## Abstract

Chemotherapy is a widely accepted method for tumour treatment. A medical doctor usually treats patients periodically with an amount of drug according to empirical medicine guides. From the point of view of cybernetics, this procedure is an impulse control system, where the amount and frequency of drug used can be determined analytically using the impulse control theory. In this paper, the stability of a chemotherapy treatment of a tumour is analysed applying the impulse control theory. The globally stable condition for prescription of a periodic oscillatory chemotherapeutic agent is derived. The permanence of the solution of the treatment process is verified using the Lyapunov function and the comparison theorem. Finally, we provide the values for the strength and the time interval that the chemotherapeutic agent needs to be applied such that the proposed impulse chemotherapy can eliminate the tumour cells and preserve the immune cells. The results given in the paper provide an analytical formula to guide medical doctors to choose the theoretical minimum amount of drug to treat the cancer and prevent harming the patients because of over-treating.

This article is part of the themed issue ‘Horizons of cybernetical physics’.

## 1. Introduction

In a healthy individual, a newly produced body cell replaces a damaged or dead one in an orderly and sustainable way. Cancer cells break this balanced order by multiplying themselves in an uncontrolled way, invading the space and demanding the nutrients of normal cells. The result is the death of the normal cells. According to the International Agency for Research on Cancer, there were 12.7 million new cancer cases in 2008; it is predicted that there will be 21.4 million cases of cancer and 13.5 million deaths by 2030 [1]. Cancer ranks as the number one killer in the world, therefore it is of great significance to explore the effective treatment techniques in order to reduce the rate of death due to cancer. It is no surprise that cancer treatment receives great attention around the scientific world [2,3].

For most types of cancers, such as chorionic carcinoma and heterogeneous tumour, a wide range of chemotherapeutic drug treatments are available [4]. Recently, there has been growing interest to understand, not only from the medical experimental point of view, but also from a theoretical perspective, the effects of the chemotherapy on the cells [5–7]. Fundamental issues involve the determination of the amount of drug used each time and the periodic interval between each drug use. From the viewpoint of cybernetics, the tumour–immune system interaction with periodical impulse chemotherapy can be considered as an impulse control procedure (or system), therefore it should be studied using impulse control theory and be treated using a cybernetics strategy.

The immune system plays an important role to identify and eliminate tumours. This is called immune surveillance. Our body’s defence against disease caused by a virus, bacteria or tumour is the destruction of infected cells or tumours by activated cytotoxic T-lymphocyte cells (CTL), also called hunter lymphocytes. CTL [8] can kill cells or make a programmed cell death. The biological activation process occurs efficiently when the CTL receive impulses generated by T-helper cells. The stimuli occur through the release of cytokines. This process involves a time delay for converting resting T-lymphocytes into CTL. The presence of the time delay makes the stability analysis become complicated in the tumour–immune interaction model. Borges *et al*. [9] proposed a tumour growth model with time delay. The authors investigated the treatment of cancer when impulse chemotherapy treatment was considered. This model is a time delay non-autonomous system, the non-autonomous nature being provided by the impulse treatment. The impulse control (treatment) of a dynamical system with delay introduces more difficulty for the cybernetic strategy design and the stability analysis of the controlled system.

In this paper, the model of Borges *et al*. [9] is extended by treating the impulsive chemotherapy as a dynamical variable. The extended system becomes a higher-dimensional delay differential system of equations concerning the tumour–immune interaction and the treatment of chemotherapy. Firstly, after some basic notations are defined in §2 and the impulse control system model is formulated in §3, the stability of the steady state (a periodic solution) of the extended system is studied in §4, which shows conditions for when the chemotherapy kills all cells. Secondly, the solution of the studied system is verified to be bounded using the Lyapunov function and comparison theorem in §5. And the periodic solution is verified to be stable in the sense of the (definition of) permanence in §6, which is guaranteed by a derived theorem (formula). Finally, a chemotherapy strategy supported by our simulations shows the correctness of the formula in §7. In conclusion, we provide a strategy to tell what parameters of the impulsive chemotherapy can eliminate tumour cells and preserve the permanence of the immune cells, i.e. they are not completely destroyed. Therefore, this study provides useful information for practical chemotherapy.

