Effects of HIV infection on CD4^{+} T-cell population based on a fractional-order model
- Sadia Arshad^{1, 6}Email authorView ORCID ID profile,
- Dumitru Baleanu^{2, 4},
- Weiping Bu^{3} and
- Yifa Tang^{1, 5}
https://doi.org/10.1186/s13662-017-1143-0
© The Author(s) 2017
Received: 30 December 2016
Accepted: 17 March 2017
Published: 29 March 2017
Abstract
In this paper, we study the HIV infection model based on fractional derivative with particular focus on the degree of T-cell depletion that can be caused by viral cytopathicity. The arbitrary order of the fractional derivatives gives an additional degree of freedom to fit more realistic levels of CD4^{+} cell depletion seen in many AIDS patients. We propose an implicit numerical scheme for the fractional-order HIV model using a finite difference approximation of the Caputo derivative. The fractional system has two equilibrium points, namely the uninfected equilibrium point and the infected equilibrium point. We investigate the stability of both equilibrium points. Further we examine the dynamical behavior of the system by finding a bifurcation point based on the viral death rate and the number of new virions produced by infected CD4^{+} T-cells to investigate the influence of the fractional derivative on the HIV dynamics. Finally numerical simulations are carried out to illustrate the analytical results.
Keywords
1 Introduction
According to WHO there were approximately 35 million people at the end of 2013 living with HIV with 2.1 million people becoming newly infected in 2013 globally with HIV. HIV belongs to the family of lentiviruses, which means being acting slowly. Lentiviruses cause diseases that progress over a long period disturbing the immune system in humans. HIV produces virus particles by converting viral RNA into DNA in the cell and then making many RNA copies. The transformation is completed with an enzyme named reverse transcriptase. The change from RNA to DNA and back to RNA is substantial and makes fighting against HIV challenging. There is a chance of the virus mutating and there being errors each time when it happens. These copies or virus particles destroy the cell after formation and infecting other cells. Although HIV attacks many different cells, it inflicts the most disorder on the CD4^{+} T-cells, the main target of CD4^{+} T-cells is to form the body’s overall immune response to external infections. HIV provides the basis to grow acquired immune deficiency syndrome (AIDS). For a person infected from HIV it can take 10-15 years to develop AIDS. On the medical frontier there have been many advances, but still there is no effective cure or vaccine available for HIV. Since the early 1980s, many mathematical models have been developed to better understand the interaction of HIV and the human immune system for the purpose of testing treatment strategies [1–8]. Silva and Torres [9] proposed a population model for TB-HIV/AIDS coinfection transmission dynamics, they applied optimal control theory to TB-HIV/AIDS model to study optimal strategies for the minimization of the number of individuals with TB and AIDS. Recently Rocha et al. [10] investigated an HIV optimal control problem with delays in both state and control variables, where the objective is to find the optimal treatment strategy that maximizes the number of CD4+ T-cells and CTL immune response cells, keeping the drug strength low. Luo et al. [11] studied bifurcations and stability of an HIV model that incorporates the immune responses. An HIV model including latent infection and antiretroviral therapy is examined by Wang et al. [12]. They obtained the global asymptotic stability of the uninfected equilibrium by constructing a Lyapunov function. We refer the reader to the excellent review paper on mathematical modeling of HIV on different phenomena of [13]. Quantitative analysis of HIV-1 replication in vivo has made significant contributions to understanding of AIDS pathogenesis and antiretroviral treatment ([4, 14]). For a detailed mathematical analysis on such models, we refer to [15] and [5].
