Multi-patch epidemic models with general exposed and infectious periods

We study   multi-patch epidemic models where individuals may migrate from one patch to another in either of the susceptible, exposed/latent, infectious and recovered states.  We assume that infections occur both locally with a rate that depends on the patch as well as ``from distance" from all the other patches. The exposed and infectious periods have general distributions, and are not affected by the possible migrations of the individuals. The migration processes in either of the three states are assumed to be Markovian, and independent of the exposed and infectious periods. We establish a functional law of large number (FLLN) and a function central limit theorem (FCLT) for the susceptible, exposed/latent, infectious and recovered processes. In the FLLN, the limit is determined by a set of Volterra integral equations. In the special case of deterministic exposed and infectious periods, the limit becomes a system of ODEs with delays. In the FCLT, the limit is given by a set of stochastic Volterra integral equations driven by a sum of independent Brownian motions and   continuous Gaussian processes with an explicit covariance structure.

Large deviations of the exit measure through a characteristic boundary for a Poisson driven SDE

Let O be the basin of attraction of a given equilibrium of a dynamical system, whose solution is the law of large numbers limit of the solution of a Poissonian SDE as the size of the population tends to +∞. We consider the law of the exit point from O of that Poissonian SDE. We adapt the approach of Day [J. Math. Anal. Appl. 147 (1990) 134–153] who studied the same problem for an ODE with a small Brownian perturbation. For that purpose, we will use the large deviations principle for the Poissonian SDE reflected at the boundary of O, studied in our recent work Pardoux and Samegni [Stoch. Anal. Appl. 37 (2019) 836–864]. The main motivation of this work is the extension of the results concerning the time of exit from the set O established in Kratz and Pardoux [Vol. 2215 of Lecture Notes in Math.. Springer (2018) 221–327] and Pardoux and Samegni [J. Appl. Probab. 54 (2017) 905–920] to unbounded open sets O. This is done in sections 4.2.5 and 4.2.7 of Britton and Pardoux [Vol. 2255 of Lecture Notes in Math. Springer (2019) 1–120], see also The SIR model with demography subsection below.

Soluble HLA-G Expression Inversely Correlates With Fetal Microchimerism Levels in Peripheral Blood From Women With Scleroderma

Women with scleroderma (SSc) maintain significantly higher quantities of persisting fetal microchimerism (FMc) from complete or incomplete pregnancies in their peripheral blood compared to healthy women. The non-classical class-I human leukocyte antigen (HLA) molecule HLA-G plays a pivotal role for the implantation and maintenance of pregnancy and has often been investigated in offspring from women with pregnancy complications. However data show that maternal HLA-G polymorphisms as well as maternal soluble HLA-G (sHLA-G) expression could influence pregnancy outcome. Here, we aimed to investigate the underlying role of maternal sHLA-G expression and HLA-G polymorphisms on the persistence of FMc. We measured sHLA-G levels by enzyme linked immunosorbent assay in plasma samples from 88 healthy women and 74 women with SSc. Male Mc was quantified by DYS14 real-time PCR in blood samples from 58 women who had previously given birth to at least one male child. Furthermore, eight HLA-G 5'URR/3'UTR polymorphisms, previously described as influencing HLA-G expression, were performed on DNA samples from 96 healthy women and 106 women with SSc. Peripheral sHLA-G was at lower concentration in plasma from SSc (76.2 ± 48.3 IU/mL) compared to healthy women (117.5 ± 60.1 IU/mL, p < 0.0001), independently of clinical subtypes, autoantibody profiles, disease duration, or treatments. Moreover, sHLA-G levels were inversely correlated to FMc quantities (Spearman correlation, p < 0.01). Finally, women with SSc had lower sHLA-G independently of the eight HLA-G 5'URR/3'UTR polymorphisms, although they were statistically more often homozygous than heterozygous for HLA-G polymorphism genotypes -716 (G/T), -201 (G/A), 14 bp (ins/del), and +3,142 (G/A) than healthy women. In conclusion, women with SSc display less sHLA-G expression independently of the eight HLA-G polymorphisms tested. This decreased production correlates with higher quantities of persisting FMc commonly observed in blood from SSc women. These results shed some lights on the contribution of the maternal HLA-G protein to long-term persistent fetal Mc and initiate new perspectives in this field.

