A dose and time response Markov model for the in-host dynamics of infection with intracellular bacteria following inhalation: with application to Francisella tularensis

Quantitative Microbial Risk Assessment of Antibiotic-Resistant E. coli, Legionella pneumophila, and Mycobacteria in Nonpotable Wastewater Reuse Applications

Antibiotic-resistant bacteria (ARB) have become a major threat to public health and modern medicine. A simple death kinetics-based dose-response model (SD-DRM) was incorporated into a quantitative microbial risk assessment (QMRA) to assess the risks of exposure to reclaimed wastewater harboring antibiotic-resistant E. coli, Legionella pneumophila, and Mycobacterium avium for multiple exposure scenarios. The fractions of ARB and trace antibiotics present in the body were incorporated to demonstrate their impact on infection risks. Both ARB and antibiotic susceptible bacteria, ASB, are assumed to have the same dose-response in the absence of antibiotics but behave differently in the presence of residual antibiotics in the body. Annual risk of L. pneumophila infection exceeded the EPA 10-4 pppy (per person per year) benchmark at concentrations in reclaimed water greater than 103-104 CFU/L, depending on parameter variation. Enteropathogenic E. coli infection risks meet the EPA annual benchmark at concentrations around 105-106 total E. coli. The results illustrated that an increase in residual antibiotics from 0 to 40% of the minimum inhibitory concentration (MIC) reduced the risk by about 1 order of magnitude for E. coli but was more likely to result in an untreatable infection.

Using mixture density networks to emulate a stochastic within-host model of Francisella tularensis infection

For stochastic models with large numbers of states, analytical techniques are often impractical, and simulations time-consuming and computationally demanding. This limitation can hinder the practical implementation of such models. In this study, we demonstrate how neural networks can be used to develop emulators for two outputs of a stochastic within-host model of Francisella tularensis infection: the dose-dependent probability of illness and the incubation period. Once the emulators are constructed, we employ Markov Chain Monte Carlo sampling methods to parameterize the within-host model using records of human infection. This inference is only possible through the use of a mixture density network to emulate the incubation period, providing accurate approximations of the corresponding probability distribution. Notably, these estimates improve upon previous approaches that relied on bacterial counts from the lungs of macaques. Our findings reveal a 50% infectious dose of approximately 10 colony-forming units and we estimate that the incubation period can last for up to 11 days following low dose exposure.

A comprehensive modelling approach to estimate the transmissibility of coronavirus and its variants from infected subjects in indoor environments

A central issue in assessing the airborne risk of COVID-19 infections in indoor spaces pertains to linking the viral load in infected subjects to the lung deposition probability in exposed individuals through comprehensive aerosol dynamics modelling. In this paper, we achieve this by combining aerosol processes (evaporation, dispersion, settling, lung deposition) with a novel double Poisson model to estimate the probability that at least one carrier particle containing at least one virion will be deposited in the lungs and infect a susceptible individual. Multiple emission scenarios are considered. Unlike the hitherto used single Poisson models, the double Poisson model accounts for fluctuations in the number of carrier particles deposited in the lung in addition to the fluctuations in the virion number per carrier particle. The model demonstrates that the risk of infection for 10-min indoor exposure increases from 1 to 50% as the viral load in the droplets ejected from the infected subject increases from 2 × 108 to 2 × 1010 RNA copies/mL. Being based on well-established aerosol science and statistical principles, the present approach puts airborne risk assessment methodology on a sound formalistic footing, thereby reducing avoidable epistemic uncertainties in estimating relative transmissibilities of different coronavirus variants quantified by different viral loads.

A Review of Using Mathematical Modeling to Improve Our Understanding of Bacteriophage, Bacteria, and Eukaryotic Interactions

Phage therapy, the therapeutic usage of viruses to treat bacterial infections, has many theoretical benefits in the ‘post antibiotic era.’ Nevertheless, there are currently no approved mainstream phage therapies. One reason for this is a lack of understanding of the complex interactions between bacteriophage, bacteria and eukaryotic hosts. These three-component interactions are complex, with non-linear or synergistic relationships, anatomical barriers and genetic or phenotypic heterogeneity all leading to disparity between performance and efficacy in in vivo versus in vitro environments. Realistic computer or mathematical models of these complex environments are a potential route to improve the predictive power of in vitro studies for the in vivo environment, and to streamline lab work. Here, we introduce and review the current status of mathematical modeling and highlight that data on genetic heterogeneity and mutational stochasticity, time delays and population densities could be critical in the development of realistic phage therapy models in the future. With this in mind, we aim to inform and encourage the collaboration and sharing of knowledge and expertise between microbiologists and theoretical modelers, synergising skills and smoothing the road to regulatory approval and widespread use of phage therapy.