## 2. Notations and definitions

In this section, we give some definitions.

### Definition 2.1 (r-order piecewise continuous function [10]).

Let *PC*(*D*,*F*) represent a piecewise continuous function mapping *D* onto *F*, where *D*⊂*R*, *F*⊂*R*. We see that *ϕ*∈*PC*(*D*,*F*), *t*∈*D*, satisfies that *ϕ* is a continuous function for *t*≠*t*_{k}, and that *ϕ* is discontinuous and left continuous for *t*=*t*_{k}=*kT*, where *T* is the impulse period, as . An *r*-order piecewise continuous function, *PC*^{r}(*D*,*F*), represents a differentiable function of *ϕ*, which satisfies *ϕ*∈*PC*(*D*,*F*) and d^{r}*ϕ*/d*t*^{r}∈*PC*(*D*,*F*), *r*∈*N*, where *R* is real and *N* is an integer.

### Definition 2.2 (upper right derivative).

For an *m*-dimensional system and a positive function *V* :*R*_{+}×*R*^{m}_{+}→*R*_{+}, where ** x**=(

*x*

_{1},

*x*

_{2},…,

*x*

_{m}). The upper right derivative of

*V*(

*t*,

**) with respect to the system is defined as**

*x*

### Definition 2.3 (boundedness).

Suppose *ϕ*(*t*)=*x*(*t*,*t*_{0},*x*(*t*_{0})) is a solution of a dynamical system with *x*(*t*_{0})=*x*_{0}; if, for any positive real *B*>0 and the initial time *t*_{0}, there exists *γ*>0, such that |*x*(*t*,*t*_{0},*x*(*t*_{0}))|≤*B* for *t*≥*γ*+*t*_{0}, then the solution is ultimately bounded.

### Definition 2.4 (positive solution).

Assume *u*_{1}(*t*),*u*_{2}(*t*),…,*u*_{m}(*t*) is a solution of an *m*-dimensional system *U*. If *u*_{i}(*t*)>0, *i*=1,2,…,*m*, then (*u*_{1}(*t*),*u*_{2}(*t*),…*u*_{m}(*t*)) is defined as a positive solution of system *U*.

### Definition 2.5 (permanence [11]).

If there exists constants *ς* and *M* such that the solution of a system, *u*_{i}(*t*), satisfies , then the system is permanence, *ς* is the ultimately lower bound and *M* is the ultimately upper bound.

## 3. Tumour growth model with impulse chemotherapy

A mathematical model describing tumour growth under a treatment of chemotherapy was proposed recently [9]. The model is based on the predator–prey system [12]. The T-lymphocyte is the predator, while the tumour cell is the prey that is being attacked. The predators can be in a hunting or a resting state. The resting cells do not kill tumour cells, but they can become hunters after activation. The chemotherapeutic agent is treated as the inducement of activation. The chemotherapeutic agent acts as a predator on both cancerous and lymphocytic cells. The model is described by
3.1where *C*, *H* and *R* are the number of cancerous, hunting and resting cells, respectively, *t* is the time and *Z* is the concentration of the chemotherapeutic agent. The values of *q*_{1}, *q*_{2}, *α*_{1}, *α*_{2}, *K*_{1}, *K*_{2}, *p*_{1}, *p*_{2}, *p*_{3}, *a*_{1}, *a*_{2}, *a*_{3}, *g*_{1}, *g*_{2}, *g*_{3}, *d*_{1}, *β*_{1}, *ξ* are shown in table 1. By Δ we represent the infusion rate of chemotherapy; *τ* is the time delay of the conversion from resting cells to hunting cells. To make a clear distinction between parameters and variables, we define *C*=*x*_{1}, *H*=*x*_{2}, *R*=*x*_{3}, *Z*=*x*_{4}. Then, the extended tumour growth model with impulsive chemotherapy as a dynamical variable is described by
3.2where Δ*x*_{i}(*t*)=*x*_{i}(*nT*^{+})−*x*_{i}(*nT*^{−}) (*i*=1,2,3,4), *T* is the period of the impulse and *n*=1,2,3,… is a positive integer. This model means that at *t*=*nT*, an impulse drug treatment is applied with amplitude Δ.

