- Research
- Open Access
HIV/HCV coinfection model: a fractional-order perspective for the effect of the HIV viral load
- Ana RM Carvalho^{1},
- Carla MA Pinto^{2}Email authorView ORCID ID profile and
- Dumitru Baleanu^{3}
https://doi.org/10.1186/s13662-017-1456-z
© The Author(s) 2018
- Received: 30 June 2017
- Accepted: 20 December 2017
- Published: 5 January 2018
Abstract
We study the burden of the HIV viremia and of treatment efficacy in the severity of the patterns of the HIV/HCV coinfection. For this, we derive a simple non-integer-order (fractional-order) model for the coinfection dynamics. Fractional-order models have been proved in the literature to provide good fits to real data from patients suffering from several diseases, such as HIV, dengue fever, and others. We have computed the basic reproduction number and the stability of the disease-free equilibrium of the model. The numerical results suggest that the HIV viral load impacts impressively the severity of the HCV infection. The treatment efficacy is also found to influence the natural progression of HCV on the HIV/HCV coinfection. The latter is repeated for all values of the order of the fractional derivative. Moreover, the fractional derivative may pave the way to better understanding the individuals’ patients’ adjustments to treatment and to viremia.
Keywords
- coinfection
- HIV
- HCV
- fractional order
- treatment
1 Introduction
The human immunodeficiency virus (HIV) and the hepatitis C virus (HCV) infections are major global public health issues. There are 37 million people infected with HIV worldwide, and about 115 million people HCV antibody positive. One-third of HIV-infected patients are infected with HCV [1]. People coinfected with HIV and HCV are prone to a faster development of HCV infection [1], presenting higher HCV viral loads, and they are more efficient in transmitting HCV [2]. Moreover, the spontaneous elimination of HCV is also decreased in untreated HIV coinfected patients [2]. One of the leading causes of death in HIV treated coinfected patients is chronic HCV infection [3], due to drug-related hepatoxicity. On the other hand, there is evidence that treatment of HIV can slow the natural progression of HCV infection and reduce HCV mortality related to liver diseases [1].
The last few decades have been rewarding in terms of the appearance of a significant diversity of useful mathematical models for the understanding of HIV and HCV coinfection. In 2012, Alexander et al. [4] proposed a model for drug resistance in patients with a chronic viral disease, such as HIV or HCV. They derived dependencies between the parameters of the system that are important factors in driving drug resistance. In 2012, Rong et al. [5] presented a mathematical model of two virus strains, one sensitive and one resistant, to HCV drugs. They provided a theoretical framework to explore the prevalence of pre-existing mutant variants and the evolution of drug resistance during HCV treatment. In 2015, Birger et al. [6] improved an existing model for HCV infection to include the dynamics of the HIV and HCV coinfection, where an immune system component for infection clearance is incorporated. They found that the progression of HCV infection is more rapid when the immune response is compromised by HIV. A better understanding of the mechanisms behind this immune impairment in coinfection may help to devise better therapeutic regimens and to identify patients more compliant to certain drugs.
1.1 Fractional calculus: brief summary
Lagrange and Leibniz were the first mathematicians exchanging letters about the possible meaning of a \(1/2\)-order derivative. They were the predecessors of non-integer-order differentiation and integration, also known as fractional calculus (FC). FC has had a huge development in the last few decades and notable work has been published in engineering, namely electronics, viscoelasticity, biology, physics, epidemiology [7–16].
