# Dynamics of an Antitumour Model with Pulsed Radioimmunotherapy

Authors: Gang Wang; Ming Yi; Sanyi Tang
Journal: Computational and Mathematical Methods in Medicine
 (2022)

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## Abstract

In this paper, an antitumour model for characterising radiotherapy and immunotherapy processes at different fixed times is proposed. The global attractiveness of the positive periodic solution for each corresponding subsystem is proved with the integral inequality technique. Then, based on the differentiability of the solutions with respect to the initial values, the eigenvalues of the Jacobian matrix at a fixed point corresponding to the tumour-free periodic solution are determined, resulting in a sufficient condition for local stability. The solutions to the ordinary differential equations are compared, the threshold condition for the global attractiveness of the tumour-free periodic solution is provided in terms of an indicatorR0, and the permanence of a system with at least one tumour-present periodic solution is investigated. Furthermore, the effects of the death rate, effector cell injection dosage, therapeutic period, and effector cell activation rate on indicator R0 are determined through numerical simulations, and the results indicate that radioimmunotherapy is more effective than either radiotherapy or immunotherapy alone.


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## Body

## 1. Introduction

Cancer is a major public health issue and the leading cause of death worldwide. According to the World Health Organization (WHO), there were nearly10 million cancer-related deaths in 2020 [1]. The global cancer burden is expected to rise to nearly 21.4 million cases and 13.5 million deaths by 2030 [2]. Although numerous effective medical treatments against cancer have been developed, cancer treatment remains a challenging problem in neoteric medicine [3]. Host cells, or normal cells, should be kept above their minimum level throughout the entire body during cancer remission. As a result, modern techniques, such as surgery, chemotherapy, and radiotherapy, fail to destroy cancerous cells due to a lack of effective treatment strategies. In addition, chemotherapy harms cells in the bone marrow (myelosuppression), hair follicles (alopecia), and digestive tract (mucositis) under normal conditions. Therefore, chemotherapy depletes the immune system of the patient, leading to dangerous infections. Therefore, many patients suffer from the adverse effects of the treatment in addition to therapeutic resistance and cancer recurrence.Novel therapeutic strategies have been investigated, and immunotherapy has been recently approved for the treatment of various types of cancer [4]. Immunotherapy includes the use of antigen- and nonantigen-specific substances, such as cytokines, as well as adoptive cellular immunotherapy (ACI) [3]. Cytokines, such as IL-2 and IFN- α, are soluble proteins that mediate cell-to-cell communication [5]. During ACI, tissue cells are cultured to enhance and expand the immune system. ACI can be administered in two ways: (i) lymphokine-activated killer (LAK) cell therapy, in which cells are extracted from patients and cultured in vitro with high concentrations of IL-2 in peripheral blood leukocytes before being injected back into the cancer site; and (ii) tumour-infiltrating lymphocyte (TIL) therapy, in which cells are extracted from lymphocytes recovered from the patient with cancer and incubated with high concentrations of IL-2 before being injected back into the patient. The use of ACI slows or stops the spread of cancer cells to other parts of the body and helps the immune system become more effective by eliminating cancer cells.Various mathematical models have been studied for cancer treatments with virotherapy, radiotherapy, chemotherapy, and immunotherapy [6–11]. Based on the inhibition model, Piantadosi model, and autostimulation model, Antonov et al. investigated impulsive tumour growth models to describe medical interventions during cancer treatments [12]. Sigal et al. modelled the effects of immunotherapy, specifically dendritic cell vaccines and T cell adoptive therapy, on tumour growth with and without chemotherapy [13]. The model demonstrated that chemotherapy increases tumorigenicity, whereas CSC-targeted immunotherapy tumorigenicity. Pratap proposed a model that describes the nonlinear dynamics between tumour cells, immune cells, and three forms of therapy: chemotherapy, immunotherapy, and radiotherapy [14]. The model was used to develop optimized combination therapy plans using optimal control theory. Feng and Navaratna demonstrated that the initial ratio between regulatory T cells and effector T cells impacts the tumour recurrence time and that the effectiveness of IL-2 use may reverse the immunotherapy outcome [15].Dong et al. investigated the role of helper T cells in the tumour immune system and proposed the following model [16]:
(1)dxdt=αx1−βx−xy,dydt=ω1xy−δ1y+ρyz,dzdt=σ2+ω2xz−δ2z,wherex,y, and z represent the populations of tumour cells (TCs), effector cells (ECs), and helper T cells (HTCs), respectively. The first equation describes the rate of change in the TC population. Here, the logistic growth term αx1−βx was chosen, where α is the maximal growth rate of the TC population, and 1/β is the carrying capacity of the TC biological environment. The second equation describes the rate of change in the EC population. ECs have an average lifespan of 1/δ1. ω1 is the EC stimulation rate by EC-lysed TC debris. ρ is the EC activation rate by the HTCs. The third equation describes the rate of change in the HTC population. σ2 is the birth rate of the HTCs produced in the bone marrow. HTCs have an average lifespan of 1/δ2. ω2 is the HTC stimulation rate in the presence of identified tumour antigens. To address the lack of biostability, Talkington et al. assumed that ω1=0 and introduced saturation into the tumour interactions [17]:
(2)dxdτ=αx1−βx−yxx+η1,dydτ=σ0−δ1y+ρyz,dzdτ=σ2−δ2z+ω2zxx+η3,where σ0>0 is the birth rate of the ECs, and η1 and η3 are half-saturation constants.As discussed above, radiotherapy is usually used in cancer treatment because it permanently damages the DNA of tumour cells, destroying these cells [18, 19]. While nearby healthy tissue cells can suffer temporary damage from this radiation, these cells can repair the DNA damage and continue to grow normally. Numerous studies have shown that radioimmunotherapy is more effective for inhibiting tumour growth than radiotherapy [4, 20]. Thus, compared to the continuous system models mentioned above, we introduce pulsed ACI and radiotherapy into system (2) and analyse the effect of the combined treatment [7, 21–23]. Our novel system is formulated as follows:
(3)dydt=−δ1y+ρyz,dzdt=σ2−δ2z+ω2zxx+η3,dxdt=αx1−βx−yxx+η1,t≠n−1T+lT,t≠nT,yt+=1−pEyt,zt+=1−pHzt,xt+=1−pTxt,t=n−1T+lT,yt+=yt+σ1,zt+=zt,xt+=xt,t=nT,wherepE, pH, and pT denote the death rates of the ECs, HTCs, and TCs due to radiotherapy at time t=n−1T+lT, respectively. Here, 0<pE,pH<pT, 0<l<1, and T>0 are the therapeutic period. σ1>0 represents the dosage of infusing the ECs with antitumour activity at time t=nT.In this article, we study the effects of impulsive perturbations on the tumour-free solution of model (3) and the threshold values of its stability conditions. In addition, the mathematical criteria for the permanence of system (3) are investigated. Numerical simulations were carried out to validate our analytical results.The article is organized as follows. In Section2, for convenience, we present some definitions and lemmas. In Section 3, the local stability and global attractiveness of the tumour-free periodic solution are studied by means of the linearized Floquet stability and comparison techniques. Several additional technical computations that were used to establish the results presented in this section are deferred to see appendix. In Section 4, it is shown that once the threshold condition is satisfied, as well as certain other conditions, system (3) is permanent, with at least one tumour-present periodic solution. Numerical simulations that confirm our theoretical findings are discussed in Section 5 and Figures 1 and 2. Finally, a discussion of the theoretical and numerical results is provided.Figure 1
Dynamic behaviours of system (3) with radiotherapy or immunotherapy alone. (a)–(c) ρ=0.01, β=0.002, pE=0.15, pH=0.1, pT=0.9619, σ1=0, and T=2; (d)–(f): ρ=0.01, β=0.002, pE=0, pH=0, pT=0, σ1=2, and T=2. The other parameters are identical to those in (101), and the initial values in (a)–(f) are 1,1,500.
(a)(b)(c)(d)(e)(f)Figure 2
Dynamic behaviours of system (3), where T=5 in (a)–(c) and T=6 in (d)–(f). The other parameters are identical to those in Figure 5, and the initial values in (a)–(f) are 0.2442,5.0060,0.0256.
(a)(b)(c)(d)(e)(f)


