On hierarchical competition through reduction of individual growth.

We consider a population organised hierarchically with respect to size in such a way that the growth rate of each individual depends only on the presence of larger individuals. As a concrete example one might think of a forest, in which the incidence of light on a tree (and hence how fast it grows) is affected by shading by taller trees. The classic formulation of a model for such a size-structured population employs a first order quasi-linear partial differential equation equipped with a non-local boundary condition. However, the model can also be formulated as a delay equation, more specifically a scalar renewal equation, for the population birth rate. After discussing the well-posedness of the delay formulation, we analyse how many stationary birth rates the equation can have in terms of the functional parameters of the model. In particular we show that, under reasonable and rather general assumptions, only one stationary birth rate can exist besides the trivial one (associated to the state in which there are no individuals and the population birth rate is zero). We give conditions for this non-trivial stationary birth rate to exist and analyse its stability using the principle of linearised stability for delay equations. Finally, we relate the results to the alternative, partial differential equation formulation of the model.

Final Size for Epidemic Models with Asymptomatic Transmission.

The final infection size is defined as the total number of individuals that become infected throughout an epidemic. Despite its importance for predicting the fraction of the population that will end infected, it does not capture which part of the infected population will present symptoms. Knowing this information is relevant because it is related to the severity of the epidemics. The objective of this work is to give a formula for the total number of symptomatic cases throughout an epidemic. Specifically, we focus on different types of structured SIR epidemic models (in which infected individuals can possibly become symptomatic before recovering), and we compute the accumulated number of symptomatic cases when time goes to infinity using a probabilistic approach. The methodology behind the strategy we follow is relatively independent of the details of the model.

On the Reproduction Number of a Gut Microbiota Model.

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.

To achieve an earlier IFN-γ response is not sufficient to control Mycobacterium tuberculosis infection in mice.

The temporo-spatial relationship between the three organs (lung, spleen and lymph node) involved during the initial stages of Mycobacterium tuberculosis infection has been poorly studied. As such, we performed an experimental study to evaluate the bacillary load in each organ after aerosol or intravenous infection and developed a mathematical approach using the data obtained in order to extract conclusions. The results showed that higher bacillary doses result in an earlier IFN-γ response, that a certain bacillary load (BL) needs to be reached to trigger the IFN-γ response, and that control of the BL is not immediate after onset of the IFN-γ response, which might be a consequence of the spatial dimension. This study may have an important impact when it comes to designing new vaccine candidates as it suggests that triggering an earlier IFN-γ response might not guarantee good infection control, and therefore that additional properties should be considered for these candidates.