1
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Yan D, He M, Tang S. Uncertainty quantification for the random HIV dynamical model driven by drug adherence. J Theor Biol 2024:111895. [PMID: 38969168 DOI: 10.1016/j.jtbi.2024.111895] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/11/2023] [Revised: 05/27/2024] [Accepted: 06/25/2024] [Indexed: 07/07/2024]
Abstract
In HIV drug therapy, the high variability of CD4+ T cells and viral loads brings uncertainty to the determination of treatment options and the ultimate treatment efficacy, which may be the result of poor drug adherence. We develop a dynamical HIV model coupled with pharmacokinetics, driven by drug adherence as a random variable, and systematically study the uncertainty quantification, aiming to construct the relationship between drug adherence and therapeutic effect. Using adaptive generalized polynomial chaos, stochastic solutions are approximated as polynomials of input random parameters. Numerical simulations show that results obtained by this method are in good agreement, compared with results obtained through Monte Carlo sampling, which helps to verify the accuracy of approximation. Based on these expansions, we calculate the time-dependent probability density functions of this system theoretically and numerically. To verify the applicability of this model, we fit clinical data of four HIV patients, and the goodness of fit results demonstrate that the proposed random model depicts the dynamics of HIV well. Sensitivity analyses based on the Sobol index indicate that the randomness of drug effect has the greatest impact on both CD4+ T cells and viral loads, compared to random initial values, which further highlights the significance of drug adherence. The proposed models and qualitative analysis results, along with monitoring CD4+ T cells counts and viral loads, evaluate the influence of drug adherence on HIV treatment, which helps to better interpret clinical data with fluctuations and makes several contributions to the design of individual-based optimal antiretroviral strategies.
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Affiliation(s)
- Dingding Yan
- School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, 710119, PR China
| | - Mengqi He
- School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, 710119, PR China
| | - Sanyi Tang
- School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, PR China.
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2
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Mainou E, Berendam SJ, Obregon-Perko V, Uffman EA, Phan CT, Shaw GM, Bar KJ, Kumar MR, Fray EJ, Siliciano JM, Siliciano RF, Silvestri G, Permar SR, Fouda GG, McCarthy J, Chahroudi A, Chan C, Conway JM. Assessing the impact of autologous neutralizing antibodies on rebound dynamics in postnatally SHIV-infected ART-treated infant rhesus macaques. BIORXIV : THE PREPRINT SERVER FOR BIOLOGY 2024:2024.06.01.596971. [PMID: 38895223 PMCID: PMC11185557 DOI: 10.1101/2024.06.01.596971] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/21/2024]
Abstract
The presence of antibodies against HIV in infected children is associated with a greater capacity to control viremia in the absence of therapy. While the benefits of early antiretroviral treatment (ART) in infants are well documented, early ART may interfere with the development of antibody responses. In contrast to adults, early treated children lack detectable HIV-specific antibodies, suggesting a fundamental difference in HIV pathogenesis. Despite this potential adverse effect, early ART may decrease the size of the latent reservoir established early in infection in infants, which can be beneficial in viral control. Understanding the virologic and immunologic aspects of pediatric HIV is crucial to inform innovative targeted strategies for treating children living with HIV. In this study, we investigate how ART initiation time sets the stage for trade-offs in the latent reservoir establishment and the development of humoral immunity and how these, in turn, affect posttreatment dynamics. We also elucidate the biological function of antibodies in pediatric HIV. We employ mathematical modeling coupled with experimental data from an infant nonhuman primate Simian/Human Immunodeficiency Virus (SHIV) infection model. Infant Rhesus macaques (RMs) were orally challenged with SHIV.C.CH505 375H dCT four weeks after birth and started treatment at different times after infection. In addition to viral load measurements, antibody responses and latent reservoir sizes were measured. We estimate model parameters by fitting viral load measurements to the standard HIV viral dynamics model within a nonlinear fixed effects framework. This approach allows us to capture differences between rhesus macaques (RMs) that develop antibody responses or exhibit high latent reservoir sizes compared to those that do not. We find that neutralizing antibody responses are associated with increased viral clearance and decreased viral infectivity but decreased death rate of infected cells. In addition, the presence of detectable latent reservoir is associated with less robust immune responses. These results demonstrate that both immune response and latent reservoir dynamics are needed to understand post-rebound dynamics and point to the necessity of a comprehensive approach in tailoring personalized medical interventions.
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Affiliation(s)
- Ellie Mainou
- Department of Biology, Pennsylvania State University, University Park, PA, USA
| | | | | | - Emilie A Uffman
- Duke Human Vaccine Institute, Duke University Medical Center, Durham, NC, USA
| | - Caroline T Phan
- Duke Human Vaccine Institute, Duke University Medical Center, Durham, NC, USA
| | - George M Shaw
- Perelman School of Medicine, University of Pennsylvania, Philadelphia, Pennsylvania, USA
| | - Katharine J Bar
- Perelman School of Medicine, University of Pennsylvania, Philadelphia, Pennsylvania, USA
| | - Mithra R Kumar
- Department of Pharmacology and Molecular Sciences, Johns Hopkins University School of Medicine, Baltimore, MD, USA
| | - Emily J Fray
- Department of Biochemistry and Molecular Biology, Johns Hopkins University School of Medicine, Baltimore, MD, USA
| | - Janet M Siliciano
- Department of Medicine, Johns Hopkins University School of Medicine, Baltimore, MD, USA
| | - Robert F Siliciano
- Department of Medicine, Johns Hopkins University School of Medicine, Baltimore, MD, USA
| | - Guido Silvestri
- Yerkes National Primate Research Center, Emory University, Atlanta, Georgia, USA
| | - Sallie R Permar
- Department of Pediatrics, Weill Cornell Medicine, New York, NY, USA
| | | | - Janice McCarthy
- Department of Biostatistics and Bioinformatics, Duke University Medical Center, Durham, NC, USA
| | - Ann Chahroudi
- Department of Pediatrics, Emory University School of Medicine, Atlanta, GA 30322, USA
| | - Cliburn Chan
- Department of Biostatistics and Bioinformatics, Duke University Medical Center, Durham, NC, USA
| | - Jessica M Conway
- Department of Mathematics, Pennsylvania State University, University Park, PA, USA
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3
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Mainou E, Ribeiro RM, Conway JM. Modeling dynamics of acute HIV infection incorporating density-dependent cell death and multiplicity of infection. PLoS Comput Biol 2024; 20:e1012129. [PMID: 38848426 PMCID: PMC11189221 DOI: 10.1371/journal.pcbi.1012129] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2023] [Revised: 06/20/2024] [Accepted: 05/02/2024] [Indexed: 06/09/2024] Open
Abstract
Understanding the dynamics of acute HIV infection can offer valuable insights into the early stages of viral behavior, potentially helping uncover various aspects of HIV pathogenesis. The standard viral dynamics model explains HIV viral dynamics during acute infection reasonably well. However, the model makes simplifying assumptions, neglecting some aspects of HIV infection. For instance, in the standard model, target cells are infected by a single HIV virion. Yet, cellular multiplicity of infection (MOI) may have considerable effects in pathogenesis and viral evolution. Further, when using the standard model, we take constant infected cell death rates, simplifying the dynamic immune responses. Here, we use four models-1) the standard viral dynamics model, 2) an alternate model incorporating cellular MOI, 3) a model assuming density-dependent death rate of infected cells and 4) a model combining (2) and (3)-to investigate acute infection dynamics in 43 people living with HIV very early after HIV exposure. We find that all models qualitatively describe the data, but none of the tested models is by itself the best to capture different kinds of heterogeneity. Instead, different models describe differing features of the dynamics more accurately. For example, while the standard viral dynamics model may be the most parsimonious across study participants by the corrected Akaike Information Criterion (AICc), we find that viral peaks are better explained by a model allowing for cellular MOI, using a linear regression analysis as analyzed by R2. These results suggest that heterogeneity in within-host viral dynamics cannot be captured by a single model. Depending on the specific aspect of interest, a corresponding model should be employed.
