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1. Identifiability and Parameter Estimation of Within-Host Model of HIV with Immune Response

   https://doi.org/10.3390/math12182837  

   Liyanage, Yuganthi R; Mirsaleh_Kohan, Leila; Martcheva, Maia; Tuncer, Necibe (September 2024, Mathematics)

   This study examines the interactions between healthy target cells, infected target cells, virus particles, and immune cells within an HIV model. The model exhibits two equilibrium points: an infection-free equilibrium and an infection equilibrium. Stability analysis shows that the infection-free equilibrium is locally asymptotically stable when R0<1. Further, it is unstable when R0>1. The infection equilibrium is locally asymptotically stable when R0>1. The structural and practical identifiabilities of the within-host model for HIV infection dynamics were investigated using differential algebra techniques and Monte Carlo simulations. The HIV model was structurally identifiable by observing the total uninfected and infected target cells, immune cells, and viral load. Monte Carlo simulations assessed the practical identifiability of parameters. The production rate of target cells (λ), the death rate of healthy target cells (d), the death rate of infected target cells (δ), and the viral production rate by infected cells (π) were practically identifiable. The rate of infection of target cells by the virus (β), the death rate of infected cells by immune cells (Ψ), and antigen-driven proliferation rate of immune cells (b) were not practically identifiable. Practical identifiability was constrained by the noise and sparsity of the data. Analysis shows that increasing the frequency of data collection can significantly improve the identifiability of all parameters. This highlights the importance of optimal data sampling in HIV clinical studies, as it determines the best time points, frequency, and the number of sample points required to accurately capture the dynamics of the HIV infection within a host. 

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2. A novel within-host model of HIV and nutrition

   https://doi.org/10.3934/mbe.2024246  

   Timsina, Archana N; Liyanage, Yuganthi R; Martcheva, Maia; Tuncer, Necibe (January 2024, Mathematical Biosciences and Engineering)

   In this paper we develop a four compartment within-host model of nutrition and HIV. We show that the model has two equilibria: an infection-free equilibrium and infection equilibrium. The infection free equilibrium is locally asymptotically stable when the basic reproduction number $$ \mathcal{R}_0 < 1 $$, and unstable when $$ \mathcal{R}_0 > 1 $$. The infection equilibrium is locally asymptotically stable if $$ \mathcal{R}_0 > 1 $$ and an additional condition holds. We show that the within-host model of HIV and nutrition is structured to reveal its parameters from the observations of viral load, CD4 cell count and total protein data. We then estimate the model parameters for these 3 data sets. We have also studied the practical identifiability of the model parameters by performing Monte Carlo simulations, and found that the rate of clearance of the virus by immunoglobulins is practically unidentifiable, and that the rest of the model parameters are only weakly identifiable given the experimental data. Furthermore, we have studied how the data frequency impacts the practical identifiability of model parameters. 

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3. Denoising of imaginary time response functions with Hankel projections

   https://doi.org/10.1103/PhysRevResearch.6.L032042  

   Yu, Yang; Kemper, Alexander F; Yang, Chao; Gull, Emanuel (August 2024, Physical Review Research)

   Imaginary-time response functions of finite-temperature quantum systems are often obtained with methods that exhibit stochastic or systematic errors. Reducing these errors comes at a large computational cost—in quantum Monte Carlo simulations, the reduction of noise by a factor of two incurs a simulation cost of a factor of four. In this paper, we relate certain imaginary-time response functions to an inner product on the space of linear operators on Fock space. We then show that data with noise typically does not respect the positive definiteness of its associated Gramian. The Gramian has the structure of a Hankel matrix. As a method for denoising noisy data, we introduce an alternating projection algorithm that finds the closest positive definite Hankel matrix consistent with noisy data. We test our methodology at the example of fermion Green's functions for continuous-time quantum Monte Carlo data and show remarkable improvements of the error, reducing noise by a factor of up to 20 in practical examples. We argue that Hankel projections should be used whenever finite-temperature imaginary-time data of response functions with errors is analyzed, be it in the context of quantum Monte Carlo, quantum computing, or in approximate semianalytic methodologies. Published by the American Physical Society2024 

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4. Order-based structure learning without score equivalence

   https://doi.org/10.1093/biomet/asad052  

   Chang, Hyunwoong; Cai, James J; Zhou, Quan (September 2023, Biometrika)

   We propose an empirical Bayes formulation of the structure learning problem, where the prior specification assumes that all node variables have the same error variance, an assumption known to ensure the identifiability of the underlying causal directed acyclic graph. To facilitate efficient posterior computation, we approximate the posterior probability of each ordering by that of a best directed acyclic graph model, which naturally leads to an order-based Markov chain Monte Carlo algorithm. Strong selection consistency for our model in high-dimensional settings is proved under a condition that allows heterogeneous error variances, and the mixing behaviour of our sampler is theoretically investigated. Furthermore, we propose a new iterative top-down algorithm, which quickly yields an approximate solution to the structure learning problem and can be used to initialize the Markov chain Monte Carlo sampler. We demonstrate that our method outperforms other state-of-the-art algorithms under various simulation settings, and conclude the paper with a single-cell real-data study illustrating practical advantages of the proposed method. 

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5. SATELLITE DRAG COEFFICIENT MODELING AND ORBIT UNCERTAINTY QUANTIFICATION USING STOCHASTIC MACHINE LEARNING TECHNIQUES

   Paul, S.N.; Sheridan, L.; Mehta, P.; Huzurbazar, S. (January 2021, American Astronomical Society meeting)

   The rapidly increasing congestion in the low Earth environment makes the modeling of uncertainty in atmospheric drag force a critical task, affecting space situational awareness (SSA) activities like the probability of collision estimation. A key element in atmospheric drag modeling is the assessment of uncertainty in the atmospheric drag coefficient estimate. While atmospheric drag coefficients for space objects with known characteristics can be computed numerically, they suffer from large computational costs for practical applications. In this work, we use cost-effective data-driven stochastic methods for modeling the drag coefficients of objects in the low Earth orbit (LEO) region. The training data is generated using the numerical Test Particle Monte Carlo (TPMC) method. TPMC is simulated with Cercignani–Lampis–Lord (CLL) gas-surface interaction (GSI) model. Mehta et al. [1] use a Gaussian process regression (GPR) model to predict satellite drag coefficient, but the authors did not estimate the predictive uncertainty. The first part of this research extends the work by Mehta et al. [1] by fitting a GPR model to the training data and performing predictive uncertainty estimation. The results of the Gaussian fit are then compared against a deep neural network (DNN) model aided by the Monte Carlo dropout approach. To the best of our knowledge, this is the first study to use the aforementioned stochastic deep learning algorithm to perform predictive uncertainty estimation of the estimated satellite drag coefficient. Apart from the accuracy of the models, we also undertake the task of calibrating the models. Simulations are carried out for a spherical satellite followed by the Champ satellite. Finally, quantification of the effect of drag coefficient uncertainty on orbit prediction is carried out for different solar activity and geomagnetic activity levels. 

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