A dose response model for Staphylococcus aureus

Dose-response models (DRMs) are used to predict the probability of microbial infection when a person is exposed to a given number of pathogens. In this study, we propose a new DRM for Staphylococcus aureus (SA), which causes skin and soft-tissue infections. The current approach to SA dose-response is only partially mechanistic and assumes that individual bacteria do not interact with each other. Our proposed two-compartment (2C) model assumes that bacteria that have not adjusted to the host environment decay. After adjusting to the host, they exhibit logistic/cooperative growth, eventually causing disease. The transition between the adjusted and un-adjusted states is a stochastic process, which the 2C DRM explicitly models to predict response probabilities. By fitting the 2C model to SA pathogenesis data, we show that cooperation between individual SA bacteria is sufficient (and, within the scope of the 2C model, necessary) to characterize the dose-response. This is a departure from the classical single-hit theory of dose-response, where complete independence is assumed between individual pathogens. From a quantitative microbial risk assessment standpoint, the mechanistic basis of the 2C DRM enables transparent modeling of dose-response of antibiotic-resistant SA that has not been possible before. It also enables the modeling of scenarios having multiple/non-instantaneous exposures, with minimal assumptions.

Dose-Response Modeling: Extrapolating From Experimental Data to Real-World Populations

Dose-response modeling of biological agents has traditionally focused on describing laboratory-derived experimental data. Limited consideration has been given to understanding those factors that are controlled in a laboratory, but are likely to occur in real-world scenarios. In this study, a probabilistic framework is developed that extends Brookmeyer's competing-risks dose-response model to allow for variation in factors such as dose-dispersion, dose-deposition, and other within-host parameters. With data sets drawn from dose-response experiments of inhalational anthrax, plague, and tularemia, we illustrate how for certain cases, there is the potential for overestimation of infection numbers arising from models that consider only the experimental data in isolation.

Stochastic dynamics of Francisella tularensis infection and replication

We study the pathogenesis of Francisella tularensis infection with an experimental mouse model, agent-based computation and mathematical analysis. Following inhalational exposure to Francisella tularensis SCHU S4, a small initial number of bacteria enter lung host cells and proliferate inside them, eventually destroying the host cell and releasing numerous copies that infect other cells. Our analysis of disease progression is based on a stochastic model of a population of infectious agents inside one host cell, extending the birth-and-death process by the occurrence of catastrophes: cell rupture events that affect all bacteria in a cell simultaneously. Closed expressions are obtained for the survival function of an infected cell, the number of bacteria released as a function of time after infection, and the total bacterial load. We compare our mathematical analysis with the results of agent-based computation and, making use of approximate Bayesian statistical inference, with experimental measurements carried out after murine aerosol infection with the virulent SCHU S4 strain of the bacterium Francisella tularensis, that infects alveolar macrophages. The posterior distribution of the rate of replication of intracellular bacteria is consistent with the estimate that the time between rounds of bacterial division is less than 6 hours in vivo.

Infecting a host cell is required for the replication of many types of bacteria and viruses. In some cases, infected cells release new infectious agents continuously over their lifetime. In others, such as the Francisella tularensis bacterium studied here, they are released in a single burst that coincides with the cell’s death. We show how a stochastic model, the birth-and-death process with catastrophe, can be used to characterise infection in a single cell, thereby allowing us to account for burst events and quantify the kinetics of pathogenesis in the lung, the initial site of infection, as well as in other organs that the infection spreads to. We learn about the parameters of the mathematical model of Francisella tularensis infection making use of the experimental measurements of bacterial loads, together with approximate Bayesian statistical inference methods. The most important parameter describing the pathogenesis is the rate of replication of intracellular bacteria.