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Affiliation(s)
- Ellie Mainou
- Department of Biology, Center for Infectious Disease Dynamics, The Pennsylvania State University, University Park, Pennsylvania, United States of America
| | - Ruy M. Ribeiro
- Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America
| | - Jessica M. Conway
- Department of Mathematics, Center for Infectious Disease Dynamics, The Pennsylvania State University, University Park, Pennsylvania, United States of America
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4
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Williams B, Carruthers J, Gillard JJ, Lythe G, Perelson AS, Ribeiro RM, Molina-París C, López-García M. The reproduction number and its probability distribution for stochastic viral dynamics. J R Soc Interface 2024; 21:20230400. [PMID: 38264928 PMCID: PMC10806437 DOI: 10.1098/rsif.2023.0400] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/12/2023] [Accepted: 12/18/2023] [Indexed: 01/25/2024] Open
Abstract
We consider stochastic models of individual infected cells. The reproduction number, R, is understood as a random variable representing the number of new cells infected by one initial infected cell in an otherwise susceptible (target cell) population. Variability in R results partly from heterogeneity in the viral burst size (the number of viral progeny generated from an infected cell during its lifetime), which depends on the distribution of cellular lifetimes and on the mechanism of virion release. We analyse viral dynamics models with an eclipse phase: the period of time after a cell is infected but before it is capable of releasing virions. The duration of the eclipse, or the subsequent infectious, phase is non-exponential, but composed of stages. We derive the probability distribution of the reproduction number for these viral dynamics models, and show it is a negative binomial distribution in the case of constant viral release from infectious cells, and under the assumption of an excess of target cells. In a deterministic model, the ultimate in-host establishment or extinction of the viral infection depends entirely on whether the mean reproduction number is greater than, or less than, one, respectively. Here, the probability of extinction is determined by the probability distribution of R, not simply its mean value. In particular, we show that in some cases the probability of infection is not an increasing function of the mean reproduction number.
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Affiliation(s)
- Bevelynn Williams
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds, UK
| | | | - Joseph J. Gillard
- CBR Division, Defence Science and Technology Laboratory, Salisbury, UK
| | - Grant Lythe
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds, UK
| | - Alan S. Perelson
- T-6, Theoretical Biology and Biophysics, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA
| | - Ruy M. Ribeiro
- T-6, Theoretical Biology and Biophysics, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA
| | - Carmen Molina-París
- T-6, Theoretical Biology and Biophysics, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA
| | - Martín López-García
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds, UK
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5
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Bandara T, Martcheva M, Ngonghala CN. Mathematical model on HIV and nutrition. JOURNAL OF BIOLOGICAL DYNAMICS 2023; 17:2287087. [PMID: 38015715 DOI: 10.1080/17513758.2023.2287087] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/03/2023] [Accepted: 11/17/2023] [Indexed: 11/30/2023]
Abstract
HIV continues to be a major global health issue, having claimed millions of lives in the last few decades. While several empirical studies support the fact that proper nutrition is useful in the fight against HIV, very few studies have focused on developing and using mathematical modelling approaches to assess the association between HIV, human immune response to the disease, and nutrition. We develop a within-host model for HIV that captures the dynamic interactions between HIV, the immune system and nutrition. We find that increased viral activity leads to increased serum protein levels. We also show that the viral production rate is positively correlated with HIV viral loads, as is the enhancement rate of protein by virus. Although our numerical simulations indicate a direct correlation between dietary protein intake and serum protein levels in HIV-infected individuals, further modelling and clinical studies are necessary to gain comprehensive understanding of the relationship.
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Affiliation(s)
- Tharusha Bandara
- Department of Mathematics, University of Florida, Gainesville, FL, USA
| | - Maia Martcheva
- Department of Mathematics, University of Florida, Gainesville, FL, USA
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6
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D’Orso I, Forst CV. Mathematical Models of HIV-1 Dynamics, Transcription, and Latency. Viruses 2023; 15:2119. [PMID: 37896896 PMCID: PMC10612035 DOI: 10.3390/v15102119] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/31/2023] [Revised: 10/10/2023] [Accepted: 10/18/2023] [Indexed: 10/29/2023] Open
Abstract
HIV-1 latency is a major barrier to curing infections with antiretroviral therapy and, consequently, to eliminating the disease globally. The establishment, maintenance, and potential clearance of latent infection are complex dynamic processes and can be best described with the help of mathematical models followed by experimental validation. Here, we review the use of viral dynamics models for HIV-1, with a focus on applications to the latent reservoir. Such models have been used to explain the multi-phasic decay of viral load during antiretroviral therapy, the early seeding of the latent reservoir during acute infection and the limited inflow during treatment, the dynamics of viral blips, and the phenomenon of post-treatment control. Finally, we discuss that mathematical models have been used to predict the efficacy of potential HIV-1 cure strategies, such as latency-reversing agents, early treatment initiation, or gene therapies, and to provide guidance for designing trials of these novel interventions.
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Affiliation(s)
- Iván D’Orso
- Department of Microbiology, University of Texas Southwestern Medical Center, Dallas, TX 75390, USA;
| | - Christian V. Forst
- Department of Genetics and Genomic Sciences, Department of Microbiology, Icahn School of Medicine at Mount Sinai, New York, NY 10029, USA
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7
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Hailegiorgis A, Ishida Y, Collier N, Imamura M, Shi Z, Reinharz V, Tsuge M, Barash D, Hiraga N, Yokomichi H, Tateno C, Ozik J, Uprichard SL, Chayama K, Dahari H. Modeling suggests that virion production cycles within individual cells is key to understanding acute hepatitis B virus infection kinetics. PLoS Comput Biol 2023; 19:e1011309. [PMID: 37535676 PMCID: PMC10426918 DOI: 10.1371/journal.pcbi.1011309] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2022] [Revised: 08/15/2023] [Accepted: 06/26/2023] [Indexed: 08/05/2023] Open
Abstract
Hepatitis B virus (HBV) infection kinetics in immunodeficient mice reconstituted with humanized livers from inoculation to steady state is highly dynamic despite the absence of an adaptive immune response. To recapitulate the multiphasic viral kinetic patterns, we developed an agent-based model that includes intracellular virion production cycles reflecting the cyclic nature of each individual virus lifecycle. The model fits the data well predicting an increase in production cycles initially starting with a long production cycle of 1 virion per 20 hours that gradually reaches 1 virion per hour after approximately 3-4 days before virion production increases dramatically to reach to a steady state rate of 4 virions per hour per cell. Together, modeling suggests that it is the cyclic nature of the virus lifecycle combined with an initial slow but increasing rate of HBV production from each cell that plays a role in generating the observed multiphasic HBV kinetic patterns in humanized mice.