Fractional-order dynamics has recently been a focus of interest because of its appearance in physics, biology and engineering applications. There is a rich literature on the theoretical research of the fractional differential equations. The book of Podlubny [16] provides an overview to the basic theory of fractional differential equations. The monograph by Samko, Kilbas, and Marichev [17] contains remarkably comprehensive theory on the topic of fractional differential equations. Recently, much work has done on modeling the HIV infection with fractional-order derivatives [18–23] and [24]. A fractional-order model retains its memory, which is one of the main characteristic of the fractional-order derivative, while the features of immune response include memory. In [22] a fractional-order time-delay model is investigated which include three types of cells, namely healthy CD4^{+} T-cells, infected CD4^{+} T-cells and free HIV virus particles. In [19] the authors introduced fractional orders to the model of HIV-1 whose components are plasma densities of uninfected CD4^{+} T-cells, they use the generalized Euler method (GEM) and homotopy analysis method (HAM) to approximate the solution. Approximate solutions of fractional-order differential system for modeling human T-cell lymphotropic virus I (HTLV-I) infection of CD4^{+} T-cells is investigated in [25] using a multi-step generalized differential transform method. Bernstein operational matrices is applied to fractional order HIV model to approximate the solution [26]. Homotopy decomposition method is given in [27] to solve a system of fractional nonlinear differential equations that arise in the model for HIV infection of CD4^{+} T-cells. Recently, Huo et al. [28] studied a fractional-order HIV model to assess the impact of vaccines in a homosexual community. They have shown that the vaccinated reproduction below unity is not a threshold of HIV eradication when effectiveness and the dosage of the vaccines are low. A new critical threshold is derived in order to eradicate the HIV. Recently Pinto et al. [29] studied a fractional-order model for HIV infection that includes latently infected cells, macrophages and CTLs. In this paper we propose a finite difference implicit scheme to solve fractional-order HIV model containing four types of populations: uninfected CD4^{+} T-cells, latently infected, actively infected CD4^{+} T-cells and HIV virus particles. Further we analyze the dynamics of fractional-order HIV by investigating bifurcation points. The benefit of fractional-order systems is that they permit greater degrees of freedom in the model.
This paper is organized as follows: Section 2 includes generalized HIV model. In Section 3, we introduce implicit scheme for solving the fractional-order HIV-1 infection model. Dynamical behavior of generalized HIV model is investigated in Section 4. In Section 5, we present numerical simulations of the model and discuss the biological significance of the results. A conclusion is given in the last section.
2 Generalized HIV model
Note that the units of the fractional differential equation are different, that is, fractional differential equations are expressed with respect to an intrinsic time variable that depends on α [32, 33] instead of the physical time. Thus a reformed parameter has been presented in model (1) to interpret the fractional order. Notice that when \(\alpha\to1\) the fractional HIV model (1) reduces to the classical HIV model.
The first two equations deal with the effects of HIV. Here \(s^{\alpha }\), corresponds to the s, the source term, from the classical HIV model, \(r^{\alpha}\) corresponding to r represents the rate of growth for the CD4^{+} cell population and \(\mu_{T}^{\alpha}\) is the analogon to the \(\mu_{T}\), the death rate of uninfected CD4^{+} T-cells. \(k_{1}^{\alpha}\) corresponding to \(k_{1}\) represents the rate at which CD4^{+} T-cell become infected by virus modeled by mass-action type of term. The dynamical behavior of actively infected T-cells is given in the third equation with \(k_{2}^{\alpha}\) corresponding to the \(k_{2}\) rate at which latently infected cells convert to being actively infected and \(\mu_{b}^{\alpha}\) corresponding to \(\mu_{b}\) represents the death rate per cell from the classical HIV model. The last equation models the free infectious virus population. Assume that an actively infected CD4^{+} T-cell produces N virus particles. The virus production rate is N times \(\mu_{b}^{\alpha}\). Free virus is expired at a rate \(k_{1}^{\alpha}VT\) by binding to uninfected CD4^{+} T-cells. Viral death from the body is represented by \(-\mu_{v}^{\alpha}V\).
3 Construction of implicit numerical scheme
In this section, we introduce an implicit numerical scheme using a finite difference approximation of the Caputo derivative. Numerous schemes have been developed for the numerical solution of the fractional differential equations [34]. A class of the fractional multi-step method is proposed by Lubich [35] and Galeone and Garrappa [36], the fractional Adams method is proposed by Diethelm et al. [37] and Odibat and Momani [3] and Grünwald-Letnikov approximation based on the Grünwald-Letnikov definition of the fractional derivative is addressed in [38] and discussed the analysis of convergence and stability. Baker [39] introduced some numerical methods for the Volterra integral and integro-differential equations.