## 2. Preliminaries

In this section, we introduce some definitions and preliminary lemmas that are useful for establishing our results.Definition 1 (see [24]).
System (3) is said to be permanent if there are constants m,M>0 (independent of the initial values) and a finite time T0 such that, for all solutions, yt,zt,xt with all initial values y0+,z0+,x0+>0, m≤yt,zt≤M and m≤xt≤M hold for all T≥T0. Here, T0 may depend on the initial value y0+,z0+,x0+.
Similar to Lemma 1 in [21], we obtain that the solution of dΨt/dt=χ1tΨt+χ2t is
(4)Ψt=Ψt0e∫t0tχ1τdτ+∫t0tχ2se−∫tsχ1τdτds.
Thus, it follows thatΨt≥0 for Ψt0≥0, with χ2t≥0 and t≥t0. Then, the following lemma is valid.Lemma 2.
R+3 is a positively invariant region for system (3).
Letxt≡0; then, system (3) can be reduced to the following system:
(5)dydt=−δ1y+ρyz,dzdt=σ2−δ2z,t≠n−1T+lT,t≠nT,yt+=1−pEyt,zt+=1−pHzt,t=n−1T+lT,yt+=yt+σ1,zt+=zt,t=nT.
According to (5), we can obtain that
(6)yt=yn−1T+e∫n−1T+t−δ1+ρzsds,n−1T<t≤n−1T+lT,yn−1T+lT+e∫n−1T+lT+t−δ1+ρzsds,n−1T+lT<t≤nT,(7)zt=σ2δ2+zn−1T+−σ2δ2e−δ2t−n−1T,n−1T<t≤n−1T+lT,σ2δ2+zn−1T+lT+−σ2δ2e−δ2t−n−1T+lT,n−1T+lT<t≤nT.
It is clear that(8)ynT+=1−pEyn−1T+e∫n−1T+nT−δ1+ρztdt+σ1,znT+=1−e−δ21−lT+1−pHe−δ21−lT−e−δ2Tσ2δ2+1−pHe−δ2Tzn−1T+.
Let(9)ynT+=yn−1T+,znT+=zn−1T+;then, we have
(10)ynT+=σ11−1−pEe∫0+T−δ1+ρztdt,znT+=1−e−δ21−lT+1−pHe−δ21−lT−e−δ2Tσ2/δ21−1−pHe−δ2T.Thus, when (15) is valid, system (5) has a unique positive periodic solution, which can be formulated as follows:
(11)y∗t=y∗0+e∫n−1T+t−δ1+ρz∗sds,n−1T<t≤n−1T+lT,y∗lT+e∫n−1T+lT+t−δ1+ρz∗sds,n−1T+lT<t≤nT,(12)z∗t=σ2δ2+z∗0+−σ2δ2e−δ2t−n−1T,n−1T<t≤n−1T+lT,σ2δ2+z∗lT+−σ2δ2e−δ2t−n−1T+lT,n−1T+lT<t≤nT,where
(13)y∗0+=σ11−1−pEe∫0+T−δ1+ρz∗tdt,y∗lT+=1−pEy∗0+e∫0+lT−δ1+ρz∗tdt,(14)z∗0+=1−e−δ21−lT+1−pHe−δ21−lT−e−δ2Tσ2/δ21−1−pHe−δ2T,z∗lT+=1−pHσ2δ2+z∗0+−σ2δ2e−δ2lT.Hence, we can obtain the following conclusion.Lemma 3.
System (5) has a unique positive periodic solution y∗t,z∗t if and only if
(15)1−pEe∫0+T−δ1+ρz∗tdt<1,and, for every solution yt,zt of (5), it follows that
(16)limt⟶+∞yt−y∗t=0,limt⟶+∞zt−z∗t=0.Proof.
It is easy to prove thatlimt⟶+∞zt−z∗t=0.
For an arbitraryε1>0, we choose an ε1>0 that is sufficiently small such that
(17)C1<1,C2eρε1T1−C1<ε14eρ∫0+Tz∗tdt,C3<ε12,where the first inequality in (17) is valid based on (15), and
(18)C1=1−pEe∫0+T−δ1+ρε1+z∗tdt,C2=ρε11−pEe∫0+T−δ1+ρε1+z∗tdt×∫0+lTy∗te−∫0+t−δ1+ρε1+z∗sdsdt+e∫lT+T−δ1+ρε1+z∗tdt×∫lT+Ty∗te−∫lT+t−δ1+ρε1+z∗sdsdt,C3=ρε1eρε1T+ρ∫0+Tz∗tdt∫0+lTy∗teδ1tdt+eρε11−lT+ρ∫lT+Tz∗tdt∫lT+Ty∗teδ1t−lTdt.
Without loss of generality, assume that(19)zt−z∗t<ε1,for t≥0.
Whent≠n−1+lT,nT and yt−y∗t≠0, it follows from (5) and (19) that
(20)dyt−y∗tdt=sgnyt−y∗t×−δ1yt−y∗t+ρytzt−y∗tz∗t=−δ1yt−y∗t+ρsgnyt−y∗tyt−y∗tzt+y∗tzt−z∗t≤−δ1+ρε1+z∗tyt−y∗t+ρε1y∗t,
which implies that(21)ddtyt−y∗te−∫t0t−δ1+ρε1+z∗sds≤ρε1y∗te−∫t0t−δ1+ρε1+z∗sds.
