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Extensive non-pharmaceutical and physical distancing measures are currently the primary interventions against coronavirus disease 2019 (COVID-19) worldwide. It is therefore urgent to estimate the impact such measures are having. We introduce a Bayesian epidemiological model in which a proportion of individuals are willing and able to participate in distancing, with the timing of distancing measures informed by survey data on attitudes to distancing and COVID-19. We fit our model to reported COVID-19 cases in British Columbia (BC), Canada, and five other jurisdictions, using an observation model that accounts for both underestimation and the delay between symptom onset and reporting. We estimated the impact that physical distancing (social distancing) has had on the contact rate and examined the projected impact of relaxing distancing measures. We found that, as of April 11 2020, distancing had a strong impact in BC, consistent with declines in reported cases and in hospitalization and intensive care unit numbers; individuals practising physical distancing experienced approximately 0.22 (0.11-0.34 90% CI [credible interval]) of their normal contact rate. The threshold above which prevalence was expected to grow was 0.55. We define the "contact ratio" to be the ratio of the estimated contact rate to the threshold rate at which cases are expected to grow; we estimated this contact ratio to be 0.40 (0.19-0.60) in BC. We developed an R package 'covidseir' to make our model available, and used it to quantify the impact of distancing in five additional jurisdictions. As of May 7, 2020, we estimated that New Zealand was well below its threshold value (contact ratio of 0.22 [0.11-0.34]), New York (0.60 [0.43-0.74]), Washington (0.84 [0.79-0.90]) and Florida (0.86 [0.76-0.96]) were progressively closer to theirs yet still below, but California (1.15 [1.07-1.23]) was above its threshold overall, with cases still rising. Accordingly, we found that BC, New Zealand, and New York may have had more room to relax distancing measures than the other jurisdictions, though this would need to be done cautiously and with total case volumes in mind. Our projections indicate that intermittent distancing measures-if sufficiently strong and robustly followed-could control COVID-19 transmission. This approach provides a useful tool for jurisdictions to monitor and assess current levels of distancing relative to their threshold, which will continue to be essential through subsequent waves of this pandemic., Competing Interests: The authors have declared that no competing interests exist.

It is a truism that antimicrobial drugs select for resistance, but explaining pathogen- and population-specific variation in patterns of resistance remains an open problem. Like other common commensals, Streptococcus pneumoniae has demonstrated persistent coexistence of drug-sensitive and drug-resistant strains. Theoretically, this outcome is unlikely. We modelled the dynamics of competing strains of S. pneumoniae to investigate the impact of transmission dynamics and treatment-induced selective pressures on the probability of stable coexistence. We find that the outcome of competition is extremely sensitive to structure in the host population, although coexistence can arise from age-assortative transmission models with age-varying rates of antibiotic use. Moreover, we find that the selective pressure from antibiotics arises not so much from the rate of antibiotic use per se but from the frequency of treatment: frequent antibiotic therapy disproportionately impacts the fitness of sensitive strains. This same phenomenon explains why serotypes with longer durations of carriage tend to be more resistant. These dynamics may apply to other potentially pathogenic, microbial commensals and highlight how population structure, which is often omitted from models, can have a large impact., (© 2017 The Authors.)

The Gram-negative bacterium Bordetella pertussis is the causative agent of whooping cough, a serious respiratory infection causing hundreds of thousands of deaths annually worldwide. There are effective vaccines, but their production requires growing large quantities of B. pertussis. Unfortunately, B. pertussis has relatively slow growth in culture, with low biomass yields and variable growth characteristics. B. pertussis also requires a relatively expensive growth medium. We present a new, curated flux balance analysis-based model of B. pertussis metabolism. We enhance the model with an experimentally-determined biomass objective function, and we perform extensive manual curation. We test the model's predictions with a genome-wide screen for essential genes using a transposon-directed insertional sequencing (TraDIS) approach. We test its predictions of growth for different carbon sources in the medium. The model predicts essentiality with an accuracy of 83% and correctly predicts improvements in growth under increased glutamate:fumarate ratios. We provide the model in SBML format, along with gene essentiality predictions.

Drug resistance to tuberculosis (TB) has become more widespread over the past decade. As such, understanding the emergence and fitness of antibiotic-resistant subpopulations is crucial for the development of new interventions. Here we use a simple mathematical model to explain the differences in the response to isoniazid (INH) of Mycobacterium tuberculosis cells cultured under two growth rates in a chemostat. We obtain posterior distributions of model parameters consistent with data using a Markov chain Monte Carlo (MCMC) method. We explore the dynamics of diverse INH-resistant subpopulations consistent with these data in a multi-population model. We find that the simple model captures the qualitative behaviour of the cultures under both dilution rates and also present testable predictions about how diversity is maintained in such cultures., (© 2016 The Authors.)

