Regularization of the semilinear sideways heat equation

A classical physical example of the sideways heat equation is represented by re-entry vehicles in the atmosphere where the temperature at the nozzle of a rocket is so high that any thermocouple attached to it would be destroyed. Instead one could measure both the temperature and heat flux, i.e. Cauchy data, at an interior boundary inward the capsule. In addition, we assume that there exists a heat source which is significantly dependent on space, time and temperature, and hence it cannot be neglected. This gives rise to a non-characteristic Cauchy inverse boundary value problem in the sense that the interior accessible boundary is overspecified, while the exterior hostile boundary is underspecified as nothing is prescribed on it. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the Cauchy data. In order to obtain a stable numerical solution, we propose two regularization methods to solve the semilinear problem in which the heat source is a Lipschitz function of temperature. We show rigourously, with error estimates provided, that the corresponding regularized solutions converge to the true solution strongly in L² uniformly with respect to the space coordinate under some a priori assumptions on the solution. These assumptions place no serious restrictions on the applicability of the results since in practice we always have some control and knowledge about how large the absolute temperature and heat flux are likely to be. Finally, in order to increase the significance of the study, numerical results are presented and discussed illustrating the theoretical findings in terms of accuracy and stability

Recovering the initial distribution for strongly damped wave equation

We study for the first time the inverse backward problem for the strongly damped wave equation. First, we show that the problem is severely ill-posed in the sense of Hadamard. Then, under the a priori assumption on the exact solution belonging to a Gevrey space, we propose the Fourier truncation method for stabilizing the ill-posed problem. A stability estimate of logarithmic type is established

Quantitative Modeling of GRK-Mediated β2AR Regulation

We developed a unified model of the GRK-mediated β2 adrenergic receptor (β2AR) regulation that simultaneously accounts for six different biochemical measurements of the system obtained over a wide range of agonist concentrations. Using a single deterministic model we accounted for (1) GRK phosphorylation in response to various full and partial agonists; (2) dephosphorylation of the GRK site on the β2AR; (3) β2AR internalization; (4) recycling of the β2AR post isoproterenol treatment; (5) β2AR desensitization; and (6) β2AR resensitization. Simulations of our model show that plasma membrane dephosphorylation and recycling of the phosphorylated receptor are necessary to adequately account for the measured dephosphorylation kinetics. We further used the model to predict the consequences of (1) modifying rates such as GRK phosphorylation of the receptor, arrestin binding and dissociation from the receptor, and receptor dephosphorylation that should reflect effects of knockdowns and overexpressions of these components; and (2) varying concentration and frequency of agonist stimulation “seen” by the β2AR to better mimic hormonal, neurophysiological and pharmacological stimulations of the β2AR. Exploring the consequences of rapid pulsatile agonist stimulation, we found that although resensitization was rapid, the β2AR system retained the memory of the previous stimuli and desensitized faster and much more strongly in response to subsequent stimuli. The latent memory that we predict is due to slower membrane dephosphorylation, which allows for progressive accumulation of phosphorylated receptor on the surface. This primes the receptor for faster arrestin binding on subsequent agonist activation leading to a greater extent of desensitization. In summary, the model is unique in accounting for the behavior of the β2AR system across multiple types of biochemical measurements using a single set of experimentally constrained parameters. It also provides insight into how the signaling machinery can retain memory of prior stimulation long after near complete resensitization has been achieved

The Diagnosis of Dengue in Patients Presenting With Acute Febrile Illness Using Supervised Machine Learning and Impact of Seasonality

BackgroundSymptomatic dengue infection can result in a life-threatening shock syndrome and timely diagnosis is essential. Point-of-care tests for non-structural protein 1 and IgM are used widely but performance can be limited. We developed a supervised machine learning model to predict whether patients with acute febrile illnesses had a diagnosis of dengue or other febrile illnesses (OFI). The impact of seasonality on model performance over time was examined.MethodsWe analysed data from a prospective observational clinical study in Vietnam. Enrolled patients presented with an acute febrile illness of &amp;lt;72 h duration. A gradient boosting model (XGBoost) was used to predict final diagnosis using age, sex, haematocrit, platelet, white cell, and lymphocyte count collected on enrolment. Data was randomly split 80/20% into a training and hold-out set, respectively, with the latter not used in model development. Cross-validation and hold out set testing was used, with performance over time evaluated through a rolling window approach.ResultsWe included 8,100 patients recruited between 16th October 2010 and 10th December 2014. In total 2,240 (27.7%) patients were diagnosed with dengue infection. The optimised model from training data had an overall median area under the receiver operator curve (AUROC) of 0.86 (interquartile range 0.84–0.86), specificity of 0.92, sensitivity of 0.56, positive predictive value of 0.73, negative predictive value (NPV) of 0.84, and Brier score of 0.13 in predicting the final diagnosis, with similar performances in hold-out set testing (AUROC of 0.86). Model performances varied significantly over time as a function of seasonality and other factors. Incorporation of a dynamic threshold which continuously learns from recent cases resulted in a more consistent performance throughout the year (NPV &amp;gt;90%).ConclusionSupervised machine learning models are able to discriminate between dengue and OFI diagnoses in patients presenting with an early undifferentiated febrile illness. These models could be of clinical utility in supporting healthcare decision-making and provide passive surveillance across dengue endemic regions. Effects of seasonality and changing disease prevalence must however be taken into account—this is of significant importance given unpredictable effects of human-induced climate change and the impact on health.</jats:sec