Threshold Dynamics for a Time-Periodic Viral Infection Model with Cell-to-Cell Transmission and Drug Treatments

Xu, Jinhu; Huang, Guokun

doi:https://doi.org/10.1155/2022/2289138

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2022 | Article ID 2289138 | https://doi.org/10.1155/2022/2289138

Threshold Dynamics for a Time-Periodic Viral Infection Model with Cell-to-Cell Transmission and Drug Treatments

Jinhu Xu¹and Guokun Huang¹

Academic Editor: Zahir Shah

Received10 Jan 2022

Accepted21 Feb 2022

Published21 Mar 2022

Abstract

In this study, a time-periodic viral infection model incorporating cell-to-cell infection and antiretroviral therapy has been investigated. The basic reproduction number has been defined as a threshold parameter which governs whether or not the disease dies out. Theoretical results indicate that the disease goes to extinction if and otherwise the disease will uniformly persist. The global stabilities of the equilibria for the corresponding autonomous model have been investigated by constructing suitable Lyapunov functions. Moreover, numerical simulations have been carried out to validate the obtained results. The results show that cell-to-cell infection mode may be a barrier to curing the viral infection and increasing the efficacy of protease inhibitors for blocking cell-to-cell infection which will benefit to weaken the severity of the viral infection.

1. Introduction

Recently, much attention and great effort have been paid on modelling of HIV, and many models have been proposed and studied on HIV spreading. Many earlier models of HIV infection models describe the interaction between virus and target cells by assuming that the infected cells produce virions instantaneously [1, 2]. However, research studies have been carried out to show that a latent period exists before the infected cells are activated to produce virus [3–5]. Therefore, it is reasonable to introduce the latent period into a model. As we know, antiretroviral drugs can effectively suppress viral replication to a low level, but cannot eradicate the virus permanently. An important reason is that HIV provirus can reside in latently infected CD4 T-cells, which can live longer and cannot be affected by antiretroviral drugs or immune responses, but can be activated to produce virus by relevant antigens [5]. Thereafter, motivated by this factor, many viral infection models with latent cells have been proposed and studied to describe this phenomena [6–13] and references therein. For example, Pankavich [6] proposed and studied the following viral infection model:where , and represent the concentration of uninfected target T-cells, latent cells, productively infected T-cells, infectious virons, and noninfectious virons at time , respectively. , , , and are the production rate of T-cells, virus-to-cell infection rate, the activation rate of latent cells, and the fraction of infections leading to latency. are the death rate of susceptible T-cells, latent cells, actively infected T-cells, and virions, respectively. denotes the burst rate of actively infected cells. and are the efficacies of RTIs (reverse transcriptase inhibitors) and PIs (protease inhibitors), respectively. The global dynamics of model (1) have been investigated in [6].

Notice that the drug efficacy in model (1) is assumed to be a constant coefficient. In fact, drugs are often administered for patients periodically. As we known, drug concentration will reach a peak value within a very short time when a dose is administrated, and the concentration down to a lower value as time goes and then reaches a peak value again when another dose is administrated [14–16]. Therefore, drug concentration may vary periodically during the dose interval. Moreover, only cell-free infection has been considered in earlier work; the cell-to-cell transmission was not considered in model (1). However, a recent research work has shown that cell-to-cell transmission may be one of the main infection mode which leads to a failed therapy and potentially contribute to viral persistence [17]. Because a better understanding of the viral dynamics is very significant in terms of applications, thus motivated by these arguments many viral infection models with cell-to-cell transmission have been proposed and studied [18–29] and references therein. Besides, drugs’ efficacy about cell-to-cell infection was not taken into consideration in model (1). However, the results obtained in [30] show that PIs can effectively block cell-to-cell spread of HIV by preventing cleavage of viral polyproteins into functional subunits leading to the formation of immature noninfectious virus particles, while RTIs are less effective inhibitors of HIV cell-to-cell spread compared to virus-to-cell infection. To the best of our knowledge, the time-periodic viral infection model with cell-to-cell infection and latency have not been studied. Hence, motivated by the abovementioned work and arguments, we consider the following time-periodic model with two infection modes:where is the infection rate of productively infected T-cells. Assume are the efficiencies of RTIs and PIs, and we assume they are continuous and periodic in time with a same period . Here, we considered two saturated incidence rates, where and are the saturation parameters and are positive constants. Other parameters have the same meaning of model (1). For convenience, we denote , , and . Since the last equation of model (2) is independent with the others. Thus, we will focus on the following reduced model:

This study is organized as follows. In Section 2, preliminary results and the definition of the basic reproduction number are studied. In Section 3, global extinction of the disease and the uniform persistence are investigated in terms of the basic reproduction number. The global asymptotic stability of the infection equilibrium to the corresponding autonomous model are discussed by applying the method of Lyapunov functions. In Section 4, some numerical simulations are carried out. A brief conclusion and discussion ends the paper.

2. The Basic Reproduction Number

In this section, we investigate the definition of the basic reproduction number for model (3) according to the work [31, 32]. The following result shows that solutions of model (3) are bounded.

