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

,e aim of this work is to present a virological model for cancer therapy that includes the innate immune response and saturation effect.,e 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.


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][2][3][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. ey 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. ey performed the stability analysis and the existence of Hopf bifurcation. ey 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. e 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: where x(t), y(t), and v(t) 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, z(t) denotes the concentration of the innate immune cells at time t. e tumor grows logistically at a rate r and K is the maximal carrying capacity of tumor cells. e 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. erefore, 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. e 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. e 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.

Positiveness, Boundedness, and Equilibria
In this section, we prove the positivity and the boundedness of solutions of model (1). 4 ) be the Banach space of continuous functions mapping the interval [− τ, 0] into IR 4 with the topology of the uniform convergence. According to the fundamental theory of functional differential equations [11], Additionally, we also assume that the initial values satisfy the following biological conditions: (2)

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 which leads to x(t) ≥ 0 and z(t) ≥ 0 for all t ≥ 0. From the second and third equations of (1), we have 2 Complexity Clearly, y(t) ≥ 0 and v(t) ≥ 0 for t ∈ [0, τ]. is procedure can be repeated on the interval [ητ, (η + 1)τ] for all η ∈ IN. en, y(t) ≥ 0 and v(t) ≥ 0 for all t ≥ 0.
rough the first equation of (1), we have Applying the comparison principle, we get For t > τ, we have where c � max T(0), rKe − mτ /ρ . is implies that T(t) is bounded on (τ, +∞). According to the continuity of T(t) on [0, τ], we concluded that T(t) is also bounded on [0, τ]. erefore, y(t) and z(t) are bounded for all t ≥ 0. By the third equation of (1) and the boundedness of y(t), we find en, v(t) is bounded. is completes the proof. Next, we discuss the existence of equilibria of model (1). Denote 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. erefore, R 0 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]. en, (1) always has two equilibria E 0 (0, 0, 0, 0) and E 1 (K, 0, 0, 0) In presence of the innate immune response, we have Since z ≥ 0, we have v ≤ δb(N − ne mτ )/μc. is indicates that there is no biological equilibrium when v > δb(N − ne mτ )/μc. Let F be a function defined on the closed interval [0, δb(N − ne mτ )/μc] as follows: Clearly, F(0) � − b(δ + pφ 2 (0))e mτ < 0 and Since When the innate immune response has not been established, we have cy 2 − b ≤ 0. en, we define the reproduction number for the innate immune response as follows: where 1/b is the average life expectancy of innate immune cells, c is the rate of immune response activation, and y 2 is the number of infected tumor cells at the steady state E 2 . Hence, R Z 1 describes the average number of innate immune cells activated by the infected tumor cells.

Model Analysis and Stability
To understand the dynamics of the proposed model, we first analyze the local asymptotic stability of equilibria. Let  E(x, y, v, z) be an arbitrary equilibrium of model (1). Hence, the characteristic equation at E is given by Theorem 3. e desired outcome therapy equilibrium E 0 (0, 0, 0, 0) is unstable.
Proof. It is not hard to see that at E 0 (0, 0, 0, 0), equation (17) becomes Since λ � r > 0 is a positive root of the above equation, we deduce that E 0 is unstable. □ Theorem 4. If R 0 < 1, then the complete failure therapy equilibrium E 1 is locally asymptotically stable and unstable if R 0 > 1.
is implies that the characteristic equation (19) has at least one positive eigenvalue when R 0 > 1. erefore, the complete failure therapy equilibrium E 1 becomes unstable as long as R 0 > 1. e following result investigates the global stability of the complete failure therapy equilibrium E 1 when R 0 ≤ 1. □ Theorem 5. If R 0 ≤ 1, then the complete failure therapy equilibrium E 1 is globally asymptotically stable for all τ ≥ 0.
Proof. We consider the following functional: Taking the derivative of V along t of the solutions of (1) delivers Seeing that limsup t⟶∞ x(t) ≤ K, we deduce that each ω-limit point satisfies x(t) ≤ K. Hence, it is sufficient to take solutions for which x(t) ≤ K. us, en, dV/dt| (1) ≤ 0 when R 0 ≤ 1. Moreover, it is easy to prove that the largest invariant subset of (x, y, v, z) | dV/ dt � 0} is the singleton E 1 . From LaSalle's invariance principle [13], we conclude that E 1 is globally asymptotically stable as long as R 0 ≤ 1. □ Next, we study the stability of E 2 . In this case, (4) becomes where Clearly, λ 1 � cy 2 − b is a root of (28). If R Z 1 > 1, then λ 1 > 0 and E 2 is unstable. However, λ 1 < 0 if R Z 1 < 1. In this case, we study the roots of the following equation: e general form of this transcendental characteristic equation was investigated by Beretta and Kuang in [14].
Remark 1. eorem 7 shows that the delay τ can cause the partial success therapy equilibrium without immune response E 2 to gain or lose its stability. In addition, periodic solutions appear when the value of this delay is equal to a critical value.