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Affiliation(s)
- Atesmachew Hailegiorgis
- The Program for Experimental & Theoretical Modeling, Division of Hepatology, Department of Medicine, Stritch School of Medicine, Loyola University Chicago, Maywood, Illinois, United States of America
| | - Yuji Ishida
- PhoenixBio Co., Ltd., Hiroshima, Japan
- Research Center for Hepatology and Gastroenterology, Graduate School of Biomedical and Health Sciences, Hiroshima University, Hiroshima, Japan
| | - Nicholson Collier
- Consortium for Advanced Science and Engineering, University of Chicago, Chicago, Illinois, United States of America
- Decision and Infrastructure Sciences, Argonne National Laboratory, Argonne, Illinois, United States of America
| | - Michio Imamura
- Research Center for Hepatology and Gastroenterology, Graduate School of Biomedical and Health Sciences, Hiroshima University, Hiroshima, Japan
| | - Zhenzhen Shi
- The Program for Experimental & Theoretical Modeling, Division of Hepatology, Department of Medicine, Stritch School of Medicine, Loyola University Chicago, Maywood, Illinois, United States of America
| | - Vladimir Reinharz
- Department of Computer Science, Université du Québec à Montréal, Montreal, Canada
| | - Masataka Tsuge
- The Program for Experimental & Theoretical Modeling, Division of Hepatology, Department of Medicine, Stritch School of Medicine, Loyola University Chicago, Maywood, Illinois, United States of America
- Research Center for Hepatology and Gastroenterology, Graduate School of Biomedical and Health Sciences, Hiroshima University, Hiroshima, Japan
- Department of Gastroenterology, Graduate School of Biomedical & Health Sciences, Hiroshima University, Hiroshima, Japan
| | - Danny Barash
- Department of Computer Science, Ben-Gurion University, Beer-Sheva, Israel
| | - Nobuhiko Hiraga
- Research Center for Hepatology and Gastroenterology, Graduate School of Biomedical and Health Sciences, Hiroshima University, Hiroshima, Japan
| | | | - Chise Tateno
- PhoenixBio Co., Ltd., Hiroshima, Japan
- Research Center for Hepatology and Gastroenterology, Graduate School of Biomedical and Health Sciences, Hiroshima University, Hiroshima, Japan
| | - Jonathan Ozik
- Consortium for Advanced Science and Engineering, University of Chicago, Chicago, Illinois, United States of America
- Decision and Infrastructure Sciences, Argonne National Laboratory, Argonne, Illinois, United States of America
| | - Susan L. Uprichard
- The Program for Experimental & Theoretical Modeling, Division of Hepatology, Department of Medicine, Stritch School of Medicine, Loyola University Chicago, Maywood, Illinois, United States of America
- The Infectious Disease and Immunology Research Institute, Stritch School of Medicine, Loyola University Chicago, Maywood, Illinois, United States of America
| | - Kazuaki Chayama
- Research Center for Hepatology and Gastroenterology, Graduate School of Biomedical and Health Sciences, Hiroshima University, Hiroshima, Japan
- RIKEN Center for Integrative Medical Sciences, Yokohama, Japan
- Collaborative Research Laboratory of Medical Innovation, Graduate School of Biomedical and Health Sciences, Hiroshima University, Hiroshima, Japan
- Hiroshima Institute of Life Sciences, Hiroshima, Japan
| | - Harel Dahari
- The Program for Experimental & Theoretical Modeling, Division of Hepatology, Department of Medicine, Stritch School of Medicine, Loyola University Chicago, Maywood, Illinois, United States of America
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8
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Burg D, Ausubel JH. Trajectories of COVID-19: A longitudinal analysis of many nations and subnational regions. PLoS One 2023; 18:e0281224. [PMID: 37352253 PMCID: PMC10289358 DOI: 10.1371/journal.pone.0281224] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/17/2023] [Accepted: 06/07/2023] [Indexed: 06/25/2023] Open
Abstract
The COVID-19 pandemic is the first to be rapidly and sequentially measured by nation-wide PCR community testing for the presence of the viral RNA at a global scale. We take advantage of the novel "natural experiment" where diverse nations and major subnational regions implemented various policies including social distancing and vaccination at different times with different levels of stringency and adherence. Initially, case numbers expand exponentially with doubling times of ~1-2 weeks. In the nations where interventions were not implemented or perhaps lees effectual, case numbers increased exponentially but then stabilized around 102-to-103 new infections (per km2 built-up area per day). Dynamics under effective interventions were perturbed and infections decayed to low levels. They rebounded concomitantly with the lifting of social distancing policies or pharmaceutical efficacy decline, converging on a stable equilibrium setpoint. Here we deploy a mathematical model which captures this V-shape behavior, incorporating a direct measure of intervention efficacy. Importantly, it allows the derivation of a maximal estimate for the basic reproductive number Ro (mean 1.6-1.8). We were able to test this approach by comparing the approximated "herd immunity" to the vaccination coverage observed that corresponded to rapid declines in community infections during 2021. The estimates reported here agree with the observed phenomena. Moreover, the decay (0.4-0.5) and rebound rates (0.2-0.3) were similar throughout the pandemic and among all the nations and regions studied. Finally, a longitudinal analysis comparing multiple national and regional results provides insights on the underlying epidemiology of SARS-CoV-2 and intervention efficacy, as well as evidence for the existence of an endemic steady state of COVID-19.
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Affiliation(s)
- David Burg
- Tel Hai Academic College, Qiryhat Shemona, Israel
- Hemdat Academic College, Netivot, Israel
- Ahskelon Academic College, Ashkelon, Israel
- Program for the Human Environment, The Rockefeller University, New York, NY, United States of America
| | - Jesse H. Ausubel
- Program for the Human Environment, The Rockefeller University, New York, NY, United States of America
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9
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Advances in Parameter Estimation and Learning from Data for Mathematical Models of Hepatitis C Viral Kinetics. MATHEMATICS 2022; 10. [DOI: 10.3390/math10122136] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/17/2022]
Abstract
Mathematical models, some of which incorporate both intracellular and extracellular hepatitis C viral kinetics, have been advanced in recent years for studying HCV–host dynamics, antivirals mode of action, and their efficacy. The standard ordinary differential equation (ODE) hepatitis C virus (HCV) kinetic model keeps track of uninfected cells, infected cells, and free virus. In multiscale models, a fourth partial differential equation (PDE) accounts for the intracellular viral RNA (vRNA) kinetics in an infected cell. The PDE multiscale model is substantially more difficult to solve compared to the standard ODE model, with governing differential equations that are stiff. In previous contributions, we developed and implemented stable and efficient numerical methods for the multiscale model for both the solution of the model equations and parameter estimation. In this contribution, we perform sensitivity analysis on model parameters to gain insight into important properties and to ensure our numerical methods can be safely used for HCV viral dynamic simulations. Furthermore, we generate in-silico patients using the multiscale models to perform machine learning from the data, which enables us to remove HCV measurements on certain days and still be able to estimate meaningful observations with a sufficiently small error.
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10
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Mondal J, Samui P, Chatterjee AN. Dynamical demeanour of SARS-CoV-2 virus undergoing immune response mechanism in COVID-19 pandemic. THE EUROPEAN PHYSICAL JOURNAL. SPECIAL TOPICS 2022; 231:3357-3370. [PMID: 35075384 PMCID: PMC8771633 DOI: 10.1140/epjs/s11734-022-00437-5] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/19/2021] [Accepted: 12/18/2021] [Indexed: 06/14/2023]
Abstract
COVID-19 is caused by the increase of SARS-CoV-2 viral load in the respiratory system. Epithelial cells in the human lower respiratory tract are the major target area of the SARS-CoV-2 viruses. To fight against the SARS-CoV-2 viral infection, innate and thereafter adaptive immune responses be activated which are stimulated by the infected epithelial cells. Strong immune response against the COVID-19 infection can lead to longer recovery time and less severe secondary complications. We proposed a target cell-limited mathematical model by considering a saturation term for SARS-CoV-2-infected epithelial cells loss reliant on infected cells level. The analytical findings reveal the conditions for which the system undergoes transcritical bifurcation and alternation of stability for the system around the steady states happens. Due to some external factors, while the viral reproduction rate exceeds its certain critical value, backward bifurcation and reinfection may take place and to inhibit these complicated epidemic states, host immune response, or immunopathology would play the essential role. Numerical simulation has been performed in support of the analytical findings.
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Affiliation(s)
- Jayanta Mondal
- Department of Mathematics, Diamond Harbour Women's University, Sarisha, West Bengal 743368 India
| | - Piu Samui
- Department of Mathematics, Diamond Harbour Women's University, Sarisha, West Bengal 743368 India
| | - Amar Nath Chatterjee
- Department of Mathematics, K. L. S. College, Nawada, Magadh University, Bodh Gaya, Bihar 805110 India
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11
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Li B, Jiao F. A delayed HIV-1 model with cell-to-cell spread and virus waning. JOURNAL OF BIOLOGICAL DYNAMICS 2020; 14:802-825. [PMID: 33084532 DOI: 10.1080/17513758.2020.1836272] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2020] [Accepted: 09/29/2020] [Indexed: 06/11/2023]
Abstract
In this paper, we propose and analyse a delayed HIV-1 model with both viral and cellular transmissions and virus waning. We obtain the threshold dynamics of the proposed model, characterized by the basic reproduction number R0 . If R0<1 , the infection-free steady state is globally asymptotically stable; whereas if R0>1 , the system is uniformly persistent. When the delays are positive, we show that the intracellular delays in both viral and cellular infections may lead to stability switches of the infected steady state. Both analytical and numerical results indicate that if the effect of cell-to-cell transmission is ignored, then the risk of HIV-1 infection will be underestimated. Moreover, the viral load of model without virus waning is higher than the one of model with virus waning. These results highlight the important role of two ways of viral transmission and virus waning on HIV-1 infection.