Theorem 1. The solutions of model (3) are uniformly and ultimately bounded, i.e., there exist an and such that , for .

Proof. From model (3), we can obtain thatwhere . Thus, there exists such that , for . It follows from the last equation of model (3); we have, for , , which implies that there exist such that , for . Let . It then follows that , for . Hence, the solutions of model (3) are uniformly and ultimately bounded. This finishes the proof.
Let be the standard ordered -dimensional Euclidean space with a norm . For , we write if , if , and if . Let be a continuous, cooperative, irreducible, and -periodic matrix function and be the fundamental solution matrix of the following linear system:Let be the spectral radius of . It follows from the Perron–Frobenius theorem that is the principal eigenvalue of in the sense that it is simple and admits an eigenvector . The following lemma comes from [33] which will be used for the discussion in the next section.

Lemma 1 (see [33]). Let . Then, there exists a positive -periodic function such that is a solution of (5).
Obviously, model (3) has a unique infection-free equilibrium , where . Linearizing (3) at yields thatDefineThen, system (6) can be written asAssume that , , is the evolution operator of the following system:Then, the matrix satisfiesLet be the ordered Banach space of all -periodic functions from to , which is equipped with the maximum norm and the positive cone . Suppose is the initial distribution of infectious cells and virus in this periodic environment; then, is the rate of new infections produced by the infected cells and virus who were introduced at time , and represents the distribution of those infected cells and virus who were newly infected at time and remain in the infected compartments at time , for . Hence,is the distribution of accumulative new infections at time produced by all those infected cells and virus introduced before .
Define the linear operator as follows:It follows from the idea in [32] that the basic reproduction number of system (3) is defined as the spectral radius of , i.e.,Moreover, the local asymptotic stability of the infection-free equilibrium follows from [32].

Theorem 2 (see [32]). The following statements are valid:(i) if and only if (ii) if and only if (iii) if and only if Thus, infection-free equilibrium of (3) is asymptotically stable if and unstable if .

3. The Threshold Dynamics

3.1. Stability and Persistence of the Disease

In this section, we will investigate the global asymptotic stability of infection-free equilibrium and the disease persistence by regarding as a threshold parameter.

Theorem 3. If , then the infection-free equilibrium is globally asymptotically stable, and it is unstable for .

Proof. It follows from Theorem 2 that if , then is locally asymptotically stable and is unstable when . Hence, it is sufficient to show that is global attractive for .
From the first equation of (3) and nonnegativity of the solutions, we then have , which implies that ; there exists such that , .
Consider the following auxiliary system:which is equivalent towhereIt then follows from Lemma 1 that there exists a positive -periodic function such that is a solution of (14), where . Choose and a real number such that , which implies thatThe comparison principle yields thatRecall Theorem 2 that if and only if . Since the continuity of the spectrum for matrices [34], then choose small enough such that , which implies that . Then, we have as . Furthermore, it follows from the first equation of model (3) and the theory of asymptotically periodic semiflows [35] that . Thus, is globally attractive.

Theorem 4. If , then there exists an such that any solution of model (3) with initial values ; the solution of (3) satisfies and admits at least one positive periodic solution.

Proof. LetDefine Poincare map , satisfying , , with as the unique solution of (3) satisfying .
We first show that is uniformly persistent with respect to . It is easy to see that and are positively invariant. Moreover, is a relatively closed set in . Recall Theorem 1 that the solutions of model (3) are uniformly and ultimately bounded; thus, the semiflow is point dissipative on , and is compact. Consequently, it follows from [36] that the semiflow admits a global attractor, which attracts every bounded set in .
DefineThen, we claim that . In fact, it is obvious that . For any , considering the following cases: (1) , (2) , (3) , (4) , (5) , and (6) . For case (1), we have which implies that , for ; then, . Similarly, for the other cases, it also has the same result; here, we omit the proof. Thus, for any , , it indicates that .
Clearly, is one fixed point of in . If is a solution of model (3) from , it then follows from that model (3) that as .
Next, we will show that if the invariant set is isolated, then is an acyclic covering. To do this, it needs to prove any solution of model (3) initiating from will remain into , which can be obtained easily. The isolated invariance of will follow proof.
Now, we need to prove that . Denote . Since the continuity of solutions with respect to the initial values, thus for , there exists such that, for all with , yieldsThen, we claim thatIf it is not true, then we haveFor some , without loss of generality, we suppose that , . Then, we can obtain thatFor any , let , where and , which is the greatest integer less than or equal to . Then, we haveSet ; then, we have , , , and , for . Then, and . It follows from model (3) thatSetIt follows from Theorem 2 that ; then, we can select small enough such that . It follows from Lemma 1 and the standard comparison principle that there exists a positive -periodic function such that , where and , which implies that ; this is a contradiction in which converges to , and hence, is acyclic in . By Theorem 1.3.1 and Remark 1.3.1 in [35], we obtain that is uniformly persistent with respect to . It then follows from Theorem 3.1.1 in [35] that the solution of (3) is uniformly persistent.
Moreover, it follows from Theorem 1.3.6 in [35] that the Poincare map has a fixed point . Then, we see that . If not, suppose , from the first equation of model (3), where satisfiesIt follows from the comparison theorem thatwhere . Then, we haveThe periodicity of implies , which is a contradiction. Thus, . Hence, is a positive -periodic solution of model (3).

