case of λ > 0, there a significant single resonant peak arises, which goes up sharply as the strength λ increases, and whose position moves to the right gradually. From Fig. 8 we can observe that the strength of correlation noise λ can increase the mean development time of the cancer cells from the extinction state to the surviving steady one greatly, the resonance peaks can greatly slow down the development speed of tumor cells. Hence, for the case of λ > 0, the appropriate multiplicative noise intensity Q plays a positive role in restraining the tumor cells to proliferate.

θ = 0.1, τ = 0.3, β = 2.6. Fig. 8. The Pst (x) as a function of the density of tumor cells x for different values of λ. The values of the system parameters are Q = 0.3, M = 0.1, τ = 0.3, θ = 0.1, β = 2.6.

In Fig. 8, we present the effect of different values of crosscorrelated noise strength λ on the SPDF Pst (x). As we can observe in Fig. 8, with λ increasing, the peak of probability at the surviving steady state xs2 rises rapidly. Meanwhile, the probability peak at the extinction state xs1 also increases drastically, which implies that the increase of λ can produce double effects on the SPDF. It can not only promote the tumor cells to diffuse, but also make contributions to killing the cancer cells. 5. Mean treatment time for a stochastic cancer growth system under the effect of noises and time delay In this section, using Eqs. (17), (18) and Eqs. (A1), (A2) of the MFPT, we analyze numerically the effects of the time delay, intensities of multiplicative and additive noises, and the correlated strength on the MFPT. Fig. 9 shows the effect of different values of λ on the MFPT T(xs1 → xs2 ) as a function of the multiplicative noise intensity Q. For the convenience of observation, we give

Fig. 10 shows the effect of the strength of cross-correlated noise

λ on the MFPT T(xs1 → xs2 ) as a function of the intensity M. The

mean treatment time is a monotonic decreasing function of M, and an increasing function of λ. The additive noise M causes a negative effect on suppressing the spread of tumor cells, and correlated noise λ just the opposite. Fig. 11 shows roughly similar properties with those of Fig. 10. During the development process of the tumor cells from the extinction to the surviving state, the increase of the time delay τ plays a passive part in delaying the evolution speed of tumor cells. Fig. 12 illustrates the effect of the multiplicative noise intensity Q on the MFPT T(xs1 → xs2 ) with different values of M. In the case of λ > 0, we can see that there exists a resonance-like peak of the MFPT in the plot. The height of peak in Fig. 12 decreases rapidly with the increase of M, giving rise to a monotonic behavior of the MFPT for M=0.07. The figure shows that small values of multiplicative noise intensity Q and additive noise intensity M can greatly extend the time of the cancer cells spreading, that is, small Q and M values can play important roles in controlling the outbreak of cancer cells. With the increase of M, the population of tumor cells tends to the explosion. Fig. 13 shows that the curves for the MFPT T(xs1 → xs2 ) versus the multiplicative noise intensity Q and the additive noise intensity

Fig. 9. The MFPT T(xs1 → xs2 ) as a function of multiplicative noise intensity Q for different values of λ(lines) and numerical simulations (symbols), and the parameters take M = 0.05, τ = 0.3, θ = 0.1, β = 2.6.

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Fig. 10. The MFPT T(xs1 → xs2 )as a function of additive noise intensity M for different values of λ(lines) and numerical simulations (symbols), and the parameters take Q = 0.3, θ = 0.1, τ = 0.3, β = 2.6.

Fig. 11. The MFPT T(xs1 → xs2 )as a function of time delay τ for different values of λ. The values of the system parameters are Q = 0.2, M = 0.05, θ = 0.1, β = 2.6.

