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ISRN Applied Mathematics
Volume 2014 (2014), Article ID 594617, 10 pages
Stochastic Model for Langerhans Cells and HIV Dynamics In Vivo
1Center for Applied Research in Mathematical Sciences, Strathmore University, P.O. Box 59857, Nairobi 00200, Kenya
2Department of Mathematics, Makerere University, P.O. Box 7062, Kampala, Uganda
Received 11 September 2013; Accepted 23 October 2013; Published 19 January 2014
Academic Editors: D. Haemmerich and X. Liu
Copyright © 2014 Waema R. Mbogo 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.
Many aspects of the complex interaction between HIV and the human immune system remain elusive. Our objective is to study these interactions, focusing on the specific roles of Langerhans cells (LCs) in HIV infection. In patients infected with HIV, a large amount of virus is associated with LCs in lymphoid tissue. To assess the influence of LCs on HIV viral dynamics during antiretroviral therapy, we present and analyse a stochastic model describing the dynamics of HIV, T cells, and LCs interactions under therapeutic intervention in vivo and show that LCs play an important role in enhancing and spreading initial HIV infection. We perform sensitivity analyses on the model to determine which parameters and/or which interaction mechanisms strongly affect infection dynamics.
HIV is a devastating human pathogen that causes serious immunological diseases in humans around the world. The virus is able to remain latent in an infected host for many years, allowing for the long-term survival of the virus and inevitably prolonging the infection process . The location and mechanisms of HIV latency are under investigation and remain important topics in the study of viral pathogenesis. Given that HIV is a blood-borne pathogen, a number of cell types have been proposed to be the sites of latency, including resting memory + T cells, peripheral blood monocytes, dendritic cells (including Langerhans cells) and macrophages in the lymph nodes, and haematopoietic stem cells in the bone marrow . This study updates the latest advances in the study of HIV interactions with Langerhans cells and highlights the potential role of these cells as viral reservoirs and the effects of the HIV-host-cell interactions on viral pathogenesis.
Despite advances in our understanding of HIV and the human immune response in the last 25 years, much of this complex interaction remains elusive. T cells are targets of HIV and are also important for the establishment and maintenance of an adaptive immune response . The skin and mucosa are the first line of defense of the organism against external agents, not only as a physical barrier between the body and the environment but also as the site of initiation of immune reactions. The immunocompetent cells which act as antigen-presenting cells are Langerhans cells (LCs). Infection of LCs by HIV is relevant to several reasons. Firstly, LCs of mucosal epithelia may be among the first cells to be infected following mucosal HIV exposure and, secondly, LCs may serve as a reservoir for continued infection of T cells, especially in lymph nodes where epidermal LCs migrate following antigenic activation .
Many indirect and/or direct experimental data have shown that LCs may be a privileged target, reservoir, and vector of dissemination for the HIV from the inoculation sites (mucosa) to lymph nodes where the emigrated infected LCs could infect T lymphocytes . Originated from the bone marrow, LCs migrate to the peripheral epithelia (skin, mucous membranes) where they play a primordial role in the induction of an immune response and are especially active in stimulating naive T lymphocytes in the primary response through a specific cooperation with -positive lymphocytes after migration to proximal lymph nodes . Apart from many plasma membrane determinants, LCs also express molecules which make them susceptible targets and reservoirs for HIV . Once infected, these cells due to their localization in areas at risk (skin, mucous membranes), their capacity to migrate from the epidermal compartment to lymph nodes, and their ability to support viral replication without major cytopathic effects could play a role of vector in the dissemination of virus from the site of inoculation to the lymph nodes and thereby contribute to the infection of T lymphocytes .
