- Research
- Open Access
Analysis and control of an age-structured HIV-1 epidemic model with different transmission mechanisms
- Xiaoyan Wang^{1}Email author,
- Junyuan Yang^{2} and
- Fei Xu^{3}
https://doi.org/10.1186/s13662-017-1455-0
© The Author(s) 2018
- Received: 3 September 2017
- Accepted: 19 December 2017
- Published: 23 January 2018
Abstract
In this paper, we propose a within-host HIV-1 epidemic model with cell-to-virus and cell-to-cell transmission. By mathematical analysis, we obtain the basic reproduction number \(\mathcal {R}_{0}\), which determines the viral persistence and the basic reproduction number \(\mathcal {R}_{0}^{\mathrm{cc}}\) with respect to cell-to-cell transmission which is not strong enough, i.e., it is less than 1. If the basic reproduction number is less than 1, then the viral-free steady state \(E_{0}\) is globally asymptotically stable, which is proved by fluctuation lemma and comparison method; if \(\mathcal {R}_{0}>1\) is greater than 1, the endemic steady state \(E^{*}\) is globally asymptotically stable, which is proved by constructing the Lyapunov functional. Antiretoviral therapy is implemented to suppress the viral replication. Protease inhibitors for cell-to-cell transmission play an important role in controlling cell-to-cell infection. Under some circumstances, the effects of the cell-to-cell infection process are more sensitive than those of cell-to-virus transmission.
Keywords
- infection age
- antiretroviral therapy
- cell-to-cell transmission
- Lyapunov functional
1 Introduction
Since the discovery of the first case of acquired immunodeficiency syndrome (AIDS), the disease has been in a major concern in global health. Nearly 43 million peopled are infected by human immunodeficiency virus (HIV) and about 29 million people died due to AIDS. Such accumulative cases are still increasing each year. Many infected individuals are receiving highly active antiretroviral therapy (HAART), an effective treatment that suppresses HIV-1 replication and progression. Even though the treatment does not lead to permanent cure of HIV infection, it extends the life of HIV-1-infected individuals and individuals under such treatment survive in asymptomatic chronic stages with low viral load. In particular, HAART is able to suppress viral replication to undetectable levels (<50 HIV-1 RNA copies/ml) in adherent patients [1, 2]. The process of in-host HIV infection is as follows. After entering the CD4^{+} T cells, HIV viruses reversely transcribe from RNA to DNA and integrate viral DNA into the hosts DNA. Then infected CD4^{+} T cells release viruses through transcription and translation. Because of the comprehensive infection process for HIV, applying antiretroviral drugs at different infection stages may have different treatment effects.
Dixit and Perelson [10] studied the decay of viral load of individuals infected with HIV under monotherapy. They showed that such scenario can exhibit complicated dynamical behaviors depending on the relative magnitudes of the pharmacokinetic, intracellular, and intrinsic viral dynamic time-scales. The investigation indicated that exponential decay dynamics can be considered as a special case of HAART. Dixit et al. [11] constructed a mathematical model with increased accuracy to investigate HIV dynamics when the drug is not 100% effective. Mathematical models have also been proposed to consider the life cycle of virus [12–14] and immune effects [15–17]. The authors compared therapeutic efficacy of medications applied at different infectious stages on the reduction of viremia. They showed that multiple stages, intracellular delay and different target cells have significant effects on viral dynamics.
In order to get some theoretical results, we make the following assumptions.
Assumption 1.1
- (1)
\(\lambda_{i},\mu_{i},\beta_{i},c>0\);
- (2)\(q_{i}(a), p_{i}(a)\in L_{+}^{\infty}\), that is, there exist essential upper bounds \(q_{i}^{+}\) and \(p_{i}^{+}\) such that$$q_{i}^{+}=\mathrm{ess}.\sup_{a\in[0,\infty)}q_{i}(a),\qquad p_{i}^{+}=\mathrm{ess}.\sup_{a\in[0,\infty)}p_{i}(a); $$
- (3)
\(q_{i}(a)\) is bounded and uniformly continuous from \([0,\infty )\) to \([0,+\infty)\);
- (4)
there exists \(\underline{\delta}_{i}>0\) such that \(\delta_{i}(\tau )\ge\underline{\delta}_{i}\) for \(\tau\in\mathbb {R}_{+}\) and \(i\in\mathbb {N}_{n}\).