In 2012 [17], Yan et al. proposed a fractional-order (FO) model for HIV infection with time delay. They compute the stability of the disease-free and of the endemic equilibria and enumerate conditions on the value of the delay, to ensure the asymptotic stability of the two equilibria. In [18] the role of treatment in a FO model was considered for HIV-1 dynamics. In [19], numerical outcomes of a FO model for HIV epidemics were fitted to data from 10 HIV patients. The FO model provides a better fit than the integer-order model. Diethelm [20] introduced a FO model for the patterns of dengue fever. Simulations of the model are fitted to real data of the 2009 dengue outbreak in Cape Verde, providing good agreement. Rihan et al. [15] proposed a FO SIRC epidemic model for the infection by Salmonella bacteria. The variation of the reproduction number was analyzed with respect to contact rate, recovery rate, and other parameters relevant parameters. An unconditionally stable numerical method to approximate the numerical solutions of the FO model was proposed. In [16], a FO model of predator-prey with type-II Holling functional response and time delay is introduced. The fractional derivative improves the stability of the solutions and provides faster transients of the solutions. The authors concluded that the FO models are more suitable to model biological systems with memory, than their integer-order counterparts. Pinto et al. [21] proposed a fractional complex-order model of drug resistance in HIV dynamics. The authors conclude that the complex-order derivative may be interpreted as the delay in the integer-order systems.
Fractional-order systems have been applied in the literature with the purpose of obtaining a deeper understanding of the complex behavioral patterns of biological systems. The memory property of the fractional models allows the integration of more information from the past, which translates in more accurate predictions for the model. With respect to the epidemiological models, this memory property may be used to devise adequate therapeutics directed to each individual, since distinct patients present different disease progression routes. The latter are associated with age, status of the immune system, and genetic profile. Clinicians can, thus, use the information (in terms of behavior’s predictions) of fractional-order systems to fit patients data with the most appropriate non-integer-order index.
With the aforementioned ideas in mind, we derive a fractional-order model for HIV and HCV coinfection, where treatment for HCV is included. The model is an adaptation of two previous integer-order models for HCV mono-infection. In Section 2, we derive the model. In Section 3, we compute the basic reproduction number of the model and the stability of the disease-free equilibrium. The simulations of the model are discussed in Section 4. Finally, the study is concluded in Section 5.
2 Model
In this section, we describe the HIV and HCV coinfection model. The population of the model is divided in six classes, namely the uninfected hepatocyte, x, the drug-sensitive infected hepatocytes, \(y_{s}\), the drug-resistance infected hepatocytes, \(y_{r}\), the drug-sensitive HCV virus, \(v_{s}\), the drug-resistance HCV virus, \(v_{r}\) and the CD4^{+} T cells, H.
The uninfected hepatocytes are produced at a rate \(\lambda^{\alpha}\) and die at a rate \(d^{\alpha}\). These cells reproduce at rate \(r_{1}^{\alpha}\). Parameter \(T_{\mathrm{max}}\) is the maximum capacity number of the hepatocytes. The uninfected hepatocytes, x, are infected when in contact with drug-sensitive virus, \(v_{s}\), and when in contact with drug-resistant virus, \(v_{r}\), at rates \(\beta_{s}^{\alpha}\) and \(\beta _{r}^{\alpha}\), respectively. The drug-sensitive and drug-resistance infected hepatocytes die, respectively, at rates \(a_{s}^{\alpha}\) and \(a_{r}^{\alpha}\). The HCV sensitive and resistant virus, \(v_{s}\) and \(v_{r}\), are produced by the corresponding infected hepatocytes classes, \(y_{s}\) and \(y_{r}\), at rates \(k_{s}^{\alpha}\) and \(k_{r}^{\alpha}\). They die at rates \(c_{s}^{\alpha}\) and \(c_{r}^{\alpha}\), respectively. The CD4^{+} T cells are recruited at rate \(s_{H}^{\alpha}\) and die at rate \(d_{H}^{\alpha }\). These cells are infected when in contact with HIV virus, \(V_{H}\), at rate \(\beta_{H}^{\alpha}\). The parameter \(\alpha_{1}\) models the dependence of the HCV clearance rate on the CD4^{+} T cells count. The dependence of the CD4^{+} T cells activation rate on the HCV infected cell count is given by the parameter γ. The mutation rates are modeled by parameters \(u_{I}\) and \(u_{P}\), where \(u_{I}\) is the mutation at the infection step and \(u_{P}\) at the virion production step.