Integrating (21) from n−1T,t or n−1T+lT,t gives
(22)yt−y∗t≤e∫n−1T+t−δ1+ρε1+z∗sdsyn−1T+−y∗n−1T++∫n−1T+tρε1y∗τe−∫n−1T+τ−δ1+ρε1+z∗sdsdτ,n−1T<t≤n−1T+lT,e∫n−1T+lT+t−δ1+ρε1+z∗sdsyn−1T+lT+−y∗n−1T+lT++∫n−1T+lT+tρε1y∗τe−∫n−1T+lT+τ−δ1+ρε1+z∗sdsdτ,n−1T+lT<t≤nT.
It should be noted that(23)yt+−y∗t+=1−pEyt−y∗t,t=n−1T+lT,yt+−y∗t+=yt−y∗t,t=nT.
Then, it follows from (22) and (23) that
(24)ynT+−y∗nT+≤C1yn−1T+−y∗n−1T++C2,which implies that
(25)ynT+−y∗nT+≤C1ny0+−y∗0+−C21−C1+C21−C1.
Then, according to (25) and (17), there exists an N1>0 such that when n≥N1, it holds that
(26)ynT+−y∗nT+<ε12eρε1T+ρ∫0+Tz∗tdt.
Hence, whent∈n−1T,nT, where n−1≥N1, it follows from (22), (26), and (17) that
(27)yt−y∗t≤eρε1T+ρ∫0+Tz∗tdtyn−1T+−y∗n−1T++C3<ε1.
Sinceε1>0 is arbitrary, we conclude that limt⟶+∞yt−y∗t=0.
This completes the proof.Similarly, we arrive at the following conclusion.Lemma 4.
For every solutionyt,zt,xt of system (3), there exist three positive constants ME,MH>0 and MT>0 such that
(28)yt≤ME,zt≤MH,xt≤MT,for sufficiently large t>0, provided that
(29)ω21/β1/β+η3<δ2,1−pEe∫0+T−δ1+ρz¯0tdt<1,where
(30)z¯0t=σ2δ2−ω21/β/1/β+η3+z¯00+−σ2δ2−ω21/β/1/β+η3e−δ2−ω21/β1/β+η3t−n−1T,n−1T<t≤n−1T+lT,σ2δ2−ω21/β/1/β+η3+z¯0lT+−σ2δ2−ω21/β/1/β+η3e−δ2−ω21/β1/β+η3t−n−1T+lT,n−1T+lT<t≤nT,with
(31)z¯00+=1−e−δ2−ω21/β/1/β+η31−lT+1−pHe−δ2−ω21/β/1/β+η31−lT−e−δ2−ω21/β/1/β+η3Tσ2/δ2−ω21/β/1/β+η31−1−pHe−δ2−ω21/β/1/β+η3T,z¯0lT+=1−pHσ2δ2−ω21/β/1/β+η3+z¯00+−σ2δ2−ω21/β/1/β+η3e−δ2−ω21/β1/β+η3lT.Proof.
On the basis of (29), we choose an εT>0 that is sufficiently small; then,
(32)ω21/β+εT1/β+εT+η3<δ2,1−pEe∫0+T−δ1+ρz¯∗tdt<1,where z¯∗t is defined in (36).
According to Lemma2 and (3), there exists a tT>0 such that when t>tT, it holds that
(33)xt≤1β+εT≜MT.
Then, consider the following system:(34)dydt=−δ1y+ρyz,dzdt=σ2−δ2−ω2MTMT+η3z,t≠n−1T+lT,t≠nT,yt+=1−pEyt,zt+=1−pHzt,t=n−1T+lT,yt+=yt+σ1,zt+=zt,t=nT,for t>tT. Similar to Lemma 3 it follows from (32) that
(35)limt⟶+∞y¯t−y¯∗t=0,limt⟶+∞z¯t−z¯∗t=0,where y¯t,z¯t is a solution of (34), and
(36)z¯∗t=σ2δ2−ω2MT/MT+η3+z¯∗0+−σ2δ2−ω2MT/MT+η3e−δ2−ω2MTMT+η3t−n−1T,n−1T<t≤n−1T+lT,σ2δ2−ω2MT/MT+η3+z¯∗lT+−σ2δ2−ω2MT/MT+η3e−δ2−ω2MTMT+η3t−n−1T+lT,n−1T+lT<t≤nT,with
(37)z¯∗0+=1−e−δ2−ω2MT/MT+η31−lT+1−pHe−δ2−ω2MT/MT+η31−lT−e−δ2−ω2MT/MT+η3Tσ2/δ2−ω2MT/MT+η31−1−pHe−δ2−ω2MT/MT+η3T,z¯∗lT+=1−pHσ2δ2−ω2MT/MT+η3+z¯∗0+−σ2δ2−ω2MT/MT+η3e−δ2−ω2MTMT+η3lT,
and(38)y¯∗t=y¯∗0+e∫n−1T+t−δ1+ρz¯∗sds,n−1T<t≤n−1T+lT,y¯∗lT+e∫n−1T+lT+t−δ1+ρz¯∗sds,n−1T+lT<t≤nT,with
(39)y¯∗0+=σ11−1−pEe∫0+T−δ1+ρz¯∗tdt,y¯∗lT+=1−pEy¯∗0+e∫0+lT−δ1+ρz¯∗tdt.
Based on Lemma2 and (3), it holds that zt≤z¯t for t>tT; thus,
(40)dydt≤−δ1y+ρyz¯,t≠n−1T+lT,t≠nT,yt+=1−pEyt,t=n−1T+lT,yt+=yt+σ1,t=nT,
fort>tT, which implies that yt≤y¯t for t>tT. Thus, for arbitrary εEH>0, there exists a tEH>tT such that when t>tEH, it holds that
(41)yt≤maxt∈0+,lT∪lT+,Ty¯∗t+εEH≜ME,zt≤maxt∈0+,lT∪lT+,Tz¯∗t+εEH≜MH.
This completes the proof.


