Abstract

In this paper, we study a model of oncolytic virus infection with two time delays, one of which is the time from the entry of viruses into tumor cells to start gene replication, and the other is the time from the entry of viruses into tumor cells to release new virus particles by infected tumor cells. In previous studies on oncolytic virus infection models, the infection rate was linear. Combined with the virus infection models, the saturated infection rate, is further considered to describe the dynamic evolution between viruses and tumor cells more objectively so as to further study the therapeutic effect of oncolytic viruses. This paper discusses the dynamics of the system under three conditions: (1) , (2) and , and (3) and , and proves the global stability and local stability of the virusfree equilibrium, the stability of the infection equilibrium, and the existence of Hopf bifurcation. Finally, the conclusions of the paper are verified by MATLAB numerical simulations.

1. Introduction

An oncolytic virus is a natural or genetically modified virus that can specifically infect and kill tumor cells without damaging healthy cells (see [1]). Oncolytic virotherapy is a very promising cancer treatment method. Initially, The Lancet reported that influenza viruses can cause tumors in patients to regress, resulting in the emerging concept of oncolytic viruses. Since then, more than 160 different oncolytic viruses have been in preclinical studies and clinical trials (see [2]). According to current development trends, oncolytic viruses can be positioned as the next major breakthrough in cancer treatment after checkpoint inhibitors in immunotherapy.

Many researchers in the fields of medicine, biology, and mathematics have been exploring and improving the related problems of oncolytic virotherapy, which has brought breakthroughs in the treatment of some malignant tumors, but there are still many problems to be solved at present (see [3]). For example, the proliferation of oncolytic viruses in host solid tumor cells is limited and the difference in tumor cell receptor expression limits the spread of the virus, which in turn affects the therapeutic effect. Immune-mediated pharmacokinetic therapy is relatively slow compared to drugs that directly kill tumor cells, so time delays need to be considered in clinical trial design and outcome evaluation. These problems have affected the antitumor effect of oncolytic viruses and failed to achieve the goal of systematic and reliable tumor elimination of oncolytic viruses. In the research process of oncolytic virotherapy, mathematical models can better understand the dynamic mechanism of tumor cells-virus-immune cells (Cancer-Virus-CTLs). It also provides an effective tool to predict the outcome of immunotherapy and optimize tumor treatment strategies.

In order to describe the dynamic changes of the immune response in viral therapy, a series of ordinary differential equation models have been constructed in the past decades (see [411]. Dingli et al. published a model based on population dynamics in the mathematical biosciences (see [12]), which focused on the basic elements of radiation viroid therapy and discussed and proved the stability of the equilibrium of complete cure, local treatment, and treatment failure. On the basis of D. Dingli et al., Kim et al. proposed a virus model that considered both virus-specific immunity and tumor-specific immunity (see [13]). In addition, since the life cycle of viruses contains many intracellular processes, the time from the entry of viruses into tumor cells to the start of gene replication and the time of the release of new virus particles by the infected tumor cells are considered to further establish mathematical models with time delays. Wang et al. (see [14]) considered the cyclic period of viruses in cells and established a time-delay differential equation. Kim et al. (see [15]) proposed a delayed ordinary differential equation model. Li et al. (see [16]) further considered the time delay of virus self-replication based on the research of Kim et al. proposed an ordinary differential equation model with two time delays to analyze the influence of the time delay on the stability of the equilibrium.

For most models of oncolytic virus infection, the infection rate of oncolytic viruses and uninfected tumor cells is bilinear, but the actual incidence is not strictly linear. Therefore, we further assume that the model has a saturated infection rate . In this paper, we study a virus infection model with a saturated infection rate .

The rest of this paper is organized as follows: the second section builds the model. The third section solves the equilibrium of the system. The fourth section discusses the stability of the virusfree equilibrium and the virus infection equilibrium, as well as the existence conditions of Hopf bifurcation of the virus infection equilibrium, and explains the dynamic properties under three conditions: (1) , (2) and , and (3) and . In the fifth section, numerical simulations are performed to verify the obtained results. Finally, the conclusions of this article are given.