In this paper, we implement the L1-scheme to approximate the Caputo fractional derivative, which was independently developed and analyzed in [40] and [41]. The L1-scheme is based on a piecewise linear approximation to the fractional derivative. We are in favor of the L1-scheme, because this scheme is derived easily and the coefficients of this scheme have good properties e.g. the representation of the coefficients is simple. The L1-scheme has been extensively used in practice and currently it is one of the most efficient numerical methods for solving the time fractional differential equations due to its ease of implementation.
A modified Jacobi matrix on Newton iteration method
1: | Given \(N_{\mathrm{iter}}\), ε, let Nd = 1 |
2: | if \(k>N_{\mathrm{iter}}\) and \(\|f(x^{k+1})\|>\|f(x^{k})\|\) |
3: | \(Nd: =\frac{1}{2}Nd\) |
4: | end if |
5: | if \(\operatorname{det}(J(x^{k}))<\varepsilon\), %ε is a sufficiently small positive number |
6: | \(J(x^{k}): =Nd\cdot(J(x^{k})+I)\), %I is an identity matrix |
7: | else |
8: | \(J(x^{k}): =Nd\cdot J(x^{k})\) |
9: | end if |
4 Dynamical behavior of generalized HIV model
Theorem 4.1
Proof
Remark 4.1
Remark 4.2
By the Routh-Hurwitz criteria for the stability of fractional-order systems [42], we have the following result for the stability of infected equilibrium.
Theorem 4.2
5 Numerical simulations
HIV model parameters
\(\boldsymbol{\mu_{T}^{\alpha}}\) | \(\boldsymbol{\mu_{b}^{\alpha}}\) | \(\boldsymbol{\mu_{V}^{\alpha}}\) | \(\boldsymbol{k_{1}^{\alpha}}\) | \(\boldsymbol{k_{2}^{\alpha}}\) | \(\boldsymbol{r^{\alpha}}\) | N | \(\boldsymbol{T_{\mathrm{max}}}\) | \(\boldsymbol{s^{\alpha}}\) |
---|---|---|---|---|---|---|---|---|
\(0.02^{\alpha}\) | \(0.24^{\alpha}\) | \(2.4^{\alpha}\) | \((2.4\times 10^{-5})^{\alpha}\) | \((3\times10^{-3})^{\alpha}\) | 0.03 | Varies | 1,500 | \(10^{\alpha}\) |
6 Conclusion
In this paper, we have investigated a fractional-order HIV model, as a generalization of an integer-order model. The advantage of the generalized model is that the fractional-order system possesses memory, which belongs to the main features of the immune response. An implicit numerical scheme has been proposed for the fractional-order HIV model using a finite difference approximation of the Caputo derivative. We showed that the system possesses two equilibrium points and derived the analytical condition for the stability of uninfected and infected equilibrium points. Further we analyzed the influence of the fractional derivative on the dynamics of system. In many AIDS patients the T-cell level can be depleted \({<}200\mbox{ mm}^{-3}\) level, on the other hand an integer-order model cannot model this fact using the parameter values in Table 1. However, with the additional degree of freedom of the fractional derivative we can obtain the low CD4^{+} cell counts seen during the late stages of the disease; see Figure 5. We established that \(\mu_{V}^{\alpha}\) and N are bifurcation parameters, transcritical bifurcation occurs as \(\mu_{V}^{\alpha}\) and N passes the point \(\mu_{\mathrm{crit}}^{\alpha}\) and \(N_{\mathrm{crit}}^{\alpha}\), respectively. We have shown that the disease can be eradicated by increasing the viral death rate greater than \(\mu_{\mathrm{crit}}^{\alpha}\) as shown in Figure 2. Another parameter that could play a vital role to control the HIV virus is the number of new virions produced by infected CD4^{+} T-cells. We found that the virus can be eliminated if N is less than \(N_{\mathrm{crit}}^{\alpha}\), that is, HIV infection will not be sustained if infected cells die without producing an adequate number of viral progeny as demonstrated in Figure 3. Mathematical models can help physicians to choose an optimal dosage and check the effects of their therapeutic action. HIV progression has many variations from patient to patient, which is difficult to capture by an integer-order derivative. The fractional derivative can be varied to best fit the real data according to the progression of different HIV patients. Thus a more reliable model can be obtained by choosing the relevant fractional index according to real data. Consequently, the clinician can recommend administration of drugs or treatment to each individual patient by using the information from the generalized model with the most relevant fractional index.