M with different large values of τ . As seen in Fig 13(a), the peak drops vertically as τ increases and the position of the maximum does not change with τ . Conversely, as shown in Fig. 13(b), the MFPT is a monotonic decreasing function of τ and M. It implies that the proper noise intensity Q could can greatly slow down the velocity of the cancer propagation . On the contrary, τ and M play prominent roles in speeding up the cancer development. In Figs. 14 and 15, we show the influence of the correlation noise intensity λ on the MFPT, when T(xs2 → xs1 ) represents the mean extinction time of cancer cells from the surviving steady state to the extinction one. In the process, we hope to shorten the treatment time as soon as possible in order to eliminate tumor cells in large quantities. By comparing Figs. 14, 15 with Fig. 9, it can be easily found that three figures show almost identical behaviors. In the case of λ < 0, the MFPT is still a monotonic decreasing function on the multiplicative noise intensity Q. In the case of λ > 0, as λ increases, a maximum of MFPT induced by the positive associated noise appears. But at present λ exhibits a negative effect on killing the large number of cancer cells. Due to the enhancement of λ, doctors need to spend more time in reducing

Fig. 12. Two-dimensional and three-dimensional curves of the MFPT T(xs1 → xs2 ) as a function of multiplicative noise intensity Q for different values of M. The values of the system parameters are λ = 0.5, τ = 0.3, θ = 0.1, β = 2.6.

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K.K. Wang et al. / Chaos, Solitons and Fractals 93 (2016) 1–13

Fig. 13. The MFPT T(xs1 → xs2 ) as a function of multiplicative noise intensity Q and additive noise intensity M for different values of τ . The values of the system parameters are λ = 0.5, M = 0.05, θ = 0.1, β = 2.6.

Fig. 14. Two-dimensional and three-dimensional curves of the MFPT T(xs2 → xs1 ) as a function of multiplicative noise intensity Q for different values of λ. The values of the system parameters are M = 0.05, τ = 0.3, θ = 0.1, β = 2.6.

Fig. 15. Two-dimensional and three-dimensional curves of the MFPT T(xs2 → xs1 )as a function of additive noise intensity M for different values of λ. The values of the system parameters are Q = 0.3, τ = 0.3, θ = 0.1, β = 2.6.

K.K. Wang et al. / Chaos, Solitons and Fractals 93 (2016) 1–13

Fig. 16. The MFPT T(xs2 → xs1 )as a function of additive noise intensity τ for different values of λ, and the other parameters take Q = 0.3, M = 0.05, θ = 0.1, β = 2.6.

the large number of tumor cells to a small one. Hence, the smaller the cross-correlation noise strength λ is, the less the time spent in lessening the density of tumor cells is. On the other hand, it can be noted that in the elimination process of tumor cells, the increase of the intensity Q can shorten greatly the decline time of the cancer cells in the case of Q > 0.01. In other words, it can lead to the rapid reduction of the tumor cells by increasing the intensity Q, which is completely in conformity with the physical behaviors shown in Figs. 2 and 5. The variation of the generalized potential function and the SPDF which are disturbed by Q displays the fact that the intensity Q can induce the cancer system to develop from the steady state of the large density to the extinction one. The physical phenomenon Fig. 16 shows is almost in agreement with those in Fig. 11, but this moment the time delay τ displays some positive meaning in the aspect of reducing the large density of tumor cells. On the contrary, the increase of λ produces a negative effect on shortening the mean treatment time. In comparison with Fig. 12, it can be found in Fig. 17 that in the case of λ > 0, the multiplicative noise intensity Q and additive noise intensity Mcan both lower the MFPT sharply like that

