BCAM Workshop Populations in epidemics and ecology Modeling and numerical simulations Bilbao, July 4-5, 2017 BOOK OF ABSTRACTS (original) (raw)

On the Reproduction Number of a Gut Microbiota Model

Bulletin of mathematical biology, 2017

A spatially structured linear model of the growth of intestinal bacteria is analysed from two generational viewpoints. Firstly, the basic reproduction number associated with the bacterial population, i.e. the expected number of daughter cells per bacterium, is given explicitly in terms of biological parameters. Secondly, an alternative quantity is introduced based on the number of bacteria produced within the intestine by one bacterium originally in the external media. The latter depends on the parameters in a simpler way and provides more biological insight than the standard reproduction number, allowing the design of experimental procedures. Both quantities coincide and are equal to one at the extinction threshold, below which the bacterial population becomes extinct. Optimal values of both reproduction numbers are derived assuming parameter trade-offs.

Basic reproduction number in a spatially structured model for gut microbiota

An ecological model for a bacterial population inside and outside an animal host is studied. We consider a linearised system for the proliferating bacteria, where they can be in three compartments: attached to the epithelial wall of the intestine or as free particles in the lumen or in the outer environment. The geometry of the intestine is reduced to a line segment where spatial densities are taken into account. We compute the next-generation operator as the composition of the cell division operator and the inverse of the transition/mortality operator. The basic reproduction number is then explicitly computed as the spectral radius of this linear operator. In addition, the extinction threshold is interpreted in terms of the expected number of bacteria coming back to the outer medium from each initial single bacterium, after travelling along the intestine. Further developments can include several hosts and/or the interaction with a bacteriophage population, and the analysis of evolutionary aspects of the model.

Reproduction numbers of infectious disease models

Infectious Disease Modelling, 2017

This primer article focuses on the basic reproduction number, ℛ 0 , for infectious diseases, and other reproduction numbers related to ℛ 0 that are useful in guiding control strategies. Beginning with a simple population model, the concept is developed for a threshold value of ℛ 0 determining whether or not the disease dies out. The next generation matrix method of calculating ℛ 0 in a compartmental model is described and illustrated. To address control strategies, type and target reproduction numbers are defined, as well as sensitivity and elasticity indices. These theoretical ideas are then applied to models that are formulated for West Nile virus in birds (a vector-borne disease), cholera in humans (a disease with two transmission pathways), anthrax in animals (a disease that can be spread by dead carcasses and spores), and Zika in humans (spread by mosquitoes and sexual contacts). Some parameter values from literature data are used to illustrate the results. Finally, references for other ways to calculate ℛ 0 are given. These are useful for more complicated models that, for example, take account of variations in environmental fluctuation or stochasticity.

A CONTINUUM MODEL OF THE WITHIN-ANIMAL POPULATION DYNAMICS OF E. COLI O157

Journal of Biological Systems, 2006

The high level of human morbidity caused by E. coli O157:H7 necessitates an improved understanding of the infection dynamics of this bacterium within the bovine reservoir. Until recently, a degree of uncertainty surrounded the issue of whether these bacteria colonize the bovine gut and as yet, only incomplete in-vivo datasets are available. Such data typically consist of bacterial counts from fecal samples. The development of a deterministic model, which has been devised to make good use of such data, is presented. A partial differential equation, which includes advection, diffusion and growth terms, is used to model the (unobserved) passage of bacteria through the bovine gut. A set of experimentally-obtained fecal count data is used to parameterize the model. Betweenanimal variability is found to be greater than between-strain variability, with some results adding further weight to the hypothesis that E. coli O157:H7 can colonize the bovine gastrointestinal tract.

Reproduction numbers for infections with free-living pathogens growing in the environment

Journal of Biological Dynamics, 2012

The basic reproduction number ℛ0 for a compartmental disease model is often calculated by the next generation matrix (NGM) approach. When the interactions within and between disease compartments are interpreted differently, the NGM approach may lead to different ℛ0 expressions. This is demonstrated by considering a susceptible–infectious–recovered–susceptible model with free-living pathogen (FLP) growing in the environment. Although the environment could play different roles in the disease transmission process, leading to different ℛ0 expressions, there is a unique type reproduction number when control strategies are applied to the host population. All ℛ0 expressions agree on the threshold value 1 and preserve their order of magnitude. However, using data for salmonellosis and cholera, it is shown that the estimated ℛ0 values are substantially different. This study highlights the utility and limitations of reproduction numbers to accurately quantify the effects of control strategies for infections with FLPs growing in the environment.

Probabilistic Model of Microbial Cell Growth, Division, and Mortality

Applied and Environmental Microbiology, 2010

After a short time interval of length δ t during microbial growth, an individual cell can be found to be divided with probability P d ( t )δ t , dead with probability P m ( t )δ t , or alive but undivided with the probability 1 − [ P d ( t ) + P m ( t )]δ t , where t is time, P d ( t ) expresses the probability of division for an individual cell per unit of time, and P m ( t ) expresses the probability of mortality per unit of time. These probabilities may change with the state of the population and the habitat's properties and are therefore functions of time. This scenario translates into a model that is presented in stochastic and deterministic versions. The first, a stochastic process model, monitors the fates of individual cells and determines cell numbers. It is particularly suitable for small populations such as those that may exist in the case of casual contamination of a food by a pathogen. The second, which can be regarded as a large-population limit of the stochastic m...

Characterizing the reproduction number of epidemics with early subexponential growth dynamics

Journal of the Royal Society, Interface, 2016

Early estimates of the transmission potential of emerging and re-emerging infections are increasingly used to inform public health authorities on the level of risk posed by outbreaks. Existing methods to estimate the reproduction number generally assume exponential growth in case incidence in the first few disease generations, before susceptible depletion sets in. In reality, outbreaks can display subexponential (i.e. polynomial) growth in the first few disease generations, owing to clustering in contact patterns, spatial effects, inhomogeneous mixing, reactive behaviour changes or other mechanisms. Here, we introduce the generalized growth model to characterize the early growth profile of outbreaks and estimate the effective reproduction number, with no need for explicit assumptions about the shape of epidemic growth. We demonstrate this phenomenological approach using analytical results and simulations from mechanistic models, and provide validation against a range of empirical di...

A size-structured model of bacterial growth and reproduction

Journal of Biological Dynamics, 2012

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