# Dynamics of a Virological Model for Cancer Therapy with Innate Immune Response

Authors: Ayoub Nouni; Khalid Hattaf; Noura Yousfi
Journal: Complexity
 (2020)

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## Abstract

The aim of this work is to present a virological model for cancer therapy that includes the innate immune response and saturation effect. The presented model combines both the evolution of a logistic growing tumor and time delay which stands for the period of the viral lytic cycle. We use the delay differential equation in order to model this time which also means the time needed for the infected tumor cells to produce new virions after viral entry. We show that the delayed model has four equilibria which are the desired outcome therapy equilibrium, the complete failure therapy equilibrium, the partial success therapy free-immune equilibrium when the innate immune response has not been established, and the partial success therapy equilibrium with immune response. Furthermore, the stability analysis of equilibria and the Hopf bifurcation are properly exhibited.


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## Body

## 1. Introduction

Combination therapy approaches have shown a serious promise to deal with cancers that are resistant to traditional therapeutic procedures. Oncolytic virotherapy, also called the selective therapy, is a developable technique that adopts replication competent viruses as a new treatment to destroy cancer cells without causing damage to normal cells [1–4]. Different researches have investigated combination strategies with oncolytic virotherapy and chemotherapeutic drugs in order to optimize both the effect of the added therapy and the viral oncolysis [5, 6]. Many mathematical models of viral infection and immune response have been established in order to study the behavior as well as the dynamics of the cancer cells. According to Phan and Tian [7], the dynamics of the system, when they considered a composed ODE’s dimensional model, is ruled by the viral burst size and some parameters in relation to the innate immune response. They have shown that getting the immune response involved in the system makes the oncolytic virotherapy more difficult by establishing more equilibria when the viral burst size is lower than a critical value, whereas the model has the same behavior like in the case when the immune response is excluded when the viral burst size is big. In 2018, Kim et al. [8] proposed a delayed mathematical model with two controls to describe cancer viral therapy dynamics and to reduce total tumor cell numbers as well as the costs of two therapies. They performed the stability analysis and the existence of Hopf bifurcation. They also inspected the optimal oncolytic immunotherapy treatment with respect to the time delay. However, it is important to notice that either these works did not incorporate the nonlinear relationship between viral dose and infection rate or they did not consider the survival probability of the infected cells during the latent period not to mention the absorbtion rate of the virus. Recently, Hattaf [9] proposed a virological model that incorporates the general infection rate for the two types of transmission, humoral immunity, and three time delays. He found that the entire behavior of the presented model is determined and ruled by the basic reproduction number and the reproduction number for humoral immunity. The study investigated the dynamical behaviors of the model including Hopf bifurcation and stability switches.Based on the above and the model for oncolytic virotherapy [10], we propose the following model:(1)dxtdt=rxt1−xt+ytK−βxtvt1+αvt,dytdt=βxt−τvt−τe−mτ1+αvt−τ−δyt−pytzt,dvtdt=Nδyt−nβxtvt1+αvt−μvt−qvtzt,dztdt=cytzz−bzt,where xt, yt, and vt have the same meaning as in [10] which are, respectively, the quantities of uninfected tumor cells, damaged tumor cells, and oncolytic virus. On the other hand, zt denotes the concentration of the innate immune cells at time t. The tumor grows logistically at a rate r and K is the maximal carrying capacity of tumor cells. The infection rate in the oncoviral therapy model [7] has been modeled by a bilinear incidence which is not reasonable in case of a high concentration of oncolytic virus. Therefore, it is very reasonable to model the infection rate by a saturated incidence of the form βxv/1+αv, where β and α are positive constants which, respectively, describe the infection process and the saturation effect. The parameters δ and μ represent, respectively, the death rate of the damaged tumor cells and the virus clearance rate while n describes the absorbtion rate of the virus in the extracellular tissue, whereas N is the number of new viruses which appear after a disruption-lysis of a damaged tumor cell. The constants p and q stand for the immune killing rate of damaged cells and viruses, respectively, while b represents the immune clearance rate and c is the incitement rate of the innate immune system. Ultimately, the delay τ expresses the time of latent period while the quantity e−mτ exemplifies the probability of getting through from time t−τ to time t, where m is the death rate for unproductive-damaged cells.Our purpose in this work is to extend our model in [10] by introducing the role of the innate immune response in oncolytic virotherapy. Additionally, the model studied in [7] is a special case of system (1) when the time period of the lytic cycle and saturation effect are not considered, that is, τ=0 and α=0. Otherwise, this paper is organized as follows. In Section 2, we give some preliminary results and we discuss the conditions of the existence of equilibria. Section 3 deals with stability analysis and provides conditions under which the system undergoes the Hopf bifurcation. Finally, Section 4 is devoted to discussion and conclusion.