3.2. Analysis of the Autonomous Model

If there no drug therapies, i.e., or drug therapies are constants, then model (3) becomes an autonomous model. Without loss of generality, we assume drug therapy is constant. Then, model (3) leads to the following autonomous model:where and . Clearly, model (31) has an infection-free equilibrium with . Furthermore, we can obtain the basic reproduction number of model (31):

Next, we will show that there exists a unique infection equilibrium . For convenience, let and . It is easy to see that satisfies the following equations:

Then, we can obtain

And satisfies the following equation:

Since , which implies , for , let

Then, we have

Thus, there exists a . Consequently, model (31) admits an infection equilibrium . Furthermore, by calculation, we havewhich implies that is decreasing at each of its root. If there exists more than one root of , then it must exists a root such that , which leads to a contradiction. Hence, we claim that there exists a unique infection equilibrium for .

By constructing suitable Lyapunov functions, we can show that the corresponding infection-free equilibrium and infection equilibrium of model (31) are globally asymptotically stable.

Theorem 5. If , then the infection-free equilibrium of model (31) is globally asymptotically stable.

Proof. Definewhere .
By a tedious computation, we haveClearly, if , then . Moreover, by LaSalle’s invariance principle, one can easy to show that the infection-free equilibrium is globally asymptotically stable.

Theorem 6. If , then the infection equilibrium of model (31) is globally asymptotically stable.

Proof. Definewhere . Then, combining (33) and by a tedious computation yieldswhere with maximum value for . Then, it follows from the LaSalle’s invariance principle that one can show is globally asymptotically stable. This completes the proof.

Remark 1. It follows from the above analysis of the model that the saturated incidence rates and can be extended to a more general form and with some conditions as in [27]. Thus, the saturated incidence rates of model (3) can be regarded as a special case.

4. Numerical Simulations

In this section, some numerical simulations are carried out to explain the obtained theoretical results. Most of these parameter values are taken from [6, 13].

Case 1. Let , , , , , , , , , , , , , , and ; then, we can obtain . The simulation shows that the infection-free equilibrium is globally asymptotically stable, which implies disease dies out. Figure 1 validates the above analysis.

Case 2. Let , , , , , , , , , , , , , , and ; then, we have . The theoretical results show that the model admits a positive periodic solution and disease keeps persistent in the host. Figure 2 confirms this conclusion.
Figure 3 shows the impact of cell-to-cell transmission on the dynamics of the model, and the other parameters are the same with Figure 2. The results imply that the peak level of the density of latent cells, infected cells, and viral load increase as increases. Hence, cell-to-cell transmission existing may contribute to viral persistence and underevaluate the risk of disease spreading for without considering cell-to-cell transmission. Thus, cell-to-cell transmission will be a barrier to curing the viral infection.
Figure 4 shows that the effects of PIs on the dynamics of the model by blocking cell-to-cell transmission, and the other parameters are the same with Figure 2. By varying the drug efficacy of PIs for blocking cell-to-cell transmission from the baseline value to and , the numerical results imply that the peak level of the density of latent cell, infected cells, and viral load decrease under PIs’ blocking cell-to-cell transmission. This implies that increase of the efficacy of PIs for blocking cell-to-cell transmission may contribute to weakening the severity of the viral infection.

5. Summary and Discussion

In this study, a time-periodic viral infection model with cell-cell transmission was investigated. We have shown that the infection-free equilibrium is globally asymptotically stable if which implies that infection will be eradicated, and the infection will persistent when . Furthermore, for the corresponding autonomous model, we have shown that the corresponding equilibria are globally asymptotically stable by applying the method of Lyapunov function. The results imply that cell-to-cell transmission existing may contribute to viral persistence and be a barrier to curing the viral infection. At the same time, increasing the efficacy of PIs for blocking cell-to-cell transmission is beneficial to weakening the severity of the viral infection.

Note that only a constant recruitment of uninfected T-cells has been considered in our model. However, T-cells can also be created by proliferation of existing T-cells with a logistic form instead of a constant recruitment [19] and the incidence rates can be a more general form and [27], which formulated a further work. Furthermore, immune response and dynamics of the drug are also a good choice to extend the current work, which will be another future work.

Data Availability

All the data used to support the study are included in the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by National Natural Science Foundation of China (#11701445, #11971379, and #11801439), Natural Science Basic Research Plan in Shaanxi Province of China (2022JM-042, 2020JQ-831, and 2021JM-320), and Scientific Research Program Funded by Shaanxi Provincial Education Department (20JK0642).

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Copyright © 2022 Jinhu Xu and Guokun Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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