Abstract
The objective of this study is to investigate the effects of immune responses on HIV replication by using a novel HIV model that incorporates immune responses including cytotoxic T lymphocytes and antibodies. In this model, the cytotoxic lymphocyte cells are stimulated by infected T cells, and the antibodies are received continuously through vectored immunoprophylaxis. In the first step, we analyze the well-posedness of our proposed model. By obtaining the basic reproduction number, we also investigate the existence of equilibrium in three cases, including infection-free equilibrium, immune-free infection equilibrium, and immune-present infection equilibrium. As a result, we demonstrate our model can admit two immune-free infection equilibria, which are dependent on the basic reproduction number. Additionally, we study their local stability and find sufficient conditions for them. In particular, we show that immune-free infection equilibrium and immune-present infection equilibrium can become unstable from stable, and then a simple Hopf bifurcation can occur. Theoretical results about backward bifurcation and forward bifurcation are further derived. In addition, simulations reveal rich dynamic behaviors, such as backward bifurcation, forward bifurcation, and Hopf bifurcation. The rich dynamics of the proposed model demonstrate the importance and complexity of immune responses when fighting HIV replication.
1. Introduction and Model
As the human immunodeficiency virus (HIV) emerged in the early 1980s, it quickly became a worldwide health issue, due to its infectiousness, mortality, and incurability. Biological studies have shown that HIV infects CD4+ T helper cells and attacks the immune system, causing the body to lose immunity. Humans are therefore susceptible to a variety of diseases, as well as malignant tumors. Therefore, revealing the HIV replication mechanism in vivo and predicting the impacts of interventions can help humans improve survival and reduce the therapy cost. Considering these two scenarios, mathematical models are imperative to reduce the cost and control virus replication. For example, the authors in [1–3] study COVID-19 and estimate the impacts of intervention strategies by mathematical models, which provides significant theoretical guidance for the process of human antiviral.
It is clear that many scholars also studied HIV replication mechanism by mathematical models. Nowak and Bangham [4, 5] proposed a basic model to describe the variation of the virus in vivo: where denotes the concentration of helper T cells, denotes the concentration of infected helper T cells, and denotes the concentration of the virus. is the production rate of new target cells. and are the death rates of uninfected cells and infected cells, respectively. is the infection coefficient, and is the burst coefficient of the virus when infected cells die. is the clearance rate of virus. The (1) model has been helpful in understanding HIV infection dynamics and developing specific treatment regimens [6, 7]. Based on this model, many mathematicians studied the dynamics of HIV infection in vivo [8–12].
Up until now, there has not been an effective way to cure HIV in the world, which implies that controlling the spread of the virus in vivo is very essential for us. It is well known that antibodies, one main component of the immune response, which are produced by B lymphocytes are used to identify and neutralize free virus particles in the blood. It then becomes a promising and effective therapy to reduce the virus in the blood in clinic experiments. In 2011, Balazs et al. [13] carried out a vectored immunoprophylaxis experiment to bring new hope to eradicating HIV. The experiment showed that the humanized mice receiving vectored immunoprophylaxis appeared to be fully protected from HIV infection. Based on this experiment, the authors in [14] presented their new model which was different from the previous one:
Here, is the concentration of antibodies in vivo. represents the neutralizing antibodies produced at a constant rate after the injection. denotes the clearance rate of antibodies. represents the loss rate of the virus under the attack of antibodies. The term depicts the loss of antibody from the effect of antibodies’ involvement with the virus. Other parameters are the same with system (1).
Another main component of the immune response is cytotoxic T lymphocytes (CTLs), which are the T cells that are capable of recognizing and killing infected cells, and they would not be infected by HIV. Some papers were devoted to investigate the HIV models under the impacts of CTLs [8, 15–17], which indicates the CTLs have a critical role on the viral infection. Moreover, many researchers have taken into account the effect of both CTLs’ response and antibodies [18–22] and the references therein. Specifically, in [20–22], the authors captured the main features of the complex interactions of HIV and immune responses and then formulated the impacts of immune responses as a term in the models instead of introducing a new variable. Their formulations make the mathematical analysis tractable. However, it cannot depict the dynamics of HIV under CTLs and antibodies in vivo exactly.
Therefore, to study the dynamics of HIV replication in vivo under CTLs and antibodies, we introduce two new variables into (1) to propose our model, which incorporates the CTLs into model (2): where denotes the concentration of CTL cells, represents the loss rate of infected cells under attack by CTLs, and is the death rate of the CTL cells. The last term in the third equation describes the loss rate of the virus because of entry into target cells. The term represents the increment of CTL cells stimulated by infected T cells. All the other parameters in (3) are the same as those in (2).
