We study the performance of a network of agents tasked with tracking an external unknown signal in the presence of stochastic disturbances and under the condition that only a limited subset of agents, known as leaders, can measure the signal directly. We investigate the optimal leader selection problem for a prescribed maximum number of leaders, where the optimal leader set minimizes total system error defined as steady-state variance about the external signal.In contrast to previously established greedy algorithms for optimal leader selection, our results rely on an expression of total system error in terms of properties of the underlying network graph. We demonstrate that the performance of any given set of noise-free leaders depends on their influence as determined by a new graph measure of centrality of a set. We define the joint centrality of a set of nodes in a network graph such that a noise-free leader set with maximal joint centrality is an optimal leader set. In the case of a single leader, we prove that the optimal leader is the node with maximal information centrality for both the nois-corrupted and noise-free leader cases. In the case of multiple leaders, we show that the nodes in the optimal noise-free leader set balance high information centrality with a coverage of the graph. For special cases of graphs, we solve explicitly for optimal leader sets. We illustrate with examples.
Introduction: “Long COVID” is a multisystem disease that lasts for 4 or more weeks following initial symptoms of COVID-19. In the UK, at least 10% of patient report symptoms at 12 weeks following a positive COVID-19 test. The aims of this quality improvement survey were to explore patients’ acute and post-acute “long” COVID-19 symptoms, their experiences of community services and their recommendations for improving these services. Methods: Seventy patients diagnosed with COVID were randomly selected from 2 large socially and ethnically diverse primary care practices. Of those contactable by telephone, 85% (41/48) agreed to participate in the quality improvement survey. They were interviewed by telephone using a semi-structured questionnaire about community services for COVID-19 patients. Interviews lasted 10 to 15 minutes. Results: Forty-nine percent of patients reported at least 1 post-acute COVID-19 symptom. The most common were severe fatigue (45%), breathlessness (30%), neurocognitive difficulties (such as poor memory), poor concentration and “brain fog” (30%), headaches (20%), and joint pain (20%). Many patients felt isolated and fearful, with scant information about community resources and little safety netting advice. Patients also expected more from primary care with over half (56%) recommending regular phone calls and follow up from healthcare staff as the most important approach in their recovery. Conclusions: In line with patients’ requests for more support, the practices now routinely refer patients with long COVID to an on-site social prescriber who explores how they are getting on, refers them to the GP or practice nurse when required, and sign posts them to support services in the community.
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