The purpose of the paper is to use analytical method and optimization tool to suggest a vaccination program intensity for a basic SIR epidemic model with limited resources for vaccination. We show that there are two different scenarios for optimal vaccination strategies, and obtain analytical solutions for the optimal control problem that minimizes the total cost of disease under the assumption of daily vaccine supply being limited. These solutions and their corresponding optimal control policies are derived explicitly in terms of initial conditions, model parameters and resources for vaccination. With sufficient resources, the optimal control strategy is the normal Bang-Bang control. However, with limited resources, the optimal control strategy requires to switch to time-variant vaccination.
Mycobacterium often presents as an abundant bacterial genus in activated sludge in many wastewater treatment plants (WWTPs), but the species-level taxonomy and functions remain poorly understood. In this study, we profiled the mycobacterial communities in eleven WWTPs from five countries by pyrosequencing the rpoB amplicons and searching against a customized database of mycobacterial rpoB sequences. Results indicated that major mycobacterial species were related to M. brumae, M. crocinum, M. sphagni, etc., most of which belong to poorly characterized rapidly-growing group. A few opportunistic pathogenic species were detected, suggesting the potential risk of mycobacteria in WWTPs. Genomic analysis of four isolates from activated sludge indicated these genomes contained genes of degradations of alkane, aromatics, steroids and a variety of cytochrome P450 families. Additionally, a few key genes responsible for cholesterol degradation were detected in a full-scale activated sludge metatranscriptomic dataset reported previously and taxonomically assigned to mycobacteria. Evidence showed that all isolates can degrade cholesterol, a major composition of sewage. Relative abundance of mycobacteria in activated sludge was enriched by 4.7 folds after adding cholesterol into the influent for one week. Our results provided the insights into mycobacterial species and functions in WWTPs.
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