Using the techniques to calculate equilibrium in time delay systems [13], the first formula of equation (3.2) has an equilibrium point given by (0, 0, 0, 0) as *t*≠*nT*. From the Jacobian matrix of system (3.2) evaluated at the equilibrium point (0, 0, 0, 0), we have
3.3implying that two eigenvalues of the Jacobian matrix have positive real part. Therefore, the equilibrium point (0, 0, 0, 0) is unstable.

## 4. The stability of periodic solutions of the chemotherapeutic agent

In this section, we study the stability of periodic solutions [14] of system (3.2), when *x*_{1}=0, *x*_{2}=0, *x*_{3}=0. Our interest is to demonstrate that the impulse perturbation creates a periodic solution in the chemotherapeutic variable *x*_{4}(*t*). For such a case, system (3.2) is described by the following equations:
4.1

### Lemma 4.1 [15]

*System* (*4.1*) *has a positive periodic solution* , *i.e. for any solution x*_{4}(*t*) *with initial condition x*_{4}(0^{+})>0, *as* , *where* .

### Proof.

Integrating the first formula of equation (4.1) on (*nT*,(*n*+1)*T*] yields
and we get
From the second formula of equation (4.1), we obtain a stroboscopic map:
This map has the only positive fixed points
or
The corresponding (4.1) has a periodic positive solution with period *T*, namely,
▪

### Theorem 4.2

*Let* (*x*_{1}(*t*),*x*_{2}(*t*),*x*_{3}(*t*),*x*_{4}(*t*)) *be any solution of* (*3.2*); *then* *is globally asymptotically stable provided* .

### Proof.

Firstly, we prove the local stability of a periodic solution by considering the behaviour of small-amplitude perturbations about the periodic solution.

Define
where (*u*(*t*),*v*(*t*),*l*(*t*),*w*(*t*)) are small perturbations. We expand system (3.2) according to Taylor’s formula, ignore higher-order terms and obtain the linearized equation
4.2Define *Φ*(*t*) to be the state transition matrix of system (4.2) (the first to fourth equations), hence
where *Φ*(*t*) satisfy
4.3and
with *Φ*(0)=*I*, where *I* is the identity matrix. The impulsive conditions of (4.2) (the fifth to eighth equations) become

Hence, if the absolute values of all eigenvalues of
are smaller than one, the periodic solution is locally stable (since [*u*(*t*),*v*(*t*), *l*(*t*),*w*(*t*)]^{T}→[0,0,0,0]^{T} for ). By calculating (4.3), we have
where , namely
then
and we have
Assume that λ_{1}, λ_{2}, λ_{3} and λ_{4} are the eigenvalues of ; then we have
The absolute values of eigenvalues *e*^{λ1}, *e*^{λ2}, *e*^{λ3}, *e*^{λ4} of *M* are less than one provided that . Therefore, according to Floquet theory, the periodic solution is locally asymptotically stable.

In the following, we prove the global stability of . Choose an *ε*>0 such that
According to the fourth equation of system (3.2), we have d*x*_{4}(*t*)/d*t*≤−*ξx*_{4}(*t*). For the following impulsive differential equation:
with initial condition
we have that *y*(*t*)≥*x*_{4}(*t*) by using the comparison theorem. Defining , we have for large enough *t*.

Let *ε*→0; we get , as .

