Uniqueness and stability for the Vlasov-Poisson system with spatial density in Orlicz spaces

In this paper, we establish uniqueness of the solution of the Vlasov-Poisson system with spatial density belonging to a certain class of Orlicz spaces. This extends the uniqueness result of Loeper (which holds for uniformly bounded density) and the uniqueness result of the second author. Uniqueness is a direct consequence of our main result, which provides a quantitative stability estimate for the Wasserstein distance between two weak solutions with spatial density in such Orlicz spaces, in the spirit of Dobrushin's proof of stability for mean-field PDEs. Our proofs are built on the second-order structure of the underlying characteristic system associated to the equation

Approximations of strongly continuous families of unbounded self-adjoint operators

The problem of approximating the discrete spectra of families of self-adjoint operators that are merely strongly continuous is addressed. It is well-known that the spectrum need not vary continuously (as a set) under strong perturbations. However, it is shown that under an additional compactness assumption the spectrum does vary continuously, and a family of symmetric finite-dimensional approximations is constructed. An important feature of these approximations is that they are valid for the entire family uniformly. An application of this result to the study of plasma instabilities is illustrated.Comment: 22 pages, final version to appear in Commun. Math. Phy

Multi-scale immune selection and the maintenance of structured antigenic diversity in the malaria parasite Plasmodium falciparum

The most virulent malaria parasite, Plasmodium falciparum, makes use of extensive antigenic diversity to maximise its transmission potential. Parasite genomes contain several highly polymorphic gene families, whose products are the target of protective immune responses. The best studied of these are the PfEMP1 surface proteins, which are encoded by the var multi-gene family and are important virulence factors. During infection, the parasite switches expression between PfEMP1 variants in order to evade adaptive immune responses and prolong infection. On the population level, parasites appear to be structured with respect to their var genes into non-overlapping repertoires, which can lead to high reinfection rates. This non-random structuring of antigenic diversity can also be found at the level of individual var gene repertoires and var genes themselves. However, not much is known about the evolutionary determinants which select for and maintain this structure at different ecological scales. In this thesis I investigate the mechanisms by which multi-scale immune selection and other ecological factors influence the evolution of structured diversity. Using a suite of theoretical frameworks I show that treating diversity as a dynamic property, which emerges from the underlying infection and transmission processes, has a major effect on the relationship between the parasite’s transmis- sion potential and disease prevalence, with important implications for monitoring control efforts. Furthermore, I show that an evolutionary trade-off between within-host and between-host fitness together with functional constraints on diversification can explain the structured diversity found at both the repertoire and parasite population level and might also account for empirically observed exposure-dependent acquisition of immunity. Together, this work highlights the need to consider evolutionary factors acting at different ecological scales to gain a more comprehensive understanding of the complex immune-epidemiology of P. falciparum malaria

Maintenance of phenotypic diversity within a set of virulence encoding genes of the malaria parasite Plasmodium falciparum

Open access article This is the final version of the article. Available from the publisher via the DOI in this record.Infection by the human malaria parasite Plasmodium falciparum results in a broad spectrum of clinical outcomes, ranging from severe and potentially life-threatening malaria to asymptomatic carriage. In a process of naturally acquired immunity, individuals living in malaria-endemic regions build up a level of clinical protection, which attenuates infection severity in an exposure-dependent manner. Underlying this shift in the immunoepidemiology as well as the observed range in malaria pathogenesis is the var multigene family and the phenotypic diversity embedded within. The var gene-encoded surface proteins Plasmodium falciparum erythrocyte membrane protein 1 mediate variant-specific binding of infected red blood cells to a diverse set of host receptors that has been linked to specific disease manifestations, including cerebral and pregnancy-associated malaria. Here, we show that cross-reactive immune responses, which minimize the within-host benefit of each additionally expressed gene during infection, can cause selection for maximum phenotypic diversity at the genome level. We further show that differential functional constraints on protein diversification stably maintain uneven ratios between phenotypic groups, in line with empirical observation. Our results thus suggest that the maintenance of phenotypic diversity within P. falciparum is driven by an evolutionary trade-off that optimizes between within-host parasite fitness and between-host selection pressure.Royal Society - University Research Fellowshi

Stability and instability in saddle point dynamics - Part I

We consider the problem of convergence to a saddle point of a concave-convex function via gradient dynamics. Since first introduced by Arrow, Hurwicz and Uzawa such dynamics have been extensively used in diverse areas, there are, however, features that render their analysis non trivial. These include the lack of convergence guarantees when the concave-convex function considered does not satisfy additional strictness properties and also the non-smoothness of subgradient dynamics. Our aim in this two part paper is to provide an explicit characterization to the asymptotic behaviour of general gradient and subgradient dynamics applied to a general concave-convex function in $C^2$ . We show that despite the nonlinearity and non-smoothness of these dynamics their ω-limit set is comprised of trajectories that solve only explicit linear ODEs characterized within the paper. More precisely, in Part I an exact characterization is provided to the asymptotic behaviour of unconstrained gradient dynamics. We also show that when convergence to a saddle point is not guaranteed then the system behaviour can be problematic, with arbitrarily small noise leading to an unbounded second moment for the magnitude of the state vector. In Part II we consider a general class of subgradient dynamics that restrict trajectories in an arbitrary convex domain, and show that when an equilibrium point exists the limiting trajectories belong to a class of dynamics characterized in part I as linear ODEs. These results are used to formulate corresponding convergence criteria and are demonstrated with examples.ERC starting grant 67977

Stability and instability in saddle point dynamics Part II: The subgradient method

In part I we considered the problem of convergence to a saddle point of a concave-convex function in $C^2$ via gradient dynamics and an exact characterization was given to their asymptotic behaviour. In part II we consider a general class of subgradient dynamics that provide a restriction in a convex domain. We show that despite the nonlinear and non-smooth character of these dynamics their ω-limit set is comprised of solutions to only linear ODEs. In particular, we show that the latter are solutions to subgradient dynamics on affine subspaces which is a smooth class of dynamics the asymptotic properties of which have been exactly characterized in part I. Various convergence criteria are formulated using these results and several examples and applications are also discussed throughout the manuscript.ERC starting grant 67977