A dose response model for quantifying the infection risk of antibiotic-resistant bacteria

Quantifying the human health risk of microbial infection helps inform regulatory policies concerning pathogens, and the associated public health measures. Estimating the infection risk requires knowledge of the probability of a person being infected by a given quantity of pathogens, and this relationship is modeled using pathogen specific dose response models (DRMs). However, risk quantification for antibiotic-resistant bacteria (ARB) has been hindered by the absence of suitable DRMs for ARB. A new approach to DRMs is introduced to capture ARB and antibiotic-susceptible bacteria (ASB) dynamics as a stochastic simple death (SD) process. By bridging SD with data from bench experiments, we demonstrate methods to (1) account for the effect of antibiotic concentrations and horizontal gene transfer on risk; (2) compute total risk for samples containing multiple bacterial types (e.g., ASB, ARB); and (3) predict if illness is treatable with antibiotics. We present a case study of exposure to a mixed population of Gentamicin-susceptible and resistant Escherichia coli and predict the health outcomes for varying Gentamicin concentrations. Thus, this research establishes a new framework to quantify the risk posed by ARB and antibiotics.

Early treatment gains for antibiotic administration and within human host time series data

As technological improvements continue to infiltrate and impact medical practice, it has become possible to non-invasively collect dense physiological time series data from individual patients in real time. These advances continue to improve physicians' ability to detect and to treat infections early. One important benefit of early detection and treatment of nascent infections is that it leads to earlier resolution. In response to current and anticipated advances in data capture, we introduce the Early Treatment Gain (ETG) as a measure to quantify this benefit. Roughly, we define the gain to be the limiting ratio: ETG=differential change in time of resolutiondifferential change in treatment time.We study the gain using standard dynamical models and demonstrate its use with time series data from Surgical Intensive Care Unit (SICU) patients facing ventilator associated pneumonia. The main conclusion from the mathematical modelling is that the ETG is always greater than one unless there is an effective immune response, in which case the ETG can be less than one. Using real patient time series data, we observe that the formula derived for a linear model can be applied and that this produces a ETG greater than one.

Microbiota and Dose Response: Evolving Paradigm of Health Triangle

SRA Dose-Response and Microbial Risk Analysis Specialty Groups jointly sponsored symposia that addressed the intersections between the "microbiome revolution" and dose response. Invited speakers presented on innovations and advances in gut and nasal microbiota (normal microbial communities) in the first decade after the Human Microbiome Project began. The microbiota and their metabolites are now known to influence health and disease directly and indirectly, through modulation of innate and adaptive immune systems and barrier function. Disruption of healthy microbiota is often associated with changes in abundance and diversity of core microbial species (dysbiosis), caused by stressors including antibiotics, chemotherapy, and disease. Nucleic-acid-based metagenomic methods demonstrated that the dysbiotic host microbiota no longer provide normal colonization resistance to pathogens, a critical component of innate immunity of the superorganism. Diverse pathogens, probiotics, and prebiotics were considered in human and animal models (in vivo and in vitro). Discussion included approaches for design of future microbial dose-response studies to account for the presence of the indigenous microbiota that provide normal colonization resistance, and the absence of the protective microbiota in dysbiosis. As NextGen risk analysis methodology advances with the "microbiome revolution," a proposed new framework, the Health Triangle, may replace the old paradigm based on the Disease Triangle (focused on host, pathogen, and environment) and germophobia. Collaborative experimental designs are needed for testing hypotheses about causality in dose-response relationships for pathogens present in our environments that clearly compete in complex ecosystems with thousands of bacterial species dominating the healthy superorganism.

Human Dose-Response Data for Francisella tularensis and a Dose- and Time-Dependent Mathematical Model of Early-Phase Fever Associated with Tularemia After Inhalation Exposure

Military health risk assessors, medical planners, operational planners, and defense system developers require knowledge of human responses to doses of biothreat agents to support force health protection and chemical, biological, radiological, nuclear (CBRN) defense missions. This article reviews extensive data from 118 human volunteers administered aerosols of the bacterial agent Francisella tularensis, strain Schu S4, which causes tularemia. The data set includes incidence of early-phase febrile illness following administration of well-characterized inhaled doses of F. tularensis. Supplemental data on human body temperature profiles over time available from de-identified case reports is also presented. A unified, logically consistent model of early-phase febrile illness is described as a lognormal dose-response function for febrile illness linked with a stochastic time profile of fever. Three parameters are estimated from the human data to describe the time profile: incubation period or onset time for fever; rise time of fever; and near-maximum body temperature. Inhaled dose-dependence and variability are characterized for each of the three parameters. These parameters enable a stochastic model for the response of an exposed population through incorporation of individual-by-individual variability by drawing random samples from the statistical distributions of these three parameters for each individual. This model provides risk assessors and medical decisionmakers reliable representations of the predicted health impacts of early-phase febrile illness for as long as one week after aerosol exposures of human populations to F. tularensis.