## 3. The Stability of the Tumour-Free Periodic Solution

LetΦt;t0,X0 denote the solution of the first three equations of (3) for initial data t=t0 and X0=y0,z0,x0T, as follows:
(42)Φt;t0,X0=yt;t0,X0,zt;t0,X0,xt;t0,X0T.Additionally, we can define the mappingsI1,I2:R3⟶R3 as follows:
(43)I1y,z,x=1−pE0001−pH0001−pTyzx,I2y,z,x=y,z,xT+σ1,0,0T,and the mapΨ:R3⟶R3 as
(44)ΨX0=I2Φ1−lT,I1ΦlT,X0=yT+;X0,zT+;X0,xT+;X0T.Theorem 5.
(i)
The tumour-free periodic solutiony∗t,z∗t,0 of system (3) is locally asymptotically stable provided that
(45)1−pTeαT−∫0Ty∗tdt/η1<1.(ii)
The tumour-free periodic solutiony∗t,z∗t,0 of system (3) is globally attractive provided that
(46)R0≜1−pTeαT−∫0+Ty∗tdt/1/β+η1<1.Proof.
(1) According to (A.12), (A.13), (15), and (45), the three eigenvalues of the Jacobian matrix of map ΨX0 at point X0∗=y∗0+,z∗0+,0 are
(47)λ1=1−pEe∫0+T−δ1+ρz∗tdt<1,λ2=1−pHe−δ2T<1,λ3=1−pTe∫0+Tα−y∗tη1dt<1,which implies that the tumour-free periodic solution y∗t,z∗t,0 is locally stable [25].
(2) Considering (46), we choose an ε2∈0,mint∈0+,lT∪lT+,Ty∗t such that
(48)ζ≜1−pTe∫0+Tα−y∗t−ε21/β+ε2+η1dt<1.
Lety^t,z^t denote the solution of (5). Then, according to Lemma 2 and (3), we have zt≥z^t; thus,
(49)dydt≥−δ1yt+ρytz^t,t≠n−1T+lT,t≠nT,yt+=1−pEyt,t=n−1T+lT,yt+=yt+σ1,t=nT,which implies that yt≥y^t [26]. Then, according to Lemmas 3 and 4, there exists a t2>0 such that
(50)yt≥y∗t−ε2>0,xt≤1β+ε2,for t>t2.
Then, fort>t2, we have
(51)dxdt≤xα−y∗t−ε21/β+ε2+η1,t≠n−1T+lT,t≠nT,xt+=1−pTxt,t=n−1T+lT,xt+=xt,t=nT,which implies that
(52)xt≤xn−1T+e∫n−1T+tα−y∗t−ε21/β+ε2+η1dt,n−1T<t≤n−1T+lT,xn−1T+lT+e∫n−1T+lT+tα−y∗t−ε21/β+ε2+η1dt,n−1T+lT<t≤nT.
Furthermore, we have that(53)xnT+≤xn−1T+ζ.
It follows from (48) and (53) that
(54)limn⟶∞xnT+=0.
Moreover, it follows from (3) that
(55)xt≤xn−1T+eαT,fort∈n−1T,nT.
Based on (54) and (55), we have limt⟶+∞xt=0.
Similar to (50), we can prove that for arbitrary ε2>0, there exists a t2>0 such that
(56)yt>y∗t−ε2,zt>z∗t−ε2,for t>t2. In addition, we can choose an ε3>0 that is sufficiently small such that
(57)ω2ε3ε3+η3<δ2,σ2ω2ε3/ε3+η3δ2δ2−ω2ε3/ε3+η3+z=∗0+z∗0+eω2ε3/ε3+η3lT−1z∗0+<ε22,σ2ω2ε3/ε3+η3δ2δ2−ω2ε3/ε3+η3+z=∗lT+z∗lT+eω2ε3/ε3+η31−lT−1z∗lT+<ε22,where z=∗0+ and z=∗lT+ are defined in (60).
Based on the fact thatlimt⟶+∞xt=0, there exists a t3>t2 such that 0<xt<ε3 for t>t3. Then, the following system is considered:
