Marios Antonios Gkogkas, Christian Kuehn, Chuang Xu
Abstract Many science phenomena are modelled as interacting particle systems (IPS) coupled on static networks. In reality, network connections are far more dynamic. Connections among individuals receive feedback from nearby individuals and make changes to better adapt to the world. Hence, it is reasonable to model myriad real-world phenomena as co-evolutionary (or adaptive) networks. These networks are used in different areas including telecommunication, neuroscience, computer science, biochemistry, social science, as well as physics, where Kuramoto-type networks have been widely used to model interaction among a set of oscillators. In this paper, we propose a rigorous formulation for limits of a sequence of co-evolutionary Kuramoto oscillators coupled on heterogeneous co-evolutionary networks, which receive both positive and negative feedback from the dynamics of the oscillators on the networks. We show under mild conditions, the mean field limit (MFL) of the co-evolutionary network exists and the sequence of co-evolutionary Kuramoto networks converges to this MFL. Such MFL is described by solutions of a generalised Vlasov equation. We treat the graph limits as signed graph measures, motivated by the recent work in [Kuehn, Xu. Vlasov equations on digraph measures, JDE, 339 (2022), 261–349]. In comparison to the recently emerging works on MFLs of IPS coupled on non-co-evolutionary networks (i.e., static networks or time-dependent networks independent of the dynamics of the IPS), our work seems the first to rigorously address the MFL of a co-evolutionary network model. The approach is based on our formulation of a generalisation of the co-evolutionary network as a hybrid system of ODEs and measure differential equations parametrised by a vertex variable, together with an analogue of the variation of parameters formula, as well as the generalised Neunzert’s in-cell-particle method developed in [Kuehn, Xu. Vlasov equations on digraph measures, JDE, 339 (2022), 261–349].
In this paper, an epidemiological model is proposed to study the dynamics of coinfection diseases (TB) and COVID-19 with the effect of vaccination. Tuberculosis (TB) and COVID-19 both are infectious diseases that pose significant global health challenges. Evidence suggests that individuals with TB have a higher risk of acquiring the COVID-19 infection. With the emergence of the COVID-19 pandemic, concerns have arisen regarding the potential impact of the concomitant presence of TB and COVID-19. The epidemiological model is qualitatively analysed using stability analysis theory. The dynamic system exhibits a stable endemic equilibrium point while R0 < 1 and unstable when R0 > 1. The Lyapunov function is used to investigate the global stability of an endemic equilibrium point. The sensitivity analysis is carried out to identify the effective parameters that have the greatest influence on the reproduction number. Numerical results are carried out to assess the effect of various biological parameters in the dyanamic of both coinfection classes of TB & COVID-19. This study aims to analyze the implications of these concurrent diseases and predict the effect of vaccination in managing their coexistence. Our simulation results show that both the coinfection disease TB and COVID-19 can be reduced by increasing rate of vaccination.
Sara Billey, Matjaz Konvalinka, T. Kyle Petersen
et al.
Parabolic subgroups WI of Coxeter systems (W,S) and their ordinary and double cosets W/WI and WI\W/WJ appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets wWI , for I ⊆ S, forms the Coxeter complex of W , and is well-studied. In this extended abstract, we look at a less studied object: the set of all double cosets WIwWJ for I,J ⊆ S. Each double coset can be presented by many different triples (I, w, J). We describe what we call the lex-minimal presentation and prove that there exists a unique such choice for each double coset. Lex-minimal presentations can be enumerated via a finite automaton depending on the Coxeter graph for (W, S). In particular, we present a formula for the number of parabolic double cosets with a fixed minimal element when W is the symmetric group Sn. In that case, parabolic subgroups are also known as Young subgroups. Our formula is almost always linear time computable in n, and the formula can be generalized to any Coxeter group.
We study the relationship between rational slope Dyck paths and invariant subsets in Z, extending the work of the first two authors in the relatively prime case. We also find a bijection between (dn, dm)–Dyck paths and d-tuples of (n, m)-Dyck paths endowed with certain gluing data. These are first steps towards understanding the relationship between the rational slope Catalan combinatorics in non relatively prime case and the geometry of affine Springer fibers and representation theory.
Motivated by a recent conjecture of R. P. Stanley we offer a lower bound for the sum of the coefficients of a Schubert polynomial in terms of $132$-pattern containment.
We show that every set $\mathcal{P}$ of $n$ non-collinear points in the plane contains a point incident to at least $\lceil\frac{n}{3}\rceil+1$ of the lines determined by $\mathcal{P}$.
We approach the problem of counting the number of walks in a digraph from three different perspectives: enumerative combinatorics, linear algebra, and symbolic dynamics.
We consider a carries process which is a generalization of that by Holte in the sense that (i) we take various digit sets, and (ii) we also consider negative base. Our results are : (i) eigenvalues and eigenvectors of the transition probability matrices, and their connection to combinatorics and representation theory, (ii) an application to the computation of the distribution of the sum of i.i.d. uniform r.v.'s on [0,1], (iii) a relation to riffle shuffle.