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Control and prevention strategies are indispensable tools for managing the spread of infectious diseases. This paper examined biological models for the post-vaccination stage of a viral outbreak that integrate two important mitigation tools: social distancing, aimed at reducing the disease transmission rate, and vaccination, which boosts the immune system. Five different scenarios of epidemic progression were considered: (ⅰ) the no control scenario, reflecting the natural evolution of a disease without any safety measures in place, (ⅱ) the reconstructed scenario, representing real-world data and interventions, (ⅲ) the social distancing control scenario covering a broad set of behavioral changes, (ⅳ) the vaccine control scenario demonstrating the impact of vaccination on epidemic spread, and (ⅴ) the both controls concurrently scenario incorporating social distancing and vaccine controls simultaneously. By comparing these scenarios, we provided a comprehensive analysis of various intervention strategies, offering valuable insights into disease dynamics. Our innovative approach to modeling the cost of control gave rise to a robust computational algorithm for solving optimal control problems associated with different public health regulations. Numerical results were supported by real data for the Delta variant of the COVID-19 pandemic in the United States.

 

A pleated membrane filter consists of a porous membrane layer, which is surrounded by two supporting layers, and the whole structure is pleated and placed into a cylindrical cartridge. Pleated membrane filters are used in a variety of industrial applications, since they offer more surface area to volume ratio that is not found in equivalent flat filters. In this work, we introduce a novel three-dimensional model of a pleated membrane filter that consists of an empty region, a pleated region, and a hollow region. The advection diffusion equation is used to model contaminant concentration in the membrane pores along with Darcy's law to model the flow within the membrane and support layers, while the Stokes equation is used for the flow in the empty region and the hollow region. We further use the key assumptions of our model based on small aspect ratios of the filter cartridge and the pleated membrane to simplify the governing equations, which can be easily solved by numerical methods. By performing these steps, we seek to discover an optimal pleat packing density to find the optimum filter performance, while not exceeding a threshold for the particle concentration at the filter outlet. 

In this work, we use the immersed boundary method with four extensions to simulate a moving liquid–gas interface on a solid surface. We first define a moving contact line model and implements a static-dynamic friction condition at the immersed solid boundary. The dynamic contact angle is endogenous instead of prescribed, and the solid boundary can be non-stationary with respect to time. Second, we simulate both a surface tension force and a Young's force with one general equation that does not involve estimating local curvature. In the third extension, we splice liquid–gas interfaces to handle topological changes, such as the coalescence and separation of liquid droplets or gas bubbles. Finally, we re-sample liquid–gas interface markers to ensure a near-uniform distribution without exerting artificial forces. We demonstrate empirical convergence of our methods on non-trivial examples and apply them to several benchmark cases, including a slipping droplet on a wall and a rising bubble. 

Flow in the inverted U-shaped tube of a conventional siphon can be established and maintained only if the tube is filled and closed, so that air does not enter. We report on siphons that operate entirely open to the atmosphere by exploiting surface tension effects. Such capillary siphoning is demonstrated by paper tissue that bridges two containers and conveys water from the upper to the lower. We introduce a more controlled system consisting of grooves in a wetting solid, formed here by pressing together hook-shaped metallic rods. The dependence of flux on siphon geometry is systematically measured, revealing behaviour different from the conventional siphon. The flux saturates when the height difference between the two container's free surfaces is large; it also has a strong dependence on the climbing height from the source container's free surface to the apex. A one-dimensional theoretical model is developed, taking into account the capillary pressure due to surface tension, pressure loss due to viscous friction, and driving by gravity. Numerical solutions are in good agreement with experiments, and the model suggests hydraulic interpretations for the observed flux dependence on geometrical parameters. The operating principle and characteristics of capillary siphoning revealed here can inform biological phenomena and engineering applications related to directional fluid transport. 

Pleated membrane filters are widely used to remove undesired impurities from a fluid in many applications. A filter membrane is sandwiched between porous support layers and then pleated and packed into an annular cylindrical cartridge with a central hollow duct for outflow. Although this arrangement offers a high surface filtration area to volume ratio, the filter performance is not as efficient as those of equivalent flat filters. In this paper, we use asymptotic methods to simplify the flow throughout the cartridge to systematically investigate how the number of pleats or pleat packing density affects the performance of the pleated membrane filters. The model is used to determine an optimal number of pleats in order to achieve a particular optimum filtration performance. Our findings show that only the “just right”—neither too few nor too many—number of pleats gives optimum performance in a pleated filter cartridge. 

Membrane filtration fouling is a very complex process and is determined by many properties such as the membrane internal morphology, membrane pore structure, flow rate and contaminant properties. In a very slow filtration process or during the late stage of filtration, when the flow rate is naturally low and Péclet number is small, particle diffusion is essential and cannot be neglected, while in typical filtration models, especially in moderate and fast filtration process, the main contribution stems from the particle advection. The objectives of this study is to formulate mathematical models that can (i) investigate how filtration process varies under possible effects of particles diffusion; and (ii) describe how membrane morphology evolves and investigate the filtration performance during the filtration process. We also compare the results with the case that diffusion is less important and make a prediction about what kind of membrane filter pore structure should be employed to achieve a particular optimum filtration performance. According to our results, the filtrate and efficiency of particle separation are found to be under the trade-off relationship, and the selection of the membrane properties depends on the requirement of the filtration.