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.

Local asymptotic stability of a system of integro-differential equations describing clonal evolution of a self-renewing cell population under mutation.

In this paper we consider a system of non-linear integro-differential equations (IDEs) describing evolution of a clonally heterogeneous population of malignant white blood cells (leukemic cells) undergoing mutation and clonal selection. We prove existence and uniqueness of non-trivial steady states and study their asymptotic stability. The results are compared to those of the system without mutation. Existence of equilibria is proved by formulating the steady state problem as an eigenvalue problem and applying a version of the Krein-Rutmann theorem for Banach lattices. The stability at equilibrium is analysed using linearisation and the Weinstein-Aronszajn determinant which allows to conclude local asymptotic stability.

Adaptive dynamics via Hamilton-Jacobi approach and entropy methods for a juvenile-adult model.

We consider a nonlinear system describing a juvenile-adult population undergoing small mutations. We analyze two aspects: from a mathematical point of view, we use an entropy method to prove that the population neither goes extinct nor blows-up; from an adaptive evolution point of view, we consider small mutations on a long time scale and study how a monomorphic or a dimorphic initial population evolves towards an Evolutionarily Stable State. Our method relies on an asymptotic analysis based on a constrained Hamilton-Jacobi equation. It allows to recover earlier predictions in Calsina and Cuadrado [A. Calsina, S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics, J. Math. Biol. 48 (2004) 135; A. Calsina, S. Cuadrado, Stationary solutions of a selection mutation model: the pure mutation case, Math. Mod. Meth. Appl. Sci. 15(7) (2005) 1091.] that we also assert by direct numerical simulation. One of the interests here is to show that the Hamilton-Jacobi approach initiated in Diekmann et al. [O. Diekmann, P.-E. Jabin, S. Mischler, B. Perthame, The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach, Theor. Popul. Biol. 67(4) (2005) 257.] extends to populations described by systems.

Asymptotic profile in selection-mutation equations: Gauss versus Cauchy distributions.

In this paper, we study the asymptotic (large time) behaviour of a selection-mutation-competition model for a population structured with respect to a phenotypic trait when the rate of mutation is very small. We assume that the reproduction is asexual, and that the mutations can be described by a linear integral operator. We are interested in the interplay between the time variable t and the rate ε of mutations. We show that depending on α > 0 , the limit ε → 0 with t = ε - α can lead to population number densities which are either Gaussian-like (when α is small) or Cauchy-like (when α is large).

Stability of equilibria of a predator-prey model of phenotype evolution.

We consider a selection mutation predator-prey model for the distribution of individuals with respect to an evolutionary trait. Local stability of the equilibria of this model is studied using the linearized stability principle and taking advantage of the (assumed) asymptotic stability of the equilibria of the resident population adopting an evolutionarily stable strategy.

Asymptotic stability of equilibria of selection-mutation equations.

We study local stability of equilibria of selection-mutation equations when mutations are either very small in size or occur with very low probability. The main mathematical tools are the linearized stability principle and the fact that, when the environment (the nonlinearity) is finite dimensional, the linearized operator at the steady state turns out to be a degenerate perturbation of a known operator with spectral bound equal to 0. An example is considered where the results on stability are applied.

Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics.

An integrodifferential equations model for the distribution of individuals with respect to the age at maturity is considered. Mutation is modeled by an integral operator. Results concerning the behaviour of the steady states and their relation to evolutionarily stable strategies when the mutation rate is small are given. The same results are obtained for a (rather) general class of models that include the one mentioned before.

Proapoptotic Fas ligand is expressed by normal kidney tubular epithelium and injured glomeruli.

Fas ligand (FasL) is a cell membrane cytokine that can promote apoptosis through activation of Fas receptors. Fas receptor activation induces glomerular cell apoptosis in vivo and participates in tubular cell death during acute renal failure. However, there is little information on the expression of FasL in the kidney. This study reports that FasL mRNA and protein are present in normal mouse and rat kidney. In situ hybridization and immunohistochemistry showed that proximal tubular epithelium is the main site of FasL expression in the normal kidney. In addition, increased total kidney FasL mRNA and de novo FasL protein expression by glomerular cells were observed in two different models of glomerular injury : rat immune-complex proliferative glumerulonephritis and murine lupus nephritis. Both full-length and soluble FasL were increased in the kidneys of the mice with nephritis. Cultured murine proximal tubular epithelial MCT cells and primary cultures of murine tubular epithelial cells expressed FasL mRNA and protein. Tubular epithelium-derived FasL induced apoptosis in Fassensitive lymphoid cell lines but not in Fas-resistant lymphoid cell lines. By contrast, MCT cells grown in the presence of the survival factors of serum were resistant to FasL, and only became partially sensitive to apoptosis induced by high concentrations (100 ng/ml) of FasL upon serum deprivation. However, MCT cells stimulated with inflammatory mediators (tumor necrosis factor-alpha, interferon-gamma, and lipopolysaccharide) increased cell surface Fas expression and were sensitized to apoptosis induced by FasL (FasL 55 +/- 5% versus control 8.3 +/- 4.1% apoptotic cells at 24 h, P < 0.05). Cytokine-primed primary cultures of tubular epithelial cells also acquired sensitivity to FasL-induced apoptosis. These results suggest that FasL expression by intrinsic renal cells may play a role in cell homeostasis in the normal kidney and during renal injury.