Under Assumption 1.1, we can obtain the nonnegativity of system (1.3).
Proposition 1.1
If \(x_{i0}\ge0\), \(y_{i0}(a)\in L_{+}^{1}\), \(v_{0}\ge0\), system (1.3) has nonnegative solution.
Proof
Lemma 1.2
- (1)
\(x_{i}(t)\le\Lambda\), \(\int_{0}^{\infty}y_{i}(t,a)\,da\le\Lambda\), \(v(t)\le\Lambda\).
- (2)
\(Q_{i}(t)\le q^{+}_{i}\Lambda\), \(P_{i}(t)\le p^{+}_{i}\Lambda\).
- (3)
\(z_{i}(t)\le\bar{\beta}\Lambda^{2}\), \(\bar{\beta}_{i}=\beta+q_{i}^{+}\).
Lemma 1.3
(Proposition 3.13, [23])
- (1)
For any \((x_{i0},y_{i0}(\cdot),v_{0})\in\Gamma\), there exists a function \(\theta:\mathbb {R}_{+}\times\mathbb {R}_{+}\rightarrow\mathbb {R}_{+}\) such that for any \(r>0\) \(\lim_{t\rightarrow+\infty}\theta(t,r)=0\) with \(\|(x_{i0},y_{i0}(\cdot),v_{0})\|_{\Gamma}\le r\), then \(\|\Phi (t,(x_{i0},y_{i0}(\cdot),v_{0}))\|_{\Gamma}\le\theta(t,r)\).
- (2)
For \(t\ge0\) and \((x_{i0},y_{i0}(\cdot),v_{0})\in\Gamma\), \(\hat{\Phi}(t,(x_{i0},y_{i0}(\cdot),v_{0}))\) maps any bounded set of Γ into compact closure in Γ.
From (2) and (3) of Assumption 1.1 and Lemma 1.2, it follows that \(x_{i}(t)\) and \(v(t)\) have Lipschitz features with Lipschitz coefficients denoted by \(L_{x_{i}}\) and \(L_{v}\). With the assistance of Proposition 5 in [24] and Proposition 2.3 in [25], \(Q_{i}(t)\) and \(P_{i}(t)\) are Lipschitz continuous with Lipschitz coefficients denoted by \(L_{P_{i}}\) and \(L_{Q_{i}}\). Combining these Lipschitz characters with Lemma 3.2.3 in [26], we can show that the semi-flow Φ is asymptotically smooth.
Proposition 1.4
Let \(\Phi(t,(x_{i0},y_{i0}(\cdot),v_{0}))\) be defined by (1.4). The solution semi-flow Φ of (1.3) in Γ is asymptotically smooth.
Proof
- (i)
The supremum of \(\int_{0}^{\infty}\tilde{y}_{i}(t,a)\,da\) is finite;
- (ii)
\(\lim_{u\rightarrow+\infty}\int_{u}^{\infty}\tilde{y}_{i}(a, t) \, da = 0\);
- (iii)
\(\lim_{h\rightarrow0^{+}}|\tilde{y}_{i}(t,a+h)-\tilde{y}_{i}(t,a)|\,da= 0\);
- (iv)
\(\lim_{h\rightarrow0^{+}}\int_{0}^{h}\tilde{y}_{i}(t,a)\,da= 0\).
The rest of this paper is organized as follows. In Section 2, we establish the existence and local stability of the steady states of (1.3). Results on global dynamics of system (1.3) are presented in Section 3. Section 4 gives the numerical simulations to illustrate the theoretical results. Conclusions and discussions are presented in Section 5.