From the first equation of (3.2), we get
4.4Integrating (4.4) on (*nT*,(*n*+1)*T*] yields
where

Thus, and *x*_{1}(*nT*)→0 as . Therefore, *x*_{1}(*t*)→0 as (since for *nT*<*t*<(*n*+1)*T*). By the same method, we can prove *x*_{2}(*t*)→0, *x*_{3}(*t*)→0 as .

Next, we prove that as if , and . For 0<*ε*_{1}<*ξ* there exist such that 0<*x*_{1}(*t*)<*ε*_{1}, 0<*x*_{2}(*t*)<*ε*_{1}, 0<*x*_{3}(*t*)<*ε*_{1} for . From the fourth equation of system (3.2), we have

Using comparison theory, we obtain *y*_{1}(*t*)≤*x*_{4}(*t*)≤*y*(*t*), , as , where *y*_{1}(*t*) are the solution of
for *nT*<*t*≤(*n*+1)*T*.

Therefore, there exists an *ε*_{1}>0 such that , for *t* being large enough. Let *ε*_{1}→0; we get . ▪

## 5. Boundedness

Now we show that all the solutions of system (3.2) are uniformly ultimately bounded.

### Lemma 5.1 [16]

*Let the function* *satisfy the following inequalities*:
*where f*(*t*), *g*(*t*)∈*C*(*R*_{+},*R*), *f*_{n}>0, *g*_{n} *and W*_{0} *are constants. Then*

### Theorem 5.2

*There exists a constant M*>0, *such that x*_{i}(*t*)≤*M, i*=1,2,3,4, *for each positive solution Ψ*(*t*)=(*x*_{1}(*t*),*x*_{2}(*t*),*x*_{3}(*t*),*x*_{4}(*t*)) *of system* (*3.2*) *with large enough t.*

### Proof.

Let *Ψ*(*t*)=(*x*_{1}(*t*),*x*_{2}(*t*),*x*_{3}(*t*),*x*_{4}(*t*)) be any positive solution of (3.2), and . Then *W*(*t*,*x*)∈*V* _{0}.

Because *Ψ*(*t*) is a positive solution of (3.2), from the third equation of system (3.2), we have . Integrating on (*t*−*τ*,*t*) yields *x*_{3}(*t*)≤*x*_{3}(*t*−*τ*)*e*^{q2τ}, and we obtain *x*_{3}(*t*−*τ*)≥*x*_{3}(*t*)*e*^{−q2τ}. Then the upper right derivative of *W*(*t*,*x*) along the solution of (3.2) is described as

For any λ>0 and *t*≠*nT*, by ignoring the third and fourth terms of the first equation of (3.2); ignoring the first, third and fourth terms of the second equation of (3.2); ignoring the third and fourth terms of the third equation of (3.2); and ignoring the second, third and fourth terms of the fourth equation of (3.2), we get
In the above equation, the second and fifth terms are positive constants. Define the sum of them as *K*, because *q*_{1}, *q*_{2}, *p*_{1}, *p*_{2}, *p*_{3}, *K*_{1}, *K*_{2}, *a*_{1}, *a*_{2}, *a*_{3}, *g*_{1}, *g*_{2}, *g*_{3} are all positive (as presented in table 1, which is determined by their biological meaning); at the same time, the first and fourth terms are negative and we have then
5.1If , for any positive solution *Ψ*(*t*) (that means that *x*_{2}>0 and *x*_{4}>0), the following equation holds:

For *t*=*nT* we obtain
where
and we have
5.2According to lemma 5.1, we have
and
then we have
The right-hand side of the inequality is *K*/λ+Δ*e*^{λT}/(*e*^{λT}−1) as .