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Affiliation(s)
- Bing Li
- School of Mathematical Science, Harbin Normal University, Harbin, People's Republic of China
| | - Feng Jiao
- Center for Applied Mathematics, Guangzhou University, Guangzhou, People's Republic of China
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12
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Reinharz V, Churkin A, Lewkiewicz S, Dahari H, Barash D. A Parameter Estimation Method for Multiscale Models of Hepatitis C Virus Dynamics. Bull Math Biol 2019; 81:3675-3721. [PMID: 31338739 PMCID: PMC7375976 DOI: 10.1007/s11538-019-00644-7] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2018] [Accepted: 07/10/2019] [Indexed: 12/11/2022]
Abstract
Mathematical models that are based on differential equations require detailed knowledge about the parameters that are included in the equations. Some of the parameters can be measured experimentally while others need to be estimated. When the models become more sophisticated, such as in the case of multiscale models of hepatitis C virus dynamics that deal with partial differential equations (PDEs), several strategies can be tried. It is possible to use parameter estimation on an analytical approximation of the solution to the multiscale model equations, namely the long-term approximation, but this limits the scope of the parameter estimation method used and a long-term approximation needs to be derived for each model. It is possible to transform the PDE multiscale model to a system of ODEs, but this has an effect on the model parameters themselves and the transformation can become problematic for some models. Finally, it is possible to use numerical solutions for the multiscale model and then use canned methods for the parameter estimation, but the latter is making the user dependent on a black box without having full control over the method. The strategy developed here is to start by working directly on the multiscale model equations for preparing them toward the parameter estimation method that is fully coded and controlled by the user. It can also be adapted to multiscale models of other viruses. The new method is described, and illustrations are provided using a user-friendly simulator that incorporates the method.
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Affiliation(s)
- Vladimir Reinharz
- Department of Computer Science, Ben-Gurion University, Beersheba, Israel
| | - Alexander Churkin
- Department of Software Engineering, Sami Shamoon College of Engineering, Beersheba, Israel
| | - Stephanie Lewkiewicz
- Department of Mathematics, University of California at Los Angeles, Los Angeles, CA, USA
| | - Harel Dahari
- Program for Experimental and Theoretical Modeling, Division of Hepatology, Department of Medicine, Loyola University Medical Center, Maywoood, IL, USA
| | - Danny Barash
- Department of Computer Science, Ben-Gurion University, Beersheba, Israel.
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13
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Sahani SK. A delayed HIV infection model with apoptosis and viral loss. JOURNAL OF BIOLOGICAL DYNAMICS 2018; 12:1012-1034. [PMID: 30462570 DOI: 10.1080/17513758.2018.1547427] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/11/2017] [Accepted: 11/06/2018] [Indexed: 06/09/2023]
Abstract
In this paper, a delayed human immunodeficiency virus (HIV) model with apoptosis of cells has been studied. Both immunological and intracellular delay have been incorporated to make the model more relevant. Firstly, the model has been investigated using local stability analysis. Next, the global stability analysis of steady states has been performed. The stability switch criteria taking the delay as the bifurcating parameter, leading to Hopf bifurcation has been studied. The transition of the system from order to chaos has been explored, and the analytical results have been verified by numerical simulations. The results thus can be used to describe the extensive dynamics exhibited by the model introduced in this article. The effects of apoptosis on viral load has been studied in the model numerically.
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Affiliation(s)
- Saroj Kumar Sahani
- a Faculty of Mathematics & Computer Science , South Asian University , New Delhi , India
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14
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Magombedze G, Marino S. Mathematical and computational approaches in understanding the immunobiology of granulomatous diseases. ACTA ACUST UNITED AC 2018. [DOI: 10.1016/j.coisb.2018.07.002] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/27/2023]
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15
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Abstract
The interplay between immune response and HIV is intensely studied via mathematical modeling, with significant insights but few direct answers. In this short review, we highlight advances and knowledge gaps across different aspects of immunity. In particular, we identify the innate immune response and its role in priming the adaptive response as ripe for modeling. The latter have been the focus of most modeling studies, but we also synthesize key outstanding questions regarding effector mechanisms of cellular immunity and development of broadly neutralizing antibodies. Thus far, most modeling studies aimed to infer general immune mechanisms; we foresee that significant progress will be made next by detailed quantitative fitting of models to data, and prediction of immune responses.
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Affiliation(s)
- Jessica M Conway
- Department of Mathematics and Center for Infectious Disease Dynamics, Pennsylvania State University, University Park PA 16802, USA
| | - Ruy M Ribeiro
- Laboratorio de Biomatematica, Faculdade de Medicina da Universidade de Lisboa, Portugal and Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
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16
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Abstract
Viruses are a main cause of disease worldwide and many are without effective therapeutics or vaccines. A lack of understanding about how host responses work to control viral spread is one factor limiting effective management. How different immune components regulate infection dynamics is beginning to be better understood with the help of mathematical models. These models have been key in discriminating between hypotheses and in identifying rates of virus growth and clearance, dynamical control by different host factors and antivirals, and synergistic interactions during multi-pathogen infections. A recent focus in evaluating model predictions in the laboratory and clinic has illuminate the accuracy of models for a variety of viruses and highlighted the critical nature of theoretical approaches in virology. Here, I discuss recent model-driven exploration of host-pathogen interactions that have illustrated the importance of model validation in establishing the model's predictive capability and in defining new biology.
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17
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Cicchese JM, Evans S, Hult C, Joslyn LR, Wessler T, Millar JA, Marino S, Cilfone NA, Mattila JT, Linderman JJ, Kirschner DE. Dynamic balance of pro- and anti-inflammatory signals controls disease and limits pathology. Immunol Rev 2018; 285:147-167. [PMID: 30129209 PMCID: PMC6292442 DOI: 10.1111/imr.12671] [Citation(s) in RCA: 145] [Impact Index Per Article: 24.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023]
Abstract
Immune responses to pathogens are complex and not well understood in many diseases, and this is especially true for infections by persistent pathogens. One mechanism that allows for long-term control of infection while also preventing an over-zealous inflammatory response from causing extensive tissue damage is for the immune system to balance pro- and anti-inflammatory cells and signals. This balance is dynamic and the immune system responds to cues from both host and pathogen, maintaining a steady state across multiple scales through continuous feedback. Identifying the signals, cells, cytokines, and other immune response factors that mediate this balance over time has been difficult using traditional research strategies. Computational modeling studies based on data from traditional systems can identify how this balance contributes to immunity. Here we provide evidence from both experimental and mathematical/computational studies to support the concept of a dynamic balance operating during persistent and other infection scenarios. We focus mainly on tuberculosis, currently the leading cause of death due to infectious disease in the world, and also provide evidence for other infections. A better understanding of the dynamically balanced immune response can help shape treatment strategies that utilize both drugs and host-directed therapies.
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Affiliation(s)
- Joseph M. Cicchese
- Department of Chemical Engineering, University of Michigan, Ann Arbor, MI, USA
| | - Stephanie Evans
- Department of Microbiology and Immunology, University of Michigan Medical School, Ann Arbor, MI, USA
| | - Caitlin Hult
- Department of Chemical Engineering, University of Michigan, Ann Arbor, MI, USA
- Department of Microbiology and Immunology, University of Michigan Medical School, Ann Arbor, MI, USA
| | - Louis R. Joslyn
- Department of Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI, USA
| | - Timothy Wessler
- Department of Chemical Engineering, University of Michigan, Ann Arbor, MI, USA
- Department of Microbiology and Immunology, University of Michigan Medical School, Ann Arbor, MI, USA
| | - Jess A. Millar
- Department of Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI, USA
| | - Simeone Marino
- Department of Microbiology and Immunology, University of Michigan Medical School, Ann Arbor, MI, USA
| | - Nicholas A. Cilfone
- Department of Chemical Engineering, University of Michigan, Ann Arbor, MI, USA
| | - Joshua T. Mattila
- Department of Infectious Diseases and Microbiology, University of Pittsburgh, Pittsburgh, PA, USA
| | | | - Denise E. Kirschner
- Department of Microbiology and Immunology, University of Michigan Medical School, Ann Arbor, MI, USA
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18
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Hill AL, Rosenbloom DIS, Nowak MA, Siliciano RF. Insight into treatment of HIV infection from viral dynamics models. Immunol Rev 2018; 285:9-25. [PMID: 30129208 PMCID: PMC6155466 DOI: 10.1111/imr.12698] [Citation(s) in RCA: 33] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/26/2022]
Abstract
The odds of living a long and healthy life with HIV infection have dramatically improved with the advent of combination antiretroviral therapy. Along with the early development and clinical trials of these drugs, and new field of research emerged called viral dynamics, which uses mathematical models to interpret and predict the time-course of viral levels during infection and how they are altered by treatment. In this review, we summarize the contributions that virus dynamics models have made to understanding the pathophysiology of infection and to designing effective therapies. This includes studies of the multiphasic decay of viral load when antiretroviral therapy is given, the evolution of drug resistance, the long-term persistence latently infected cells, and the rebound of viremia when drugs are stopped. We additionally discuss new work applying viral dynamics models to new classes of investigational treatment for HIV, including latency-reversing agents and immunotherapy.