2 Existence and local stability of steady states
Theorem 2.1
Let \(\mathcal {R}_{0}^{\mathrm{cc}}<1\) hold. If \(\mathcal{R}_{0}< 1\), then the only steady state is the viral-free steady state \(E_{0}\); if \(\mathcal{R}_{0}>1\), then besides the virus-free steady state \(E_{0}\), there exists an endemic steady state \(E^{*}\).
Proof
Obviously, \(g(\bar{v})=0\) has a unique positive solution if and only if \(\mathcal{R}_{0}>1\). Then system (1.3) admits a unique endemic steady state \(E^{*}=(\bar{x}_{i}, \overline{ y_{i}(0)}\pi_{i}(\tau),\bar{v})\). □
In what follows, we study the local stability of the steady states. The steady state is locally (asymptotically) stable if all eigenvalues of the corresponding characteristic equations have negative real parts and it is unstable if at least one eigenvalue has a positive real part (see [18]).
Theorem 2.2
Suppose \(\mathcal {R}_{0}^{\mathrm{cc}}<1\). If \(\mathcal{R}_{0}<1\), then the virus-free steady state \(E_{0}\) is locally asymptotically stable and if \(\mathcal{R}_{0}>1\), the unique endemic steady state \(E^{*}\) is locally asymptotically stable.
Proof
3 Global stability analysis
Theorem 3.1
Suppose \(\mathcal {R}_{0}^{\mathrm{cc}}<1\). If \(\mathcal {R}_{0}<1\), then the virus-free steady state \(E_{0}=(x_{i}^{0},0,0)\) is globally asymptotically stable.
Proof
Definition 3.1
([27])
For \((x_{i0},y_{i0},v_{0})\in\Gamma_{0}\), if there exists an \(\varepsilon >0\), independent of the initial conditions, such that \(\limsup_{t\to\infty} \rho(\Phi(t,(x_{i0},y_{i0}(\cdot),v_{0})))>\varepsilon \), then (1.3) is said to be uniformly weakly ρ-persistent; while if there exists a positive ε such that \(\liminf_{t\to\infty} \rho(\Phi(t,(x_{i0},y_{i0}(\cdot),v_{0})))>\varepsilon \), then (1.3) is said to be uniformly strongly ρ-persistent.
In order to prove the persistence of system (1.3) (see [27, Theorem 4.2]), we divided the proof process into two steps: Step 1, we show that system (1.3) is uniformly weakly ρ-persistent; Step 2, by the relative compactness of the orbit \(\Phi (t,(x_{i0},y_{i0}(\cdot),v_{0}))\), system (1.3) is uniformly strongly ρ-persistent.
Proposition 3.2
Suppose \(\mathcal {R}_{0}^{\mathrm{cc}}<1\). If \(\mathcal{R}_{0}>1\), then system (1.3) is uniformly weakly ρ-persistent.
Proof
Combining Propositions 1.4 and Proposition 3.2 with Theorems 4.2 in [27] and Theorem 3.2 in [28], we immediately have the following theorem.
Theorem 3.3
System (1.3) is uniformly strongly ρ-persistent if \(\mathcal{R}_{0}>1\) and \(\mathcal {R}_{0}^{\mathrm{cc}}<1\).
Proposition 3.4
For a total trajectory \(X(\cdot)\) in Γ, \(x_{i}(t)\) is strictly positive and either \(Q_{i}(t)\) and \(v(t)\) are identically zero or \(Q_{i}(t)\) and \(v(t)\) are strictly positive.
Proof
By the definition of the total trajectory, for any \(s\in \mathbb {R}\) the function \(X_{s}(t)=X(s+t)\) is a semi-trajectory of system (1.3) with initial condition \(X_{s}(0)=X(s)\in\Gamma\).
If \(x_{i}(s)=0\) for some s, then the first equation of system (3.12) implies that \(\frac{d x_{i}(s)}{ds}>0\). From the continuity of the solution, for sufficiently small \(\varepsilon >0\), we have \(x_{i}(s-\varepsilon )<0\), which is a contradiction with \(X(\cdot)\in\Gamma\).