A Novel Stochastic Multi-Scale Model of Francisella tularensis Infection to Predict Risk of Infection in a Laboratory

We present a multi-scale model of the within-phagocyte, within-host and population-level infection dynamics of Francisella tularensis, which extends the mechanistic one proposed by Wood et al. (2014). Our multi-scale model incorporates key aspects of the interaction between host phagocytes and extracellular bacteria, accounts for inter-phagocyte variability in the number of bacteria released upon phagocyte rupture, and allows one to compute the probability of response, and mean time until response, of an infected individual as a function of the initial infection dose. A Bayesian approach is applied to parameterize both the within-phagocyte and within-host models using infection data. Finally, we show how dose response probabilities at the individual level can be used to estimate the airborne propagation of Francisella tularensis in indoor settings (such as a microbiology laboratory) at the population level, by means of a deterministic zonal ventilation model.

Multiplicity of Mathematical Modeling Strategies to Search for Molecular and Cellular Insights into Bacteria Lung Infection

Even today two bacterial lung infections, namely pneumonia and tuberculosis, are among the 10 most frequent causes of death worldwide. These infections still lack effective treatments in many developing countries and in immunocompromised populations like infants, elderly people and transplanted patients. The interaction between bacteria and the host is a complex system of interlinked intercellular and the intracellular processes, enriched in regulatory structures like positive and negative feedback loops. Severe pathological condition can emerge when the immune system of the host fails to neutralize the infection. This failure can result in systemic spreading of pathogens or overwhelming immune response followed by a systemic inflammatory response. Mathematical modeling is a promising tool to dissect the complexity underlying pathogenesis of bacterial lung infection at the molecular, cellular and tissue levels, and also at the interfaces among levels. In this article, we introduce mathematical and computational modeling frameworks that can be used for investigating molecular and cellular mechanisms underlying bacterial lung infection. Then, we compile and discuss published results on the modeling of regulatory pathways and cell populations relevant for lung infection and inflammation. Finally, we discuss how to make use of this multiplicity of modeling approaches to open new avenues in the search of the molecular and cellular mechanisms underlying bacterial infection in the lung.

A systematic review and meta-analysis on the incubation period of Campylobacteriosis

Accurate knowledge of pathogen incubation period is essential to inform public health policies and implement interventions that contribute to the reduction of burden of disease. The incubation period distribution of campylobacteriosis is currently unknown with several sources reporting different times. Variation in the distribution could be expected due to host, transmission vehicle, and organism characteristics, however, the extent of this variation and influencing factors are unclear. The authors have undertaken a systematic review of published literature of outbreak studies with well-defined point source exposures and human experimental studies to estimate the distribution of incubation period and also identify and explain the variation in the distribution between studies. We tested for heterogeneity using I 2 and Kolmogorov-Smirnov tests, regressed incubation period against possible explanatory factors, and used hierarchical clustering analysis to define subgroups of studies without evidence of heterogeneity. The mean incubation period of subgroups ranged from 2·5 to 4·3 days. We observed variation in the distribution of incubation period between studies that was not due to chance. A significant association between the mean incubation period and age distribution was observed with outbreaks involving only children reporting an incubation of 1·29 days longer when compared with outbreaks involving other age groups.

Comparison among the Quantification of Bacterial Pathogens by qPCR, dPCR, and Cultural Methods

