Particle size is an essential factor when considering the fate and transport of virus-containing droplets expelled by human, because it determines the deposition pattern in the human respiratory system and the evolution of droplets by evaporation and gravitational settling. However, the evolution of virus-containing droplets and the size-dependent viral load have not been studied in detail. The lack of this information leads to uncertainties in understanding the airborne transmission of respiratory diseases, such as the COVID-19. In this study, through a set of differential equations describing the evolution of respiratory droplets and by using the SARS-CoV-2 virus as an example, we investigated the distribution of airborne virus in human expelled particles from coughing and speaking. More specifically, by calculating the vertical distances traveled by the respiratory droplets, we examined the number of viruses that can remain airborne and the size of particles carrying these airborne viruses after different elapsed times. From a single cough, a person with a high viral load in respiratory fluid (2.35 × 10 9 copies per ml) may generate as many as 1.23 × 10 5 copies of viruses that can remain airborne after 10 seconds, compared to 386 copies of a normal patient (7.00 × 10 6 copies per ml). Masking, however, can effectively block around 94% of the viruses that may otherwise remain airborne after 10 seconds. Our study found that no clear size boundary exists between particles that can settle and can remain airborne. The results from this study challenge the conventional understanding of disease transmission routes through airborne and droplet mechanisms. We suggest that a complete understanding of the respiratory droplet evolution is essential and needed to identify the transmission mechanisms of respiratory diseases.

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In this paper, we consider an interesting generalization of the weighted vertex cover problem, called the Facility Terminal Cover (FTC) problem. In the FTC problem, each vertex is associated with a positive weight, each edge is associated with a positive demand, and the objective is to determine a subset of vertices and a capacity for each selected vertex so that the demand of each edge is covered by the capacity of one of its two endpoints and the total weighted capacity of all selected vertices is minimized. The FTC problem is motivated by several key network optimization problems, such as the power assignment problem in ad hoc networks, and could be used as a subroutine to solve such problems. No quality-guaranteed solution is previously known for the FTC problem. In this paper, we present two linear time approximation algorithms for this problem. Our first algorithm achieves deterministically an approximation ratio of 8 by using an interesting rounding technique and a lower-bounding technique. Based on interesting randomization techniques, our second algorithm further improves the approximation ratio to 2e, where e is the natural logarithmic base. The second algorithm can be easily derandomized in quadratic time. Our algorithms are relatively simple and can be easily implemented for networking applications. Experiments show that the two algorithms behave rather similarly, especially in largesize graphs, indicating that the solutions yielded by one or both algorithms are much closer to the optimum.