Pertussis, or whooping cough, is an important respiratory infection causing considerable infant mortality worldwide. Recently, incidence has risen in countries with strong vaccine programmes and there are concerns about antigenic shift resulting in vaccine evasion. Interactions between pertussis and non-vaccine-preventable strains will play an important role in the evolution and population dynamics of pertussis. In particular, if we are to understand the role strain replacement plays in vaccinated settings, it will be essential to understand how strains or variants of pertussis interact. Here we explore under what conditions we would expect strain replacement to be of concern in pertussis. We develop a dynamic transmission model that allows for coinfection between Bordetella pertussis (the main causative agent of pertussis) and a strain or variant unaffected by the vaccine. We incorporate both neutrality (in the sense of ecological/population genetic neutrality) and immunity into the model, leaving the specificity of the immune response flexible. We find that strain replacement may be considerable when immunity is non-specific. This is in contrast to previous findings where neutrality was not considered. We conclude that the extent to which models reflect ecological neutrality can have a large impact on conclusions regarding strain replacement. This will likely have onward consequences for estimates of vaccine efficacy and cost-effectiveness., (Copyright © 2015 Elsevier Inc. All rights reserved.)

High resolution tests for genetic variation reveal that individuals may simultaneously host more than one distinct strain of Mycobacterium tuberculosis. Previous studies find that this phenomenon, which we will refer to as "mixed infection", may affect the outcomes of treatment for infected individuals and may influence the impact of population-level interventions against tuberculosis. In areas where the incidence of TB is high, mixed infections have been found in nearly 20% of patients; these studies may underestimate the actual prevalence of mixed infection given that tests may not be sufficiently sensitive for detecting minority strains. Specific reasons for failing to detect mixed infections would include low initial numbers of minority strain cells in sputum, stochastic growth in culture and the physical division of initial samples into parts (typically only one of which is genotyped). In this paper, we develop a mathematical framework that models the study designs aimed to detect mixed infections. Using both a deterministic and a stochastic approach, we obtain posterior estimates of the prevalence of mixed infection. We find that the posterior estimate of the prevalence of mixed infection may be substantially higher than the fraction of cases in which it is detected. We characterize this bias in terms of the sensitivity of the genotyping method and the relative growth rates and initial population sizes of the different strains collected in sputum., (Copyright © 2015 The Authors. Published by Elsevier Ltd.. All rights reserved.)

Human populations are interconnected through a variety of different networks. The complex interactions of diverse populations of individuals and their interconnected network structures affect the diffusion of processes through the population. For example, the different modes of transmission of HIV and TB mean that they are transmitted along very different contact networks: HIV via sexual contact and TB via respiratory contact. In addition, co-infection with HIV raises the risk of progressing to active TB and reduces the response to TB treatment, potentially causing increasing incidence of TB. Here we extend existing network theory to find the effect of multiple networks and multiple host types on epidemic thresholds. We first analyse how transmission of a pathogen via an additional network affects its epidemic threshold. We then use the theory behind branching processes to study how multiple host types in a population affect its threshold. The formulation we obtain enables modellers to determine how multiple networks and host heterogeneity (i.e. vaccination, behavioural change, mixing patterns, etc.) affect the epidemic threshold. We apply the results to the example of HIV and TB to illustrate how the interactions of the diseases can substantially alter the epidemic threshold of TB., (Copyright © 2012 Elsevier Ltd. All rights reserved.)

Background: Bacteria have evolved the ability to efficiently and resourcefully adapt to changing environments. A key means by which they optimize their use of available nutrients is through adjustments in gene expression with consequent changes in enzyme activity. We report a new method for drawing environmental inferences from gene expression data. Our method prioritizes a list of candidate carbon sources for their compatibility with a gene expression profile using the framework of flux balance analysis to model the organism's metabolic network., Principal Findings: For each of six gene expression profiles for Escherichia coli grown under differing nutrient conditions, we applied our method to prioritize a set of eighteen different candidate carbon sources. Our method ranked the correct carbon source as one of the top three candidates for five of the six expression sets when used with a genome-scale model. The correct candidate ranked fifth in the remaining case. Additional analyses show that these rankings are robust with respect to biological and measurement variation, and depend on specific gene expression, rather than general expression level. The gene expression profiles are highly adaptive: simulated production of biomass averaged 94.84% of maximum when the in silico carbon source matched the in vitro source of the expression profile, and 65.97% when it did not., Conclusions: Inferences about a microorganism's nutrient environment can be made by integrating gene expression data into a metabolic framework. This work demonstrates that reaction flux limits for a model can be computed which are realistic in the sense that they affect in silico growth in a manner analogous to that in which a microorganism's alteration of gene expression is adaptive to its nutrient environment.