If \(y_{i}(s,\cdot)\) and \(v(s)\) are both equal to zero for some \(s\in \mathbb {R}\) and any \(t< s\), we have \(0= y_{i}(s,s-t)=y_{i}(t,0)\pi _{i}(s-t)=x_{i}(t)(\beta_{i}v(t)+Q_{i}(t))\pi_{i}(s-t)\). By the Assumption 1.1, \(\pi_{i}(s-t)\) remains positive for \(t\in\mathbb {R}\), \(i\in \mathbb {N}_{n}\). Thus, \(v(t)\) and \(Q_{i}(t)\) are identically zero for all \(t< s\). For \(t>s\), it follows from (3.1) and Gronwall inequality that \(v(t)\) and \(Q_{i}(t)\) are both equal to zero.
Now we assume that \(y_{i}(s,\cdot)\) is non-zero for each \(s\in\mathbb {R}\). Since \(X(\cdot)\in\Gamma\), there exists a sequence \(\{s_{n}\}\) such that \(y_{i}(s_{n},\cdot)\) is non-zero for each n. For each n, there exists a sequence \(\{a_{n}\}\) such that \(0\neq y_{i}(s_{n},a_{n})=y_{i}(s_{n}-a_{n},0)\pi _{i}(a)\). This implies that \(y_{i}(s_{n}-a_{n},0)\neq0\) as \(s_{n}\) goes to −∞. Solving the last equation of system (3.12), we obtain \(v(s)\neq0\) for all \(s\in\mathbb {R}\). □
Corollary 3.5
Suppose \(\mathcal{R}_{0}>1\) and \(\mathcal {R}_{0}^{\mathrm{cc}}<1\). Let \((x_{i}(s),y_{i}(s,\cdot),v(s))\) be a total trajectory in \(\mathcal{A}_{1}\). Then there exists an \(\varepsilon_{0}>0\) such that \(x_{i}(s), v(s)>\varepsilon_{0}\) and \(y_{i}(s,\tau)>\varepsilon_{0}\pi_{i}(\tau )\), for all \(s\in\mathbb{R}\).
Proof
The following lemma is used to cancel some terms in the proof of Theorem 3.7.
Lemma 3.6
Proof
Employing Lemma 3.6 and Lyapunov functional methods, we obtain the global stability of the endemic steady state \(E^{*}\).
Theorem 3.7
Suppose \(\mathcal {R}_{0}^{\mathrm{cc}}<1\). If \(\mathcal{R}_{0}>1\), then the endemic steady state \(E^{*}\) is globally asymptotically stable in \(\Gamma_{0}\).
Proof
By Theorem 2.2 and Proposition 1.4, it suffices to show \(\mathcal{A}_{1}=\{E^{*}\}\). Let \(X(t)=(x_{i}(t),y_{i}(t,\cdot),v(t))\) be a total trajectory in \(\mathcal {A}\). By Corollary 3.5, there exists \(\varepsilon _{0}>0\) for any \(t\in\mathbb{R}\) and \(\tau\in\mathbb {R}_{+}\) such that \(0\le\varphi (m)<\varepsilon _{0}\) for \(m=\frac{x_{i}(t)}{x_{i}^{*}}\), \(\frac{y_{i}(t,\tau )}{y_{i}^{*}(\tau)}\), and \(\frac{v(t)}{v^{*}}\).