Declarations
Acknowledgements
The authors would like to thank the anonymous referees for their valuable suggestions. This research is supported by the State Key Program of National Natural Science Foundation of China (Grant No. 11631013), the National Natural Science Foundation of China (Grant No. 11371357) and by the ITER-China Program (Grant No. 2014GB124005). The third author extend his appreciation to the National Natural Science Foundation of China (No. 11601460) and the Research Foundation of Education Commission of Hunan Province of China (No. 16C1540) for supporting this research work.
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
References
- Nowak, MA, Bangham, CR: Population dynamics of immune responses to persistent viruses. Science 272, 74-79 (1996) View ArticleGoogle Scholar
- Nowak, MA, Bonhoeffer, S, Shaw, GM, May, RM: Anti-viral drug treatment: dynamics of resistance in free virus and infected cell populations. J. Theor. Biol. 184, 203-217 (1997) View ArticleGoogle Scholar
- Odibat, Z, Momani, S: An algorithm for the numerical solution of differential equations of fractional order. J. Appl. Math. Inform. 26, 15-27 (2008) MATHGoogle Scholar
- Perelson, AS: Modelling viral and immune system dynamics. Nat. Rev. Immunol. 2, 28-36 (2002) View ArticleGoogle Scholar
- Perelson, AS, Nelson, PW: Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev. 41, 3-44 (1999) MathSciNetView ArticleMATHGoogle Scholar
- Perelson, AS, Essunger, P, Cao, Y, Vesanen, M, Hurley, A, Saksela, K, Markowitz, M, Ho, DD: Decay characteristics of HIV-1-infected compartments during combination therapy. Nature 387, 188-191 (1997) View ArticleGoogle Scholar
- Perelson, AS, Essunger, P, Ho, DD: Dynamics of HIV-1 and CD4+ lymphocytes in vivo. AIDS 11, S17-S24 (1997) Google Scholar
- Perelson, AS, Neumann, AU, Markowitz, M, Leonard, JM, Ho, DD: HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science 271, 1582-1586 (1996) View ArticleGoogle Scholar
- Silva, CJ, Torres, DFM: A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete Contin. Dyn. Syst. 35(9), 4639-4663 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Rocha, D, Silva, CJ, Torres, DFM: Stability and optimal control of a delayed HIV model. Math. Methods Appl. Sci. (2016). doi:10.1002/mma.4207 Google Scholar
- Luo, J, Wang, W, Chen, H, Fu, R: Bifurcations of a mathematical model for HIV dynamics. J. Math. Anal. Appl. 434, 837-857 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Wang, Y, Liu, J, Liu, L: Viral dynamics of an HIV model with latent infection incorporating antiretroviral therapy. Adv. Differ. Equ. 2016, Article ID 225 (2016) MathSciNetView ArticleGoogle Scholar
- Rong, L, Perelson, AS: Modeling HIV persistence, the latent reservoir, and viral blips. J. Theor. Biol. 260, 308-331 (2009) MathSciNetView ArticleGoogle Scholar
- Finzi, D, Siliciano, R: Viral dynamics in HIV-1 infection. Cell 93, 665-671 (1998) View ArticleGoogle Scholar
- Kirschner, DE: Using mathematics to understand HIV immune dynamics. Not. Am. Math. Soc. 43, 191-202 (1996) MathSciNetMATHGoogle Scholar
- Podlubny, I: Fractional Differential Euations. Acdemic Press, San Diego (1999) MATHGoogle Scholar
- Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Yverdon (1993) MATHGoogle Scholar
- Arafa, AAM, Rida, SZ, Khalil, M: Fractional modeling dynamics of HIV and CD4^{+} T-cells during primary infection. Nonlinear Biomed. Phys. 6, Article ID 1 (2012) View ArticleGoogle Scholar
- Arafa, AAM, Rida, SZ, Khalil, M: The effect of anti-viral drug treatment of human immunodeficiency virus type 1 (HIV-1) described by a fractional order model. Appl. Math. Model. 37, 2189-2196 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Gökdogana, A, Yildirim, A, Merdana, M: Solving a fractional order model of HIV infection of CD4^{+} T cells. Math. Comput. Model. 54, 2132-2138 (2011) MathSciNetView ArticleGoogle Scholar