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Affiliation(s)
- Alison L. Hill
- Program for Evolutionary DynamicsHarvard UniversityCambridgeMassachusetts
| | - Daniel I. S. Rosenbloom
- Department of PharmacokineticsPharmacodynamics, & Drug MetabolismMerck Research LaboratoriesKenilworthNew Jersey
| | - Martin A. Nowak
- Program for Evolutionary DynamicsHarvard UniversityCambridgeMassachusetts
| | - Robert F. Siliciano
- Department of MedicineJohns Hopkins University School of MedicineBaltimoreMaryland
- Howard Hughes Medical InstituteBaltimoreMaryland
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19
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Abstract
Influenza virus infections are a leading cause of morbidity and mortality worldwide. This is due in part to the continual emergence of new viral variants and to synergistic interactions with other viruses and bacteria. There is a lack of understanding about how host responses work to control the infection and how other pathogens capitalize on the altered immune state. The complexity of multi-pathogen infections makes dissecting contributing mechanisms, which may be non-linear and occur on different time scales, challenging. Fortunately, mathematical models have been able to uncover infection control mechanisms, establish regulatory feedbacks, connect mechanisms across time scales, and determine the processes that dictate different disease outcomes. These models have tested existing hypotheses and generated new hypotheses, some of which have been subsequently tested and validated in the laboratory. They have been particularly a key in studying influenza-bacteria coinfections and will be undoubtedly be useful in examining the interplay between influenza virus and other viruses. Here, I review recent advances in modeling influenza-related infections, the novel biological insight that has been gained through modeling, the importance of model-driven experimental design, and future directions of the field.
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Affiliation(s)
- Amber M Smith
- University of Tennessee Health Science CenterMemphisTNUSA
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20
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Smith AP, Moquin DJ, Bernhauerova V, Smith AM. Influenza Virus Infection Model With Density Dependence Supports Biphasic Viral Decay. Front Microbiol 2018; 9:1554. [PMID: 30042759 PMCID: PMC6048257 DOI: 10.3389/fmicb.2018.01554] [Citation(s) in RCA: 32] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2018] [Accepted: 06/22/2018] [Indexed: 01/13/2023] Open
Abstract
Mathematical models that describe infection kinetics help elucidate the time scales, effectiveness, and mechanisms underlying viral growth and infection resolution. For influenza A virus (IAV) infections, the standard viral kinetic model has been used to investigate the effect of different IAV proteins, immune mechanisms, antiviral actions, and bacterial coinfection, among others. We sought to further define the kinetics of IAV infections by infecting mice with influenza A/PR8 and measuring viral loads with high frequency and precision over the course of infection. The data highlighted dynamics that were not previously noted, including viral titers that remain elevated for several days during mid-infection and a sharp 4–5 log10 decline in virus within 1 day as the infection resolves. The standard viral kinetic model, which has been widely used within the field, could not capture these dynamics. Thus, we developed a new model that could simultaneously quantify the different phases of viral growth and decay with high accuracy. The model suggests that the slow and fast phases of virus decay are due to the infected cell clearance rate changing as the density of infected cells changes. To characterize this model, we fit the model to the viral load data, examined the parameter behavior, and connected the results and parameters to linear regression estimates. The resulting parameters and model dynamics revealed that the rate of viral clearance during resolution occurs 25 times faster than the clearance during mid-infection and that small decreases to this rate can significantly prolong the infection. This likely reflects the high efficiency of the adaptive immune response. The new model provides a well-characterized representation of IAV infection dynamics, is useful for analyzing and interpreting viral load dynamics in the absence of immunological data, and gives further insight into the regulation of viral control.
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Affiliation(s)
- Amanda P Smith
- Department of Pediatrics, University of Tennessee Health Science Center, Memphis, TN, United States
| | - David J Moquin
- Department of Internal Medicine, University of Tennessee Health Science Center, Memphis, TN, United States
| | | | - Amber M Smith
- Department of Pediatrics, University of Tennessee Health Science Center, Memphis, TN, United States
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21
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Reinharz V, Dahari H, Barash D. Numerical schemes for solving and optimizing multiscale models with age of hepatitis C virus dynamics. Math Biosci 2018; 300:1-13. [PMID: 29550297 PMCID: PMC5992100 DOI: 10.1016/j.mbs.2018.03.011] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2018] [Accepted: 03/07/2018] [Indexed: 12/16/2022]
Abstract
Age-structured PDE models have been developed to study viral infection and treatment. However, they are notoriously difficult to solve. Here, we investigate the numerical solutions of an age-based multiscale model of hepatitis C virus (HCV) dynamics during antiviral therapy and compare them with an analytical approximation, namely its long-term approximation. First, starting from a simple yet flexible numerical solution that also considers an integral approximated over previous iterations, we show that the long-term approximation is an underestimate of the PDE model solution as expected since some infection events are being ignored. We then argue for the importance of having a numerical solution that takes into account previous iterations for the associated integral, making problematic the use of canned solvers. Second, we demonstrate that the governing differential equations are stiff and the stability of the numerical scheme should be considered. Third, we show that considerable gain in efficiency can be achieved by using adaptive stepsize methods over fixed stepsize methods for simulating realistic scenarios when solving multiscale models numerically. Finally, we compare between several numerical schemes for the solution of the equations and demonstrate the use of a numerical optimization scheme for the parameter estimation performed directly from the equations.
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Affiliation(s)
- Vladimir Reinharz
- Department of Computer Science, Ben-Gurion University, Beer-Sheva 84105, Israel.
| | - Harel Dahari
- The Program for Experimental & Theoretical Modeling, Division of Hepatology, Department of Medicine, Loyola University Medical Center, Maywood, IL 60153, USA
| | - Danny Barash
- Department of Computer Science, Ben-Gurion University, Beer-Sheva 84105, Israel.
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22
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Cangelosi RA, Schwartz EJ, Wollkind DJ. A Quasi-Steady-State Approximation to the Basic Target-Cell-Limited Viral Dynamics Model with a Non-Cytopathic Effect. Front Microbiol 2018; 9:54. [PMID: 29445361 PMCID: PMC5797985 DOI: 10.3389/fmicb.2018.00054] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/01/2017] [Accepted: 01/10/2018] [Indexed: 11/27/2022] Open
Abstract
Analysis of previously published target-cell limited viral dynamic models for pathogens such as HIV, hepatitis, and influenza generally rely on standard techniques from dynamical systems theory or numerical simulation. We use a quasi-steady-state approximation to derive an analytic solution for the model with a non-cytopathic effect, that is, when the death rates of uninfected and infected cells are equal. The analytic solution provides time evolution values of all three compartments of uninfected cells, infected cells, and virus. Results are compared with numerical simulation using clinical data for equine infectious anemia virus, a retrovirus closely related to HIV, and the utility of the analytic solution is discussed.
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Affiliation(s)
| | - Elissa J Schwartz
- Department of Mathematics and Statistics, Washington State University, Pullman, WA, United States.,School of Biological Sciences, Washington State University, Pullman, WA, United States
| | - David J Wollkind
- Department of Mathematics and Statistics, Washington State University, Pullman, WA, United States
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23
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Abstract
Some HIV-infected patients (the so-called post-treatment controllers) can control the virus after cessation of antiretroviral therapy. A small fraction of patients can even naturally maintain undetectable viral load without therapy (they are called elite controllers). The immune response may play an important role in viral control in these patients. In this paper, we analyze a within-host model including immune response to study the virus dynamics in HIV-infected patients. We derived two threshold values for the immune cell proliferation parameter. Below the lower immune proliferation rate, the model has a stable immune-free steady state, which predicts that patients have a high viral load. Above the higher immune proliferation rate, the model has a stable low infected steady state, which indicates that patients are under elite control. Between the two immune thresholds, the model exhibits the dynamic behavior of bistability, which suggests that patients either undergo viral rebound after treatment termination or achieve the post-treatment control. These results may explain the different virus dynamics in HIV-infected patients.