## 2. Positiveness, Boundedness, and Equilibria

In this section, we prove the positivity and the boundedness of solutions of model (1).LetC=C−τ,0,IR4 be the Banach space of continuous functions mapping the interval −τ,0 into IR4 with the topology of the uniform convergence. According to the fundamental theory of functional differential equations [11], model (1) has a unique solution xt,yt,vt,zt with respect to initial values x0,y0,v0,z0∈C. Additionally, we also assume that the initial values satisfy the following biological conditions:(2)x0θ≥0,y0θ≥0,v0θ≥0,z0θ≥0,θ∈−τ,0.Theorem 1.
Each solution of model (1) starting from initial condition (2) remains positive and bounded for all t≥0.Proof.
From the first and the fourth equations of (1), we get(3)xt=x0e∫0tr1−xs+ys/K−βvs/1+αvsds,zt=z0e−bt+∫0tcysds,which leads to xt≥0 and zt≥0 for all t≥0. From the second and third equations of (1), we have(4)yt=y0e−δt−∫0tpzsds+e−mτ−δt−∫0tpzsds∫0tβxs−τvs−τ1+αvs−τeδs+∫0spzududs,vt=v0e−∫0tnβxs/1+αvs+qzsds+Nδ∫0tyueμu−∫utnβxs/1+αvs+qzsdsdue−μt.
Clearly,yt≥0 and vt≥0 for t∈0,τ. This procedure can be repeated on the interval ητ,η+1τ for all η∈IN. Then, yt≥0 and vt≥0 for all t≥0.
Through the first equation of (1), we have(5)dxtdt≤rxt1−xtK.
Applying the comparison principle, we get(6)limsupt⟶+∞xt≤K.
Therefore,xt is bounded. Let(7)Tt=xt−τe−mτ+yt+pczt.
Fort>τ, we have(8)dTtdt=rxt−τ1−xt−τ+yt−τKe−mτ−δyt−pbczt≤rK1−xt−τKe−mτ−δyt−pbczt≤rKe−mτ−ρTt,where ρ=minr,δ,b. Consequently,(9)Tt≤γ,where γ=maxT0,rKe−mτ/ρ. This implies that Tt is bounded on τ,+∞. According to the continuity of Tt on 0,τ, we concluded that Tt is also bounded on 0,τ. Therefore, yt and zt are bounded for all t≥0.
By the third equation of (1) and the boundedness of yt, we find(10)limsupt⟶+∞vt≤rKNδe−mτμρ.
Then,vt is bounded. This completes the proof.
Next, we discuss the existence of equilibria of model (1). Denote(11)ℛ0=NKβnβK+μe−mτ=1δ×NδnβK+μ×Kβ×e−mτ,where 1/δ is the average life expectancy of the infected tumor cells, Nδ/nβK+μ is the viral quantity generated from one infected cell during its survival period, K the number of uninfected tumor cells at the beginning of the infection, and e−mτ is the probability of surviving from time t−τ to time t. Therefore, ℛ0 is the basic reproduction number of model (1) which biologically describes the average number of the newly infected tumor cells generated from one infected cell at the beginning of the infection.
In absence of the innate immune response, system (1) reduced to the model in [10]. Then, (1) always has two equilibria E00,0,0,0 and E1K,0,0,0 if ℛ0≤1. However, model (1) has another equilibrium E2x2,y2,v2,0 if ℛ0>1, where(12)x2=μr+KδN−nemτrα−β+Δ2rN−nemτβe−mτ+αδ,y2=rx2K−x2rx2+Kδemτ,v2=δN−nemτμy2,with Δ=KδN−nemτβ−rα−μr2+4μδKremτN−nemτβe−mτ+δα.
In presence of the innate immune response, we have(13)x=crKα−brα−Kcβv+rcK−brc1+αv≔φ1v,y=bc,z=bδN−nemτ−μcvqcv+nbpemτ≔φ2v,βxv1+αv=bcemτδ+pz.
Sincez≥0, we have v≤δbN−nemτ/μc. This indicates that there is no biological equilibrium when v>δbN−nemτ/μc. Let F be a function defined on the closed interval 0,δbN−nemτ/μc as follows:(14)Fv=βφ1vv1+αv−bcemτδ+pφ2v.
Clearly,F0=−bδ+pφ20emτ<0 and(15)F′v=βcK−bc1+αv2−bcpemτφ2′v.
Sinceφ2′v=−bcμnpemτ+qδN−nemτ/qcv+nbpemτ2, we have F′v>0.
When the innate immune response has not been established, we havecy2−b≤0. Then, we define the reproduction number for the innate immune response as follows:(16)ℛ1Z=cy2b,where 1/b is the average life expectancy of innate immune cells, c is the rate of immune response activation, and y2 is the number of infected tumor cells at the steady state E2. Hence, ℛ1Z describes the average number of innate immune cells activated by the infected tumor cells.
Ifℛ1Z<1, then y2<b/c, v2<δbN−nemτ/μc, and FδbN−nemτ/μc<δb/cβx2N−nemτ/μ1+αv2−emτ=0. Then, there is no equilibrium when ℛ1Z<1.
Ifℛ1Z>1, then y2>b/c, v2>δbN−nemτ/μc, and FδbN−nemτ/μc>δb/cβx2N−nemτ/μ1+αv2−emτ=0. Therefore, model (1) has a unique equilibrium with immune response E3x3,y3,v3,z3, where v3∈0,δbN−nemτ/μc, x3=crKα−brα−Kcβv3+rcK−b/rc1+αv3, y3=b/c, and z3=bδN−nemτ−μcv3/qcv3+nbpemτ.
By rearranging the above discussions, we have the following theorem.Theorem 2.
(i)
Ifℛ0≤1, then model (1) has uniquely two equilibria that are the desired outcome therapy equilibrium E00,0,0,0 and the complete failure therapy equilibrium E1K,0,0,0(ii)
Ifℛ0>1, then model (1) has a unique partial success therapy equilibrium without immune response E2x2,y2,v2,0 besides E0 and E1, where x2=μr+KδN−nemτrα−β+Δ/2rN−nemτβe−mτ+αδ, y2=rx2K−x2/rx2+Kδemτ, and v2=δN−nemτ/μy2(iii)
Ifℛ1Z>1, then model (1) has a unique partial success therapy equilibrium with immune response E3x3,y3,v3,z3 besides E0, E1, and E2, where v3∈0,δbN−nemτ/μc, x3=crKα−brα−Kcβv3+rcK−b/rc1+αv3, y3=b/c, and z3=bδN−nemτ−μcv3/qcv3+nbpemτ.


