Functional limit theorems for random regular graphs

Consider d uniformly random permutation matrices on n labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices. We consider limit theorems for various combinatorial and analytical properties of this graph (or the matrix) as n grows to infinity, either when d is kept fixed or grows slowly with n. In a suitable weak convergence framework, we prove that the (finite but growing in length) sequences of the number of short cycles and of cyclically non-backtracking walks converge to distributional limits. We estimate the total variation distance from the limit using Stein's method. As an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix. A key step in this latter derivation is an extension of the Kahn-Szemer\'edi argument for estimating the second largest eigenvalue for all values of d and n.Comment: Added Remark 27. 39 pages. To appear in Probability Theory and Related Field

Immunogenicity and Protective Capacity of a Virosomal Respiratory Syncytial Virus Vaccine Adjuvanted with Monophosphoryl Lipid A in Mice

Respiratory Syncytial Virus (RSV) is a major cause of viral brochiolitis in infants and young children and is also a significant problem in elderly and immuno-compromised adults. To date there is no efficacious and safe RSV vaccine, partially because of the outcome of a clinical trial in the 1960s with a formalin-inactivated RSV vaccine (FI-RSV). This vaccine caused enhanced respiratory disease upon exposure to the live virus, leading to increased morbidity and the death of two children. Subsequent analyses of this incident showed that FI-RSV induces a Th2-skewed immune response together with poorly neutralizing antibodies. As a new approach, we used reconstituted RSV viral envelopes, i.e. virosomes, with incorporated monophosphoryl lipid A (MPLA) adjuvant to enhance immunogenicity and to skew the immune response towards a Th1 phenotype. Incorporation of MPLA stimulated the overall immunogenicity of the virosomes compared to non-adjuvanted virosomes in mice. Intramuscular administration of the vaccine led to the induction of RSV-specific IgG2a levels similar to those induced by inoculation of the animals with live RSV. These antibodies were able to neutralize RSV in vitro. Furthermore, MPLA-adjuvanted RSV virosomes induced high amounts of IFNγ and low amounts of IL5 in both spleens and lungs of immunized and subsequently challenged animals, compared to levels of these cytokines in animals vaccinated with FI-RSV, indicating a Th1-skewed response. Mice vaccinated with RSV-MPLA virosomes were protected from live RSV challenge, clearing the inoculated virus without showing signs of lung pathology. Taken together, these data demonstrate that RSV-MPLA virosomes represent a safe and efficacious vaccine candidate which warrants further evaluation