Langerhans cells (LCs), which are members of the dendritic cells family and are professional antigen-presenting cells, reside in epithelial surfaces such as the skin and act as one of the primary, initial targets for HIV infection . They specialize in antigen presentation and belong to the skin immune system (SIS) and play a major role in HIV pathogenesis. As part of the normal immune response, LCs capture virions at the site of transmission in the mucosa (peripheral tissues) and migrate to the lymphoid tissue where they present to naive T cells and hence are responsible for large-scale infection of T lymphocytes . These cells play an important role in the transmission of HIV to cells ; thus, LC- cell interactions in lymphoid tissue, which are critical in the generation of immune responses, are also a major catalyst for HIV replication and expansion. This replication independent mode of HIV transmission, known as trans-infection, greatly increases T cell infection in vitro and is thought to contribute to viral dissemination in vivo .
The Langerhans cell is named after Paul Langerhans, a German physician and anatomist, who discovered the cells at the age of 21 in 1868 while he was a medical student . The uptake of HIV by professional antigen-presenting cells (APCs) and subsequent transfer of virus to T cells can result in explosive levels of virus replication in the T cells. This could be a major pathogenic process in HIV infection and development of AIDS. This process of trans- (Latin; to the other side) infection of virus going across from the APC to the T cell is in contrast to direct, cis- (Latin; on this side) infection of T cells by HIV . Langerhans cell results in a burst of virus replication in the T cells that is much greater than that resulting from direct, cis infection of either APC or T cells, or trans-infection between T cells. This consequently shows that Langerhans cells may be responsible for the quick spread of HIV infection.
The individual cells of the immune system are highly interactive, and the overall function of the system is a product of this multitude of interactions. The interplay between HIV and the immune system is particularly complicated, as HIV directly interacts with many immune cells, altering their functions, ultimately subverting the system at its core . Because of this complexity, the immune response and its interaction with HIV are naturally suited to a mathematical modelling approach. Elucidating the mechanisms of LC-HIV- T cells interactions is crucial in uncovering more details about host-HIV dynamics during HIV infection. To explore the role of LCs in HIV infection, we first develop a stochastic model of HIV dynamics in vivo before therapy. Next, we introduce therapeutic intervention and finally investigate which parameters and/or which interaction mechanisms strongly affect the infection dynamics.
The organization of this paper is as follows. In Section 2, we formulate our stochastic model describing the interaction of HIV and the immune system and obtain a partial differential equation for the joint probability generating function of the numbers of healthy immune cells, the HIV infected immune cells, and the free HIV particles at any time . The marginal probability distributions for the variables and the population measures which include the expectation of the variables are also derived in Section 2. HIV dynamic model incorporating therapy is derived and model analysis is discussed in Section 3. Some concluding remarks follow in Section 4.
2. The Role of LC in HIV Infection In Vivo
In HIV infection, LCs play a dual role in promoting immunity while also facilitating infection. During antigen presentation, LC-associated viruses migrate to the lymphoid tissue where they present to naive T cells and hence facilitate infection of T cells . Taken together, these interactions suggest that LC dynamics are particularly important to HIV infection. Several mechanisms have been proposed to trigger progression from the chronic phase of infection to AIDS. Many have hypothesized that progressive alteration of the immune system results in the transition to AIDS [16, 17].
To study the roles of LCs during HIV infection, we present a mathematical model of HIV infection and accompanying immune response. Specifically, we develop a stochastic model focusing on the HIV-LC- cell dynamics in vivo. Our model analysis allows us to predict the importance of LC mechanisms and their role in triggering progression to AIDS. In particular, our model predicts which mechanisms of LC dysfunction are most significant in the transition to AIDS. A typical life cycle of HIV virus and immune system interaction is shown in Figure 1.
2.1. Variables and Parameters for the Model
From Table 1 and by using the population change scenarios and parameters in Table 2 and applying probability arguments, we now summarize the events that occur during the interval together with their transition probabilities in Table 3.
The change in population size during the time interval , which is assumed to be sufficiently small to guarantee that only one such event can occur in , is governed by the following conditional probabilities: Rearranging (1), then dividing through by , and taking the limit as we have the following difference differential equation: This is also called the Master equation or the forward Kolmogorov partial differential equation for , with initial condition For detailed description and derivation of the model we refer the reader to .
