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Abstract This manuscript introduces a new Erlang-distributed SEIR model. The model incorporates asymptomatic spread through a subdivided exposed class, distinguishing between asymptomatic ($$\hbox {E}_a$$ ${\text{E}}_a$) and symptomatic ($$\hbox {E}_s$$ ${\text{E}}_s$) cases. The model identifies two key parameters: relative infectiousness,$$\beta _{{SA}}$$ ${\beta }_{\mathrm{SA}}$, and the percentage of people who become asymptomatic after being infected by a symptomatic individual,$$\kappa $$ $\kappa$. Lower values of these parameters reduce the peak magnitude and duration of the infectious period, highlighting the importance of isolation measures. Additionally, the model underscores the need for strategies addressing both symptomatic and asymptomatic transmissions. 

Single-particle tracking (SPT) is a powerful technique for probing the diverse physical properties of the cytoplasm. Genetically encoded nanoparticles provide an especially convenient tool for such investigations, as they can be expressed and tracked in cells via fluorescence. Among these, 40-nm genetically encoded multimerics (GEMs) provide a unique opportunity to explore the cytoplasm. Their size corresponds to that of ribosomes and big protein complexes, allowing us to investigate the effects of the cytoplasm on the diffusivity of these objects while excluding the influence of chemical interactions during stressful events and pathological conditions. However, the effects of GEM expression levels on the measured cytoplasmic diffusivity remain largely uncharacterized in mammalian cells. To optimize the GEMs tracking and assess expression level effects, we developed a doxycycline-inducible GEM expression system and compared it with a previously reported constitutive expression system. The inducible GEM expression system reduced the number of GEM particles from 2000 to as low as 5–500 per average 2D cell cytoplasmic area, depending on doxycycline concentration and incubation time. This optimization enabled adjustment of particle density for imaging and improved homogeneity across the cell population. Moreover, we enhanced the analysis of GEM diffusivity by incorporating an effective diffusion coefficient that accounts for the type of motion and by quantifying motion heterogeneity through standard deviations of particle displacements within and between cells. 

We present a novel method for identifying topological features of chromatin domains in live cells using single-particle tracking and topological data analysis (TDA). By applying TDA to particle trajectories, we can effectively detect complex spatial patterns, such as loops, that are often missed by traditional time series analysis. Using simulations of polymer bead–spring chains, we have validated the accuracy of our method and determined its limitations for detecting loops. Our approach offers a promising avenue for exploring the topological complexity of chromatin in living cells using TDA techniques. 

This article provides an introductory review of the mathematical modeling of viscoelastic fluids (VEFs), which exhibit both viscous and elastic behaviors critical to applications in biological and industrial processes. Focusing on the interplay between microscopic dynamics and macroscopic properties, the paper explores constitutive modeling challenges for VEFs, particularly polymeric fluids. It discusses three levels of description: stochastic differential equations (SDEs) for individual microstructural dynamics, Fokker-Planck equations for ensemble behavior, and macroscopic partial differential equations (PDEs) for continuum flow fields. Using bead-spring models, such as Hookean and FENE dumbbell models, the review illustrates how microstructural configurations influence macroscopic stress responses. Key mathematical challenges, including numerical convergence, equation well-posedness, and closure approximations, are highlighted, with specific attention to the Upper Convected Maxwell and FENE-P models. The article underscores the balance between computational feasibility and physical accuracy in modeling VEFs, offering insights into ongoing research and future directions in rheology and applied mathematics. 

Modeling the flow of polymer solutions requires knowledge at various length and time scales. The macroscopic behavior is described by the overall velocity, pressure, and stress. The polymeric contribution to the stress requires knowledge of the evolution of polymer chains. In this work, we use a microscopic model, the finitely extensible nonlinear elastic (FENE) model, to capture the polymer’s behavior. The benefit of using microscopic models is that they remain faithful to the polymer dynamics without information loss via averaging. Their downside is the computational cost incurred in solving the thousands to millions of differential equations describing the microstructure. Here, we describe a multiscale flow solver that utilizes GPUs for massively parallel, efficient simulations. We compare and contrast the microscopic model with its macroscopic counterpart under various flow conditions. In particular, significant differences are observed under nonlinear flow conditions, where the polymers become highly stretched and oriented. 

We explore the effects of cross-diffusion dynamics in epidemiological models. Using reaction–diffusion models of infectious disease, we explicitly consider situations where an individual in a category will move according to the concentration of individuals in other categories. Namely, we model susceptible individuals moving away from infected and infectious individuals. Here, we show that including these cross-diffusion dynamics results in a delay in the onset of an epidemic and an increase in the total number of infectious individuals. This representation provides more realistic spatiotemporal dynamics of the disease classes in an Erlang SEIR model and allows us to study how spatial mobility, due to social behavior, can affect the spread of an epidemic. We found that tailored control measures, such as targeted testing, contact tracing, and isolation of infected individuals, can be more effective in mitigating the spread of infectious diseases while minimizing the negative impact on society and the economy. 

The Langevin equations (LE) and the Fokker–Planck (FP) equations are widely used to describe fluid behavior based on coarse-grained approximations of microstructure evolution. In this manuscript, we describe the relation between LE and FP as related to particle motion within a fluid. The manuscript introduces undergraduate students to two LEs, their corresponding FP equations, and their solutions and physical interpretation.