9

in Fig. 12. As shown in Figs. 17(a) and (b), it can greatly shorten the time of the cancer cells shifting from a large density state to a small one to increase Q and M. Consequently, we should increase the intensities of Q and Mbecause at this time they play a positive part in controlling and disrupting the spread of cancer cells. It should be pointed out that the intensity M can not only accelerate the spread of the tumor cells in the cancer control process (see Fig 12), but also speed up significantly the decline and fall of the tumor cells in the elimination process of the cells (see Fig. 17(b)), which reflects the double characters of the intensity M on the tumor cells. It is in agreement with the fact in Figs. 3 and 7 that M can simultaneously weaken the stability of the large steady state and the extinction one. Analogous to the behavior shown in Fig. 13, two peaks of MFPT T(xs2 → xs1 ) in Fig. 18 fall vertically with the increase of τ together. Whereas, there is something different, in Fig. 18(a), as τ increases, the mean treatment time decreases considerably. Comparatively, in Fig. 18(b), the decrease of τ on the MFPT, which is a function of the additive noise M, is not significant. In brief, in the process of destroying the cancer cells, it plays a critical role in inhibiting and eliminating the tumor cells to extend the time delay. Fig. 19 exhibits that the MFPT T(xs1 → xs2 ) and T(xs2 → xs1 ) are both monotonic increasing functions on the correlation noise strength λ. The increase of λ can not only prolong the time of the cancer cells developing and diffusing, but also add the time that doctors spends in eradicating the large number of tumor cells. The correlation noise strength λ plays a dual role in two processes of treating and controlling the tumor cells. Combining Figs. 9 and 14 with Fig. 19, we can easily that due to the existence of resonantlike peak of the MFPT, it is difficult for the state particle of the tumor cells system to shift from one steady state to the other. In other words, the stability of the two steady states are both enhanced greatly, which coincides with the physical behaviors of the generalized potential function and the SPDF impacted by λ shown in Figs. 4 and 8. 6. Conclusions In summary, we focus on investigating the transition properties of two steady states of a cancer growth model including the term of time delay subjected to the correlated Gaussian white noises. By applying the method of small time delay and Fokker–Planck equation to the steady probability density, we have acquired the gen-

Fig. 17. The MFPT T(xs2 → xs1 ) as functions of multiplicative noise intensity Q and additive noise intensity M, and other parameters take λ = 0.5, τ = 0.3, θ = 0.1, β = 2.6.

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K.K. Wang et al. / Chaos, Solitons and Fractals 93 (2016) 1–13

Fig. 18. Two-dimensional and three-dimensional curves of the MFPT as functions of multiplicative noise intensity Q and additive noise intensity M for different values of τ . The values of the system parameters are λ = 0.5, θ = 0.1, β = 2.6. (a) M = 0.03 (b) Q = 0.3.

Fig. 19. The MFPT T(xs1 → xs2 ) and T(xs2 → xs1 ) as functions of correlation noise strength λ. The values of the system parameters are Q = 0.3, M = 0.05, τ = 0.3, θ = 0.1, β = 2.6.

eralized potential function. By use of the steepest-descent approximation method, we get the expressions of the MFPT, and analyze in detail the influence of the time delay, the multiplicative, additive noises and the correlation between two types of noises on the stability and the MFPT of the cancer growth system. We have great interest in how the cancer development system is induced to shift from one stable state to the other because of the noises. Some conclusions are drawn as follows: The multiplicative noise and the correlation between two noises can induce cancer cells to shift from the surviving stable state to the extinction state, while the additive noise may simultaneously weaken the stability of two stable states for the tumor cells system. No matter that the strength of the correlation between two noises is positive or negative, the intensity of multiplicative noises Q and time delay τ can both play a positive role in controlling the development of the tumor cells. In particular, the time delay τ can even lead to the disappearance of tumor cells. On the other hand, it can decrease the probabilities at the extinction state and surviving steady one simultaneously by

K.K. Wang et al. / Chaos, Solitons and Fractals 93 (2016) 1–13

increasing the intensity M. It means that M causes beneficial effect on the controlling the overflow of the cancer cells. Inversely, the strength of cross-correlated noise λ can enhance the possibility that tumor cells diffuse and die out at the same time. Our research shows that multiplicative noise and time delay are both in favor of restraining the growth of the tumor cells. Therefore, in order to achieve our goal to induce the beneficial variation of systematic parameters, we should increase the dose of anti-cancer drugs, extend the action time of the drugs and improve the intensity of the ray in radiotherapy. With respect to the mean treatment time, we summarize the following laws:

11

2. During the process of controlling the development of the number of cancer cells from the extinction state to a surviving steady state, in the case of λ > 0, we should improve the strength of the correlation noise λ, and reduce the intensity of M to achieve our aim to restrain the outburst of the tumor cells. Meanwhile, the appropriate intensity of Q can exert a positive effect on holding down the spread of tumor cells. The increase of time delay τ can lead to acceleration of the cancer development. 3. In the destruction process of a large number of cancer cells, it is beneficial to the extinction of cancer cells to increase the time delay, intensities of multiplicative and additive noises and reduce the strength of the correlation noise.

