Analysis and Simulation of a Fractional Order Optimal Control Model for HBV

Yang, Liangli; Su, Yongmei; Li, Xiaoke; Ye, Yongan

doi:https://doi.org/10.1155/2020/8175217

Journal of Function Spaces

On this page

Abstract Introduction Conclusion and Discussion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Special Issue

Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2020

View this Special Issue

Research Article | Open Access

Volume 2020 | Article ID 8175217 | https://doi.org/10.1155/2020/8175217

Analysis and Simulation of a Fractional Order Optimal Control Model for HBV

Liangli Yang,¹Yongmei Su,¹Xiaoke Li,²and Yongan Ye²

Guest Editor: Lishan Liu

Received27 Mar 2020

Revised13 May 2020

Accepted25 May 2020

Published23 Jun 2020

Abstract

In this paper, we propose a fractional optimal control model of anti-HBV infection based on saturation incidence and logistic proliferation of uninfected cells for the first time. We derive the basic reproduction number and the cytotoxic T lymphocyte immune response reproductive number and give the stable analysis based on and . We analyse the optimal control condition and give two optimal control strategies about entecavir monotherapy and combination therapy of traditional Chinese medicine and entecavir with different fractional orders by simulation. The simulation shows that combination therapy may be a good choice in anti-HBV infection therapy. We also compare the objective function values of the optimal control strategies with other constant control strategies; the comparison shows that the optimal therapy can get similar or better treatment effect with less drug dose and side effects.

1. Introduction

Hepatitis B virus infection is a major global public health problem. Based on the information published by the World Health Organization on World Hepatitis Day 2019, hepatitis B is the second deadly epidemic, and the number of people infected with the hepatitis virus is 9 times that of HIV [1]. The prevalence of chronic hepatitis B has brought a huge medical burden on society. At present, the most widely used drugs for chronic hepatitis B are nucleot(s)ide analogues (NUCs), such as tenofovir and entecavir, which are used to inhibit viral DNA polymerase and reverse transcriptase activity [2]. The combined therapy of NUCs plus Chinese herbal medicine (CHM) is widely accepted in China, which has been recognized as a prospective alternative approach [3, 4].

In order to better understand the transmission mechanism of various infectious diseases, many mathematical models were set up to enhance our understanding of the dynamics of infectious diseases and chronic viral infections [5–8]. Mathematic models are also used in the study of anti-HBV infection treatment. The initial model of HBV infection was proposed in 1996 by Zeuzem et al. [9]: where represent the concentration of uninfected cells, infected cells, and viruses at time , respectively.Uninfected cells are assumed to be produced at the constant rate , die at the rate of , and become infected at the rate of . Infected cells are thus produced at the rate of and die at the rate . Free virions are generated from infected cells at the rate of and decay at the rate of .

The infection rate in model (1) is assumed to be linear with respect to the virus. However, the basic reproduction number of model (1) is given by , where represents the number of total liver cells, which implies that an individual with a larger liver will be more difficult to be cured than a person with a smaller one. Paper [10] changed in (1) to and gave the following model:

The reproductive number of (2) is which seems more reasonable because it is no more dependent on the total number of liver cells. On the other hand, when modelling virus infection, the host’s immune response should not be ignored; Nowak and Bangham [11] proposed the following immune model: where represents the number of CTL cells, means the activated rate of the immune cells, and indicates the rate at which infected liver cells are eliminated by CTL immune cells. Then, Su et al. changed into and gave the model as follows [12]:

Perelson and Nelson [13] added logistic function in his HIV model, that is . Based on paper [13], Song and Neumann proposed a HIV viral infection model with logistic function and saturated mass action incidence [14]. Then, Yu et al. gave a HBV model with logistic function and standard mass action incidence [15]:

In recent years, more and more fractional order models were used in the biological immune system, because the fractional order model has the memory, while the characteristic of the immune response contains the memory [16, 17]. So, when we set the virus immune model, fractional mathematical models have become an important choice. Many fractional order HIV infection models were set up [18, 19].

