The long-run analysis of COVID-19 dynamic using random evolution, peak detection and time series.

It is now almost three years that COVID-19 has been the cause of misery for millions of people around the world. Many countries are in process of vaccination. Due to the social complexity of the problem, the future of decisions is not clear. As such, there is a need for the mathematical modeling to predict the long-run behavior of the COVID-19 dynamic for the decision-making with regard to the result of the pandemic on the economy, health, and others. In this paper, we have studied the short and long-run behavior of COVID-19. In a novel way, random evolution (Trichotomous and Dichotomous Markov Noise) is used to model and analyze the long-run behavior of the pandemic in different phases of the pandemic in different countries. On the given conditions, the random evolution model can help us establish the long-run asymptotic behaviour of the pandemic. This allows us to consider different phases of the pandemic as well as the effect of vaccination and other measures taken. The simplicity of the model makes it a practical tool for decision-making based on the long-run behavior of the pandemic. As such, we have established a criterion for the comparison of different regions and countries in different phases. In this regard, we have used real pandemic data from different countries to validate our results.

A Mathematical Model for Amyloid-𝜷 Aggregation in the Presence of Metal Ions: A Timescale Analysis for the Progress of Alzheimer Disease.

The aggregation of amyloid-𝛽 (A𝛽) proteins through their self-assembly into oligomers, fibrils, or senile plaques is advocated as a key process of Alzheimer's disease. Recent studies have revealed that metal ions play an essential role in modulating the aggregation rate of amyloid-𝛽 (A𝛽) into senile plaques because of high binding affinity between A𝛽 proteins and metal ions. In this paper, we proposed a mathematical model as a set of coupled kinetic equations that models the self-assembly of amyloid-𝛽 (A𝛽) proteins in the presence of metal ions. The numerical simulations capture four timescales in the A𝛽 dynamics associated with three important events which include the formation of the amyloid-metal complex, the homogeneous aggregation of the amyloid-metal complexes, and the non-homogeneous aggregation of the amyloid-metal complexes. The method of singular perturbation is used to identify these timescales in the framework of slow-fast systems.

Novel EM based ML Kalman estimation framework for superresolution of stochastic three-states microtubule signal.

BACKGROUND: Recent research has found that abnormal functioning of Microtubules (MTs) could be linked to fatal diseases such as Alzheimer's. Hence, there is an imminent need to understand the implications of MTs for disease- diagnosis. However, studies of cellular processes like MTs are often constrained by physical limitations of their data acquisition systems such as optical microscopes and are vulnerable to either destruction of the specimen or the probe. In addition, study of MTs is challenged with non-uniform sampling of the MT dynamic instability phenomenon relative to its time-lapse observation of the cellular processes. Thus, the above caveats limit the overall period of time that the MT data can be collected, thereby causing limited data availability scenario. RESULTS: In this work, two novel superresolution frameworks based on Expectation Maximization (EM) based Maximum Likelihood (ML) estimation using Kalman filters (MLK) technique are proposed to address the issues of non-uniform sampling and limited data availability of MT signals. The proposed MLK methods optimizes prediction of missing observations in the MT signal through information extraction using correlation-based patch processing and principal component analysis -based mutual information. Experimental results prove that the proposed MLK-based superresolution methods outperformed nonlinear interpolation and compressed sensing methods. CONCLUSIONS: This work aims to address limited data availability and data/observation loss incurred due to non-uniform sampling of biological signals such as MTs. For this purpose, statistical modelling of stochastic MT signals using EM based ML driven Kalman estimation (MLK) is considered as a fundamental framework for prediction of missing MT observations. It was experimentally validated that the proposed superresolution methods provided superior overall performance, better MT signal estimation using fewer samples, high SNR, low errors, and better MT parameter estimation than other methods.

Mathematical models for the effects of hypertension and stress on kidney and their uncertainty.