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Affiliation(s)
- SHAOLI WANG
- School of Mathematics and Statistics, Henan University, Kaifeng, Henan 475001, P. R. China
| | - FEI XU
- Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario N2L 3C5, Canada
| | - LIBIN RONG
- Department of Mathematics, University of Florida, Gainesville FL 32611, USA
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24
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Reinharz V, Churkin A, Dahari H, Barash D. A Robust and Efficient Numerical Method for RNA-Mediated Viral Dynamics. FRONTIERS IN APPLIED MATHEMATICS AND STATISTICS 2017; 3:20. [PMID: 30854378 PMCID: PMC6404971 DOI: 10.3389/fams.2017.00020] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
The multiscale model of hepatitis C virus (HCV) dynamics, which includes intracellular viral RNA (vRNA) replication, has been formulated in recent years in order to provide a new conceptual framework for understanding the mechanism of action of a variety of agents for the treatment of HCV. We present a robust and efficient numerical method that belongs to the family of adaptive stepsize methods and is implicit, a Rosenbrock type method that is highly suited to solve this problem. We provide a Graphical User Interface that applies this method and is useful for simulating viral dynamics during treatment with anti-HCV agents that act against HCV on the molecular level.
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Affiliation(s)
- Vladimir Reinharz
- Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel
| | - Alexander Churkin
- Department of Software Engineering, Sami Shamoon College of Engineering, Beer-Sheva, Israel
| | - Harel Dahari
- Program for Experimental and Theoretical Modeling, Division of Hepatology, Department of Medicine, Loyola University Medical Center, Maywood, IL, United States
| | - Danny Barash
- Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel
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25
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Abstract
Viral latency is a major barrier to curing HIV infection with antiretroviral therapy, and consequently, for eliminating the disease globally. The establishment, maintenance, and potential clearance of latent infection are complex dynamic processes and can be best understood and described with the help of mathematical models. Here we review the use of viral dynamics models for HIV, with a focus on applications to the latent reservoir. Such models have been used to explain the multiphasic decay of viral load during antiretroviral therapy, the early seeding of the latent reservoir during acute infection and the limited inflow during treatment, the dynamics of viral blips, and the phenomenon of posttreatment control. In addition, mathematical models have been used to predict the efficacy of potential HIV cure strategies, such as latency-reversing agents, early treatment initiation, or gene therapies, and to provide guidance for designing trials of these novel interventions.
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26
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Ad26/MVA therapeutic vaccination with TLR7 stimulation in SIV-infected rhesus monkeys. Nature 2016; 540:284-287. [PMID: 27841870 PMCID: PMC5145754 DOI: 10.1038/nature20583] [Citation(s) in RCA: 220] [Impact Index Per Article: 27.5] [Reference Citation Analysis] [Abstract] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/03/2016] [Accepted: 11/01/2016] [Indexed: 02/07/2023]
Abstract
The development of immunologic interventions that can target the viral reservoir in HIV-1-infected individuals is a major goal of the HIV-1 cure field1,2. However, little evidence exists that the viral reservoir can be sufficiently targeted to improve virologic control following discontinuation of antiretroviral therapy (ART). Here we show that Ad26/MVA3,4 therapeutic vaccination with toll-like receptor 7 (TLR7) stimulation improves virologic control and delays viral rebound following ART discontinuation in SIV-infected rhesus monkeys that initiated ART during acute infection. Ad26/MVA therapeutic vaccination resulted in a dramatic increase in the magnitude and breadth of SIV-specific cellular immune responses in virologically suppressed, SIV-infected monkeys. TLR7 agonist administration led to innate immune stimulation and cellular immune activation. The combination of Ad26/MVA vaccination and TLR7 stimulation resulted in decreased levels of viral DNA in lymph nodes and peripheral blood, as well as improved virologic control and delayed viral rebound following ART discontinuation. Cellular immune breadth correlated inversely with setpoint viral loads and correlated directly with time to viral rebound. These data demonstrate the potential of therapeutic vaccination with innate immune stimulation as a strategy aimed at an HIV-1 functional cure.
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27
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Mayer BT, Matrajt L, Casper C, Krantz EM, Corey L, Wald A, Gantt S, Schiffer JT. Dynamics of Persistent Oral Cytomegalovirus Shedding During Primary Infection in Ugandan Infants. J Infect Dis 2016; 214:1735-1743. [PMID: 27651417 DOI: 10.1093/infdis/jiw442] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/02/2016] [Accepted: 09/12/2016] [Indexed: 01/06/2023] Open
Abstract
Cytomegalovirus (CMV) infection occurs frequently in young children, who, when infected, are then a major source of transmission. Oral CMV shedding by 14 infants with primary infection was comprehensively characterized using quantitative polymerase chain reaction weekly for ≥9 months. Three phases of oral shedding were identified: expansion, transition, and clearance. Viral expansion occurred over a median of 7 weeks, with a median doubling time of 3 days. During the transition phase, expansion slowed over a median of 6 weeks before peak viral load was reached. Clearance was slow (22-day median half-life), and shedding did not resolve during observation for any infant. Mathematical modeling demonstrated that prolonged oral CMV expansion is explained by a low within-host reproduction number (median, 1.63) and a delayed immune response that only decreases the infected cell half-life by 44%. Thus, the prolonged oral CMV shedding observed during primary infection can be explained by slow viral expansion and inefficient immunologic control.
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Affiliation(s)
| | | | - Corey Casper
- Vaccine and Infectious Disease Division.,Clinical Research Division.,Public Health Science Division, Fred Hutchinson Cancer Research Center.,Department of Medicine.,Department of Epidemiology.,Department of Global Health
| | | | - Lawrence Corey
- Vaccine and Infectious Disease Division.,Clinical Research Division.,Department of Medicine.,Department of Laboratory Medicine, University of Washington, Seattle
| | - Anna Wald
- Vaccine and Infectious Disease Division.,Department of Medicine.,Department of Epidemiology.,Department of Laboratory Medicine, University of Washington, Seattle
| | - Soren Gantt
- Vaccine and Infectious Disease Division.,BC Children's Hospital, University of British Columbia, Vancouver, Canada
| | - Joshua T Schiffer
- Vaccine and Infectious Disease Division.,Clinical Research Division.,Department of Medicine
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28
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Vaidya NK, Ribeiro RM, Perelson AS, Kumar A. Modeling the Effects of Morphine on Simian Immunodeficiency Virus Dynamics. PLoS Comput Biol 2016; 12:e1005127. [PMID: 27668463 PMCID: PMC5036892 DOI: 10.1371/journal.pcbi.1005127] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2015] [Accepted: 08/25/2016] [Indexed: 12/20/2022] Open
Abstract
Complications of HIV-1 infection in individuals who utilize drugs of abuse is a significant problem, because these drugs have been associated with higher virus replication and accelerated disease progression as well as severe neuropathogenesis. To gain further insight it is important to quantify the effects of drugs of abuse on HIV-1 infection dynamics. Here, we develop a mathematical model that incorporates experimentally observed effects of morphine on inducing HIV-1 co-receptor expression. For comparison we also considered viral dynamic models with cytolytic or noncytolytic effector cell responses. Based on the small sample size Akaike information criterion, these models were inferior to the new model based on changes in co-receptor expression. The model with morphine affecting co-receptor expression agrees well with the experimental data from simian immunodeficiency virus infections in morphine-addicted macaques. Our results show that morphine promotes a target cell subpopulation switch from a lower level of susceptibility to a state that is about 2-orders of magnitude higher in susceptibility to SIV infection. As a result, the proportion of target cells with higher susceptibility remains extremely high in morphine conditioning. Such a morphine-induced population switch not only has adverse effects on the replication rate, but also results in a higher steady state viral load and larger CD4 count drops. Moreover, morphine conditioning may pose extra obstacles to controlling viral load during antiretroviral therapy, such as pre-exposure prophylaxis and post infection treatments. This study provides, for the first time, a viral dynamics model, viral dynamics parameters, and related analytical and simulation results for SIV dynamics under drugs of abuse.
