The four-dimensional Kirschner-Panetta type cancer model: How to obtain tumor eradication?

  • Received: 03 January 2017 Revised: 06 March 2018 Published: 01 October 2018
  • MSC : Primary: 58F35, 58F17; Secondary: 53C35

  • In this paper we examine ultimate dynamics of the four-dimensional model describing interactions between tumor cells, effector immune cells, interleukin -2 and transforming growth factor-beta. This model was elaborated by Arciero et al. and is obtained from the Kirschner-Panetta type model by introducing two various treatments. We provide ultimate upper bounds for all variables of this model and two lower bounds and, besides, study when dynamics of this model possesses a global attracting set. The nonexistence conditions of compact invariant sets are derived. We obtain bounds for treatment parameters s1,2 under which all trajectories in the positive orthant tend to the tumor-free equilibrium point. Conditions imposed on under which the tumor population persists are presented as well. Finally, we compare tumor eradication/ persistence bounds and discuss our results.

    Citation: Alexander P. Krishchenko, Konstantin E. Starkov. The four-dimensional Kirschner-Panetta type cancer model: How to obtain tumor eradication?[J]. Mathematical Biosciences and Engineering, 2018, 15(5): 1243-1254. doi: 10.3934/mbe.2018057

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  • In this paper we examine ultimate dynamics of the four-dimensional model describing interactions between tumor cells, effector immune cells, interleukin -2 and transforming growth factor-beta. This model was elaborated by Arciero et al. and is obtained from the Kirschner-Panetta type model by introducing two various treatments. We provide ultimate upper bounds for all variables of this model and two lower bounds and, besides, study when dynamics of this model possesses a global attracting set. The nonexistence conditions of compact invariant sets are derived. We obtain bounds for treatment parameters s1,2 under which all trajectories in the positive orthant tend to the tumor-free equilibrium point. Conditions imposed on under which the tumor population persists are presented as well. Finally, we compare tumor eradication/ persistence bounds and discuss our results.


    1. Introduction

    The Kirschner-Panetta equations [2] have a great influence on the modelling tumor dynamics under immunotherapy. These equations describe dynamics of interactions between tumor cells, effector immune cells and interleukin-2 (IL-2). One of the promising generalizations of this model is obtained by incorporating in these equations yet another differential equation characterizing the dynamics of the suppressor cytokine, transforming growth factor-β (TGF-β), [1]. It is well-known [16] that TGF-β may display both inhibitory activity and stimulating activity on the growth of most of cells depending on type of cells, their differentiation and activation state. The production of TGF-β by tumor cells greatly challenges the immune system through the promotion of angiogenesis, enhancing tumor growth and metastasis. Tumors can evade immune surveillance by secreting various immuno-suppressive factors including (interleukin-10) IL-10 and TGF-β, [8,17]. It was indicated in [17] that activation of the immunosuppressed immune system by cytokine IL-2 therapy is a possible strategy to limit the malignant immuno-modulatory activities of TGF-β. This type of therapy may be applied alone or in combination with other immunotherapeutic approaches and is used in the clinical practice, [16]. This possible approach to cancer treatment is formalized in this paper by introducing into the model from [1] two treatment parameters s1,2. Parameters s1,2 are included in the equations for the same cells populations as in the Kirschner-Panetta (KP)- model. We recall that s1 is the treatment term that represents an external source of IL-2 injected into the system; s2 is the treatment term that represents an external source of effector cells such as LAK (lymphocytes activated killer cells) or TIL (tumor-infiltrating lymphocytes). Following to [1] we consider that s1,2 are constants. So we come to the following model in the non-dimensional form

    ˙x=rx(1xk)awx1+x+p2xz1+z,˙y=y+p3xw(g4+x)(1+αz)+s2,˙z=z+p4x2x2+τ2,˙w=cx1+γzμ1w+wy1+y(p1q1zq2+z)+s1. (1)

    In equation (1) x(t) describes the number of tumor cells at the moment t; y(t) describes the concentration of effector molecules at the moment t; z(t) describes TGF-β's immuno-suppressive and growth stimulatory effects in the single tumor-site compartment; w(t) describes the number of immune cells at the moment t.