We consider CTLs and antibodies as two major components of the immune response in our model (3), which describes HIV replication in vivo under the protection of the immune response. Compared to models that only consider CTLs or antibodies, the results are more informative. It can also be more accurate than those works in [20–22], where formulate the impacts of CTLs and antibodies as a term of the model. On the other hand, model (2) is proposed to investigate the viral dynamics for the introduction of vectored immunoprophylaxis antibodies in the experiment of [13]. As we mentioned before, our model (3) incorporates the CTLs into model (2), which can describe the dynamics of HIV under the CTLs in the vectored immunoprophylaxis experiment. It can provide the theoretical results for the impacts of CTLs in this experiment, which can be further extended to study the clinical possibilities of the experiment. Clearly, this model and its results cannot be derived from [18–22] and the references therein, where CTLs and antibodies are also considered in their models.
The organization of this paper is as follows: In the next section, we study the model (3). We study the well-posedness of solutions in Section 2. In Section 3, we study the existence of equilibria and then present the stability analysis of infection-free equilibrium in Section 4. We conduct the numerical simulation of this model in Section 5. A brief summary and discussion are shown in the last section.
2. Well-Posedness of Solutions
For the sake of brevity, we offer the following scaling:
Dropping the bars, model (3) becomes
In this section, we show that all solutions of the system (5), for any given nonnegative initial conditions, are nonnegative.
Theorem 1. Any solution of system (5), , is nonnegative and ultimately bounded for with any provided nonnegative initial conditions.
Proof. . Firstly, we show that is positive for . Assume that is the first time such that . Then, according to the first equation of system (5), we have , which implies that always holds for with . It is similar to show that and always hold true for for any given positive initial conditions.
To prove for , we argue it by contradiction. Let be the first time to make , then we have . Solving from the second equation of system (5) yields which means that . Then we have and then for . Furthermore, we can also get
It shows that is nonnegative for .
Now we show that solutions of system (5) are ultimately uniformly bounded for . Firstly, from system (5), we have which means that and . Adding the former two equations yields where , which means that . It follows that and then . Therefore, we obtain that where , which means that .
Therefore, we get the following feasible region:
It can be verified that is positively invariant with respect to system (5).
3. The Existence of Equilibria
It is easy to obtain that the infection-free equilibrium (IFE) is . In virus dynamics, the basic reproduction number is a basic concept, which denotes the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual [23]. Obviously, whether the basic reproduction number exceeds unit one is an important factor to determine the spread or extinction of the infection, biologically. Hence, it is necessary and reasonable for us to derive the basic reproduction number of model (5). By the method introduced in [23], we rewrite (5) as , where .
By computing , where denotes the spectral radius of a matrix we can obtain that the basic reproduction number of virus in (5) is
3.1. The Existence of the Immune-Free Infection Equilibrium
It is obvious that model (5) has an immune-free infection equilibrium . It is clear that the existence of this immune-free infection equilibrium is equivalent to study the existence of the infection equilibrium of the following system:
Then, we study the existence of infection equilibrium of system (16). An infection equilibrium satisfies
Obviously, we have where satisfies the equation in which
From , we know that system (16) has a unique infection equilibrium , when , where
As in the case of , note that
We define that
Theorem 2. System (16) has an infection equilibria if . Hence, system (5) admits an immune-free equilibria if .
Lemma 3. Let and hold. We have the following conclusions: (1)System (16) has two infection equilibria and when (2)System (16) has a unique infection equilibrium when (3)System (16) has no infection equilibrium when
Proof. Clearly, (19) has no positive root when , two positive roots when and , has a positive root when and , and no positive root when . So, we study the case that .
To determine the signs of in terms of the basic reproduction number, we denote by so that
It is easy to obtain
where are given in (22). Since , we get
Suppose
then . Let ; then we have
If , we can get , and then
It follows that admits a root , which is defined in (22), in interval (, 1). Clearly, we have when ; when and when .
According to Lemma 3, we can conclude the following theorem.
Theorem 4. Let and hold. (1)System (5) has two immune-free equilibria and when (2)System (5) has a unique immune-free equilibrium when (3)System (5) has no immune-free equilibrium when
3.2. The Existence of the Immune-Present Infection Equilibrium of System (5)
An immune-present infection equilibrium satisfies
Direct computation leads to where satisfies the equation in which
To ensure the existence of a positive equilibrium, we let and , which is equivalent to
By the mathematical analysis, we know that must have a unique positive solution when . When , it is easy to get . Then has also a unique positive solution when . Therefore, we have the following theorem.
Theorem 5. System (5) has a unique immune-present infection equilibrium when , where is the positive solution of . Otherwise, system (5) has no immune-present infection equilibrium.
4. The Stability of Equilibra
The Jacobian matrix of (5) at an equilibrium is
Theorem 6. The infection-free equilibrium is asymptotically stable if and is unstable if . The immune-free equilibrium is unstable when it exists.