## 3. Model Analysis and Stability

To understand the dynamics of the proposed model, we first analyze the local asymptotic stability of equilibria. LetEx,y,v,z be an arbitrary equilibrium of model (1). Hence, the characteristic equation at E is given by(17)r1−2x+yK−βv1+αv−λ−rxK−βx1+αv20βv1+αve−m+λτ−δ−pz−λβx1+αv2e−m+λτ−py−nβv1+αvNδ−nβx1+αv2−μ−qz−λ−qv0cz0cy−b−λ=0.Theorem 3.
The desired outcome therapy equilibriumE00,0,0,0 is unstable.Proof.
It is not hard to see that atE00,0,0,0, equation (17) becomes(18)r−λδ+λμ+λb+λ=0.
Sinceλ=r>0 is a positive root of the above equation, we deduce that E0 is unstable.Theorem 4.
Ifℛ0<1, then the complete failure therapy equilibrium E1 is locally asymptotically stable and unstable if ℛ0>1.Proof.
AtE1, (17) can be written as follows:(19)r+λb+λλ2+nβK+μ+δλ+δnβK+μ1−ℛ0e−λτ=0.
Obviously,λ1=−r and λ2=−b are two negative roots of equation (19). Then, we consider the following transcendental equation:(20)λ2+nβK+μ+δλ+δnβK+μ1−ℛ0e−λτ=0.
Forτ=0 and ℛ0<1, we have δnβK+μ1−ℛ0>0. Thus, the entire roots of (20) have negative real parts. Afterward, we set iψψ>0 to be a purely imaginary root of (20). Then,(21)−ψ2+δnβK+μ=δnβK+μℛ0cosψτ,nβK+μ+δψ=−δnβK+μℛ0sinψτ,which leads to(22)ψ4+δ2+nβK+μ2ψ2+δ2nβK+μ21−ℛ02=0.
DenoteS=ψ2. Then, the previous equation becomes(23)S2+δ2+nβK+μ2S+δ2nβK+μ21−ℛ02=0,which has no positive root when ℛ0<1. This implies that E1 is locally asymptotically stable if ℛ0<1. In fact, (23) having no positive roots implies that equation (20) does not exhibit any stability switch [12]. This means that the stability of E1 for τ=0 is the same as that for τ≥0, implying that E1 is asymptotically stable for all τ≥0
Forℛ0>1, we consider the following function:(24)fλ=λ2+nβK+μ+δλ+δnβK+μ1−ℛ0e−λτ.
We havef0=δnβK+μ1−ℛ0<0 and limλ⟶+∞fλ=+∞. Then, the equation fλ=0 has at least one positive root when ℛ0>1. This implies that the characteristic equation (19) has at least one positive eigenvalue when ℛ0>1. Therefore, the complete failure therapy equilibrium E1 becomes unstable as long as ℛ0>1.
The following result investigates the global stability of the complete failure therapy equilibriumE1 when ℛ0≤1.Theorem 5.
Ifℛ0≤1, then the complete failure therapy equilibrium E1 is globally asymptotically stable for all τ≥0.Proof.
We consider the following functional:(25)Vt=emτyt+emτNvt+∫t−τtβxsvs1+αvsds+pemτcz.
Taking the derivative ofV along t of the solutions of (1) delivers(26)dVdt1=1−nemτNβxv1+αv−μemτNv−qN+pbczemτ.
Seeing thatlimsupt⟶∞xt≤K, we deduce that each ω-limit point satisfies xt≤K. Hence, it is sufficient to take solutions for which xt≤K. Thus,(27)dVdt1≤emτNnβK+μℛ0−1v−qN+pbczemτ.