- Kou, CH, Yan, Y, Liu, J: Stability analysis for fractional differential equations and their applications in the models of HIV-1 infection. Comput. Model. Eng. Sci. 39, 301-317 (2009) MathSciNetMATHGoogle Scholar
- Liu, Z, Lu, P: Stability analysis for HIV infection of CD4^{+} T-cells by a fractional differential time-delay model with cure rate. Adv. Differ. Equ. 2014, Article ID 298 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Pinto, CMA, Carvalho, ARM: Fractional modeling of typical stages in HIV epidemics with drug-resistance. Prog. Fract. Differ. Appl. 1(2), 111-122 (2015) Google Scholar
- Yan, Y, Kou, C: Stability analysis for a fractional differential model of HIV infection of CD4^{+} T-cells with time delay. Math. Comput. Simul. 82, 1572-1585 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Ertürk, VS, Odibat, ZM, Momanic, S: An approximate solution of a fractional order differential equation model of human T-cell lymphotropic virus I (HTLV-I) infection of CD4^{+} T-cells. Comput. Math. Appl. 62, 996-1002 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Alipour, M, Arshad, S, Baleanu, D: Numerical and bifurcations analysis for multi-order fractional model of HIV infection of CD4^{+}T-cells. Sci. Bull. “Politeh.” Univ. Buchar., Ser. A, Appl. Math. Phys. 78(4), 243-258 (2016) Google Scholar
- Atangana, A, Alabaraoye, E: Solving a system of fractional partial differential equations arising in the model of HIV infection of CD4^{+} cells and attractor one-dimensional Keller-Segel equations. Adv. Differ. Equ. 2013, Article ID 94 (2013) View ArticleGoogle Scholar
- Huo, J, Zhao, H, Zhu, L: The effect of vaccines on backward bifurcation in a fractional order HIV model. Nonlinear Anal., Real World Appl. 26, 289-305 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Pinto, CMA, Carvalho, ARM: A latency fractional order model for HIV dynamics. J. Comput. Appl. Math. 312, 240-256 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Abramowitz, M, Stegun, IA (ed.): Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55. U.S. Government Printing Office, Washington (1964) MATHGoogle Scholar
- Perelson, AS, Kirschner, DE, De Boer, R: Dynamics of HIV infection of CD4 T-cells. Math. Biosci. 114, 81-125 (1993) View ArticleMATHGoogle Scholar
- Arenas, AJ, González-Parrab, G, Chen-Charpentierc, BM: Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order. Math. Comput. Simul. 121, 48-63 (2016) MathSciNetView ArticleGoogle Scholar
- Diethelm, K: A fractional calculus based model for the simulation of an outbreak of dengue fever. Nonlinear Dyn. 71, 613-619 (2013) MathSciNetView ArticleGoogle Scholar
- Weilbeer, M: Efficient numerical methods for fractional differential equations and their analytical background. Ph.D. thesis, Technischen Universität Braunschweig (2005) Google Scholar
- Lubich, C: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704-719 (1986) MathSciNetView ArticleMATHGoogle Scholar
- Galeone, L, Garrappa, R: On multistep methods for differential equations of fractional order. Mediterr. J. Math. 3, 565-580 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Diethelm, K, Ford, NJ, Freed, AD: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3-22 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Scherer, R, Kalla, SL, Tang, Y, Huang, J: The Grünwald-Letnikov method for fractional differential equations. Comput. Math. Appl. 62, 902-917 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Baker, CTH: A perspective on the numerical treatment of Volterra equations. J. Comput. Appl. Math. 125, 217-249 (2000) MathSciNetView ArticleMATHGoogle Scholar
- Lin, Y, Xu, C: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225(2), 1533-1552 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Sun, Z, Wu, X: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193-209 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Ahmed, E, El-Sayed, AMA, El-Saka, HAA: On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems. Phys. Lett. A 358, 1-4 (2006) MathSciNetView ArticleMATHGoogle Scholar