(58)dzdt=σ2−δ2−ω2ε3ε3+η3z,t≠n−1T+lT,t≠nT,zt+=1−pHzt,t=n−1T+lT,zt+=zt,t=nT.
Fort>t3, we obtain the following positive periodic solution:
(59)z=∗t=σ2δ2−ω2ε3/ε3+η3+z=∗0+−σ2δ2−ω2ε3/ε3+η3e−δ2−ω2ε3ε3+η3t−n−1T,n−1T<t≤n−1T+lT,σ2δ2−ω2ε3/ε3+η3+z=∗lT+−σ2δ2−ω2ε3/ε3+η3e−δ2−ω2ε3ε3+η3t−n−1T+lT,n−1T+lT<t≤nT,with
(60)z=∗0+=1−e−δ2−ω2ε3/ε3+η31−lT+1−pHe−δ2−ω2ε3/ε3+η31−lT−e−δ2−ω2ε3/ε3+η3Tσ2/δ2−ω2ε3/ε3+η31−1−pHe−δ2−ω2ε3/ε3+η3T,z=∗lT+=1−pHσ2δ2−ω2ε3/ε3+η3+z=∗0+−σ2δ2−ω2ε3/ε3+η3e−δ2−ω2ε3ε3+η3lT.
Similar to Lemma3, it follows from the first inequality of (57) that limn⟶∞z=t−z=∗t=0, where z=t is a solution of (58). Thus, there exists a t′3>t3 such that
(61)zt≤z=t<z=∗t+ε22,fort>t′3.
Whent>t′3, it follows from (59), (12), and (57) that
(62)z=∗t≤z∗t+σ2ω2ε3/ε3+η3δ2δ2−ω2ε3/ε3+η3+maxz=∗0+z∗0+eω2ε3/ε3+η3lT−1z∗0+,z=∗lT+z∗lT+eω2ε3/ε3+η31−lT−1z∗lT+<z∗t+ε22.
Therefore, based on (61), (62), and (56), we can infer that
(63)zt−z∗t<ε2,fort>t′3.
Sinceε2>0 is arbitrary, we conclude that limt⟶+∞zt−z∗t=0.
To prove thatlimt⟶+∞yt−y∗t=0, we choose an ε4∈0,δ1/ρ such that
(64)1−pEe∫0+Tρε4−δ1+ρz∗tdt<1,maxt∈0+,lTy∗ty=∗0+y∗0+eρε4lT−1<ε22,maxt∈lT+,Ty∗ty=∗lT+eρε4Ty∗lT+−1<ε22,where the first inequality results from (15), and the expressions of y=∗0+ and y=∗∗lT+ are defined in (67).
Based on the fact thatlimt⟶+∞zt−z∗t=0, there exists a t4>t2 such that zt<z∗t+ε4 for t>t4. Then, the following system is considered:
(65)dydt=yρε4−δ1+ρz∗t,t≠n−1T+lT,t≠nT,yt+=1−pEyt,t=n−1T+lT,yt+=yt+σ1,t=nT.
Fort>t4, we obtain the following positive periodic solution:
(66)y=∗t=y=∗0+e∫n−1T+tρε4−δ1+ρz∗sds,n−1T<t≤n−1T+lT,y=∗lT+e∫n−1T+lT+tρε4−δ1+ρz∗sds,n−1T+lT<t≤nT,
with(67)y=∗0+=σ11−1−pEe∫0+Tρε4−δ1+ρz∗tdt,y=∗lT+=1−pEy=∗0+e∫0+lTρε4−δ1+ρz∗tdt.
According to (66), (11), and (64), when t>t4, it holds that
(68)y=∗t≤y∗t+maxmaxt∈0+,lTy∗ty=∗0+y∗0+eρε4lT−1,maxt∈lT+,Ty∗ty=∗lT+y∗lT+eρε4T−1<y∗t+ε22.
Similar to Lemma3, we can prove that limt⟶+∞y=t−y=∗t=0, where y=t is a solution of (65). Thus, there exists a t′4>t4 such that
(69)yt≤y=t<y∗t+ε2,fort>t′4.
Therefore, based on (69) and (56), we can infer that
(70)yt−y∗t<ε2,fort>t′4.
Sinceε2>0 is arbitrary, we conclude that limt⟶+∞yt−y∗t=0.
This completes the proof.Remark 6.
Since (45) can be inferred from (46), it follows from Theorem 5 that the tumour-free periodic solution y∗t,z∗t,0 of system (3) is globally asymptotically stable provided that (46) holds.