    All parameters are supposed to be positive excepting s1,2 which are nonnegative. In the first equation r is the cancer growth rate; a is the cancer clearance term. The proliferation of tumor cells due to the response to TGF-β is denoted by p2 and is modeled by Michaelis-Menten kinetics. In the second equation p3 is the rate of IL-2 production in the presence of effector cells; g4 is half-saturation constant; α is a measure of inhibition. In the third equation p4 is maximal rate of TGF-β production; τ is the critical tumor cells population at which angiogenesis switch occurs. In the fourth equation c is known as the antigenicity of the tumor which measures the ability of the immune system to recognize tumor cells; γ is inhibitory parameter; μ1 is the death rate of immune cells; p1 is the proliferation rate of immune cells; q1 is the rate of anti-proliferative effect of TGF-β; q2 is half-saturation constant. More details concerning these parameters are contained in [1].

    We notice that if we put z=0,p4=0 in (1) we get the KP-model which was created for studying the immune response to tumors under special types of immunotherapy.

    Dynamics of the KP-model has been studied in [1,2,3,14] and some others. In particular, ultimate upper and lower bounds for state variables of the KP- model have been derived in different cases. The main result of [14] consists in global asymptotic tumor clearance conditions obtained under various assumptions imposed on the ratio between the proliferation rate of the immune cells and their mortality rate. To the best of the authors' knowledge, up to now there have not been published any results concerning rigorous dynamical analysis of (1). We notice that the system (1) introduced for the case s1=s2=0 was explored in [1] only by means of numerical simulations of its dynamics.

    In this paper we consider the tumor growth system pertained to the broad class of life sciences models which possess the following characteristic feature: there is a tumor-free equilibrium point, which is the most preferable state of the system. From the biological point of view some deviations from this equilibrium point can be dangerous and cause fatal outcomes.

    Therefore, the control goal is to return the system to the indicated equilibrium point and keep it in a sufficiently small neighborhood of this state. It can be done by different treatment injections.

    This article establishes the existence conditions for the positively invariant polytope that has a biological meaning. Further, two types of conditions are found: the first one is the tumor persistence (impossibility to achieve the control goal), another one is the tumor eradication (possibility of global asymptotical stabilization at the tumor-free equilibrium point).

    The goal and the novelty of this work consists in studies of ultimate dynamics (1) in case of applied treatments si,i=1,2. Namely, we find upper and lower ultimate bounds for all variables of the system (1) and establish conditions under which (1) is dissipative in the Levinson sense; 2) we propose the nonexistence conditions of compact invariant sets in the positive orthant; 3) we deduce the global asymptotic tumor eradication conditions; 4) we describe the tumor persistence conditions.

    In other words, the three most important results for understanding of any dynamical system's behavior are actually proved: the existence conditions of the global system's attractor; the coincidence of this attractor with the tumor-free equilibrium point; the presence of a local attractor of the system that does not contain the tumor-free equilibrium point, but attracts almost all ("perturbed") trajectories of the system.

    Our approach is based on the localization method of compact invariant sets in which the first order extremum conditions are utilized, see [4,5,6]. We also mention that earlier this method has been successfully utilized in studies of various cancer tumor growth models, see e.g. [7,9,10,11,12,13,14,15] and references therein.