Treatment is considered at two steps of the replication cycle. A first drug blocks the infection of target cells, through reverse transcriptase or integrase inhibitors, which reduce the successful infection rate of the sensitive strain by a factor \(\epsilon_{I}\), called the efficacy. A second drug, such as a protease inhibitor, prevents the production of viable virions, with efficacy \(\epsilon_{P}\).
2.1 Non-negative solutions
In this section we prove the positivity of the solutions of model (1).
Let \(R_{+}^{6}=\lbrace w\in R^{6}\mid w\geq0\rbrace\) and \(w(t)=(x(t),y_{s}(t),y_{r}(t),v_{s}(t),v_{r}(t),H(t))^{T}\).
To prove the main theorem, we need the following generalized mean value theorem [22] and corollary.
Lemma 1
([22])
Corollary 2
Suppose that \(f(w)\in C[a,b]\) and \(D_{a}^{\alpha}f(w)\in C(a,b]\), for \(0<\alpha\leq1\). If \(D_{a}^{\alpha} f(w)\geq0\), \(\forall w\in(a,b)\), then \(f(w)\) is non-decreasing for each \(w\in[a,b]\). If \(D_{a}^{\alpha }\leq0\), \(\forall w\in(a,b)\), then \(f(w)\) is non-increasing for each \(w\in[a,b]\).
We now prove the main theorem.
Theorem 3
There is a unique solution \(w(t)=(x(t),y_{s}(t),y_{r}(t),v_{s}(t),v_{r}(t),H(t))^{T}\) to model (1) on \(t\geq0\) and the solution will remain in \(R_{+}^{6}\).
Proof
Thus, by Corollary 2, the solution of model (1) will remain in \(R_{+}^{6}\). □
3 Reproduction numbers and local stability of the disease-free equilibrium
In this section, we compute the reproduction number of model (1), \(R_{0}\), and the local stability of its disease-free equilibrium. The basic reproduction number is defined as the number of secondary infections due to a single infection in a completely susceptible population.
We begin by considering two sub-models of model (1). Model (4) arises from model (1) by setting the variables concerning resistance dynamics (\(y_{r}\) and \(v_{r}\)) to zero, and model (10) follows from model (1) by setting the variables concerning sensitive dynamics (\(y_{s}\) and \(v_{s}\)) to zero.
Stability of \(P_{0}^{1}\) can be determined using the following lemmas.
Lemma 4
(Theorem 2, [25])
Lemma 5
The disease-free equilibrium \(P_{0}^{1}\) of the system (4) is unstable if \(R_{s}>1\).
Proof
Similarly arguments of the roots of the equation \(\Lambda^{M\alpha }+d_{H}^{\alpha}+\beta_{H}^{\alpha}V_{H}=0\) are all greater than \(\frac{\pi}{2M}\).
Stability of \(P_{0}^{2}\) can be determined using the following lemmas.
Lemma 6
(Theorem 2, [25])
Lemma 7
The disease-free equilibrium \(P_{0}^{2}\) of the system (10) is unstable if \(R_{r}>1\).
Proof
Similarly the arguments of the roots of the equation \(\Lambda^{M\alpha }+d_{H}^{\alpha}+\beta_{H}^{\alpha}V_{H}=0\) are all greater than \(\frac{\pi}{2M}\).
The stability of \(P_{0}\) can be determined using the following lemmas.
Lemma 8
(Theorem 2, [25])
Lemma 9
The disease-free equilibrium \(P_{0}\) of the system (1) is unstable if \(R_{0}>1\).
Proof
Similarly the arguments of the roots of the equation \(\Lambda^{M\alpha }+d_{H}^{\alpha}+\beta_{H}^{\alpha}V_{H}=0\) are all greater than \(\frac{\pi}{2M}\).