Then,dV/dt1≤0 when ℛ0≤1. Moreover, it is easy to prove that the largest invariant subset of x,y,v,zdV/dt=0 is the singleton E1. From LaSalle’s invariance principle [13], we conclude that E1 is globally asymptotically stable as long as ℛ0≤1.Next, we study the stability ofE2. In this case, (4) becomes(28)cy2−b−λλ3+a1λ2+a2λ+a3+b1λ+b2e−λτ=0,where(29)a1=μ+δ+rx2K+nβx21+αv22,a2=δμ+μ+δrx2K+nδβx21+αv22+rnβx22K1+αv22−nβ2x2v21+αv23,a3=rμδx2K+rnβδx22K1+αv22−nβ2δx2v21+αv23,b1=βx21+αv2rv2K−Nδ1+αv2e−mτ,b2=βx21+αv2rμv2K+Nβδv21+αv22−Nδrx2K1+αv2e−mτ.Clearly,λ1=cy2−b is a root of (28). If ℛ1Z>1, then λ1>0 and E2 is unstable. However, λ1<0 if ℛ1Z<1. In this case, we study the roots of the following equation:(30)λ3+a1λ2+a2λ+a3+b1λ+b2e−λτ=0.The general form of this transcendental characteristic equation was investigated by Beretta and Kuang in [14].Forτ=0, (30) becomes(31)λ3+a1λ2+a2+b1λ+a3+b2=0.Sincea1>0 and a3+b2=N−nβ2δx2v2/1+αv23+rμαδ+βx2v2/K1+αv2>0, we deduce by applying the Routh–Hurwitz criterion that E2 is locally asymptotically stable if a1a2+b1−a3+b2>0.Letiϑϑ>0 be a root of (30). Then,(32)a1ϑ2−a3=b1ϑsinϑτ+b2cosϑτ,−ϑ3+a2ϑ=−b1ϑcosϑτ+b2sinϑτ.Hence,(33)ϑ6+a12−2a2ϑ4+a22−2a1a3−b12ϑ2+a32−b22=0,which reduces to(34)gS≔S3+q2S2+q1S+q0=0,where S=ϑ2, q2=a12−2a2, q1=a22−2a1a3−b12, and q0=a32−b22. By an analogical discussion as in [10], let Δ=q22−3q1 and S∗=Δ−q2/3. Hence, we consider the following assertions:(i1)
q0≥0 and Δ≤0(i2)
q0≥0, Δ>0, and S∗≤0(i3)
q0≥0, Δ>0, and gS∗>0Therefore, we have the following result.Theorem 6.
Assumeℛ0>1.(1)
Ifℛ1Z<1, a1a2+b1−a3+b2>0, and one of the conditions (i1)–(i3) holds, then the partial success therapy equilibrium without immune response E2 is locally asymptotically stable for any time delay τ≥0(2)
Ifℛ1Z>1, then E2 is unstable.
Assume that equation (34) has positive roots. Without loss of generality, we assume that (34) has three positive solutions named S1, S2, and S3 which are ordered as follows: S1<S2<S3. It follows that equation (33) admits three positive solutions that are(35)ϑ1=S1,ϑ2=S2andϑ3=S3.
By (32), we get(36)τkj=1ϑjarccosb2a1ϑj2−a3+b1ϑj2ϑj2−a2b22+b12ϑj2+2kπϑj,where j=1,2,3 and k∈IN. Therefore, ±iϑj is a pair of purely imaginary roots of (30) with τ=τkj. Let(37)τ0=τ0j0=minj∈1,2,3τ0j,ϑ0=ϑj0.
We setλτ=ςτ+iϑτ to be the root of equation (30) satisfying ςτkj=0 and ϑτkj=ϑj. Differentiating (30) with respect to τ, we get(38)dλdτ−1=3λ2+2a1λ+a2+b1e−λτλb1λ+b2e−λτ−τλ.
Thus,(39)Redλdτ−1τ=τkj=3ϑj4+2a12−2a2ϑj2+a22−2a1a3−b12b12ϑj2+b22,=g′ϑj2b12ϑj2+b22.
It is not difficult to find out thatg′ϑj2≠0 for all j=1,2,3. Then, the transversality condition holds and we get the following result.Theorem 7.
Assumeℛ1Z<1<ℛ0 and a1a2+b1−a3+b2>0 hold.
If eitherq0<0 or q0≥0, Δ>0, S∗>0, and gS∗≤0, then E2 is locally asymptotically stable for all τ∈0,τ0 and becomes unstable when τ>τ0. Moreover, model (1) undergoes a Hopf bifurcation at E2 when τ=τkj, for j=1,2,3 and k∈IN.Remark 1.