Hence, *W*(*t*) is ultimately bounded for any positive solution of system (3.2). ▪

## 6. Permanence of the solution

### Theorem 6.1

*System* (*3.2*) *is permanent if β*_{1}*K*_{2}*e*^{(−q2τ)}>*α*_{2}*K*_{1} *and* (*p*_{2}/*a*_{2}(*β*_{1}*K*_{2}*e*^{(−q2τ)}−*α*_{2}*K*_{1}))Δ/*ξ*+*g*_{3}/(*a*_{3}+*K*_{2})+*g*_{2}/*a*_{2}+*g*_{1}/(*a*_{1}+*K*_{1}), *where K*_{1}, *K*_{2} *are parameters of* (*3.2*).

### Proof.

Suppose that *x*(*t*) is a solution of (3.2) with *x*(0)>0. From theorem 5.2, we can assume *x*_{4}(*t*)≤*M*. According to the first equation of (3.2), we get d*x*_{1}(*t*)/d*t*≤*q*_{1}*x*_{1}(*t*)(1−*x*_{1}(*t*)/*K*_{1}) for any positive solution of the system.

Considering the following comparison equation:
and
we have *x*_{1}(*t*)≤*w*(*t*) and *w*(*t*)→*K*_{1} as . Similarly, we can get the comparison equation for the second equation of (3.2):
and
and the comparison equation for the third equation of (3.2):
and

Thus, there exists an *ε*_{1}>0, such that *x*_{1}(*t*)<*K*_{1}+*ε*_{1} for large enough *t*. Without loss of generality, we assume *x*_{2}(*t*)<*ε*_{2}, *x*_{3}(*t*)<*K*_{2}+*ε*_{3}(*t*>0).

Let *m*_{4}=Δ*e*^{−ξT}/(1−*e*^{−ξT})−*ε*_{4}>0, *ε*_{4}>0. According to the comparison theorem, we have *x*_{4}(*t*)>*m*_{4} for large enough *t*. In the following, we want to find , , , such that , *x*_{2}(*t*)≥, for large enough *t*. We will do it in the following two steps.

*Step I*: Let *m*_{1}>0, *m*_{2}>0, *m*_{3}>0; we will prove that there exist *t*_{1}, *t*_{2}, , such that *x*_{1}(*t*_{1})≥*m*_{1}, *x*_{2}(*t*_{2})≥*m*_{2}, *x*_{3}(*t*_{3})≥*m*_{3}.

Firstly, we prove that there exist , such that *x*_{1}(*t*_{1})≥*m*_{1}. We use proof by contradiction and suppose that, for any , *x*_{1}(*t*_{1}) ≤*m*_{1}. ▪

### Proof.

Let *ε*_{1}>0 small enough so that
According to the above assumption, we get
According to the comparison theorem, we have *x*_{4}(*t*)≤*y*_{3}(*t*). By lemma 4.1, we get as , where *y*_{3}(*t*) is the solution of
6.1

Similarly to the periodic solution of equation (4.1), we have
for *t*∈(*nT*,(*n*+1)*T*].

Thus, there exists *T*_{1}>0 such that . In the first equation of system (3.2), replace *x*_{4} with , *x*_{2} with *ε*_{2}, and *x*_{1} with *m*_{1}. For *t*≥*T*_{1} we have
6.2Let *N*_{1}∈*Z*_{+} be a positive integer, and *N*_{1}*T*≥*T*_{1}. Integrating (6.2) on (*nT*,(*n*+1)*T*] (for *n*≥*N*_{1}), we get
where
similarly to the above case, for ,
6.3which is a contradiction to the boundedness of the solution. We conclude that there exists a *t*_{1} (*t*_{1}>0) such that *x*_{1}(*t*)≥*m*_{1}. In the same way, we can get similar conclusions for *x*_{2}(*t*), *x*_{3}(*t*).

From the above discussion, we get that there exist such that *x*_{1}(*t*_{1})≥*m*_{1}, *x*_{2}(*t*_{2})≥*m*_{2}, *x*_{3}(*t*_{3})≥*m*_{3}.