    The remainder of the paper is organized as follows. In Section 2 we briefly present useful results. In Section 3 under some condition we obtain formulae for a polytope containing all compact invariant sets. This polytope provides us ultimate upper and lower bounds for all variables of the system (1), see Theorem 3.4. In Section 4 under the same condition as in Theorem 3.4 we show in Theorem 4.1 that this polytope contains the attracting set of the system (1). In Section 5 we present the nonexistence conditions of compact invariant sets in R4+,0{x>0}, see Theorem 5.1. Further, in Section 6 using results of Theorems 3.4; 4.1; 5.1 we derive conditions of global asymptotic stability with respect to the tumor-free equilibrium point (TFEP), see Theorem 6.1. In the latter theorem we provide bounds for treatments parameters si,i=1,2, for which the global asymptotic tumor eradication process may be observed. In Section 7 we describe the persistence tumor conditions which are compared with tumor eradication bounds of Theorem 7.1 in Section 8. Section 9 contains the concluding remarks.

    In what follows, we examine dynamics of (1) in the positive orthant

    R4+={(w,x,y,z)TR4,w;x;y;z>0};

    let R4+,0 be the closure of R4+.


    2. Some useful results

    We consider a nonlinear system

    ˙x=v(x) (2)

    where v is a C1differentiable vector field; xRn is the state vector. Let h(x) be a C1differentiable function such that h is not the first integral of (2). By h|B we denote the restriction of h on a set BRn. By S(h) we denote the set {xRnLvh(x)=0}, where Lvh(x) is a Lie derivative of h(x) with respect to v.

    Assume that we are interested in the localization of all compact invariant sets contained in the set U. Further, we define

    S(h;U):=S(h)U={xULvh(x)=0};hinf(U):=inf{h(x)xS(h;U)};hsup(U):=sup{h(x)xS(h;U)}.

    Assertion 1 [5,6] For any h(x)C1(Rn) all compact invariant sets of the system (2) located in U are contained in the localization set K(h;U) defined by the formula

    K(h;U)={xUhinf(U)h(x)hsup(U)}

    as well. If US(h)= then there are no compact invariant sets located in U.

    Assertion 2. Let U be a positively invariant set; ˙h(x)|H<0, where

    H={xUhsup(U)<h(x)}.

    Then for any τ0>0 the extended localization set

    ˆK(h;U;τ0):={xUh(x)hsup(U)+τ0} (3)

    is positively invariant.

    Assertion 3. Let U be a positively invariant set; τ0>0. If for any τ10 exists c>0: ˙h|H1c, where

    H1={xUhsup(U)+τ0h(x)hsup(U)+τ0+τ1},

    then every trajectory of the system (2) goes into the set ˆK(h;U;τ0) in finite time.


    3. Formulae for a polytope containing all compact invariant sets

    Here we find localization sets for the system (1). Let f be a vector field of this system.

    Lemma 3.1. All compact invariant sets in R4+,0 are located in the set

    K1=K(x,R4+,0)={0xxmax:=k(p2+r)r}R4+,0. (4)

    Proof. We apply the function h1=x and get that

    S(h1)={r(1xk)aw1+x+p2z1+z=0}{x=0}

    and h1,inf(R4+,0)=0. On the set S(h1)R4+,0{x>0} the inequality

    r(1xk)+p2z1+z=aw1+x0

    holds. Using the last inequality to calculate the supremum we obtain

    h1,sup(R4+,0)=k(p2+r)r.

    Let η=min(g4;1)ap13.

    Lemma 3.2. All compact invariant sets in R4+,0 are located in the set

    K2={ymin:=s2yymax:=(1+r+p2)2k4rη+s2}R4+,0. (5)

    Proof. We apply the function h2=x+ηy and compute

    Lfh2=xw(p3η(g4+x)(1+αz)a1+x)ηy+rxrx2k+p2xz1+z+ηs2=xw(p3η(g4+x)(1+αz)a1+x)+xh2+rxrx2k+p2xz1+z+ηs2.

    In R4+,0 we get the inequality

    Lfh2xh2+rxrx2k+p2x+ηs2

    because z1+z1 and η was chosen earlier such that

    p3η(g4+x)(1+αz)a1+x0.