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Affiliation(s)
- Naveen K. Vaidya
- Department of Mathematics and Statistics, University of Missouri-Kansas City, Missouri, United States of America
- Division of Pharmacology, School of Pharmacy, University of Missouri-Kansas City, Missouri, United States of America
| | - Ruy M. Ribeiro
- Theoretical Biology and Biophysics Group, Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America
| | - Alan S. Perelson
- Theoretical Biology and Biophysics Group, Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America
| | - Anil Kumar
- Division of Pharmacology, School of Pharmacy, University of Missouri-Kansas City, Missouri, United States of America
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29
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Li B, Chen Y, Lu X, Liu S. A delayed HIV-1 model with virus waning term. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2016; 13:135-157. [PMID: 26776264 DOI: 10.3934/mbe.2016.13.135] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
In this paper, we propose and analyze a delayed HIV-1 model with CTL immune response and virus waning. The two discrete delays stand for the time for infected cells to produce viruses after viral entry and for the time for CD8+ T cell immune response to emerge to control viral replication. We obtain the positiveness and boundedness of solutions and find the basic reproduction number R0. If R0 < 1, then the infection-free steady state is globally asymptotically stable and the infection is cleared from the T-cell population; whereas if R0 > 1, then the system is uniformly persistent and the viral concentration maintains at some constant level. The global dynamics when R0 > 1 is complicated. We establish the local stability of the infected steady state and show that Hopf bifurcation can occur. Both analytical and numerical results indicate that if, in the initial infection stage, the effect of delays on HIV-1 infection is ignored, then the risk of HIV-1 infection (if persists) will be underestimated. Moreover, the viral load differs from that without virus waning. These results highlight the important role of delays and virus waning on HIV-1 infection.
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Affiliation(s)
- Bing Li
- Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, 3041#, 2 Yi-Kuang street, Harbin, 150080, China.
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30
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Bibi Z, Ahmad J, Ali A, Siddiqa A, Shahzad S, Tareen SHK, Janjua HA, Khusro S. On the modeling and analysis of the biological regulatory network of NF-$${\kappa }$$B activation in HIV-1 infection. ACTA ACUST UNITED AC 2016. [DOI: 10.1186/s40294-015-0013-4] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/09/2023]
Abstract
Abstract
Purpose
The complex interactions between genetic machinery of HIV-1 and host immune cells mediate dynamic adaptive responses leading to Autoimmune Deficiency Syndrome. These interactions are captured as Biological Regulatory Network (BRN) which acts to maintain the viability of host cell machinery through feedback control mechanism which is a characteristic of complex adaptive systems. In this study, the BRN of immune response against HIV-1 infection is modeled to investigate the role of NF-κB and TNF-α in disease transmission using qualitative (discrete) and hybrid modeling formalisms.
Methods
Qualitative and Hybrid modeling approaches are used to model the BRN for the dynamic analysis. The qualitative model is based on the logical parameters while the hybrid model is based on the time delay parameters.
Results
The qualitative model gives useful insights about the physiological condition observed as the homeostasis of all the entities of the BRN as well as pathophysiological behaviors representing high expression levels of NF-κB, TNF-α and HIV. Since the qualitative model is time abstracted, so a hybrid model is developed to analyze the behavior of the BRN by associating activation and inhibition time delays with each entity. HyTech tool synthesizes time delay constraints for the existence of homeostasis.
Conclusion
Hybrid model reveals various viability constraints that characterize the conditional existence of cyclic states (homeostasis). The resultant relations suggest larger cycle period of HIV-1 than the cycle periods of the other two entities (NF-κB and TNF-α) to maintain a homeostatic expressions of these entities.
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31
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Miranda PJ, Delgobo M, Marino GF, Paludo KS, da Silva Baptista M, de Souza Pinto SE. The Oral Tolerance as a Complex Network Phenomenon. PLoS One 2015; 10:e0130762. [PMID: 26115356 PMCID: PMC4483238 DOI: 10.1371/journal.pone.0130762] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2015] [Accepted: 05/23/2015] [Indexed: 11/18/2022] Open
Abstract
The phenomenon of oral tolerance refers to a local and systemic state of tolerance induced in the gut after its exposure to innocuous antigens. Recent findings have shown the interrelationship between cellular and molecular components of oral tolerance, but its representation through a network of interactions has not been investigated. Our work aims at identifying the causal relationship of each element in an oral tolerance network, and also to propose a phenomenological model that's capable of predicting the stochastic behavior of this network when under manipulation. We compared the changes of a "healthy" network caused by "knock-outs" (KOs) in two approaches: an analytical approach by the Perron Frobenius theory; and a computational approach, which we describe within this work in order to find numerical results for the model. Both approaches have shown the most relevant immunological components for this phenomena, that happens to corroborate the empirical results from animal models. Besides explain in a intelligible fashion how the components interacts in a complex manner, we also managed to describe and quantify the importance of KOs that hasn't been empirically tested.
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Affiliation(s)
| | - Murilo Delgobo
- Department of Biology, State University of Ponta Grossa, Ponta Grossa, Paraná, Brazil
| | - Giovani Favero Marino
- Department of Biology, State University of Ponta Grossa, Ponta Grossa, Paraná, Brazil
| | - Kátia Sabrina Paludo
- Department of Structural Biology, Molecular and Genetics, State University of Ponta Grossa, Ponta Grossa, Paraná, Brazil
| | - Murilo da Silva Baptista
- Institute for Complex Systems and Mathematical Biology, SUPA, University of Aberdeen, Aberdeen, United Kingdom
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The majority of CD4+ T-cell depletion during acute simian-human immunodeficiency virus SHIV89.6P infection occurs in uninfected cells. J Virol 2014; 88:3202-12. [PMID: 24390339 DOI: 10.1128/jvi.03428-13] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022] Open
Abstract
UNLABELLED Untreated human immunodeficiency virus (HIV) infection is characterized by depletion of CD4(+) T cells, ultimately leading to the impairment of host immune defenses and death. HIV-infected CD4(+) T cells die from direct virus-induced apoptosis and CD8 T-cell-mediated elimination, but a broader and more profound depletion occurs in uninfected CD4(+) T cells via multiple indirect effects of infection. We fit mathematical models to data from experiments that tested an HIV eradication strategy in which five macaques with a proportion of CD4(+) T cells resistant to simian-human immunodeficiency virus (SHIV) entry were challenged with SHIV89.6P, a highly pathogenic dual-tropic chimeric SIV-HIV viral strain that results in rapid loss of both SHIV-susceptible and SHIV-resistant CD4(+) T cells. Our results suggest that uninfected (bystander) cell death accounts for the majority of CD4(+) T-lymphocyte loss, with at least 60% and 99% of CD4(+) T cell death occurring in uninfected cells during acute and established infection, respectively. Mechanisms to limit the profound indirect killing effects associated with HIV infection may be associated with immune preservation and improved long-term survival. IMPORTANCE HIV infection induces a massive depletion of CD4(+) T cells, leading to profound immunodeficiency, opportunistic infections, and eventually death. While HIV induces apoptosis (programmed cell death) by directly entering and replicating in CD4(+) T cells, uninfected CD4(+) T cells also undergo apoptosis due to ongoing toxic inflammation in the region of infection. In this paper, we use mathematical models in conjunction with data from simian-human immunodeficiency virus SHIV89.6P infection in macaques (a model of HIV infection in humans) to estimate the percentage of cell death that occurs in uninfected cells during the initial period of infection. We reveal that the vast majority of cell death occurs in these cells, which are not infected. The "bystander effects" that lead to enormous reductions in the number of uninfected CD4(+) T cells may be a target for future interventions that aim to limit the extent of damage caused by HIV.