*Step II*: If *x*_{1}(*t*)≥*m*_{1} for all *t*≥*t*_{1}, then our aim is obtained. Otherwise, *x*_{1}(*t*)<*m*_{1} for some *t*≥*t*_{1}.

Setting , we have *x*_{1}(*t*)≥*m*_{1} for *t*∈[*t*_{1},*t**). It is easy to see that *x*_{1}(*t**)=*m*_{1}, since *x*_{1}(*t*) is continuous at *t**∈(*n*_{1}*T*,(*n*_{1}+1)*T*] for *n*_{1}∈*Z*_{+}. Select *n*_{2}, *n*_{3}∈*Z*_{+} such that
and
where
Setting *T*′=*n*_{2}*T*+*n*_{3}*T*, we claim that there must exist *t*′∈((*n*_{1}+1)*T*,(*n*_{1}+1)*T*+*T*′], such that *x*_{1}(*t*′)≥*m*_{1}. Otherwise, *x*_{1}(*t*)<*m*_{1} (for *t*∈((*n*_{1}+1)*T*,(*n*_{1}+1)*T*+*T*′]); considering (6.1) and *y*_{3}((*n*_{1}+1)*T*^{+})=*x*_{4}((*n*_{1}+1)*T*^{+}), we have
for *t*∈(*nT*,(*n*+1)*T*], *n*_{1}+1≤*n*≤*n*_{1}+1+*n*_{2}+*n*_{3}.

According to *y*_{3}((*n*_{1}+1)*T*^{+})=*y*_{3}((*n*_{1}+1)*T*^{−})+Δ and *x*_{4}(*t*)≤*M*, we get
where
and for (*n*_{1}+1+*n*_{2})*T*≤*t*≤(*n*_{1} +1)*T*+*T*′, which implies that (6.2) holds for (*n*_{1}+1+*n*_{2}) *T*≤*t*≤(*n*_{1}+1)*T*+*T*′. Similarly to (6.3), we have

There are two possible cases for *t*∈(*t**,(*n*_{1}+1)*T*]:

*Case* (*1*) (*x*_{1} has an upper bound for a finite time in ((*t**,(*n*_{1}+1)*T*]).

If *x*_{1}(*t*)<*m*_{1} for *t*∈(*t**,(*n*_{1}+1)*T*], then *x*_{1}(*t*)<*m*_{1} for all *t*∈(*t**,(*n*_{1}+1+ *n*_{2})*T*]. According to system (3.2), we have
6.4Integrating (6.4) on (*t**,(*n*_{1}+1+*n*_{2})*T*] yields
Then
which is a contradiction to the boundedness of *x*_{1}(*t*). Therefore, the assumption *x*_{1}(*t*)<*m*_{1} for all *t*∈(*t**,(*n*_{1}+1)*T*] is invalid.

Set ; then and (6.4) holds if only . Then integrating (6.4) on yields
for , a similar deduction can be made (since ≥*m*_{1}) to have for all *t*>*t*_{1}.

*Case* (*2*) (*x*_{1} still has an upper bound when a finite time in ((*t**,(*n*_{1}+1)*T*]) is smaller than Case (1)).

There exists a *t*^{′′}∈(*t**,(*n*_{1}+1)*T*] such that *x*_{1}(*t*^{′′})≥*m*_{1}. Let ; then *x*_{1}(*t*)<*m*_{1} for *t*∈ and . By integrating (6.4) on , we have
This process can be continued since and we have for all *t*≥*t*_{1}.