    After calculating the supremum we obtain that

    h2|S(h2,R4+,0)h2,sup(R4+,0)=supx0((1+r+p2)xrkx2)+ηs2=(1+r+p2)2k4r+ηs2.

    Now let us use the function h3=y and compute

    h3,inf(K)=s2;h3,sup(K)supKy(1+r+p2)2k4rη+s2.

    Therefore,

    K(h3,K){s2y(1+r+p2)2k4rη+s2}K=K2

    and we come to the desirable conclusion.

    Lemma 3.3. All compact invariant sets in R4+,0 are located in the set

    K3=K(z,R4+,0)={0zzmax:=p4}R4+,0. (6)

    Proof. We apply the function h4=z and get that

    S(h4)={z=p4x2x2+τ2},h4,sup(R4+,0)=p4

    which leads to the desirable conclusion.

    Let M=K1K2K3.

    Theorem 3.4. If

    μ1>μpol1=μpol1(s2):=p1ymax1+ymax,ymax:=(1+r+p2)2k4rη+s2 (7)

    then all compact invariant sets are located in the polytope

    Π={0xxmax;yminyymax;0zzmax;wminwwmax},

    where

    wmin=s1μ1+q1p4q2+p4ymax1+ymaxp1s21+s2,wmax=cxmax+s1μ1μpol1(s2).

    Proof. We apply the function h5=w and transform equation Lfh5=0 in the equality

    w(μ1+y1+y(q1zz+q2p1))=cx1+γz+s1.

    Therefore,

    h5|S(h5,M)(μ1p1ymax1+ymax)cxmax+s1;h5,sup(M)wmax

    and

    s1h5|S(h5,M)(μ1+y1+yq1zq2+zp1y1+y)h5|S(h5,M)(μ1+ymax1+ymaxq1zmaxq2+zmaxp1ymin1+ymin);
    h5|S(h5,M)h5,inf(M)wmin,

    because

    μ1+ymax1+ymaxq1zmaxq2+zmaxp1ymin1+ymin>p1ymax1+ymaxp1ymin1+ymin0,

    and all compact invariant sets are located in the set {wminwwmax}M=Π.

    Corollary 1. If

    μ1>μM1=μM1(s2):=q1p4q2+p4ymax1+ymax+p1s21+s2 (8)

    then all compact invariant sets are located in the set

    M1:={wminw}M=={0xxmax;yminyymax;0zzmax;wminw}.

    4. On the dissipativity in the sense of Levinson

    Below we shall establish conditions under which the system (1) is dissipative in the sense of Levinson. Here we recall that the system (2) is called dissipative in the sense of Levinson if there exists r>0 such that for any xRn we have that

    limtsup|φ(x,t)|<r;

    here |φ(x,t)| is the Euclidean norm of the solution φ(x,t) of the system (2) starting in time t=0 at the point xRn.

    In this case there exists a bounded set which attracts any trajectory in Rn.

    Theorem 4.1. If condition (7) is fulfilled then the system (1) is dissipative in sense of Levinson in R4+,0

    Proof. Firstly, we note that extended localization sets

    ˆK1={h1=xˆxmax:=xmax+τ1}R4+,0;
    ˆK2={h2=x+ηyˆh2,sup(R4+,0):=h2,sup(R4+,0)+τ2}R4+,0;
    ˆK3={h4=zˆzmax:=zmax+τ3}R4+,0,

    where τi>0, i=1,2,3, have the form of the set (3) and, by Assertions 2, 3 (see remark below), are positively invariant and every trajectory goes into these sets in finite time. Therefore, the intersection of these sets ˆM=ˆK1ˆK2ˆK3 is a positively invariant set and every trajectory goes into this set in finite time.

    Next, if condition (7) is fulfilled then for some sufficiently small τ2>0 we get

    μ1>p1ˆymax1+ˆymax,ˆymax=ymax+τ2η.