2.2. The Probability Generating Function
Now we apply the generating function method. Multiplying (2) by and summing over , and , then applying the properties of generating function, we obtain On simplification we have This is called Lagrange partial differential equation for the probability generating function (pgf) .
2.3. The Marginal Generating Functions
Recall that Assuming and solving (5), we obtain the marginal partial generating function for healthy cells: Assuming and solving (5), we obtain the marginal partial generating function for healthy Langerhans cells: Assuming and solving (5), we obtain the marginal partial generating function for infected cells: Assuming and solving (5), we obtain the marginal partial generating function for infected Langerhans cells:
Assuming and solving (5), we obtain the marginal partial generating function for virus population:
2.4. Numbers of Cells and the Virions
As we know from probability generating function, Letting , we have Differentiating the partial differential equation of the pgf , we get the moments of , and .
Differentiating (5) with respect to and setting , we have Differentiating (5) with respect to and setting , we have Differentiating (5) with respect to and setting , we have Differentiating (5) with respect to and setting , we have Differentiating (5) with respect to and setting , we have Therefore the moments of , and from the pgf before introduction of treatment are
3. HIV Dynamics under Therapeutic Intervention
Now we introduce the treatment effect on the HIV-immune cell interaction. If we let be the reverse transcriptase inhibitor and let be the protease inhibitor drug effects, then introducing these treatment effects on our model, we have the following forward Kolmogorov partial differential equations (Master equation) for : with initial condition Solving the Master equation using generating function, the Lagrange partial differential equation becomes Solving the pgf of the in-host HIV dynamics with therapeutic intervention, we have the moments of , and :
3.1. Simulation of the Models
3.1.1. Variables and Parameter Values
Using the parameter values and initial conditions defined in Tables 4 and 5, we illustrate the general dynamics of the T cells and HIV virus by performing sensitivity analysis on the effect of intracellular delay and drug efficacy.
From the simulations in Figures 2 and 3, it is clear that, in the primary stage of HIV infection, drastic decrease in the levels of healthy immune cells occurs but the number of free virions and infected LCs increases with time.
In this study, we derived and analysed a stochastic model describing the dynamics of HIV, T cells, and LCs interactions under therapeutic intervention in vivo. This model included dynamics of five compartments—the number of healthy cells, the number of infected cells, the number of healthy Langerhans cells, the number of infected Langerhans cells, and the HIV virions. The model describes HIV infection before and during therapy. We derived equations for the joint probability generating function of the numbers of healthy immune cells, the HIV infected immune cells, and the free HIV particles at any time and obtained the moment structures of the healthy immune cells, infected immune cells, and the virus particles over time . We simulated the mean number of the healthy immune cells, the infected immune cells, and the virus particles before and after combined therapeutic treatment at any time .
Our analysis shows that eradication of HIV is not possible without clearance of latently infected Langerhans cells. Therefore, understanding HIV therapeutic treatment dynamics in vivo is critical to eliminate the virus, which shows that LCs are important in determining the disease progression. Our model analysis suggests that therapies should be developed to block the binding of HIV onto Langerhans cells. Such therapies will have the potential to dramatically accelerate viral decay. We conclude that, to control the concentrations of the virus and the infected cells in HIV infected person, a strategy should aim to improve the drug efficacy; hence the efficacy of the protease inhibitor and the reverse transcriptase inhibitor and also the intracellular delay play crucial role in preventing the progression of HIV . These findings illustrate the role of LCs as a central hub of interaction and information exchange during HIV infection. Our model produces interesting feature that classification of HIV disease states should not be based on cells as the only immune cells infected by the virus, but the most reliable HIV state classification criteria should be classification using clinical signs (the CDC/WHO classification).
In our work, the dynamics of mutant virus were not considered and also our study only included dynamics of only five compartments (healthy immune cells, infected immune cells, and free virus particles ignoring the latency of infected immune cells) of which extensions are recommended for further extensive research. In a follow-up work, we intend to obtain real data in order to test the efficacy of our models as we have done here with parameter values from the literature.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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