Phenotypic lag and population extinction in the moving-optimum model: insights from a small-jumps limit

Continuous environmental change-such as slowly rising temperatures-may create permanent maladaptation of natural populations: Even if a population adapts evolutionarily, its mean phenotype will usually lag behind the phenotype favored in the current environment, and if the resulting phenotypic lag becomes too large, the population risks extinction. We analyze this scenario using a moving-optimum model, in which one or more quantitative traits are under stabilizing selection towards an optimal value that increases at a constant rate. We have recently shown that, in the limit of infinitely small mutations and high mutation rate, the evolution of the phenotypic lag converges to an Ornstein-Uhlenbeck process around a long-term equilibrium value. Both the mean and the variance of this equilibrium lag have simple analytical formulas. Here, we study the properties of this limit and compare it to simulations of an evolving population with finite mutational effects. We find that the "small-jumps limit" provides a reasonable approximation, provided the mean lag is so large that the optimum cannot be reached by a single mutation. This is the case for fast environmental change and/or weak selection. Our analysis also provides insights into population extinction: Even if the mean lag is small enough to allow a positive growth rate, stochastic fluctuations of the lag will eventually cause extinction. We show that the time until this event follows an exponential distribution, whose mean depends strongly on a composite parameter that relates the speed of environmental change to the adaptive potential of the population.

Binary Trees, Exploration Processes, and an Extended Ray-Knight Theorem

We study the bijection between binary Galton-Watson trees in continuous time and their exploration process, both in the subcritical and in the supercritical cases. We then take the limit over renormalized quantities, as the size of the population tends to ∞. We thus deduce Delmas' generalization of the second Ray-Knight theorem.

HLA-DRB1 genotypes and the risk of developing anti citrullinated protein antibody (ACPA) positive rheumatoid arthritis

OBJECTIVE

To provide a table indicating the risk for developing anti citrullinated protein antibody (ACPA) positive rheumatoid arthritis (RA) according to one's HLA-DRB1 genotype.

METHODS

We HLA-DRB1 genotyped 857 patients with ACPA positive RA and 2178 controls from South Eastern and Eastern France and calculated Odds Ratios (OR) for developing RA for 106 of 132 possible genotypes accounting for 97% of subjects.

RESULTS

HLA-DRB1 genotypic ORs for developing ACPA positive RA range from 28 to 0.19. HLA-DRB1 genotypes with HLA-DRB1*04SE (HLA-DRB1*0404, HLA-DRB1*0405, HLA-DRB1*0408), HLA-DRB1*04∶01, HLA-DRB1*01 are usually associated with high risk for developing RA. The second HLA-DRB1 allele in genotype somewhat modulates shared epitope associated risk. We did not identify any absolutely protective allele. Neither the Reviron, nor the du Montcel models accurately explains our data which are compatible with the shared epitope hypothesis and suggest a dosage effect among shared epitope positive HLA-DRB1 alleles, double dose genotypes carrying higher ORs than single dose genotypes.

CONCLUSION

HLA-DRB1 genotypic risk for developing ACPA positive RA is influenced by both HLA-DRB1 alleles in genotype. We provide an HLA-DRB1 genotypic risk table for ACPA positive RA.

Evolution of the ancestral recombination graph along the genome in case of selective sweep

We consider the genome of a sample of n individuals taken at the end of a selective sweep, which is the fixation of an advantageous allele in the population. When the selective advantage is high, the genealogy at a locus under selective sweep can be approximated by a comb with n teeth. However, because of recombinations during the selective sweep, the hitchhiking effect decreases as the distance from the selected site increases, so that far from this locus, the tree can be approximated by a Kingman coalescent tree, as in the neutral case. We first give the distribution of the tree at a given locus. Then we focus on the evolution of this tree along the genome. Since this tree-valued process is not Markovian, we study the evolution of the Ancestral Recombination Graph along the genome in case of selective sweep.

On the Height and Length of the Ancestral Recombination Graph

The goal of this paper is to provide formulae for the expectation and variance of the height and length of the ancestral recombination graph (ARG). While the formula for the expectation of the height is known (see, e.g. Krone and Neuhauser (1997)), the other formulae seem to be new. We obtain in particular (see Theorem 4.1) a very simple formula which expresses the expectation of the length of the ARG as a linear combination of the expectations of both the length of the coalescent tree and the height of the ARG. Finally, we study the speed at which the ARG comes down from infinity.