    We fix such value of τ2 and find the localization set

    K:={wh5,sup(ˆM)¯wmax}ˆM,¯wmax=cˆxmax+s1μ1p1ˆymax1+ˆymax

    (see the proof of Theorem 3.4). By Assertions 2, 3 (see remark below), the bounded set

    ˆK4:={h5=wˆwmax:=¯wmax+τ4}ˆM,τ4>0,

    is a positively invariant set and every trajectory goes into this set in finite time. As a result, the polytope ˆM contains the attracting set of the system (1).

    Remark 1. The conditions of Assertions 2, 3 are fulfilled for localizing functions h1; h2; h4; h5, because the next estimations for their derivatives are correct:

    in the set {h1ˆxmax}R4+,0 the localizing function h1=x is equal to ˆxmax+Δ1>0, where Δ10, and therefore,

    ˙h1=(ˆxmax+Δ1){rrk(xmax+τ1+Δ1)aw1+x+p2z1+z}(xmax+τ1+Δ1)rk(τ1+Δ1)(xmax+τ1)rkτ1<0;

    in the set {h2ˆh2,sup(R4+,0)}R4+,0 the localizing function h2=x+ηy is equal to ˆh2,sup(R4+,0)+Δ2, where Δ20, and therefore,

    ˙h2τ2Δ2τ2<0;

    in the set {h4ˆzmax}R4+,0 the localizing function h4=z is equal to ˆzmax+Δ3, where Δ30, and therefore,

    ˙h4τ3Δ3τ3<0;

    in the set {h5ˆwmax}ˆM the localizing function h5=w is equal to ˆwmax+Δ4>0, where Δ40, and therefore,

    ˙h5cˆxmax+s1+(¯wmax+τ4+Δ4)(μ1+p1ˆymax1+ˆymax)=(τ4+Δ4)(μ1+p1ˆymax1+ˆymax)
    τ4(μ1p1ˆymax1+ˆymax)<0.

    5. The nonexistence conditions of compact invariant sets in R4+,0{x>0}

    Under condition (8) all compact invariant sets lying in the set R4+,0{x>0} are contained in the set O1:=M1{x>0} (see Corollary~1). Below we apply localizing function h1=x and show that its derivative is negative in the set O1 if some inequality holds. Therefore, in the case (8) this inequality is a nonexistence condition of compact invariant sets in the set R4+,0{x>0}. This condition may hold both in case of the existence of the TFEP and its nonexistence. It means the nonexistence of bounded tumor persistence dynamics, for example, the tumor dormancy. As a corollary, we describe the property of the nonexistence of periodic orbits and tumor persistence equilibrium points (TPEPs) in some range of model and treatment parameters.

    Let us denote

    C1:=μ1p1s2s2+1+q1ymaxp4(ymax+1)(q2+p4);C2:=r+rk+p2p41+p4.

    Theorem 5.1. Suppose that (8) and

    s1>satt1=satt1(s2):=w0aC1, (9)

    where

    w0={r+p2p41+p4,ifr(11k)+p2p41+p40,kC224r,ifr(11k)+p2p41+p4>0,

    hold. Then there are no compact invariant sets in the set R4+,0{x>0}.

    Proof. Let us apply the function h1=x and find that Lfh1|O1<0 in the set O1 if

    r(1xk)aw1+x+p2z1+z<0,

    i.e.

    awmin>maxx[0;xmax](1+x)(rrxk+p2zmax1+zmax)=w0.

    In order to find w0 we consider

    ˆη(x)=(1+x)(rrxk+p2p41+p4)

    and get

    ˆη(0)=r+p2p41+p4>0;ˆη(xmax)<0;ˆη(0)=r(11k)+p2p41+p4;
    ˆη(x)=0   if   x=k212+kp2p42r(1+p4);ˆη(x)=k4rC22.

    Therefore if ˆη(0)0 then w0=ˆη(0) and if ˆη(0)>0 then w0=ˆη(x).