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Vidurupola SW, Allen LJS. Basic stochastic models for viral infection within a host. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2012; 9:915-935. [PMID: 23311428 DOI: 10.3934/mbe.2012.9.915] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Stochastic differential equation (SDE) models are formulated for intra-host virus-cell dynamics during the early stages of viral infection, prior to activation of the immune system. The SDE models incorporate more realism into the mechanisms for viral entry and release than ordinary differential equation (ODE) models and show distinct differences from the ODE models. The variability in the SDE models depends on the concentration, with much greater variability for small concentrations than large concentrations. In addition, the SDE models show significant variability in the timing of the viral peak. The viral peak is earlier for viruses that are released from infected cells via bursting rather than via budding from the cell membrane.
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Affiliation(s)
- Sukhitha W Vidurupola
- Texas Tech University, Department of Mathematics and Statistics, Lubbock, Texas 79409-1042, United States.
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34
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Uncertainty quantification in simulations of epidemics using polynomial chaos. COMPUTATIONAL AND MATHEMATICAL METHODS IN MEDICINE 2012; 2012:742086. [PMID: 22927889 PMCID: PMC3426294 DOI: 10.1155/2012/742086] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/08/2012] [Revised: 06/25/2012] [Accepted: 07/03/2012] [Indexed: 11/17/2022]
Abstract
Mathematical models based on ordinary differential equations are a useful tool to study the processes involved in epidemiology. Many models consider that the parameters are deterministic variables. But in practice, the transmission parameters present large variability and it is not possible to determine them exactly, and it is necessary to introduce randomness. In this paper, we present an application of the polynomial chaos approach to epidemiological mathematical models based on ordinary differential equations with random coefficients. Taking into account the variability of the transmission parameters of the model, this approach allows us to obtain an auxiliary system of differential equations, which is then integrated numerically to obtain the first-and the second-order moments of the output stochastic processes. A sensitivity analysis based on the polynomial chaos approach is also performed to determine which parameters have the greatest influence on the results. As an example, we will apply the approach to an obesity epidemic model.
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35
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Kinetic model of HIV infection including hematopoietic progenitor cells. Math Biosci 2012; 236:36-43. [DOI: 10.1016/j.mbs.2012.01.003] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/20/2010] [Revised: 01/10/2012] [Accepted: 01/13/2012] [Indexed: 12/16/2022]
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36
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Optimal control of drug therapy: Melding pharmacokinetics with viral dynamics. Biosystems 2012; 107:174-85. [DOI: 10.1016/j.biosystems.2011.11.011] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/14/2011] [Revised: 11/26/2011] [Accepted: 11/28/2011] [Indexed: 11/19/2022]
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37
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Stochastic models for virus and immune system dynamics. Math Biosci 2011; 234:84-94. [PMID: 21945381 DOI: 10.1016/j.mbs.2011.08.007] [Citation(s) in RCA: 37] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/03/2011] [Revised: 06/10/2011] [Accepted: 08/26/2011] [Indexed: 11/23/2022]
Abstract
New stochastic models are developed for the dynamics of a viral infection and an immune response during the early stages of infection. The stochastic models are derived based on the dynamics of deterministic models. The simplest deterministic model is a well-known system of ordinary differential equations which consists of three populations: uninfected cells, actively infected cells, and virus particles. This basic model is extended to include some factors of the immune response related to Human Immunodeficiency Virus-1 (HIV-1) infection. For the deterministic models, the basic reproduction number, R0, is calculated and it is shown that if R0<1, the disease-free equilibrium is locally asymptotically stable and is globally asymptotically stable in some special cases. The new stochastic models are systems of stochastic differential equations (SDEs) and continuous-time Markov chain (CTMC) models that account for the variability in cellular reproduction and death, the infection process, the immune system activation, and viral reproduction. Two viral release strategies are considered: budding and bursting. The CTMC model is used to estimate the probability of virus extinction during the early stages of infection. Numerical simulations are carried out using parameter values applicable to HIV-1 dynamics. The stochastic models provide new insights, distinct from the basic deterministic models. For the case R0>1, the deterministic models predict the viral infection persists in the host. But for the stochastic models, there is a positive probability of viral extinction. It is shown that the probability of a successful invasion depends on the initial viral dose, whether the immune system is activated, and whether the release strategy is bursting or budding.
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Quantification of the relative importance of CTL, B cell, NK cell, and target cell limitation in the control of primary SIV-infection. PLoS Comput Biol 2011; 7:e1001103. [PMID: 21408213 PMCID: PMC3048377 DOI: 10.1371/journal.pcbi.1001103] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/10/2010] [Accepted: 01/28/2011] [Indexed: 01/22/2023] Open
Abstract
CD8+ cytotoxic T lymphocytes (CTLs), natural killer (NK) cells, B cells and target cell limitation have all been suggested to play a role in the control of SIV and HIV-1 infection. However, previous research typically studied each population in isolation leaving the magnitude, relative importance and in vivo relevance of each effect unclear. Here we quantify the relative importance of CTLs, NK cells, B cells and target cell limitation in controlling acute SIV infection in rhesus macaques. Using three different methods, we find that the availability of target cells and CD8+ T cells are important predictors of viral load dynamics. If CTL are assumed to mediate this anti-viral effect via a lytic mechanism then we estimate that CTL killing is responsible for approximately 40% of productively infected cell death, the remaining cell death being attributable to intrinsic, immune (CD8+ T cell, NK cell, B cell) -independent mechanisms. Furthermore, we find that NK cells have little impact on the death rate of infected CD4+ cells and that their net impact is to increase viral load. We hypothesize that NK cells play a detrimental role in SIV infection, possibly by increasing T cell activation.
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Drylewicz J, Guedj J, Commenges D, Thiébaut R. Modeling the dynamics of biomarkers during primary HIV infection taking into account the uncertainty of infection date. Ann Appl Stat 2010. [DOI: 10.1214/10-aoas364] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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40
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Saccomani MP. An effective automatic procedure for testing parameter identifiability of HIV/AIDS models. Bull Math Biol 2010; 73:1734-53. [PMID: 20953911 DOI: 10.1007/s11538-010-9588-2] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2010] [Accepted: 09/24/2010] [Indexed: 11/26/2022]
Abstract
Realistic HIV models tend to be rather complex and many recent models proposed in the literature could not yet be analyzed by traditional identifiability testing techniques. In this paper, we check a priori global identifiability of some of these nonlinear HIV models taken from the recent literature, by using a differential algebra algorithm based on previous work of the author. The algorithm is implemented in a software tool, called DAISY (Differential Algebra for Identifiability of SYstems), which has been recently released (DAISY is freely available on the web site http://www.dei.unipd.it/~pia/ ). The software can be used to automatically check global identifiability of (linear and) nonlinear models described by polynomial or rational differential equations, thus providing a general and reliable tool to test global identifiability of several HIV models proposed in the literature. It can be used by researchers with a minimum of mathematical background.
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Viral dynamics during primary simian immunodeficiency virus infection: effect of time-dependent virus infectivity. J Virol 2010; 84:4302-10. [PMID: 20147390 DOI: 10.1128/jvi.02284-09] [Citation(s) in RCA: 39] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/21/2022] Open
Abstract
A recent experiment involving simian immunodeficiency virus (SIV) infection of macaques revealed that the infectivity of this virus decreased over the first few months of infection. Based on this observation, we introduce a viral dynamic model in which viral infectivity varies over time. The model is fit to viral load data from eight (donor) monkeys infected by intravaginal inoculation of SIVmac251, three monkeys infected by intravenous inoculation of virus isolated from the donors during the ramp-up phase of acute infection, and three monkeys infected by intravenous inoculation of virus isolated at the viral set-point. Although we only analyze data from 14 monkeys, the new model with time-dependent infectivity seems to fit the data significantly better than a widely used model with constant infectivity (P = 2.44 x 10(-11)). Our results indicate that plasma virus infectivity on average decays approximately 8-fold (95% confidence interval [CI] = 5.1 to 10.3) over the course of acute infection, with the decay occurring exponentially with an average rate of 0.28 day(-1) (95% CI = 0.14 to 0.42 day(-1)). The decay rate in set point plasma virus recipient animals is approximately 16 times slower than in ramp-up plasma virus recipient animals and approximately 6 times slower than in donor animals. Throughout acute infection up to the set-point, the infection rate is higher in ramp-up plasma virus recipient animals than in set-point plasma virus recipient animals. These results show that the infectivity depends upon the source of viral infection.
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