The demand for rapid methods for the quantification of pathogens is increasing. Among these methods, those based on nucleic acids amplification (quantitative PCRs) are the most widespread worldwide. Together with the qPCR, a new approach named digital PCR (dPCR), has rapidly gained importance. The aim of our study was to compare the results obtained using two different dPCR systems and one qPCR in the quantification of three different bacterial pathogens: Listeria monocytogenes, Francisella tularensis, and Mycobacterium avium subsp. paratuberculosis. For this purpose, three pre-existing qPCRs were used, while the same primers and probes, as well as PCR conditions, were transferred to two different dPCR systems: the QX200 (Bio-Rad) and the Quant Studio 3D (Applied Biosystems). The limits of detection and limits of quantification for all pathogens, and all PCR approaches applied, were determined using genomic pure DNAs. The quantification of unknown decimal suspensions of the three bacteria obtained by the three different PCR approaches was compared through the Linear Regression and Bland and Altman analyses. Our results suggest that, both dPCRs are able to quantify the same amount of bacteria, while the comparison among dPCRs and qPCRs, showed both over and under-estimation of the bacteria present in the unknown suspensions. Our results showed qPCR over-estimated the amount of M. avium subsp. paratuberculosis and F. tularensis cells. On the contrary, qPCR, compared to QX200 dPCR, under-estimated the amount of L. monocytogenes cells. However, the maximum difference among PCRs approaches was <0.5 Log₁₀, while cultural methods underestimated the number of bacteria by one to two Log₁₀ for Francisella tularensis and Mycobacterium avium subsp. paratuberculosis. On the other hand, cultural and PCRs methods quantified the same amount of bacteria for L. monocytogenes, suggesting for this last pathogen, PCRs approaches can be considered as a valid alternative to the cultural ones.

A human time dose response model for Q fever

The causative agent of Q fever, Coxiella burnetii, has the potential to be developed for use in biological warfare and it is classified as a bioterrorism threat agent by the Centers for Disease Control and Prevention (CDC) and as a category B select agent by the National Institute of Allergy and Infectious Diseases (NIAID). In this paper we focus on the in-host properties that arise when an individual inhales a dose of C. burnetii and establish a human time-dose response model. We also propagate uncertainty throughout the model allowing us to robustly estimate key properties including the infectious dose and incubation period. Using human study data conducted in the 1950's we conclude that the dose required for a 50% probability of infection is about 15 organisms, and that one inhaled organism of C. burnetti can cause infection in 5% of the exposed population. In addition, we derive a low dose incubation period of 17.6 days and an extracellular doubling time of half a day. In conclusion this paper provides a framework for detailing the parameters and approaches that would be required for risk assessments associated with exposures to C. burnetii that might cause human infection.

A review of back-calculation techniques and their potential to inform mitigation strategies with application to non-transmissible acute infectious diseases

Back-calculation is a process whereby generally unobservable features of an event leading to a disease outbreak can be inferred either in real-time or shortly after the end of the outbreak. These features might include the time when persons were exposed and the source of the outbreak. Such inferences are important as they can help to guide the targeting of mitigation strategies and to evaluate the potential effectiveness of such strategies. This article reviews the process of back-calculation with a particular emphasis on more recent applications concerning deliberate and naturally occurring aerosolized releases. The techniques can be broadly split into two themes: the simpler temporal models and the more sophisticated spatio-temporal models. The former require input data in the form of cases' symptom onset times, whereas the latter require additional spatial information such as the cases' home and work locations. A key aspect in the back-calculation process is the incubation period distribution, which forms the initial topic for consideration. Links between atmospheric dispersion modelling, within-host dynamics and back-calculation are outlined in detail. An example of how back-calculation can inform mitigation strategies completes the review by providing improved estimates of the duration of antibiotic prophylaxis that would be required in the response to an inhalational anthrax outbreak.

Modeling early events in Francisella tularensis pathogenesis

Computational models can provide valuable insights into the mechanisms of infection and be used as investigative tools to support development of medical treatments. We develop a stochastic, within-host, computational model of the infection process in the BALB/c mouse, following inhalational exposure to Francisella tularensis SCHU S4. The model is mechanistic and governed by a small number of experimentally verifiable parameters. Given an initial dose, the model generates bacterial load profiles corresponding to those produced experimentally, with a doubling time of approximately 5 h during the first 48 h of infection. Analytical approximations for the mean number of bacteria in phagosomes and cytosols for the first 24 h post-infection are derived and used to verify the stochastic model. In our description of the dynamics of macrophage infection, the number of bacteria released per rupturing macrophage is a geometrically-distributed random variable. When combined with doubling time, this provides a distribution for the time taken for infected macrophages to rupture and release their intracellular bacteria. The mean and variance of these distributions are determined by model parameters with a precise biological interpretation, providing new mechanistic insights into the determinants of immune and bacterial kinetics. Insights into the dynamics of macrophage suppression and activation gained by the model can be used to explore the potential benefits of interventions that stimulate macrophage activation.