    6. Global asymptotic stability respecting the TFEP

    If

    μ1>μTFEP1=μTFEP1(s2):=p1s21+s2 (10)

    the system (1) has the TFEP

    E1=(0,s2,0,w1)T,

    where

    w1=s1(1+s2)μ1+μ1s2p1s2=s1μ1p1s21+s2.

    The TFEP is asymptotically stable if r<aw1 i.e.

    s1>sst1=sst1(s2):=ra(μ1p1s21+s2). (11)

    Theorem 6.1. If conditions (7); (9) and (11) hold then the TFEP attracts all trajectories in R4+,0.

    Proof. If conditions of this theorem are fulfilled then all trajectories of the system (1) go into bounded positively invariant set ˆK4; the TFEP exists and is asymptotically stable. Therefore, in order to prove Theorem 6.1 it is sufficient to show that the TFEP is the unique compact invariant set of the system.

    The system (1) has no compact invariant sets in R4+,0{x>0} (see Theorem 5.1). The TFEP is the only compact invariant set of the system in the invariant plane {x=0}. Indeed, let us consider the system restricted on this plane

    ˙y=y+s2,˙z=z,˙w=μ1w+wy1+y(p1q1zq2+z)+s1. (12)

    Next, applying localizing functions x; y; w we obtain localization sets for compact invariant set of the system (12)

    K(y,R3)=Y:={y=s2};K(z,R3)=Z:={z=0};K(w,YZ)={(s2,0,w1)}.

    Now let us prove that the system (1) has no compact invariant set C for which CR4+,0{x>0} and C{x=0}. Indeed, otherwise C{x=0}=E1 and there exists a point PCR4+,0{x>0}. In this case the α-limit set of the trajectory starting at the point P is a nonempty compact invariant set D and E1D because otherwise the point E1 is not stable. Therefore, the nonempty compact invariant set D is a subset of R4+,0{x>0} and the statement of the theorem follows from this contradiction with Theorem 5.1.


    7. Tumor persistence conditions

    Theorem 7.1. Suppose that condition (7) holds and ωmax<r/a i.e.

    s1<sper1=sper1(s2):=ra(μ1p1ymax1+ymax)ck(p2r+1). (13)

    Then in R3+,0{x>0} each trajectory goes into the bounded positively invariant set

    P:=ˆK4{xx+τ5}

    where sufficiently small τ5>0;

    x+:=k12+(k1)24+kakrwmax

    in finite time.

    Proof. In the set R3+,0{x>0} each trajectory goes into bounded positively invariant set ˆK4{x>0} in finite time (see the proof of Theorem 4.1). In the set ˆK4{x>0} we have that

    ˙xx(rrkxaˆwmax1+x)=rxk(1+x)Q(x),Q(x)=x2+x(1k)+akrˆwmaxk.

    We note that ˆwmaxwmax under τ1,τ2,τ40. Therefore, if the condition (13) holds there exist sufficiently small τ1, τ2, τ4 for which ˆwmax<ra. Hence, then for any τ5>0, τ5<<min{x+;1}, the inequality Q(x)<0 is fulfilled in

    {0<xx+τ5}ˆK4.

    For any x1(0;x+τ5) the derivative ˙x is separated from zero in compact set {x1xx+τ5}ˆK4 and we come to the statement of the theorem.


    8. Comparison of bounds

    Now we describe how some features of ultimate dynamics depend on values s1 and s1 under condition that all other parameters are fixed. We recall the formulas

    μTFEP1(s2)=p1s21+s2(see (10));μM1(s2)=q1p4q2+p4ymax1+ymax+p1s21+s2(see (8));μpol1(s2)=p1ymax1+ymax(see ( 7)).