2. Mathematical Model Formulation

On the basis of the research (see [16]), this chapter considers the impact of saturated infection rate and two time delays on the virus, and the following model is shown:where , and represent the number of uninfected tumor cells, infected tumor cells, oncolytic viruses, and CTL immune cells, respectively. represents the maximum growth rate of uninfected tumor cells. represents the average rate at which the infected tumor cells release viruses. and represent the rate of CTL immune cells regeneration. represents the maximum carrying capacity of the human environment to tumor cells. represents the delay from viruses entering tumor cells to the beginning of gene replication. indicates the time from the entry of viruses into tumor cells to the release of new virus particles by infected tumor cells. Obviously, there is . and represent the rate of CTL immune cells clearing uninfected tumor cells and infected tumor cells, respectively. , and indicate the natural mortality of infected tumor cells, oncolytic viruses, and immune cells, respectively. The rupture of tumor cells releases tumor antigens, thereby inducing a systemic antitumor immune response, which may enhance the cytolytic activity of viruses.

The initial condition of system (1) iswhere . . Note , define the norm , where is a continuous function, constitutes a Banach space.

3. The Existence of the Equilibrium

Through simple calculations, can be obtained from the first equation of system (1). The virusfree equilibrium of system (1) is , where and .

Let the infection equilibrium of system (1) be , which is the positive solution of the following equations. Based on the model, there are reasonable assumptions as follows: , and . All the above parameters are positive constants.

Through simple calculations, the following conclusions are obtained:where

When , we get . Substituting (4)–(6) into the first equation of (3), we obtain the quadratic equation of one variable with respect to .where

When , equation (8) has a unique positive root . Therefore, is called as the basic regeneration number of system (1), when and , system (1) has a unique positive equilibrium . is the number of basic regeneration, and is the number of immune response regeneration. and together ensure the existence of positive equilibrium .

4. Stability Analysis of the Equilibrium

In order to study the stability of the equilibrium, we consider the linear approximation equation of system (1) at the equilibrium .where ,

Letwhere is a fourth-order identity matrix.

4.1. Global and Local Stability of Virusfree Equilibrium

Theorem 1. If , the virusfree equilibrium is globally asymptotically stable for any and .

Proof. Assuming , we define the Lyapunov functionalFurther, calculating the derivative along with system (1)When , is obtained, therefore . It is obvious to obtain that if and only if . Assuming that M̃ is the largest invariant set contained in , it is easy to get M̃ =  . According to the LaSalle’s invariant principle, the virusfree equilibrium is globally asymptotically stable.

Theorem 2. If , the virusfree equilibrium is locally asymptotically stable for any and . If , is unstable for any and .

Proof. According to equation (12), the characteristic equation of virusfree equilibrium is as follows:Through certain calculations, it follows thatConsidering the following two equations separately(i)Equation (17) can be written as . Because and , the roots of equation (15) have negative real parts for all and .(ii)To discuss the case of the roots of equation (18), first of all, we consider , so it is easy to get equation (18) as follows:When , it follows that , therefore, all the roots of (19) have negative real parts. When , is a single root of (18), and the remaining roots have negative real parts. When , , therefore, (19) has a root of positive real parts.
Secondly, we consider and . Assuming that for some and , is the roots of (18), substituting into (18) and separating the real and imaginary parts to get thatBased on , it follows thatwhereIf , because of the following inequalityEquation (21) has no positive root. Thus, when , (18) has no pure imaginary root. Consequently, it can be seen from (i) and (ii) that when , is locally asymptotically stable for any and . When , is continuous with respect to , and , respectively. For any , we get and . Therefore, there exists such that holds. In summary, when , the equilibrium is unstable for any and .

4.2. Local Stability Analysis of the Virus Infection Equilibrium

Suppose that the positive equilibrium of system (1) is , and the characteristic equation corresponding to the positive equilibrium of equation (12) is as follows:

which is equivalent to the following equation:where

The case and .

When and , the characteristic equation (25) becomeswhere

(H1) .

The following theorem is obtained by the Routh–Hurwitz criterion.

Theorem 3. Suppose and , if and condition (H1) holds, then the equilibrium is locally asymptotically stable.

4.2.1. The Case and

When and , the characteristic equation (25) is written as follows:where

Assuming that is a pure imaginary root of (29), substituting into (29) and separating the real and the imaginary parts to obtain that

According to , we can get thatwhere

Let , it follows from (32) that

Let be the positive roots of (32), then (29) has pure imaginary roots , where . From (31), the following solutions are obtained:where , . The distribution of the roots of (34) is discussed next, and the following conditions are given:(S1) (1) and . (2) and .It is sufficient if any one of the above conditions is satisfied.(S2) (1) and .(2) and .Meeting any one of the above conditions can be established.(S3) and .(S4) and .(S5) (1) and .(2) and .(3) and .(4) and .