## 4. A Sufficient Condition for the Permanence of System (3)

In this section, we present some conditions for evaluating the permanence of system (46).Theorem 7.
System (3) is permanent with at least one positive periodic solution provided that (15), (29) and
(71)1−pTe∫0+Tα−y∗tη1dt>1
hold.Proof.
(29) implies that Lemma 4 holds; that is, there exist three positive constants ME,MH>0 and MT>0 such that yt<ME, zt<MH and xt<MT for all sufficiently large t. Without loss of generality, assume that yt<ME, zt<MH and xt<MT for t≥0.
Moreover, (15) indicates that Lemma 3 holds. Thus, similar to (56), we can prove that
(72)yt>mint∈0+,lT∪lT+,Ty∗t−ε¯E≜mE,zt>mint∈0+,lT∪lT+,Tz∗t−ε¯H≜mH,for sufficiently large t, where
(73)ε¯E∈0,mint∈0+,lT∪lT+,Ty∗t,ε¯H∈0,mint∈0+,lT∪lT+,Tz∗t.
According to Definition1, we only need to find mT>0 such that xt≥mT for sufficiently large t. We can divide the process of determining mT into two steps for convenience:Step 1.
According to (8) and (41), we can choosem0>0 and ε5∈0,δ1/ρ such that
(74)ω2m0m0+η3<δ2,1−pEe∫0+Tρε5−δ1+ρz~∗tdt<1,ηm0,ε5>1,where z~∗t and y~∗t are defined in (77) and (81), respectively, and
(75)ηm0,ε5≜1−pTe∫0+Tα−αβm0−y~∗t+ε5η1dt.
Next, we consider the following system:(76)dzdt=σ2−δ2−ω2m0m0+η3z,t≠n−1T+lT,t≠nT,zt+=1−pHzt,t=n−1T+lT,zt+=zt,t=nT.
Similar to Lemma3, it follows from the first inequality of (74) that limt⟶+∞z~t−z~∗t=0, where z~t is the solution of (76), and
(77)z~∗t=σ2δ2−ω2m0/m0+η3+z~∗0+−σ2δ2−ω2m0/m0+η3e−δ2−ω2m0m0+η3t−n−1T,n−1T<t≤n−1T+lT,σ2δ2−ω2m0/m0+η3+z~∗lT+−σ2δ2−ω2m0/m0+η3e−δ2−ω2m0m0+η3t−n−1T+lT,n−1T+lT<t≤nT,
with(78)z~∗0+=1−e−δ2−ω2m0/m0+η31−lT+1−pHe−δ2−ω2m0/m0+η31−lT−e−δ2−ω2m0/m0+η3Tσ2/δ2−ω2m0/m0+η31−1−pHe−δ2−ω2m0/m0+η3T,z~∗lT+=1−pHσ2δ2−ω2m0/m0+η3+z~∗0+−σ2δ2−ω2m0/m0+η3e−δ2−ω2m0m0+η3lT.
Therefore, there exists at5>0 such that when t>t5, it holds that
(79)zt≤z~t<z~∗t+ε5.
We show thatxt<m0 cannot hold for all t>t5. In contrast, assume that xt<m0 holds for all t>t5. Then, consider the following system:
(80)dydt=yρε5−δ1+ρz~∗t,t≠n−1T+lT,t≠nT,yt+=1−pEyt,t=n−1T+lT,yt+=yt+σ1,t=nT.
Fort>t5, we obtain the following positive periodic solution:
(81)y~∗t=y~∗0+e∫n−1T+tρε5−δ1+ρz~∗sds,n−1T<t≤n−1T+lT,y~∗lT+e∫n−1T+lT+tρε5−δ1+ρz~∗sds,n−1T+lT<t≤nT,with
(82)y~∗0+=σ11−1−pEe∫0+Tρε5−δ1+ρz~∗tdt,y~∗lT+=1−pEy~∗0+e∫0+lTρε5−δ1+ρz~∗tdt.
Similar to Lemma3, we can prove that limt⟶+∞y~t−y~∗t=0, where y~t is the solution of (80). Thus, based on (79) and (3), there exists a t′5>t5 such that
(83)yt≤y~t<y~∗t+ε5,fort>t′5,which implies that
(84)dxdt≥xα−αβm0−y~∗t+ε5η1,t≠n−1T+lT,t≠nT,xt+=1−pTxt,t=n−1T+lT,xt+=xt,t=nT,for t>t′5.
LetN5∈Z+ such that N5−1T+lT≥t′5. By integrating (84) on n−1T+lT,nT+lT, where n≥N5, we obtain
(85)xnT+lT+≥xn−1T+lT+ηm0,ε5.
Then, based on (85) and (74), we have that xN5−1+lT+nT+≥xN5−1T+lT+ηm0,ε5n⟶+∞ as n⟶∞, which contradicts the boundedness of xt. Thus, there exists a t6>t5>0 such that xt6≥m0.Step 2.