So far, some fractional order models about HBV have been set up [20–25]. Papers [20–22] considered the epidemic transmission SEIR model of HBV. In paper [23], a within-host fractional order HBV model was proposed:

In this model, , and have the same meaning of (1); infected hepatocytes are cured by noncytolytic processes at a constant rate per cell. But the model still used the bilinear mass action incidence . Paper [24] also considered the HBV models based on the bilinear mass action incidence . Paper [25] added to the first equation in (4) and changed it to fractional order. The model is given as follows:

On the other hand, in the process of anti-HBV treatment, we hope not only to lower the levels of HBV and the infected hepatocytes during and at the end of therapy but also to minimize the therapeutic side effects and the cost of drugs, so it is very important to use optimal control theory to study the anti-HBV treatment model.

Paper [26] provided a control model, which uses the linear incidence. Here, means the control and is the efficacy of the antiviral drug:

For the fractional order model, paper [27] gave the analysis of the control model. Here, , , and have the same meaning of (1) and represent the drug effect on HBV by interferon (IFN) and lamivudine (LAM):

Based on the above discussion, according to the clinical anti-HBV combination therapy of traditional Chinese and western medicine [4], we consider a HBV fractional order model as follows:

Here, have the same meaning as those in model (3). represents the treatment effect of entecavir (ETV), which can block the replication of virus. represents the treatment effect of Tiaogan Jianpi Jiedu Granules (TGJPJD), which can accelerate the death of virus. represents the treatment effect of Tiaogan-Yipi Granule (TGYP) which can enhance immunity.

The structure of this paper is given as follows. In Section 2, we show some definitions of fractional order and related lemmas. In Section 3, the existence and uniqueness of the positive solution are discussed. In Section 4, we give the stable analysis of our system. The necessary conditions for an optimal control are derived in Section 5. The numerical simulation and the conclusion are given, respectively, in Sections 6 and 7.

2. Basic Concepts and Lemmas

First of all, we give some basic definitions and inferences about fractional order. There are several definitions of fractional derivatives, but we only consider the Caputo derivation used in this paper.

Definition 1. The Caputo fractional derivative of order of a function is given: Here, , and is the Euler gamma function. In this paper, we only discuss the situation that . Function (11) will become

Lemma 2 (see [28] (Laplace transform). The Laplace transform of formula (12) is

Lemma 3 (see [29] (Generalized mean value theorem). Suppose that and , for , we have Notice. Suppose that and , for , we have the following: (1)If , then neighbourhood satisfies (2)If , then neighbourhood satisfies

Lemma 4 (see [30]). For a fractional order system (15) of which and , Assume satisfies the following conditions: (1) is Lebesgue measurable with respect to on (2) and are continuous for all (3); here, are two positive constantsThen, the initial problem (15) has unique and positive solution on .

Lemma 5 (see [31]). For the system (15), the equilibrium point is locally asymptotically stable, if all eigenvalues of Jacobian matrix evaluated at the equilibrium point satisfies

Lemma 6. For the discriminant of polynomial equation is defined by . Here, represents the determinant for the corresponding Sylvester matrix.

Lemma 7. For the polynomial equation the conditions which make all the roots of (18) satisfy (16) are displayed as follows: (1)For , the condition is (2)For , the conditions are either Routh-Hurwitz conditions or For (a)If the discriminant of (18) is positive, then Routh-Hurwitz conditions are the necessary and sufficient conditions, that is (b)If , and , then (16) holds when (c)If , and , then (16) holds when (d)If , and , then (16) holds for all (4)When , suppose is the Routh-Hurwitz discriminant, that is, For , (16) is sufficient but not necessary.

3. The Existence and Uniqueness of Positive Solutions

We first analyse the system (10) without control, that is, .

Theorem 8. The solution of system (10) is always nonnegative.

Proof. For system (10), we can get that Based on Lemma 2, we can prove that for . For , assume that there exists satisfying that We can find that and , which implies . This result is contradicted with the assumption, so for any .

Theorem 9. There exists an , such that , for any .

Proof. Let Calculate the -order derivatives on both sides, respectively, Here, and . So, we can get is completely monotonic for [32], and . Here, we assume So, , . Similarly, we can have the following inequation: Assume that , .
Let so .

Theorem 10. System (10) has a unique positive solution on .

