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Abstract The elementary method of Balog and Ruzsa and the large sieve of Linnik are utilized to investigate the behaviour of the norm of an exponential sum over the primes. A new proof of a lower bound due to Vaughan for the norm of an exponential sum formed with the von Mangoldt function is furnished. 

There is an ongoing debate on the different transmission modes of SARS-CoV-2 and their relative contributions to the pandemic. In this paper, we employ a simple mathematical model, which incorporates both the human-to-human and environment-to-human transmission routes, to study the transmission dynamics of COVID-19. We focus our attention on the role of airborne transmission in the spread of the disease in a university campus setting. We conduct both mathematical analysis and numerical simulation, and incorporate published experimental data for the viral concentration in the air to fit model parameters. Meanwhile, we compare the outcome to that of the standard SIR model, utilizing a perturbation analysis in the presence of multiple time scales. Our data fitting and numerical simulation results show that the risk of airborne transmission for SARS-CoV-2 strongly depends on how long the virus can remain viable in the air. If the time for this viability is short, the airborne transmission route would be inconsequential in shaping the overall transmission risk and the total infection size. On the other hand, if the infectious virus can persist in aerosols beyond a few hours, then airborne transmission could play a much more significant role in the spread of COVID-19. 

We present a new mathematical model to investigate the transmission dynamics of cholera under disease control measures that include education programs and water sanitation. The model incorporates the impact of education programs into the disease transmission rates and that of water sanitation into the environmental pathogen dynamics. We conduct a detailed analysis to the autonomous system of the model and establish the local and global stabilities of its equilibria that characterize the threshold dynamics of cholera. We then perform an optimal control study on the general model with time-dependent controls and explore effective approaches to implement the education programs and water sanitation while balancing their costs. Our analysis and simulation highlight the complex interaction among the direct and indirect transmission pathways of the disease, the intrinsic growth of the environmental pathogen and the impact of multiple control measures, and their roles in collectively shaping the transmission dynamics of cholera. 

We characterize by boundary conditions the Krein-von Neumann extension of a strictly positive minimal operator corresponding to a regular even order quasi-differential expression of Shin-Zettl type. The characterization is stated in terms of a specially chosen basis for the kernel of the maximal operator and employs a description of the Friedrichs extension due to Möller and Zettl.