Testing individuals for pathogens can affect the spread of epidemics. Understanding how individual-level processes of sampling and reporting test results can affect community- or population-level spread is a dynamical modeling question. The effect of testing processes on epidemic dynamics depends on factors underlying implementation, particularly testing intensity and on whom testing is focused. Here, we use a simple model to explore how the individual-level effects of testing might directly impact population-level spread. Our model development was motivated by the COVID-19 epidemic, but has generic epidemiological and testing structures. To the classic SIR framework we have added a per capita testing intensity, and compartment-specific testing weights, which can be adjusted to reflect different testing emphases—surveillance, diagnosis, or control. We derive an analytic expression for the relative reduction in the basic reproductive number due to testing, test-reporting and related isolation behaviours. Intensive testing and fast test reporting are expected to be beneficial at the community level because they can provide a rapid assessment of the situation, identify hot spots, and may enable rapid contact-tracing. Direct effects of fast testing at the individual level are less clear, and may depend on how individuals’ behaviour is affected by testing information. Our simple model shows that under some circumstances both increased testing intensity and faster test reporting can reduce the effectiveness of control, and allows us to explore the conditions under which this occurs. Conversely, we find that focusing testing on infected individuals always acts to increase effectiveness of control.

The k-token graph T k (G) is the graph whose vertices are the k-subsets of vertices of a graph G, with two vertices of T k (G) adjacent if their symmetric difference is an edge of G. We explore when T k (G) is a well-covered graph, that is, when all of its maximal independent sets have the same cardinality. For bipartite graphs G, we classify when T k (G) is well-covered. For an arbitrary graph G, we show that if T 2 (G) is well-covered, then the girth of G is at most four. We include upper and lower bounds on the independence number of T k (G), and provide some families of well-covered token graphs.

Given a finite simple undirected graph [Formula: see text] there is a simplicial complex [Formula: see text], called the independence complex, whose faces correspond to the independent sets of [Formula: see text]. This is a well-studied concept because it provides a fertile ground for interactions between commutative algebra, graph theory and algebraic topology. In this paper, we consider a generalization of independence complex. Given [Formula: see text], a subset of the vertex set is called [Formula: see text]-independent if the connected components of the induced subgraph have cardinality at most [Formula: see text]. The collection of all [Formula: see text]-independent subsets of [Formula: see text] form a simplicial complex called the [Formula: see text]-independence complex and is denoted by [Formula: see text]. It is known that when [Formula: see text] is a chordal graph the complex [Formula: see text] has the homotopy type of a wedge of spheres. Hence, it is natural to ask which of these complexes are shellable or even vertex decomposable. We prove, using Woodroofe’s chordal hypergraph notion, that these complexes are always shellable when the underlying chordal graph is a tree. Using the notion of vertex splittable ideals we show that for caterpillar graphs the associated [Formula: see text]-independence complex is vertex decomposable for all values of [Formula: see text]. Further, for any [Formula: see text] we construct chordal graphs on [Formula: see text] vertices such that their [Formula: see text]-independence complexes are not sequentially Cohen–Macaulay.

Importance: Differential use of therapies for respiratory failure according to patient race/ethnicity may represent health inequity and could impact patient survival. Objective: Measure the association between patient race/ethnicity and the use of invasive ventilation, and the impact of any association on survival. Design: Retrospective cohort analysis using a Bayesian multistate model that adjusted for baseline covariates and time-varying severity. Setting: Multicenter study using the Medical Information Mart for Intensive Care IV (MIMIC-IV) and Phillips eICU (eICU) databases from the USA. Participants: Non-intubated adults receiving oxygen within the first 24 hours of ICU admission. Exposure: Patient race/ethnicity (Asian, Black, Hispanic, white). Main outcomes and measures: Primary output was the cause-specific hazard ratio (HR) of invasive ventilation for patient race/ethnicity. Secondary output was change in 28-day survival mediated by differences in invasive ventilation rate. We reported posterior means and 95% credible intervals (CrI). Results: We studied 38,263 patients, 52% (20,033) from MIMIC-IV and 48% (18,230) from eICU, 2% Asian (892), 11% Black (4,289), 5% Hispanic (1,964), and 81% white (31,923). Invasive ventilation occurred in 3,511 (9.2%), and 2,869 (7.5%) died. The rate of invasive ventilation was lower in Asian (HR 0.82, CrI 0.70 to 0.95), Black (HR 0.78, CrI 0.71 to 0.86), and Hispanic (HR 0.70, CrI 0.61 to 0.79) patients as compared to white patients. For the average patient, lower rates of invasive ventilation did not mediate differences in survival. For a reference patient with inspired oxygen (FiO2) varied from 0.5 to 1.0, the change in survival mediated by lower rates of invasive ventilation ranged from probable benefit (probability 0.82 for Asian patients, 0.91 for Black patients, and 0.93 for Hispanic patients) at FiO2 0.5 to probable harm (probability 0.87 for Asian patients, 0.92 for Black patients, and 0.97 for Hispanic patients) at FiO2 1.0, although the mean absolute changes in mortality were all less than 1.5%. Conclusions: Asian, Black, and Hispanic patients had a lower rate of invasive ventilation than white patients. The changes in 28-day survival mediated by this difference ranged from slight benefit at lower inspired oxygen fractions to slight harm at inspired oxygen fraction of 1.0, and there was no difference in survival for the average patient.

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