Proof. Let . Suppose , system (10) can be transformed into the following form: Obviously, (27) satisfies conditions (1) and (2) of Lemma 4. We only prove that system (33) satisfies condition (3) of Lemma 4. Let So, we can get Here, , and is a bound value mentioned in Theorem 9. By Lemma 4, system (10) has a unique solution on .

4. Stable Analysis

In this section, we will discuss the stability of the system (10). The system always has an infection-free equilibrium , where

Here, we have the basic reproduction number as

When , the system (10) will have an immune-absence equilibrium ; here, and which means that the infected cells and virus coexist but the immune response is not activated yet, that is, . Further, we will have the cytotoxic T lymphocyte immune response reproductive number

When , it means that immune response is activated; the system (10) will have an immune-response equilibrium , where

Here are given as follows:

Theorem 11. For the model of (10): when , the infection-free equilibrium point is locally asymptotically stable; when , is unstable.

Proof. The Jacobi matrix for equilibrium is shown as So, the characteristic equation for is Suppose that , we can have the four characteristic roots: , , . When , , from Lemma 5, is locally asymptotically stable. When , , from Lemma 5, is unstable.

Theorem 12. When , if , is locally asymptotically stable for If , then is locally asymptotically stable for . If , is unstable.

Proof. The characteristic equation for is given as follows: Let When , the characteristic root .
Let , we have From Lemma 6, the discriminant of the polynomial equation is as follows:

According to Lemma 7, when , if is locally asymptotically stable for If , then is locally asymptotically stable for . If , the equilibrium is unstable.

Theorem 13. When , is locally asymptotically stable for

Proof. The characteristic equation for is given as follows: Here, let , When we have , which ensures . So,

According to Lemma 7, Theorem 13 can be proofed.

5. Optimal Control Problem for Combination of Traditional Chinese and Western Medicines

In this section, we analyse the optimal control problem for system (10). Considering the high cost and side effects of long-term use of the highest dose drugs and in order to achieve the treatment effect, we need to analyse the optimal control model to find optimal treatment strategy. Here, represent the effects of the drugs ETV, TGJPJD, and TGYP, respectively. be the endpoint of treatment, we choose the objective function of the optimal control as follows [33]:

Here, represent the corresponding weight of each variable. Let, then (9) can be rewritten as follows:

Then, its Hamilton function of can be expressed as

Using Pontryagin’s minimum principle, the necessary conditions for the existence of the optimal solution of system (10) are shown as follows:

So, we get the optimal control as

6. Simulation

In this section, we use the numerical method provided in [34] to simulate our system (10). The parameters are given in Table 1.

According to the clinical trials [4], we choose 108 weeks as a treatment period, that is, the initial time and the end time (days). , by which 0 means no medicine treatment and 1 means the medicine is fully functional. Due to the fact that the human body cannot fully absorb all of the medicine, we assume a maximum drug effect of 0.98. In this simulation, we assume that there is a positive correlation between the efficacy and the dose in a certain range. As the influence of memory decreases [37]. Here, we choose to see the difference between different fractional orders with the lower memory effect.

In the following part, we will give optimal control strategies according to different treatment protocols.

Strategy 1. ETV monotherapy .
In this case, the objective function (45) is transformed to

The initial condition is . We give the simulation with different orders ; the corresponding dynamic route simulation is shown in Figure 1, and the optimal control strategy is shown in Figure 2.

(a)

(b)

(c)

(d)

From the simulation in Figure 1, we can see that the change rate and the terminal value of infected cells, virus, and CTL are obviously different with different order even with the same initial condition and the same change trend, which is consistent with the clinical fact that there do exist individual difference, so individual difference may be reflected by different , and the fractional order model should be a good tool to describe HBV infection.

From the simulation in Figure 2, we find that the maximum dose should only be used at the earlier stage of treatment. And the drug dose can be lowered after a period of high-dose treatment. But the time point of lowering the drug dose is different with different order . If we use constant control (maximum effect) and (minimum effect), we give the objective function value of the optimal control and constant control with different orders in Table 2. We can see that the objective function values are lower than those of the maximum and minimum constant control, which shows that the optimal therapy can get similar or better treatment effect with less drug dose and side effects.