1. In the case of λ > 0, the mean development time of cancer cells T(xs1 → xs2 ) is a non-monotonic function on the multiplicative noise, and a decreasing function of additive noises, the mean extinction time of tumor cells T(xs2 → xs1 ) is always the non-monotonic function of two noises; In the case of λ < 0, the MFPT is consistently the monotonic decreasing function on them. The increase of the initial density xs0 of tumor cells can speed up the spread of cancer cells towards the surviving steady state. Appendix The effective potential function U(x) is given by



U (x ) =



ln(Q x2 − 2λ QMx − 2λ QMx2 + M + 2Mx + Mx2 ) × A1

arctan +

√

B2 x+M−λ B3

B1 ×

B22

QM

B1 × B32 × A3

× B3

Where



arctan +

B1 ×



√

B2 x+M−λ B3

B32

QM

+

× A4

× B3

ln(1 + 2θ xτ ) × A2 B22 × B1 × (2θ τ − 1 )

+

1 x2 × A5 1 x2 ln(1 + x ) × A6 + + 2 , 2 2 B2 × τ 4 B2 × τ B2 × ( 2θ τ − 1 )





1 3 3 3 2 A1 = 3Q 2 M2 − Q 2 θ τ β M2 + 4θ τ λ3 M  QM + Q θ τ λ QM − 2 Q β − 2β M λ QM 3 2 4 2 2 2 2 2 − 8Q θ τ λ M + 15Q θ τ M λ Q M − 2M θ τ λ Q M − 2 Q M − Q M + 2Q 3 M 3 2 3 3/2 3 + 2Q M + 2M λ (QM ) − 4Q λ QM − Q 2 Mθ 2 τ + 3Q M2 θ 2 τ − θ τ β M3 2 2 − 2λ QMM θ τ − 2Q θ τ β M2 − 3Q 2 θ τ M2 − 3Q θ τ M3 + 4Q θ τ βλM QM 2 + 4Q τ βλM Q M − Q τ M4 + 12Q λ2 M3 + 12 β M3 − 4θ τ M2 λ2 (Q M )3/2 + 2Q 2 θ M − 18Q θ τ λ2 M3 − 2Q θ M2 + 2Q λ3 (QM )3/2 − 4Q θ τ βλ2 M2 + 4Q θ τ βλ2 M3





1 2 /2 2 2 2 3 + 4λ3 (QM )3 M + λθ M2 QM  − 2Q βλM − 2 Q β M+ 2Q βλ QM − Q θ τ M 2 2 2 2 2 + 9Q θ τ λM QM + 4Q Mλ Q M + λM Q M − λQ Q M − 2λ M Q + 2Q 2 θ τ M





2 3 3/2 + 2Q M2 θ τ − 6Q Mλθ τ Q M − 6Q 3 θ τ Mλ + 7θ τ M3 λ QM  + 4Q θ τ λ (QM  ) 2 2 2 2 2 3 − Q 2 θ λ QM − 24 Q θ τ λ M − 12 Q λ M QM − 12 Q M λ QM − 4 M λ QM     − 18Q 2 λ3 M QM − 2λ3 M3 QM − 20Q λ3 M2  QM + 2Q βλ2 M2 + 8Q 2 θ τ λ3 M QM + 12Q 3 λ2 M + 8Q 2 λ3 M2 + 12 M4 + 20Q θ τ λ3 M2 Q M + 12 Q 4 + 12 Q 3