Ifxt≥m0 for all t≥t6, then our goal is obtained. Otherwise, xt<m0 for some t>t6. We set t∗=inftt>t6,xt<m0; then, we have t∗≥t6>t5>0, which implies that (45) holds on the interval t∗,+∞. Thus, we can consider the following two cases for t∗:Case 1.
There exists ann1∈Z+ such that t∗=n1T+lT.
According to the definition of the infimumt∗=inftt>t6,xt<m0, we know that xt≥m0 holds for all t∈t6,t∗ and xt∗+≤m0.
We choosen2,n3∈Z+ such that
(86)n2>lnε5/MEeρ∫0+Tz~∗tdtln1−pEe∫0+Tρε5−δ1+ρz~∗tdt,1−pTn2en2+1η∧Tηm0,ε5n3>1,where η^=α−αβm0−maxME,αη1/η1<0.We claim that there exists at7∈t∗,t∗+n2+n3T such that xt7>m0. Otherwise, we can assume that xt≤m0 is valid for t∈t∗,t∗+n2+n3T. Then, similar to Lemma 3, when t∈t∗,t∗+n2+n3T, it holds that
(87)y~t=y~n−1T+e∫n−1T+tρε5−δ1+ρz~∗sds,n−1T<t≤n−1T+lT,y~n−1T+lT+e∫n−1T+lT+tρε5−δ1+ρz~∗sds,n−1T+lT<t≤nT,where y~t is the solution of (80). Thus, we obtain
(88)y~nT+lT+−y~∗nT+lT+=1−pEe∫0+Tρε5−δ1+ρz~∗tdtn−n1×y~t∗+−y~∗t∗+.Then, whent∈t∗+n2T,t∗+n2+n3T, it follows from (87), (81), (88), (74), and (86) that
(89)y~t−y~∗t≤yt∗+1−pEe∫0+Tρε5−δ1+ρz~∗tdtn2×eρ∫0+Tz~∗tdt<ε5,which implies that
(90)yt≤y~t<y~∗t+ε5,for t∈t∗+n2T,t∗+n2+n3T. Similar to (85), we have
(91)xt∗+n2+n3T≥xt∗+n2Tηm0,ε5n3.On the other hand, in the intervalt∗,t∗+n2T, the following is valid:
(92)dxdt≥xtη^,t≠n−1T+lT,t≠nT,xt+=1−pTxt,t=n−1T+lT,xt+=xt,t=nT,since xt≤m0 and yt<ME hold for t∈t∗,t∗+n2T. Integrating (92) on the interval t∗,t∗+n2T yields the following:
(93)xt∗+n2T≥xt∗1−pTeη∧Tn2≥1−pTn2m0en2η∧T.It follows from (91), (93), and (86) that
(94)xt∗+n2+n3T>m0,which is a contradiction.Lett∗∗=inftt>t∗,xt>m0; then, t∗∗∈t∗,t∗+n2+n3T, and xt≤m0 holds for t∈t∗,t∗∗. Suppose that there exists an n4∈Z+ such that t∗∗=n4T+lT; then, according to xt∗∗+<m0, there exists a δ0>0 such that when t∈t∗∗,t∗∗+δ0, xt<m0 holds, which contradicts the definition of the infimum t∗∗=inftt>t∗,xt>m0. Thus, there is no n4∈Z+ such that t∗∗=n4T+lT. Since xt is continuous at t=t∗∗, we thus have that xt∗∗=m0.Fort∈t∗,t∗∗, assume that t∗+n5−1T<t≤t∗+n5T, where n5∈1,2,⋯,n2+n3. Then, similar to (93), we have that
(95)xt≥1−pTn5xt∗et−t∗η∧≥m1,where
(96)m1=1−pTn2+n3m0en2+n3η∧T.Fort>t∗∗, the same arguments can be continued since xt∗∗=m0.Case 2.
There exists non∈Z+ such that t∗=nT+lT.
It is clear thatxt≥m0 for t∈t6,t∗ and xt∗=m0. Suppose that t∗∈n6−1T+lT,n6T+lT, where n6∈Z+.Case 3.
There exists somet∈t∗,n6T+lT such that xt>m0.
Lett∗∗∗=inftt>t∗,xt>m0; then, t∗∗∗∈t∗,n6T+lT. For t∈t∗,t∗∗∗, it follows that xt≤m0; thus, xt∗∗∗=m0. Similar to (93), when t∈t∗,t∗∗∗, we have
(97)xt≥xt∗eη∧t−t∗≥m2,where
(98)m2=m0eη∧T.
Fort>t∗∗∗, the same arguments can be continued since xt∗∗∗=m0.Case 4.
For allt∈t∗,n6T+lT, xt≤m0.
Lett∗∗∗∗=inftt>t∗,xt>m0; then, t∗∗∗∗∈n6T+lT,n6T+lT+n2+n3T and xt∗∗∗∗+=m0. Similar to Cases 1 and 3, we can prove that when t∈t∗,t∗∗∗∗, it holds that
(99)xt≥mT,where
(100)mT=1−pTn2+n3m0en2+n3+1η∧T,where mT<m1<m2<m0.
We repeat the above procedure to prove thatxt≥mT for t≥t6.
Furthermore, according to Schauder’s fixed point theorem, there exists a tumour-present periodic solution for system (3).
This completes the proof.