Strategy 2. Combination of ETV and CHM

In this section, we will give the optimal control strategies of combination therapy of ETV and Chinese medicine with different orders as . We choose the same initial condition and the same parameters as those of Strategy 1 and the objective function (45). As mentioned before, ETV can block the replication of virus; we use to represent ETV treatment effect, and represents the treatment effect of TGJPJD, which can accelerate the death of virus. represents the treatment effect of TGYP which can enhance immunity. The simulations are shown in Figures 3 and 4.

(a)

(b)

(c)

(d)

From the simulation in Figure 3, we can see that dynamic routes are similar as that in Figure 1. But the terminal values of infected cells and virus are obviously lower than those in Figure 1, which show that the Chinese medicine can enhance the death of infected cells and virus. The simulation can also show that the individual difference may be reflected by different .

From the simulation in Figure 4, we also find that the maximum dose should only be used at the earlier stage of treatment. The drug dose can be lowered after a period of high-dose treatment with different order . And the time point of lowering the drug dose is also different with different order α. We also find that the time point of lowering the drug dose is different with different medicine; that is, ETV can be lowered at the earliest time, and then, TGJPJD and TGYP need to be taken at the maximum dose for the longest time which shows the importance of enhancing the immune in anti-HBV therapy. Moreover, for Strategy 1, the time point of lowering the ETV drug dose is about 100 days, but for strategy 2, the time point is only about 60 days.

We also compared the objective function values of constant control (maximum effect) and (minimum effect) with the optimal control. The results are shown in Table 3. We can see that the optimal objective function values are always the lowest for different orders. By comparing the optimal objective function values in Table 2 with those in Table 3, we find that combination of ETV and CHM can greatly reduce objective function values, which show that combination of ETV and CHM may be a good choice in anti-HBV infection therapy.

7. Conclusion and Discussion

In this paper, based on the combination therapy of traditional Chinese medicine and Western medicine, we proposed a fractional optimal control model of anti-HBV infection based on saturation incidence and logistic proliferation of uninfected cells for the first time. After giving the stable analysis of our system, we discussed the necessary conditions of the optimal control problem. Two optimal control strategies about ETV monotherapy and combination therapy of ETV and CHM with different fractional orders were given by simulation.

In the simulation, we suppose that there is a positive correlation between drug use and effectiveness. From the simulations, we know that the dynamic routes were different with different orders even with the same parameters, which may show that the individual difference could be reflected by different .

By comparing the dynamic routes between ETV monotherapy and combination therapy of ETV and CHM, we found that combination therapy can not only obtain better treatment effect but also reduce the taking time and dose of ETV. The simulation shows that combination therapy may be a good choice in anti-HBV infection therapy. We also compared the objective function values of the optimal control strategies with other constant control strategies; the comparison showed the optimal therapy can get similar or better treatment effect with less drug dose and side effects.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is jointly supported by the 2015 National Traditional Medicine Clinical Research Base Business Construction Special Topics (JDZX2015299) and the Fundamental Research Funds for the Central Universities FRF-BR-16-019A.