Individuals living with HIV experience a much higher risk of progression from latent M. tuberculosis infection to active tuberculosis (TB) disease relative to individuals with intact immune systems. A several-month daily course of a single drug during latent infection (i.e. isoniazid preventive therapy (IPT)) has proved in clinical trials to substantially reduce an HIV-infected individual's risk of TB disease. As a result of these findings and ongoing studies, the World Health Organization has produced strong guidelines for implementing IPT on a community-wide scale for individuals with HIV at risk of TB disease. To date, there has been limited use of IPT at a community-wide level. In this paper, we present a new co-network model for HIV and TB co-epidemics to address questions about how the population-level impact of community-wide IPT may differ from the individual-level impact of IPT offered to selected individuals. In particular, we examine how the effect of clustering of contacts within high-TB incidence communities may affect the rates of re-infection with TB and how this clustering modifies the expected population-level effects of IPT. We find that populations with clustering of respiratory contacts experience aggregation of TB cases and high numbers of re-infection events. While, encouragingly, the overall population-level effects of community-wide IPT appear to be sustained regardless of network structure, we find that in populations where these contacts are highly clustered, there is dramatic heterogeneity in the impact of IPT: in some sub-regions of these populations, TB is nearly eliminated, while in others, repeated re-infection almost completely undermines the effect of IPT. Our findings imply that as IPT programmes are brought to scale, we should expect local heterogeneity of effectiveness as a result of the complex patterns of disease transmission within communities.

Technologies for strain differentiation and typing have made it possible to detect genetic diversity of pathogens, both within individual hosts and within communities. Coinfection of a host by more than one pathogen strain may affect the relative frequency of these strains at the population level through complex within- and between-host interactions; in infectious diseases that have a long latent period, interstrain competition during latency is likely to play an important role in disease dynamics. We show that SEIR models that include a class of latently coinfected individuals can have markedly different long-term dynamics than models without coinfection, and that coinfection can greatly facilitate the stable coexistence of strains. We demonstrate these dynamics using a model relevant to tuberculosis in which people may experience latent coinfection with both drug sensitive and drug resistant strains. Using this model, we show that the existence of a latent coinfected state allows the possibility that disease control interventions that target latency may facilitate the emergence of drug resistance.

While bacillus Calmette-Guérin vaccination plays an important role in reducing the morbidity of tuberculosis (TB) infection during childhood, new tuberculosis vaccines are necessary to disrupt the transmission of disease and improve global control of this pathogen. Growing evidence of the presence of meaningful Mycobacterium tuberculosis strain diversity, coupled with the possibility that new vaccines may differentially protect against infection or disease with circulating M. tuberculosis strains, suggest that these vaccines may have complicated effects on disease dynamics. We use a mathematical model to explore the potential effects of strain diversity on the performance of vaccines and find that vaccines offer great promise for improving tuberculosis control, but the expected benefits of mass vaccination will be eroded if strain replacement with M. tuberculosis variants that are not effectively targeted by vaccines occurs. Determining the likelihood of strain replacement will require additional knowledge of the strain specificities of current vaccine candidates, and an improved understanding of the mechanisms of strain interaction, which are responsible for maintaining the diversity of M. tuberculosis within communities.

Infection with Mycobacterium tuberculosis leads to tuberculosis (TB) disease by one of the three possible routes: primary progression after a recent infection; re-activation of a latent infection; or exogenous re-infection of a previously infected individual. Recent studies show that optimal TB control strategies may vary depending on the predominant route to disease in a specific population. It is therefore important for public health policy makers to understand the relative frequency of each type of TB within specific epidemiological scenarios. Although molecular epidemiologic tools have been used to estimate the relative contribution of recent transmission and re-activation to the burden of TB disease, it is not possible to use these techniques to distinguish between primary disease and re-infection on a population level. Current estimates of the contribution of re-infection therefore rely on mathematical models which identify the parameters most consistent with epidemiological data; these studies find that exogenous re-infection is important only when TB incidence is high. A basic assumption of these models is that people in a population are all equally likely to come into contact with an infectious case. However, theoretical studies demonstrate that the social and spatial structure can strongly influence the dynamics of infectious disease transmission. Here, we use a network model of TB transmission to evaluate the impact of non-homogeneous mixing on the relative contribution of re-infection over realistic epidemic trajectories. In contrast to the findings of previous models, our results suggest that re-infection may be important in communities where the average disease incidence is moderate or low as the force of infection can be unevenly distributed in the population. These results have important implications for the development of TB control strategies.

Several hematological diseases are characterised by oscillations of various blood cell populations. Two of these are a variant of chronic myelogenous leukemia (CML) and cyclical neutropenia (CN). These oscillations typically have long periods ranging from 20 to 60 days, despite the fact that the stem cell cycling time is thought to be of the order of 2-3 days. Clinical data from humans and laboratory data from the grey collie animal model of CN is suggestive of the idea that these long period oscillations may also contain higher frequency spiky oscillations. We show how such oscillations can be understood in the context of slow periodic stem cell oscillations, by analysing a two component differential-delay equation model of stem cell and neutrophil populations.