As shown previously, there is exactly one sign change for each polynomial when \(R_{s}>1\) and \(R_{r}>1\), respectively. Since \(R_{0}=\operatorname{max}(R_{s},R_{r})\), the disease-free equilibrium \(P_{0}\) of the system (1) is unstable for \(R_{s}>1\) or \(R_{r}>1\). □
4 Numerical simulations
Parameters used in the numerical simulations of model ( 1 )
Parameter | Value (Units) | Reference |
---|---|---|
λ | 7.5 × 10^{5} (mL^{−1} day^{−1}) | [5] |
d | 1.06 × 10^{−3} (day^{−1}) | [6] |
\(\beta_{s}\) | 7.3 × 10^{−5} (mL day^{−1}) | Estimated |
\(\epsilon_{I}\) | 0.8 | [6] |
\(\beta_{r}\) | 9.3 × 10^{−7} (mL day^{−1}) | [27] |
\(r_{1}\) | 2.7 (day^{−1}) | [6] |
\(T_{\mathrm{max}}\) | 4.016 × 10^{6} (day^{−1}) | [6] |
\(u_{I}\) | 1.2 × 10^{−4} | [4] |
\(a_{s}\) | 0.14 (day^{−1}) | [5] |
\(\alpha_{1}\) | 5 × 10^{−3} | [6] |
\(a_{r}\) | 0.14 (day^{−1}) | [5] |
\(k_{s}\) | 45 (day^{−1}) | [27] |
\(\epsilon_{P}\) | 0.8 | [6] |
\(u_{P}\) | 1.2 × 10^{−4} | [4] |
\(k_{r}\) | 45 (day^{−1}) | [27] |
\(c_{s}\) | 6.2 (day^{−1}) | [5] |
\(c_{r}\) | 6.2 (day^{−1}) | [5] |
\(s_{H}\) | 10^{4} (mL^{−1} day^{−1}) | [28] |
γ | 2 × 10^{−8} | [6] |
\(d_{H}\) | 9 × 10^{−3} (day^{−1}) | [6] |
\(\beta_{H}\) | 4.1 × 10^{−6} (mL day^{−1}) | [6] |
\(V_{H}\) | 10^{5} (mL day^{−1}) | [6] |
5 Conclusions
We derive a simple non-integer-order model for the coinfection of HIV and HCV, with treatment for HCV. We compute the basic reproduction number and the stability of the disease-free equilibrium. The simulations of the model reveal a strong dependency of the HCV infection progression on the HIV viral load. Higher HIV viral loads are associated with reduced immune response, which in turn translates in higher HCV viral loads. We have also considered the influence of the protease drug efficacy on the dynamics of the coinfection. We find that smaller values of this parameter are associated with a higher number of infected hepatocytes. The results of the models suggest that specific measures should be implemented, by the policy makers, in order to reduce HIV viral load (preventive measures and treatment), and to treat HCV infection. The order of the fractional derivative seems to increase the severity of the disease, translated by higher HIV viral loads and infected hepatocytes. The results of the model are biologically reasonable.
Declarations
Acknowledgements
The authors were partially funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT - Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2013. The research of AC was partially supported by a FCT grant with reference SFRH/BD/96816/2013.