References

World Health Organization, 2019 hepatitis B, World Health Organization Fact Sheet N204, 2019, http://www.who.int/mediacentre/factsheets/fs204/en/index.html.
M.-F. Yuen, W.-K. Seto, J. Fung, D. K.-H. Wong, J. C.-H. Yuen, and C.-L. Lai, “Three years of continuous entecavir therapy in Treatment-Naïve chronic hepatitis B patients: viral suppression, viral resistance, and clinical safety,” American Journal of Gastroenterology, vol. 106, no. 7, pp. 1264–1271, 2011.
View at: Publisher Site | Google Scholar
M. Z. Song, “Effective observation of 56 cases with hepatitis B by combined medicine,” Journal of Henan University of science & Technology, vol. 25, no. 2, pp. 117-118, 2007.
View at: Google Scholar
Y. A. Ye, X. K. Li, D. Q. Zhou et al., “Chinese herbal medicine combined with entecavir for HBeAg positive chronic hepatitis B: study protocol for a multi-center, double-blind randomized-controlled trial,” Chinese Journal of Integrative Medicine, vol. 24, no. 9, pp. 653–660, 2018.
View at: Publisher Site | Google Scholar
T. Zhang, J. Wang, Y. Li, Z. Jiang, and X. Han, “Dynamics analysis of a delayed virus model with two different transmission methods and treatments,” Advances in Difference Equations, vol. 2020, no. 1, 2020.
View at: Publisher Site | Google Scholar
F. Li, S. Zhang, and X. Meng, “Dynamics analysis and numerical simulations of a delayed stochastic epidemic model subject to a general response function,” Computational and Applied Mathematics, vol. 38, no. 2, p. 95, 2019.
View at: Publisher Site | Google Scholar
T. Feng, X. Meng, T. Zhang, and Z. Qiu, “Analysis of the predator–prey interactions: a stochastic model incorporating disease invasion,” Qualitative Theory of Dynamical Systems, vol. 19, no. 2, p. 55, 2020.
View at: Publisher Site | Google Scholar
H. Qi, X. Meng, and Z. Chang, “Markov semigroup approach to the analysis of a nonlinear stochastic plant disease model,” Electronic Journal of Differential Equations, vol. 2019, no. 116, pp. 1–19, 2019.
View at: Google Scholar
S. Zeuzem, R. A. de Man, P. Honkoop, W. K. Roth, S. W. Schalm, and J. M. Schmidt, “Dynamics of hepatitis B virus infection _in vivo_,” Journal of Hepatology, vol. 27, no. 3, pp. 431–436, 1997.
View at: Publisher Site | Google Scholar
Y. Ji, L. Q. Min, Y. M. Su, and Y. Zheng, “Global stability of a viral infection model with saturation incidence,” Journal of Biomathematics, vol. 25, no. 2, pp. 267–272, 2010.
View at: Google Scholar
M. A. Nowak and C. R. M. Bangham, “Population dynamics of immune Responses to persistent viruses,” Science, vol. 272, no. 5258, pp. 74–79, 1996.
View at: Publisher Site | Google Scholar
Y. Su, L. Zhao, and L. Min, “Analysis and simulation of an Adefovir anti-hepatitis B virus infection therapy immune model with alanine aminotransferase,” IET Systems Biology, vol. 7, no. 5, pp. 205–213, 2013.
View at: Publisher Site | Google Scholar
A. S. Perelson and P. W. Nelson, “Mathematical analysis of HIV-1 dynamics in vivo,” SIAM Review, vol. 41, no. 1, pp. 3–44, 1999.
View at: Publisher Site | Google Scholar
X. Song and A. U. Neumann, “Global stability and periodic solution of the viral dynamics,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 281–297, 2007.
View at: Publisher Site | Google Scholar
Y. Ji, L. Min, and Y. Ye, “Global analysis of a viral infection model with application to HBV infection,” Journal of Biological Systems, vol. 18, no. 2, pp. 325–337, 2011.
View at: Publisher Site | Google Scholar
A. A. M. Arafa, S. Z. Rida, and M. Khalil, “A fractional-order model of HIV infection with drug therapy effect,” Journal of the Egyptian Mathematical Society, vol. 22, no. 3, pp. 538–543, 2014.
View at: Publisher Site | Google Scholar
H. Ertik, A. E. Calik, and H. Sirin, “Investigation of electrical RC circuit within the framework of fractional calculus,” Revista Mexicana de Físisca, vol. 61, no. 1, pp. 58–63, 2015.
View at: Google Scholar
A. Gökdoǧan, A. Yildirim, and M. Merdan, “Solving a fractional order model of HIV infection of CD4⁺ T cells,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2132–2138, 2011.
View at: Publisher Site | Google Scholar
Y. Wang, L. Liu, X. Zhang, and Y. Wu, “Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection,” Applied Mathematics and Computation, vol. 258, pp. 312–324, 2015.
View at: Publisher Site | Google Scholar