    It is easy to see that if s2>0 then the double inequality

    μM1(s2)<μTFEP1(s2)<μpol1(s2)

    is fulfilled. We notice that the value s2 determines the existence of:

    (ⅰ) the localization set ˆM1 for compact invariant sets of (1) under condition μM1(s2)<μ1;

    (ⅱ) TFEP, with μTFEP1(s2)<μ1;

    (ⅲ) bounded localization set Π for compact invariant sets of the system and global attractor, with μpol1(s2)<μ1.

    Let us consider the case when our system has the TFEP, i.e. μ1>μTFEP1(s2). In this case the value s1 determines the behavior of system trajectories relating to the TFEP. Indeed, we have introduced above functions

    sper1(s2)=ra(μ1p1ymax1+ymax)ck(p2r+1)==ra(μ1μpol1(s2))ck(p2r+1)(see (13));
    sst1(s2)=ra(μ1s21+s2)=ra(μ1μTFEP1(s2))(see (11));
    satt1(s2)=w0a(μ1p1s2s2+1+q1ymaxp4(ymax+1)(q2+p4))=w0a(μ1μM1(s2))(see (9)).

    It is easy to see that

    sper1(s2)<sst1(s2)<satt1(s2), s2>0.

    If s1<sst1(s2) holds then the TFEP is not stable and under additional conditions

    s1<sper1(s2);μpol1(s2)<μ1

    we have the tumor persistence.

    If s1>sst1(s2) holds then the TFEP is asymptotically stable and under additional conditions

    s1>satt1(s2);μpol1(s2)<μ1

    all trajectories in R4+ tend to the TFEP.


    9. Concluding remarks

    The main contribution of the present paper lies in the rigorous dynamical analysis of the four-dimensional system (1) and in obtaining global tumor clearance conditions via the localization method of compact invariant sets. We have studied various aspects of the ultimate dynamics of (1) describing interactions of cancer cells, TGF-β and immune cells under two types of the treatment. This research includes the following parts.

    1. Under condition (7) we have found all upper bounds for variables of the state vector of the system (1). Moreover, in this case it was shown that (1) has the property of the dissipativity in the sense of Levinson, because there exists the positively invariant polytope.

    2. Further, we provide conditions (8) and (9) under which there are no compact invariant sets in the set R4+,0{x>0}. As a result, there is no conditions for tumor dormancy. In particular, the system (1) has neither TPEPs nor periodic orbits.

    3. We find conditions (7); (9) and (11) under which the TFEP attracts all trajectories in R4+,0. The biological sense of this behavior consists in asymptotic eradication of tumor cells which means that after a while the tumor cells population will be under control.

    4. Tumor eradication and tumor persistence bounds are compared in Section 8. One can point to the following essential difference of dynamics of (1) in cases s1,2=0, [1], and in case s1,2>0 under varying antigenicity c. Namely, it was noticed in [1] that there are many negative scenarios including uncontrolled tumor growth and damped oscillations around the TPEP, which corresponds to tumor dormancy. In our work, because of the proper assignment of treatment parameters s1,2 satisfying Theorem 6.1 for given model parameters a,k,r,μ1,g4,p1,p2,p3,p4 one may achieve tumor eradication regardless the value of c, where c>0. However, the tumor persistence bound sper1 depends on parameter c.

    All assertions are formulated in terms of simple algebraic inequalities imposed on parameters of the model and treatments. These inequalities are stable for sufficiently small perturbations caused by imprecise knowledge of parameters' values which is convenient in applications.


    Acknowledgments

    1. We thank anonymous referees for the careful reading of this manuscript and helpful comments. 2. The work of the first author is supported by the Ministry of Education and Science of the Russian Federation (project Scientific Research Organization no. 1.4769.2017/6.7) and by the Russian Foundation of Basic Research (projects 16-07-00927 and 16-07-00902). The work of the second author is supported by the CONACYT project N 219614 "Analisis de sistemas con dinamica compleja en las areas de medicina matematica y fisica utilizando los metodos de localizacion de conjuntos compactos invariantes", Mexico.


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