The above analysis indicates that the ω-limit set of \(X(\cdot)\) consists of just the endemic steady state \(E^{*}\) and hence \(V(X(t))\ge V(E^{*})\) for all \(t\in\mathbb{R}\). Thus \(\mathcal {A}_{1}=\{E^{*}\}\). □
4 Numerical simulations
List of parameters
Parameters | Biological meaning | Values | Source |
---|---|---|---|
\(\beta_{1}\) | Transmission rate from virus-to-cell | 4.6 × 10^{−6} | [5] |
\(q_{1}\) | Transmission rate from cell-to-cell | 8 × 10^{−5} | [30] |
\(\delta_{0}\) | Background death rate | 0.05 | [12] |
\(\delta_{m}\) | Extra death rate | 0.35 | [5] |
c | Clearance rate of virus | 23 | [5] |
\(\mu_{1}\) | Death rate for lymphoblasts | 0.01 | [31] |
\(\mu_{2}\) | Death rate for macrophages | 0.024 | [31] |
\(a_{1}\) | Age at which reverse transcription is completed | 0.25 | [12] |
\(a_{2}\) | Window in viral reproduction | 0.5 | [12] |
\(p_{1\mathrm{max}}\) | Maximum production rate for lymphoblasts | 850 | [12] |
4.1 Effects of drug inhibitors
4.2 Decay dynamics of the system
From initial HAART treatment, viremia decays and becomes undetectable (<50 HIV-1 RNA copies/ml) in adherent patients. During the first antiretroviral therapy period, the evolution of viremia strongly depends on the distributions of the infected target cells.
Second, if protease inhibitors completely block the production of infected virion, i.e. \(\varepsilon _{i}^{\mathrm{PI}p}=100\%\), we then have \(v(t)=v_{0}e^{-ct}\). The decay of viremia is at the clearance rate.
5 Conclusion and discussion
As popular antireviral therapies, reverse-transcriptase (RT) inhibitor and protease inhibitor (PI) suppress the reproduction of virion particles. Recent investigations showed that the persistence of latent viral reservoirs is responsible for viral rebound. Such a reservoir is insensitive to HAART and able to self-re-establish. It is necessary to evaluate the effects that the cell-to-cell infection has on viral dynamics. HIV viruses weaken and damage human immune systems, and invade many target cells. In this article, we propose a mathematical model with infection age to investigate the viral dynamics of HIV with different therapies. Cell-to-cell and multi-target-cell infections are both integrated into the model to consider their effects on the evolution of the virus under treatments. We obtain the basic reproduction number of the model, which determines the persistence of the disease. We show that when the basic reproduction number is less than one, the virus-free equilibrium is globally stable. On the other hand, if it is greater than one, then the endemic equilibrium is globally asymptotically stable. It follows from the expression of the basic reproduction number that multi-target-cell and cell-to-cell infections contribute positively to the value of the basic reproduction number. Such type of infections was underestimated. Revealing the consequences of multi-target-cell and cell-to-cell infections provides insights into the development of optimal therapy to control the disease.
We consider the decay dynamics of our model and analyze the effects of the reverse-transcriptase inhibitors and protease inhibitors. If the protease inhibitor is effective enough [4], the decay rate of the virus only depends on the clearance rate of the virion particles. On the other hand, if the reverse-transcriptase inhibitors are effective enough, the decay rate of the virus depends on the distribution of initially infected cells. Through the comparison with the two kinds of inhibitors, it is easy to show that protease inhibitors play a more effective role in controlling cell-to-cell transmission than other therapies.
In order to investigate global behaviors of our model with multi-target-cell and cell-to-cell transmissions, we simplified the input rate of uninfected target cells as a constant. In the literature, logistic growth [34] and mixed growth forms [35] have been used in modeling such input rate. We thus change the input mechanism in our model and the model may display complex dynamical behaviors. The drift phenomenon for free virus often happens in the virion disperse process. Fickian diffusion term models biologically meaningful scenario in virus dynamics and as such incorporating the diffusion into HIV model is necessary [36]. Therefore, we will incorporate such growth rates and diffusions into HIV disease model to investigate viral dynamics under various antiretroviral treatments.
Declarations
Acknowledgements
XW is partially supported by the National Natural Science Foundation of China (No. 61203228) and Startup foundation for Personal of Shanxi University of Finance and Economics. JY is partially supported by the National Natural Science Foundation of China (No. 61203228, No. 61573016), Shanxi Scholarship Council of China (2015-094), Shanxi Scientific Data Sharing Platform for Animal Diseases, and Startup foundation for High-level Personal of Shanxi University.
Authors’ contributions
XW designed the modeling process and analyzed theoretical results. JY carried out numerical algorithms and simulation parts. FX conceived of the study and helped to draft the manuscript. All the authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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