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Here, we consider a regularized mean-field game model that features a low-order regularization. We prove the existence of solutions with positive density. To do so, we combine a priori estimates with the continuation method. In contrast with highorder regularizations, the low-order regularizations are easier to implement numerically. Moreover, our methods give a theoretical foundation for this approach.

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Viral spread has been intermittently threatening human life over time. Characterizing the viral concentration and modelling the viral transmission are, therefore, considered major milestones for enhancing viral detection capabilities. This paper addresses the problem of viral aerosol detection based on the exhaled breath in a bounded environment, e.g., a bounded room. The paper models the exhaled breath as a cloud which is emitted through the room continuously, and analyzes the temporal-spatial virus concentration by accounting for partial absorption and reflection at each side of the room. The paper first derives a closed form expression of the temporal-spatial virus concentration. It then considers the deployment of a receiver composed of an air sampler and a bio-sensor to detect the viral existence of a specific virus. We, therefore, assess the detection capabilities of the proposed system via evaluating the viral miss-detection probability as a function of the sampling volume and the detection time-instance at the receiver side. Our numerical simulations verify the validity of the analytical results, and illustrate the ability of the proposed system to detect viruses in indoor environments. The results further characterize the impacts of several system parameters on the miss-detection probability.