Theorem7 shows that the delay τ can cause the partial success therapy equilibrium without immune response E2 to gain or lose its stability. In addition, periodic solutions appear when the value of this delay is equal to a critical value.
Finally, we discuss the stability of the partial success therapy equilibrium with immune responseE3 when ℛ1Z>1. In this case, (17) becomes(40)λ4+c3λ3+c2λ2+c1λ+c0+d2λ2+d1λ+d0e−λτ=0,where(41)c3=μ+δ+p+qz3+rx3K+nβx31+αv32,c2=bpz3+μ+δ+p+qz3+nβx31+αv32rx3K+δ+pz3μ+qz3+nβx31+αv32−nβ2x3v31+αv33,c1=bpz3μ+qz3+nβx31+αv32+rx3Kbpz3+δ+pz3μ+qz3+nβx31+αv32−nβ2x3v31+αv33δ+pz3,c0=bpz3rx3Kμ+qz3+nβx31+αv32−nβ2x3v31+αv33,d2=βx31+αv3rv3K−Nδ1+αv3e−mτ,d1=βNδx31+αv32βv31+αv3−rx3K+βx3v31+αv3rKμ+qz3+qcz31+αv3e−mτ,d0=βqcx3v3z31+αv32rx3K−βv31+αv3e−mτ.
The above equation is the same as that analyzed by Hattaf in [9]. Then, let iϕϕ>0 be a root of (41). We have(42)ϕ4−c2ϕ2+c0=d2ϕ2−d0cosϕτ−b1ϕsinϕτ,−c3ϕ3+c1ϕ=−d1ϕcosϕτ−d2ϕ2−d0sinϕτ,
which can be reduced to(43)ϕ8+c32−2c2ϕ6+c22−d22+2c0−2c1c3ϕ4+c12−d12−2c0c2+2d0d2ϕ2+c02−d02=0.
Letu=ϕ2. Then, (43) becomes(44)hu≔u4+p3u3+p2u2+p1u+p0=0,where p3=c32−2c2, p2=c22−d22+2c0−2c1c3, p1=c12−d12−2c0c2+2d0d2, and p0=c02−d02. Clearly, if p0<0, equation (44) admits at least one positive root. Additionally, we have(45)h′u=4u3+3p3u2+2p2u+p1=0.
According to Cardano’s formula, the cubic roots of (45) can be written as follows:(46)u1=−Q2+Δ¯3+−Q2+Δ¯3−p34,u2=j−Q2+Δ¯3+j2−Q2−Δ¯3−p34,u3=j2−Q2+Δ¯3+j−Q2−Δ¯3−p34,where(47)P=8p2−3p3216,Q=p33−4p3p2+8p132,Δ¯=P33+Q22,j=−12+i32.
Hence, we discuss the existence of real positive roots of (44).(i)
WhenΔ¯>0, (44) has only a real root u1 and the other two roots are conjugate complex numbers. Thus,(48)h′u=4u−u1u2−2Reu2z+u22.It follows thath admits a unique strict global minimum at u=u1, and it is because u2−2Reu2z+u22>0 for all u∈IR.(ii)
WhenΔ¯=0, all roots are real with u1=3Q/P−p3/4 and u2=u3=3Q/2P−p3/4. Hence,(49)h′u=4u−u1u−u22.Thus,h reaches its strict global minimum at u=u1. We conclude that if p0≥0 and Δ¯≥0, then equation (44) has a positive root if and only if u1>0 and hu1≤0.(iii)
WhenΔ¯<0, the entire roots are real and distinct. In this case, we have(50)h′u=4u−u1u−u2u−u3.By analogical reasoning, we deduce that ifp0≥0 and Δ¯<0, then equation (44) has positive root if and only if there exists at least one u∗∈u1,u2,u3 verifying u∗>0 and hu∗≤0.A summary of the above analysis leads to the following lemma.Lemma 1.
(i)
Ifp0<0, then equation (44) has at least one positive root(ii)
Ifp0≥0 and Δ¯<0, then equation (44) has a positive root if and only if u1≥0 and hu1≤0(iii)