M. Farman, A. Ahmad, M. U. Saleem, and A. Hafeez, “A mathematical analysis and modelling of hepatitis B model with non-integer time fractional derivative,” Communications in Mathematics and Applications, vol. 10, no. 3, pp. 571–584, 2019.
View at: Publisher Site | Google Scholar
S. Ullah, M. A. Khan, M. Farooq, T. Gul, and F. Hussain, “A fractional order HBV model with hospitalization,” Discrete and Continuous Dynamical Systems-series S, vol. 13, no. 3, pp. 957–974, 2020.
View at: Publisher Site | Google Scholar
T. Khan, A. R. Seadawy, G. Zaman, and A. Abdullah, “Optimal control of the mathematical viral dynamic model of different hepatitis B infected individuals with numerical simulation,” International Journal of Modern Physics B, vol. 33, no. 26, article 1950310, 2019.
View at: Publisher Site | Google Scholar
S. M. Salman and A. M. Yousef, “On a fractional-order model for HBV infection with cure of infected cells,” Journal of the Egyptian Mathematical Society, vol. 25, no. 4, pp. 445–451, 2017.
View at: Publisher Site | Google Scholar
L. C. Cardoso, F. L. P. Dos Santos, and R. F. Camargo, “Analysis of fractional-order models for hepatitis B,” Computational and Applied Mathematics, vol. 37, no. 4, pp. 4570–4586, 2018.
View at: Publisher Site | Google Scholar
S. Yong-mei, L. Liu, Y. Yong-an, and L. Xiao-ke, “Stability analysis and simulation of a fractional-order HBV infection model based on saturation incidence,” DEStech Transactions on Computer Science and Engineering, 2018.
View at: Publisher Site | Google Scholar
M. Sheikhan and S. A. Ghoreishi, “Application of covariance matrix adaptation–evolution strategy to optimal control of hepatitis B infection,” Neural Computing and Applications, vol. 23, no. 3-4, pp. 881–894, 2013.
View at: Publisher Site | Google Scholar
R. Shi, T. Lu, and C. Wang, “Dynamic analysis of a fractional-order model for hepatitis B virus with Holling II functional response,” Complexity, vol. 2019, Article ID 1097201, 13 pages, 2019.
View at: Publisher Site | Google Scholar
A. K. Shukla and J. C. Prajapati, “On a generalization of Mittag-Leffler function and its properties,” Journal of Mathematical Analysis and Applications, vol. 336, no. 2, pp. 797–811, 2007.
View at: Publisher Site | Google Scholar
Z. M. Odibat and N. T. Shawagfeh, “Generalized Taylor's formula,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 286–293, 2007.
View at: Publisher Site | Google Scholar
A. Boukhouima, K. Hattaf, and N. Yousfi, “Dynamics of a fractional order HIV infection model with specific functional response and cure rate,” International Journal of Differential Equations, vol. 2017, Article ID 8372140, 8 pages, 2017.
View at: Publisher Site | Google Scholar
E. Ahmed, A. M. A. El-Sayed, and H. A. A. El-Saka, “On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rossler, Chua and Chen systems,” Physics Letters A, vol. 358, no. 1, pp. 1–4, 2006.
View at: Publisher Site | Google Scholar
H. Pollard, “The completely monotonic character of the Mittag-Leffler function $E_a,” Bulletin of the American Mathematical Society, vol. 54, no. 12, pp. 1115–1117, 1948.
View at: Publisher Site | Google Scholar
H. M. Ali, F. L. Pereira, and S. M. A. Gama, “A new approach to the Pontryagin maximum principle for nonlinear fractional optimal control problems,” Mathematical Methods in the Applied Sciences, vol. 39, no. 13, pp. 3640–3649, 2016.
View at: Publisher Site | Google Scholar
L. Zhang, G. Huang, A. Liu, and R. Fan, “Stability analysis for a fractional HIV infection model with nonlinear incidence,” Discrete Dynamics in Nature and Society, vol. 2015, no. 3, Article ID 563127, 11 pages, 2015.
View at: Publisher Site | Google Scholar
M. A. Nowak and R. M. May, Viral Dynamics, Oxford University Press, Oxford, 2000.
A. S. Perelson, “Modelling viral and immune system dynamics,” Nature Reviews Immunology, vol. 2, no. 1, pp. 28–36, 2002.
View at: Publisher Site | Google Scholar
F. A. Rihan and G. Velmurugan, “Modelling viral and immune system dynamics,” Chaos, Solitons and Fractals, vol. 132, article 109592, 2020.
View at: Google Scholar

Copyright

Copyright © 2020 Liangli Yang et al. 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.

PDF Download Citation

Download other formats

Order printed copies

Views

309

Downloads

666

Citations