## 5. Numerical Analysis

We are interested in how the key factors (i.e., the killing ratespE, pH, and pT, the dosage of infusing the ECs (σ1), the therapeutic period T, and the activation rate of the ECs (ρ)) affect the threshold value R0 defined in (46). Since ∫0+Ty∗tdt is independent of pT, R0 decreases monotonically with respect to pT, indicating that the strong tumour cell killing effect of radiotherapy can increase tumour cell death.First, we set the parameters as follows [16, 17]:
(101)δ1=0.3473,σ2=0.38,δ2=0.055,ω2=0.02,η3=0.1,α=1.636,η1=0.1,l=0.5..Figure3(a) shows that when pH (or pE) decreases to the threshold value TpH=0.1440 (or TpE=0.0377) of the tumour-free periodic solution, R0 monotonically decreases to 0. In addition, when TpH<pE,pH<0.2, R0pH is much smaller than R0pE, where R0pE>1. Therefore, in this case, the optimal control strategy is achieved when pH and pE are sufficiently small, and, compared to parameter pE, a smaller parameter pH is more beneficial for tumour control. Similarly, Figures 3(b) and 3(c) show that R0 monotonically decreases as σ1 increases or T decreases, indicating that a higher dosage of infusing the ECs or more frequent radioimmunotherapy can accelerate the eradication of tumour cells. In addition, Figure 3(d) shows that when ρ increases to the threshold value Tρ=0.2003 of the tumour-free periodic solution, R0 rapidly decreases to 0. Thus, strong activation of the ECs by the HTCs is beneficial for tumour control.Figure 3
Simulations of the effects ofpE, pH, σ1, T, and ρ on R0. The baseline parameters are ρ=0.16, β=0.3, pE=0.2, pH=0.2, pT=0.85, σ1=0.2, and T=2, and the other relevant parameters are identical to those in (101).
(a)(b)(c)(d)Moreover, whenσ1=0 and
(102)R0=1−pTeαT>1,it follows from Theorem 5 (i) that the tumour-free periodic solution is unstable for radiotherapy alone. For example, if we fix the parameters as those shown in Figures 1(a)–1(c), the tumour cell population oscillates as a periodic cycle. Similar results are observed for the case of immunotherapy alone (see Figures 1(d)–1(f)). On the other hand, if we set the parameter values as those shown in Figures 4(a)–4(c), the tumour cells are eventually eradicated with radioimmunotherapy. Thus, we can say that radioimmunotherapy is more effective than therapy regimes with radiotherapy or immunotherapy alone.Figure 4
Dynamic behaviours of system (2) with radioimmunotherapy. (a)–(c)ρ=0.01, β=0.002, pE=0.15, pH=0.1, pT=0.9619, σ1=2, and T=2; (d)–(f) ρ=0.01, β=0.002, pE=0.2, pH=0.1, pT=0.9625, σ1=0.2, and T=5. The other parameters are identical to those in (101), and the initial values in (a)–(f) are 1,1,500.
(a)(b)(c)(d)(e)(f)Furthermore, Figure5 displays bifurcation diagrams for system (3). The dynamical behaviour of system (3) is dominated by tumour-free and tumour-present periodic solutions. When T is smaller than the threshold value Ta=2.0076, the global attractiveness of the tumour-free periodic solution can be validated (see Figures 4(a)–4(c)). However, when Ta<T<Tp=5.733, where Tp is the threshold value for the permanence of system (3), the emergence of a tumour-present periodic solution leads to the local stability of the tumour-free periodic solution (see Figures 4(d)–4(f) and Figures 2(a)–2(c)). In addition, when T>Tp=5.733, all three cell populations oscillate periodically, which indicates that system (3) is permanent and has a tumour-present periodic solution (see Figures 2(d)–2(f)). In particular, the complex patterns shown in Figures 2, 4, and 5 demonstrate that a properly designed control period T is crucial for successful tumour control.Figure 5
Bifurcation diagrams for system (3) with respect to T, where ρ=0.01, β=0.002, pE=0.2, pH=0.1, pT=0.9625, and σ1=0.2. The other parameters are identical to those in (101).
(a)(b)(c)


## 6. Discussion

In this paper, we develop a tumour-immune model with pulsed treatments to show how radiotherapy and immunotherapy affect the dynamics of tumour treatments. It is assumed that the radiotherapy and immunotherapy are administered with the same periodicity but not simultaneously. Additionally, it is assumed that fixed proportions of tumour cells, effector cells, and helper T cells are degraded each time the radiotherapy is administered.Similar to the proof of the continuity of the solution with respect to the right-hand side of the ordinary differential equations, we proved thatlimt⟶+∞yt−y∗t=0 in Lemma 3 by using the integral inequality technique. Then, based on the differentiability of the solution with respect to the initial values, we determined the eigenvalues of the Jacobian matrix at the fixed point corresponding to the tumour-free periodic solution, which was used to obtain the local stability threshold condition. Furthermore, the indicator R0 is provided as a sufficient condition for the global attractiveness of the tumour-free periodic solution. We emphasize that a comparison of the solutions of the ordinary differential equations is critical for proving this claim. Biologically speaking, (46) indicates that the tumour cells have been completely eradicated throughout the body, indicating the ultimate success of our treatment strategy. Similarly, we proved that system (3) is permanent with at least one tumour-present periodic solution under certain conditions, suggesting that tumour cells, effector cells, and helper T cells coexist indefinitely in the tumour-present periodic solution. It is clear that t=t5 is an important threshold value for our proof since zt<z~∗∗∗t+ε5 for t>t5.Our results demonstrate that the effectiveness of radioimmunotherapy, the therapeutic period, and the activation rate of theECs by the HTCs are all crucial for tumour depression and resurgence. The numerical results presented in Section 5 indicate that R0 is sensitive to small changes in the killing rates pE, pH, and pT, the dosage σ1 of infusing the ECs, the therapeutic period T, and the activation rate ρ of the ECs; that is, the smaller (or larger) the parameters pE, pH, and T (or pT, σ1, and ρ) are, the smaller the indicator R0 is. In particular, decreases in pH are more beneficial for tumour control than decreases in pE, and radioimmunotherapy is more effective than either radiotherapy or immunotherapy alone.Furthermore, we performed one-parameter bifurcation analyses on the threshold valueR0, as shown in Figure 5. Figure 5 shows the impact of the period T on the threshold value R0. The tumour-free periodic solution is locally stable for Ta<T<Tp, whereas system (3) is permanent with a tumour-present periodic solution for T>Tp. Figure 2 shows that if T is increased from 5 to 6, the permanence of system (3) causes the tumour-free periodic solution to lose its local stability. These results demonstrate that the parameters pE, pH, pT, σ1, T, and ρ are crucial for tumour control. This information may help doctors in designing and determining the optimum therapeutic approaches for tumour control.As a comparison to other relevant studies, we note the following highlights of our study: (i) note that system (3) in [24] includes only one impulsive control strategy (injecting the effector cells into the body), while system (4) in [24] and system (1) in [27] include only the tumour cells and effector cells. However, system (3) includes not only helper T cells but also two impulsive control strategies that are not implemented simultaneously. (ii) In contrast to the small amplitude perturbation method used in [24], we use a more rigorous method to prove the local stability of the tumour-free periodic solution of system (2) with. (iii) Although the proofs of the permanence of the corresponding impulsive systems in [24, 27] are omitted, we include the proof of the permanence of our system in this study.Similar to [28, 29], the existence of tumour-present periodic solutions for system (3) was investigated in our study. Furthermore, based on the techniques used in [30–34], we hypothesize that the existence and global attractiveness of the tumour-present periodic solution of system (3) can be proven by using Lyapunov’s second method. This proof will be demonstrated in our future research.



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