Ifp0≥0 and Δ¯<0, then equation (44) has a positive root if and only if there exists at least one u∗∈u1,u2,u3 verifying u∗>0 and hu∗≤0.Based on Lemma1, we set the following conditions:(H1)p0<0(H2)p0≥0, Δ¯≥0, u1>0, and hu1≤0(H3)p0≥0, Δ¯<0, and there exists at least one u∗∈u1,u2,u3 verifying u∗>0 and hu∗≤0,If conditions (H1)–(H3) are not fulfilled, then equation (44) has no positive solutions. Thus, equation (40) has no purely imaginary roots. Consequently, the partial success therapy equilibrium with immune response E3x3,y3,v3,z3 is locally asymptotically stable for all τ≥0. In this case, the presence of Hopf bifurcation is not achievable.Next, we assume that one of the conditions (H1)–(H3) is fulfilled; then equation (44) admits at least one positive solution. Let d∈1,2,3,4 be the number of positive roots of (44). Denote these d positive roots by uk. Then, equation (43) has d positive solutions ϕk=uk, k=1,2,…,d. Therefore, from (42), we obtain(51)τηk=1ϕkarccosϕk4−c2ϕk2+c0b2ϕk2−b0+b1ϕk2c3ϕk4−c1b1ϕk2+b2ϕk2−b02+2ηπϕk,where k=1,2,…,d and η∈IN. We deduce that ±iϕk is a pair of purely imaginary roots of (40) with τ=τηk. Define(52)τ0=τ0k0=mink∈1,2,…,dτ0k,ϕ0=ϕk0.We setλτ=ςτ+iϕτ to be the root of equation (40) such that ςτηk=0 and ϕτηk=ϕk. Considering ς as a function of τ and differentiating both sides of (40) with respect to τ lead to(53)dλdτ−1=4λ3+3c3λ2+2c2λ+c1+2d2λ+d1e−λτλd2λ2+d1λ+d0e−λτ−τλ.This implies that(54)Redλdτ−1τ=τηk=4ϕk6+3p3ϕk4+2p2ϕk2+p1b1ϕk2+b2ϕk2−b02,=h′ϕk2b12ϕk2+b22.SincesigndReλ/dττ=τηk=signRedλ/dτ−1τ=τηk, we obtain(55)signdReλ/dττ=τηk=signh′uk=sign4ϕk6+3p3ϕk4+2p2ϕk2+p1.Therefore, based on the above analysis, we claim the following result.Theorem 8.
Assume thatℛ1Z>1.(i)
If conditions (H1)–(H3) are not fulfilled, then the partial success therapy equilibrium with immune response E3 is locally asymptotically stable for all τ≥0.(ii)
If one of the conditions (H1)–(H3) is fulfilled, then the partial success therapy equilibrium with immune response E3 is locally asymptotically stable for any time delay τ∈0,τ0 and becomes unstable when τ>τ0. Furthermore, if h′ϕ02≠0, then the transversality condition holds and model (1) undergoes a Hopf bifurcation at E3 when τ=τ0.
Based on Lemma 4.3 in [9], we easily deduce the following theorem.Theorem 9.
Assume thatℛ1Z>1.(i)
If equation (44) admits only one positive and simple root u1, then E3 is locally asymptotically stable for τ∈0,τ01 and becomes unstable for τ>τ01. Furthermore, a Hopf bifurcation appears when τ=τη1, η∈IN.(ii)
If equation (44) admits only two positive and simple roots u1, u2 which are ordered as u2<u1, then there exist a finite number of intervals such that if the delay τ is fixed in these intervals, the equilibrium E3 is locally asymptotically stable, while unstable if τ not belonging to ones. In this case, E3 switches from stability to instability.(iii)
If equation (44) admits at least three positive and simple roots, then there exists a least one stability switch.Remark 2.
Theorems8 and 9 show that when the delay τ is considered, the partial success equilibrium with immune response E3 can lose or gain its stability and rich dynamical behaviors occur including Hopf bifurcation and stability switches.