Authors’ contributions
The authors have contributed equally to this manuscript. They 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
References
- Hernandez, MD, Sherman, KE: Hiv/hcv coinfection natural history and disease progression, a review of the most recent literature. Curr. Opin. HIV AIDS 6(6), 478-482 (2011) View ArticleGoogle Scholar
- Platt, L, Easterbrook, P, Gower, E, McDonald, B, Sabin, K, McGowan, C, Yanny, I, Razavi, H, Vickerman, P: Prevalence and burden of hcv co-infection in people living with hiv: a global systematic review and meta-analysis. Lancet Infect. Dis. 16(7), 797-808 (2016) View ArticleGoogle Scholar
- Taylor, LE, Swan, T, May, KH: Hiv coinfection with hepatitis c virus: evolving epidemiology and treatment paradigms. Clin. Infect. Dis. 55(Suppl. 1), 33-42 (2012) View ArticleGoogle Scholar
- Alexander, HK, Bonhoeffer, S: Pre-existence and emergence of drug resistance in a generalized model of intra-host viral dynamics. Epidemics 4, 187-202 (2012) View ArticleGoogle Scholar
- Rong, L, Ribeiro, RM, Perelson, AS: Modeling quasispecies and drug resistance in hepatitis c patients treated with a protease inhibitor. Bull. Math. Biol. 74(8), 1789-1817 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Birger, R, Kouyos, R, Dushoff, J, Grenfell, B: Modeling the effect of hiv coinfection on clearance and sustained virologic response during treatment for hepatitis c virus. Epidemics 12, 1-10 (2015) View ArticleGoogle Scholar
- Baleanu, D, Muslih, SI, Rabei, EM: On fractional Euler-Lagrange and Hamilton equations and the fractional generalization of total time derivative. Nonlinear Dyn. 53(1-2), 67-74 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Pinto, CMA, Machado, JAT: Complex order van der Pol oscillator. Nonlinear Dyn. 65(3), 247-254 (2010) MathSciNetView ArticleGoogle Scholar
- Pinto, CMA, Machado, JAT: Complex-order forced van der Pol oscillator. J. Vib. Control 18(14), 2201-2209 (2012) MathSciNetView ArticleGoogle Scholar
- Pinto, CMA, Machado, JAT: Fractional model for malaria transmission under control strategies. Comput. Math. Appl. 66(5), 908-916 (2013) MathSciNetView ArticleGoogle Scholar
- Pinto, CMA, Carvalho, ARM: The role of synaptic transmission in a hiv model with memory. Appl. Math. Comput. 292, 76-95 (2017) MathSciNetGoogle Scholar
- Baleanu, D, Magin, RL, Bhalekar, S, Daftardar-Gejji, V: Chaos in the fractional order nonlinear Bloch equation with delay. Commun. Nonlinear Sci. Numer. Simul. 25(1-3), 41-49 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Caputo, M, Fabrizio, M: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2, 73-85 (2015) Google Scholar
- Atangana, A, Koca, I: Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos Solitons Fractals 89, 447-454 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Rihan, FA, Baleanu, D, Lakshmanan, S, Rakkiyappan, R: On fractional sirc model with salmonella bacterial infection. Abstr. Appl. Anal. 2014, 9 (2014) MathSciNetView ArticleGoogle Scholar
- Rihan, FA, Lakshmanan, S, Hashish, AH, Rakkiyappan, R, Ahmed, E: Fractional order delayed predator-prey systems with Holling type-ii functional response. Nonlinear Dyn. 80(1), 777-789 (2015) MathSciNetView ArticleMATHGoogle 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
- 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. Applied Mathematical Modelling 37, 2189-2196 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Arafa, AAM, Rida, SZ, Khalil, M: A fractional-order model of hiv infection: numerical solution and comparisons with data of patients. Int. J. Biomath. 7(4), 1450036 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Diethelm, K: A fractional calculus based model for the simulation of an outbreak of dengue fever. Nonlinear Dyn. 71(4), 613-619 (2013) MathSciNetView ArticleGoogle 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
- Odibat, ZM, Shawagfeh, NT: Generalized Taylor’s formula. Appl. Math. Comput. 186, 286-293 (2007) MathSciNetMATHGoogle Scholar
- Lin, W: Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl. 332, 709-726 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Driessche, P, Watmough, P: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29-48 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Tavazoei, MS, Haeri, M: Chaotic attractors in incommensurate fractional order systems. Physica D 237, 2628-2637 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Scherer, R, Kalla, SL, Tang, Y, Huang, J: The Grünwald-Letnikov method for fractional differential equations. Comput. Math. Appl. 62(3), 902-917 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Debroy, S, Bolker, BM, Martcheva, M: Bistability and long-term cure in a within-host model of hepatitis c. J. Biol. Syst. 19, 533-550 (2011) MathSciNetView ArticleGoogle Scholar
- Hadjiandreou, MM, Conejeros, R, Wilson, DI: Long-term hiv dynamics subject to continuous therapy and structured treatment interruptions. Chem. Eng. Sci. 64, 1600-1617 (2009) View ArticleGoogle Scholar