Proof. The characteristic equation of the Jacobian matrix of (5) at , , in is
It follows that all the characteristic roots have negative real parts when , and one characteristic root is positive when . Thus, we know that is asymptotically stable when and is unstable when .
The characteristic polynomial of in is then
where
We note that
It then implies that has at least a positive real eigenvalue, which means that is unstable.
As for the stability of , we can also obtain that the characteristic polynomial of in is then
where
Hence, according to Routh-Hurwitz criterion, we obtain the following theorem.
Theorem 7. When system (5) admits the immune-free equilibrium , it is locally stable when , , , , and .
Obviously, the characteristic polynomial of in is then
where
According to Routh-Hurwitz criterion, we can also obtain the following theorem.
Theorem 8. When system (5) admits the immune-present equilibrium , it is locally stable when , , , and .
5. Bifurcations and Numerical Simulation
In this section, we study bifurcations of model (3) on the basis of Section 3 and Section 4. When crosses unit one, the IFE loses stability, which results in a bifurcation where a curve of endemic equilibria emerges. The bifurcation is forward if the endemic curve occurs when is slightly larger than one, and there is no positive equilibrium near the for . In contrast, the bifurcation is backward if a positive equilibrium occurs when . Now, we derive the condition for the backward bifurcation from of model (3). Notice that Theorem 4 implies that (24) is the existent condition of the backward bifurcation of model (5). By (4), we get an equivalent equation of (27):
Thus, we show the proposition below.
Proposition 9. System (3) admits a backward bifurcation as crosses unity if and admits a forward bifurcation if
In this section, we implement numerical simulations on the basis of the MatCont package [24, 25] to testify the theoretical results above and explore more patterns of dynamical behaviors of model (3). We select , , , , , and from [8, 14].
If , we have and a backward bifurcation exists from Proposition 9. We derive a bifurcation figure shown in Figure 1. To further address the results of Figure 1 at length, we get evolutionary results, as presented by Figures 2 and 3.
(a)
(b)
(a)
(b)
From Figure 1, we find a Hopf bifurcation point (H) at , a fold bifurcation point (LP) at , and a branch point (BP) at . It illustrates that two unstable immune-free infection equilibria coexist with a stable infection-free equilibrium when ; a stable immune-free infection equilibrium, an unstable immune-free infection equilibrium, and a stable infection-free equilibrium coexist when (see Figure 2); and an unstable immune-free infection equilibrium coexists with a stable immune-present infection equilibrium when (see Figure 3(a)). Further numerical simulations indicate that any solution tends to an infection-free equilibrium when (see Figure 3(b)), which means that the subthreshold of is required to eliminate the infection. It is different to the classical backward bifurcation that needs to eliminate the infection.
When , we have , and a forward bifurcation exists from Proposition 9. When we get a bifurcation figure shown in Figure 4, the detailed evolutionary processes are, correspondingly, illustrated in Figure 5.
(a)
(b)
From Figure 4, a branch point (BP) is observed at . It demonstrates a stable infection-free equilibrium exists when ; a stable immune-free infection equilibrium, and an unstable infection-free equilibrium coexist when (see Figure 5(a)), and an unstable immune-free infection equilibrium coexists with a stable immune-present infection equilibrium when (see Figure 5(b)).
Note that the qualitative difference between Figures 1 and 4 results from the fact that the value of in Figure 1 is higher than it is in Figure 4. This indicates that a stronger loss of antibodies’ involvement with a virus can lead to a backward bifurcation, which supports the result of Proposition 9.
6. Conclusion and Discussion
In this paper, we focused on the modeling analysis of HIV epidemics, which incorporated immune systems based on CTLs and vectored immunoprophylaxis for controlling HIV replication. To address this role of immune response in detail, we integrated the immune response as a term into the model (3).
Specifically, we gave the analytical condition of the existence of equilibria. We further studied the local stability of the existing equilibria. According to these theoretical results, we described the bifurcation figures of the given model. We noted that there exists a backward bifurcation at the infection-free equilibrium, thereby, demonstrating that driving the basic reproduction number () below 1 is not enough to eliminate HIV. More significantly, we observed a Hopf point at , which indicates the classical strategy to drive below a certain value (limit point) maybe unnecessary. Due to these findings, we noted that it is feasible to extinguish the infection even when is below . Meanwhile, a forward bifurcation and an immune-present infection equilibrium exist in this model. Therefore, any strategy to drive below 1 is adequate to clear the infection away, according to our results.
Using numerical simulations and mathematical analyses of the proposed model, we can gain insights into the mechanisms of viral infection under the conditions of vectored immunoprophylaxis and immune response. With the introduction of the immune response, the model exhibits rich dynamical characteristics. As a result of the above findings, vectored immunoprophylaxis may be an effective therapy for HIV, and immune responses have a great influence on viral infection in vivo, which is overlooked in [14].
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
We are very grateful to all the referees for careful reading and valuable comments which led to important improvements of our original manuscript.