## 4. Discussion and Conclusion

In this paper, we have proposed and analyzed a virological model for cancer therapy with effects of saturation, innate immune response, and delay that biologically represents the time needed for infected tumor cells to produce new virions after viral entry. We first proved the positivity and the boundedness of solutions and discussed the existence of equilibria by means of two threshold parameters that are the basic reproduction number denoted byℛ0 and the reproduction number for innate immune response labeled by ℛ1Z which represents the average number of innate immune cells activated by damaged tumor cells. More accurately, the proposed model has uniquely (i) two equilibria, the desired outcome therapy equilibrium E0 and the complete failure therapy equilibrium E1 if ℛ0≤1; (ii) three equilibria: E0, E1, and the partial success therapy equilibrium without immune response E2 if ℛ1Z≤1<ℛ0; and (iii) four equilibria: E0, E1, E2, and the partial success therapy equilibrium with immune response E3 if ℛ1Z>1. We have demonstrated that E0 is always unstable and E1 is globally asymptotically stable if ℛ0≤1 and becomes unstable if ℛ0>1. Additionally, the stability of E2 and E3, Hopf bifurcation, and stability switches are analyzed rigorously. Furthermore, our model generalizes those in [7, 10] and our analytical results show that the delay in infection with oncolytic viruses can lead to the loss or stability of both equilibria E2 and E3.



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*Source: 8694821-2020-09-27.xml*