For both cases, we conclude for all *t*≥*t*_{1}. Similarly, we can prove for all *t*≥*t*_{2} and for all *t*≥*t*_{3}. ▪

### Theorem 6.2.

*Let (x*_{1}*(t),x*_{2}*(t),x*_{3}*(t),x*_{4}*(t)) be any solution of (3.2); then x*_{2}*, x*_{3} *and x*_{4} *are permanence and x*_{1}*(t)→0 as* *provided that β*_{1}*K*_{2}*e*^{(−q2τ)}*>α*_{2}*K*_{1} *and* *, (p*_{3}*/a*_{3}*q*_{2}*)Δ/ξ+g*_{3}*/a*_{3}*+g*_{2}*/a*_{2}*+g*_{1}*/(a*_{1}*+K*_{1}*)}<T<(p*_{1}*/a*_{1}*q*_{1}*)Δ.*

### Proof.

By the proving process of theorem 4.2, when *σ*=*q*_{1}*T*+(*p*_{1}*ε*/*a*_{1})*T*− *p*_{1}Δ/*a*_{1}<0, we have
Integrating (4.4) on *nT*<*t*<(*n*+1)*T*, we get
where
Then , and *x*_{1}(*nT*)→0 as . Therefore, *x*_{1}(*t*)→0 as (since ) (for *nT*<*t*< (*n*+1)*T*). By the proving process of theorem 6.1, we get *x*_{1}(*t*)>*m*_{1}, and according to the permanence condition, let , *m*_{1}→0, *ε*→0, *ε*_{1}→0, *ε*_{2}→0. From the proof process of theorems 4.2 and 6.1, we derive the conclusion of theorem 6.2. ▪

## 7. Simulation

Considering the parameters in table 1 for system (3.2), [11] gives the dashed line in figure 1 (numerically obtained) to show the relationship of the time interval *T* of the pulsed chemotherapy and the minimum value of Δ for which cancer can be suppressed. According to theorem 6.2, we know that the infusion rate Δ is linearly related to the period *T* of the impulsive chemotherapy to suppress the cancer. When *T* increases, it is necessary to increase the intensity of the chemotherapy to obtain cancer suppression. According to theorem 6.2 and parameters in table 1, we obtain the solid line in figure 1 by considering the upper bound of theorem 6.2, i.e. Δ=(*a*_{1}*q*_{1}/*p*_{1})*T*. The solid line is below the dashed line, which indicates that the infusion rate of chemotherapy given by theorem 6.2 is lower than that given in [11].

Using the parameters determined by the principle of theorem 6.2, we obtain the simulation results shown in figure 2, where the parameters are Δ=0.23 and *P*=12, marked by the point in figure 1.

## 8. Conclusion

Tumour chemotherapy procedure is a cybernetical system using impulse control in the field of cybernetic physics. In this paper, we investigate the stability of a tumour growth model with time delay and impulse chemotherapy using impulse control theory. We show the stability of the equilibrium point (chemotherapy kills all cells), the stability of the periodic oscillation of the chemotherapeutic agent (so the impulse chemotherapy function has a well-defined shape), the permanence of the immune cells (i.e. they are not completely destroyed by the chemotherapy), and the condition under which the chemotherapy can eliminate the cancer cells and preserve the immune cells. The relationship between the impulse treatment period and the intensity of the drug is given by the proposed theorem, which can be used by a doctor to determine the minimum amount of drugs administered to a patient to eliminate the cancer and at the same time minimize the harm to the immune cells and patient’s body.

## Authors' contributions

H.-P.R. designed the study, and carried out theoretical analysis, paper writing and editing. Y.Y. performed equation deduction and numerical simulation and drafted the manuscript. M.S.B. provided some background knowledge and edited the paper. C.G. gave valuable pieces of advice in discussion and edited the paper. All authors read and approved the manuscript.

## Competing interests

The authors declare that they have no competing interests.

## Funding

H.-P.R. was supported in part by NSFC (60804040), Fok Ying Tong Education Foundation Young Teacher Foundation (111065), Innovation Research Team of Shaanxi Province (2013KCT-04) and the Key programme of Natural Science Foundation of Shaanxi Province (2016ZDJC-01). M.S.B. was supported in part by the EPSRC (EP/I032606/1).

## Footnotes

One contribution of 15 to a theme issue ‘Horizons of cybernetical physics’.

- Accepted November 28, 2016.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.