A2 = 2θ τ β Q 2 + 2θ τ β M2 − 2θ MQ − 4θ τ QM − 16τ 2 θ 2 λ3 (QM )3/2 − 4τ θ Q 2 M −  2τ θ Q M 2 2 2 2 2 2 3 + 3M − Qθ − 2τ θ Q − M  Q − 2M θ τ − 2Q θ τ + 4θ Mλ QM + 4Qθ λ Q M + 8θ τ λM QM + 8 θ τ λ Q QM −8λ2 Q θ τ M − 4λ2 θ MQ − 4τ 2 θ 2 λQ 2 QM  2 2 2 2 2 2 2 2 2 2 2 − 8τ 2 θ 2 λMQ Q  QM − 4τ θ λM QM + 16τ θ λ MQ + 16τ θ λ M  2 2 3 2 2 2 + 10λτ θ Q 2 QM − 16 λ MQ τ θ − 8 λ M Q τ θ − 8 λ M Q τ θ + 2 τ θ λ M QM     3 + 4τ θ β QM + 12τ θ λ QMQM + 8τ θ λ QMMQ − 8τ θ β Q λ Q M− 8τ θ β Mλ Q M + 8τ θ βλ2 QM + 6QM − 2β Q 2 − 2β M2 − 4β QM + 12λ2 MQ − 12λ QMQ



− 8βλ2 QM − 12λ QMM + 8βλQ



QM + 8βλM



QM + 3Q 2 + τBθ4 + Bτ5 + τB26θ + τ B2 θ7 2 + τ B3 θ8 2

A3 = 2Q 2 M2 θ 2 τ − 6Q θ M4 τ − 6Q 2 θ τ M3 + 2Q M2 − 2Q 3 θ τ M + 4λ3 M3 (QM )3/2 2 − 4Q M2 θ τ − 2Q 2 θ λ2 M3 − 2Q 2 θ τ M2 + 4Q 3 M2 + 2Q θ τ M3 + 4Q 2 θ τ M 2 3 3 2 3 3 2 3/2 2 2 3 3 3 + 4λ M QM 8λ Q M + λQ QM  + 6Q M − 4λ M (QM ) −  + 2θ τ β M + 4λM2 θ 2 τ QM + 4Q θ βτ M2 − 8Q θ βτ Mλ QM − 8Q βτ M2 λ QM

(A1)

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K.K. Wang et al. / Chaos, Solitons and Fractals 93 (2016) 1–13

/2 − 6λ2 Q 3 M + 2θ τ M4 − 18λ2 Q M3 − β M3 + 8θ τ M2 λ3 (QM )3 + 28Q θ τ λ2 M3 2 2 3 3/2 2 2 2 − Q θ M + 3Q θ M + 4Q λ (QM ) + 8Q θ τ λ M − 2λM θ QM − Q 2 β M − 24Q 2 λ2 M2 − 2Q β M2 − 8 λ4 Q 2 M2 + 4Q λ2 θ 2 τ M3 + 2Q 2 M − M4 − Q 3 M − 3Q M3 − 3 Q 2 M2 − 6λQ M Q M − 2λM2  Q M + 4λ2 Q M2 − 2Q 2 Mθ τ − 2Q M2 θ τ 2 2 2 3 2 4 2 + 8Q θ τ Mλ Q M − 8Q θ τ Mλ + 4M θ τ λ QM + 4Q θ τ M + 16Qθ τ λ M 3 3/2 3 2 3 2 − 24λ θ τ M ( QM ) + 4Q θ τ λ M − 12θ τ λM  QM − 8Q θ τ λM Q M 3 − 4Q β M2 λ Q M − 2Q 2 θτ β M2 + 12Q θ τ  βλM2 Q M + 16Q 2 θ τ λ2 M2  2 + 15Q M2 λ  QM + 7λM3 QM + 8λ3 Q 2 M QM  + 4β M λ QM +4Q β Mλ Q M 2 2 3 3 2 + 20Q λ3 M2 QM − 4Q βλ2 M  − 16Q θ τ λ M QM − 8Q θ τ λ M QM 3 3 2 − 8θ τ λ M QM + 9Q Mλ QM







3 A4 = 12λ2 Q M3 τ θ + 4Q θ M2 λ Q M − 6Q 2 θ Mλ Q M+6 Q M3 θ 2 τ 2Q M λ QM + − 30Q 2 M2 λQM − 28Q M3 λ QM + 8λ3 M2 (QM )3/2 − 12Q 3 Mλ QM  − 24Q 3 λ3 M QM − 64Q λ3  M3 QM + 4Q λ3 M (QM )3/2 −84Q 2 M2 λ3 QM  3 4 − 16Q 2 M2 λ5 QM + Q 3 βλ QM − 2θ τ M 5 − 4Q 2 τ θ M λ QM + 8θ τ λ M  QM 3 2 2 2 3 4 − 4M θ τ λ QM + 24Q τ θ M λ QM + 36Q τ θ M λ Q M + 16θ τ M λ Q M