Determining the future state of individual patients, with regard to the susceptibility to life-threatening condition is a crucial problem in the public health. Chronic Kidney Disease (CKD) is a perilous disease, which is characterized by gradual loss of kidney function. This paper presents a set of mathematical models for CKD by applying both deterministic and stochastic approaches. Specifically, hypertension and stress are considered as the main causes of damaging and removing of nephrons thereby reducing the Glomerular Filtration Rate (GFR), which leads to irreparable damages to the kidney function and may result in renal failure, or even kidney loss. This paper analyzes the equilibria for both deterministic and stochastic models. In this regard, mathematical theorems related to stability and stochastic stability of the models are provided. The other important issue, which is discussed in the presented work, is the uncertainty problem for stochastic models. We have used simulations in MATLAB to calculate the uncertainty of the stochastic models. In addition, we have provided a framework for the future prediction of proposed models in specific cases. Finally, the models and the theoretical results are verified in application by applying them to the real clinical data.

An Îto stochastic differential equations model for the dynamics of the MCF-7 breast cancer cell line treated by radiotherapy.

In this paper, a new mathematical model is proposed for studying the population dynamics of breast cancer cells treated by radiotherapy by using a system of stochastic differential equations. The novelty of the model is essentially in capturing the concept of the cell cycle in the modeling to be able to evaluate the tumor lifespan. According to the cell cycle, each cell belongs to one of three subpopulations G, S, or M, representing gap, synthesis and mitosis subpopulations. Cells in the M subpopulation are highly radio-sensitive, whereas cells in the S subpopulation are highly radio-resistant. Therefore, in the process of radiotherapy, cell death rates of different subpopulations are not equal. In addition, since flow cytometry is unable to detect apoptotic cells accurately, the small changes in cell death rate in each subpopulation during treatment are considered. Subsequently, the proposed model is calibrated using experimental data from previous experiments involving the MCF-7 breast cancer cell line. Consequently, the proposed model is able to predict tumor lifespan based on the number of initial carcinoma cells. The results show the effectiveness of the radiation under the condition of stability, which describes the decreasing trend of the tumor cells population.

The use of compressive sensing and peak detection in the reconstruction of microtubules length time series in the process of dynamic instability.

Microtubules (MTs) are intra-cellular cylindrical protein filaments. They exhibit a unique phenomenon of stochastic growth and shrinkage, called dynamic instability. In this paper, we introduce a theoretical framework for applying Compressive Sensing (CS) to the sampled data of the microtubule length in the process of dynamic instability. To reduce data density and reconstruct the original signal with relatively low sampling rates, we have applied CS to experimental MT lament length time series modeled as a Dichotomous Markov Noise (DMN). The results show that using CS along with the wavelet transform significantly reduces the recovery errors comparing in the absence of wavelet transform, especially in the low and the medium sampling rates. In a sampling rate ranging from 0.2 to 0.5, the Root-Mean-Squared Error (RMSE) decreases by approximately 3 times and between 0.5 and 1, RMSE is small. We also apply a peak detection technique to the wavelet coefficients to detect and closely approximate the growth and shrinkage of MTs for computing the essential dynamic instability parameters, i.e., transition frequencies and specially growth and shrinkage rates. The results show that using compressed sensing along with the peak detection technique and wavelet transform in sampling rates reduces the recovery errors for the parameters.

Existence and stability of steady states of a reaction convection diffusion equation modeling microtubule formation.

We generalize the Dogterom-Leibler model for microtubule dynamics (Dogterom and Leibler in Phys Rev Lett 70(9):1347-1350, 1993) to the case where the rates of elongation as well as the lifetimes of the elongating shortening phases are a function of GTP-tubulin concentration. We analyze also the effect of nucleation rate in the form of a damping term which leads to new steady-states. For this model, we study existence and stability of steady states satisfying the boundary conditions at x=0. Our stability analysis introduces numerical and analytical Evans function computations as a new mathematical tool in the study of microtubule dynamics.