Any one of the above conditions can be satisfied, where

Next, we discuss the relationship between the roots of the characteristic equation and the coefficient. According to conclusions (see [17]), the following lemma is obtained.

Lemma 1. For equation (34), the following conclusions are drawn:(i)If (S1) holds, then equation (34) has only one root of the positive real parts. Satisfying condition (1), the equation has one positive root and three negative roots. Satisfying condition (2), there is a positive root, a negative root, and a pair of conjugate virtual roots.(ii)If (S2) holds, then equation (34) has two roots with positive real parts. Satisfying condition (1), the equation has two positive roots and two negative roots. Satisfying condition (2), there are two positive roots and a pair of conjugate virtual roots.(iii)If (S3) holds, then equation (34) has three different positive real roots and one negative root.(iv)If (S4) holds, then equation (34) has four different positive real roots.(v)If (S5) holds, then equation (34) has no positive real root. That is, if (1) or (2) is satisfied, the equation has two pairs of different conjugate virtual roots. Satisfying condition (3), equation (34) has four different negative roots. Satisfying condition (4), there are two negative roots and a pair of conjugate virtual roots.

According to Lemma 1, the following lemma is obtained:

Lemma 2. For equation (29), the following conclusions are drawn:(i)If (S1) holds and , then equation (29) has a pair of pure imaginary roots , where and .(ii)If (S2) holds and , then equation (29) has two pairs of pure imaginary roots , where and .(iii)If (S3) holds and , then equation (29) has three pairs of pure imaginary roots , where and .(iv)If (S4) holds and , then equation (29) has four pairs of pure imaginary roots , where and .(v)If (S5) holds and , then equation (29) has no pure imaginary root.

Lemma 3. If is a solution to equation (32), then

Proof. Assuming that is the solution of equation (25), and both sides of the equation are differentiated from at the same time, it follows thatwhere , and denote the derivation of , respectively. According to , we obtain thatThrough further calculations, we get thatWhen , , it follows that , and then substitute into to get the following equation:Therefore,The proof is completed.

According to Lemmas 13, the following theorem is obtained.

Theorem 4. For equation (29), the following conclusions are established:(i)If (S1) is true, then .(ii)If (S2) is true, then and .(iii)If (S3) is true, then , (iv)If (S4) is true, then , .

Proof. (i)If (S1) is true, then equation (34) has one positive root and three negative roots or one positive root, one negative root, and a pair of conjugate imaginary roots, and is monotonically increasing at the positive root , so we get . Therefore, the conclusion is true.(ii)If (S2) is true, then equation (34) has two positive roots ( and ) and two negative roots or has two positive roots and a pair of conjugate imaginary roots. Function decreases monotonically at the positive root and increases monotonically at the positive root , so and are obtained. Therefore, the conclusion is true.(iii)If (S3) is true, equation (34) has three different positive real roots ( and ) and one negative root. Function increases monotonically at , decreases monotonically at , and increases monotonically at . So, we get , and . Therefore, the conclusion is true.(iv)If (S4) holds, equation (34) has four different positive real roots ( and ). Function is monotonically decreasing at and , and monotonically increasing at and . So we obtain , , and . Therefore, the conclusion is true.

According to the analysis in references (see [1821]), Lemma 4 and Theorem 5 are obtained on the basis of Theorems 3 and 4.

Lemma 4. For equation (25), when and , the following conclusions are obtained:(i)If (H1) and (S1) are true, then for , all the roots have a negative real part. There is a pair of pure imaginary roots at . There is at least one root with a positive real part at .(ii)If (H1) and (S2) are true, then there exists a positive integer that makes . This indicates that the system goes through stability switches from stable state to an unstable state. All the roots have negative real parts at . There are two pairs of pure imaginary roots at or , and the other roots have negative real parts. There is at least one root with a positive real part at or .(iii)If (S2) is true and (H1) is not, then there exists a positive integer that makes . This means that the system goes through stability switches from an unstable state to a stable state. There is at least one root with a positive real part at or . There are two pairs of pure imaginary roots at or , and the other roots have negative real parts. All the roots have negative real parts at .(iv)If (H1) and (S3) are true, or (H1) and (S4) are true, then the system has at least one stability switch.(v)If (H1) and (S5) are true, then for any , all roots have negative real parts.

Theorem 5. Suppose and .(i)If (H1) and (S1) are true, then the equilibrium is locally asymptotically stable for any . When , system (1) appears Hopf bifurcation at the equilibrium .(ii)If (H1) and (S2) are true, when , the equilibrium is locally asymptotically stable. When or , the equilibrium is unstable. When or