− 4Q 2 βλ2 M2 − 2Q 3 τ θ M2 − 9M4 λ QM − 4λ3 M4 QM + 8Q βλ2 M3 Q M 2 + 32Q 3 λ4 M2 − 8Q 2 θ τ β M2 λ2 − 4Q 3 β Mλ2 + 2Q 3 θ λ2 M + 12Q 2 M2 θ τ λ2 − 4Q θ M 4 2 3 4 2 3 2 4 2 3 3 − 2Q θ τ λ M  − 24Q θ τ λ M +2Q λ M + 8Q λ  M − 5β M λ QM− 4θ τ M λ QM 2 2 2 − 16Q τ θ M2 λQ M + 5Q 2 β Mλ Q M − Q β M λ  Q M + 2QMθ τ λ Q M  3 3 2 3 + 10θ τ β M λ QM − 4Q θ τ β M + 11Q M λ QM +  2M λ QM + 4Q M2 λ3 QM 3 3 3/2 4 2 3 3 3 − 8Q τ θ M λ (Q M ) − 50Q τ θ M  λ + 76Q θ τ M λ Q M − 16Q θ τ βλ2 M 3 2 + 8Q τ θ βλ3 M2 Q M + 4Q 2 βλ3 M  Q M + 4Q τ θ λM Q M + 2Q θ τ βλ M Q M − 56Q 2 θ τ λ4 M3 + 16Q 2 θ τ λ5 M2 QM − 30Q 3 θ τ λ2 M2 + 12Q 3 θ τ λ3 M QM − 2θ τ M4 β





+ 8Q θ τ λ3 M (Q M )3/2 − 78 Q 2 θ τ λ2 M3 +64Q 2 θ τ λ3 M2 Q M − 8Q θ τ λ3 M2 Q M 2 2 + 2Q θ τ β M + 4Q θ τ λM QM − λQ 3 QM + Q β M3 − 6Q λ2 M3 + 4Q 2 θ M2 + β M4



+ 32Q λ2 M4 + 48Q 3 λ2 M3 − λQ 4 QM − Q 3 β M − Q 2 β M2 + 48Q 2 λ4 M3 + 72Q 2 λ2 M3 + 4Q M 4 + Q 4 M + M 5 + Q 3 M − 3Q M 3 − 2Q 2 M 2 A5 =

A6 =

xQ

θ

+

xM

θ

4λ3 (QM ) Q

B1 = Q + 4λθ τ



1 xQ + 2λx QM + − 2 τθ 3/2

 λx QM 1 xM + − xQ τθ 2 τθ



− 4Q λ Q M + 4Q λ2 M − 4λQ M





Q M − 2Q M − 4λ3 M







(λ2 − 1 )QM



3 2 B4 = 2βλ2 Q M − 6λ2 Q M + 6λQ Q M −  2 Q − 3Q M + + β MQ − 2βλQ QM − 2βλM QM



QM − 6λM

B6 = −3λ2 QM −

Q M + Q 2 + M2

QM − 4Mθ τ − 2λ QM + M + 4Mθ 2 τ 2

B2 = Q − 2λ QM + M, B3 =

B5 = −6λQ





QM −

1 2

 β Q 2 + 12 β M2 + 6λM Q M

3 2 3 Q + 3QM + 6λ2 QM + M2 2 2

  3 3 3 MQ − Q 2 + 3λM QM + 3λQ QM − M2 2 4 4

B7 =

 1 1 1 MQ − λM QM + Q 2 + λ2 QM + M2 2 4 4

B8 =

  1 1 1 1 2 1 1 λ QM + MQ − λM QM − λQ QM − Q 2 + M2 2 4 2 2 8 8

Remark: B4 − B8 are included in the last terms of A2 .

(A2)

K.K. Wang et al. / Chaos, Solitons and Fractals 93 (2016) 1–13

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