arXiv subjects

Florian Nill

Publications and source records attributed to Florian Nill.

15 recordsLinked to original sources

arXiv2024

The replacement number dynamics in SIR-type epidemic models I: From SSISS to RND picture

In SIR-type epidemic models time derivative of prevalence $I$ can always be cast into the form $\dot{I}=(X-1)I$, where $X$ is the replacement number and recovery rate is normalized to one. Assuming $\dot{X}=f(X,I)$ for some smooth function $f$ defines a "replacement number dynamics" (RND). Choosing transmission coefficients $\beta_1>\beta_2$, any such system uniquely maps to an isomorphic "SSISS model", i.e. an abstract SIR-type 3-compartment model. Extending to negative values $\beta_i<0$ takes care of demographic dynamics with compartment dependent birth and death rates. Fixing $f$ and varying $\beta_i$ generates a family of isomorphic SSISS systems, the "SSISS fiber" $\cal{F}(f)$}. A symmetry group $G_\sigma$ acts freely and transitively on fibers $\cal{F}(f)$, so SSISS systems become a principal $G_\sigma$-fiber bundle over the space of RND systems. Choosing two specific 6-parameter polynomials $f$ at most quadratic in $(X,I)$ covers a large class of models in the literature, with in total up to 17 (largely redundant!) parameters, including also reactive social behavior models. Epidemiological admissibility conditions guarantee forward boundedness and absence of periodic solutions in all these models. Part II of this work will prove existence and stability properties of endemic equilibria, which in RND picture simply boils down to analyzing zeros of the parabolas $f|_{I=0}$ and $f|_{X=1}$. This will cover and extend well known results for a wide range of models by a unifying approach while also closing some open issues in the literature.

q-bio.PE↗

arXiv2023

Scaling symmetries and parameter reduction in epidemic SI(R)S models

Symmetry concepts in parametrized dynamical systems may reduce the number of external parameters by a suitable normalization prescription. If, under the action of a symmetry group G, parameter space A becomes a (locally) trivial principal bundle, A ~ A/G x G, then the normalized dynamics only depends on the quotient A/G. In this way, the dynamics of fractional variables in homogeneous epidemic SI(R)S models, with standard incidence, absence of R-susceptibility and compartment independent birth and death rates, turns out to be isomorphic to (a marginally extended version of) Hethcote's classic endemic model, first presented in 1973. The paper studies a 10-parameter master model with constant and I-linear vaccination rates, vertical transmission and a vaccination rate for susceptible newborns. As recently shown by the author, all demographic parameters are redundant. After adjusting time scale, the remaining 5-parameter model admits a 3-dimensional abelian scaling symmetry. By normalization we end up with Hethcote's extended 2-parameter model. Thus, in view of symmetry concepts, reproving theorems on endemic bifurcation and stability in such models becomes needless.

q-bio.PE↗

arXiv2023

On the redundancy of birth and death rates in homogenous epidemic SIR models

The dynamics of fractional population sizes y_i=Y_i/N in homogeneous compartment models with time dependent total population N is analyzed. Assuming constant per capita birth and death rates the vector field Y_i'=V_i(Y) naturally projects to a vector field F_i(Y) tangent to the leaves of constant population N. A universal formula for the projected field F_i is given. In this way, in many SIR-type models with standard incidence all demographic parameters become redundant for the dynamical system y_i'=F_i(y). They may be put to zero by shifting remaining parameters appropriately. Normalizing eight examples from the literature this way, they unexpectedly become isomorphic for corresponding parameter ranges. Thus, some recently published results turn out to be already covered by papers 20 years ago.

q-bio.PE↗

arXiv2023

Symmetries and normalization in 3-compartment epidemic models I: The replacement number dynamics

As shown recently by the author, constant population SI(R)S models map to Hethcote's classic endemic model originally proposed in 1973. This unifies a whole class of models with up to 10 parameters, all being isomorphic to a simple 2-parameter master model for endemic bifurcation. In this work this procedure is extended to a 14-parameter SSISS Model, including social behavior parameters, a (diminished) susceptibility of the R-compartment and unbalanced constant per capita birth and death rates, thus covering many prominent models in the literature. Under mild conditions, in the dynamics for fractional variables in this model all vital parameters become redundant at the cost of possibly negative incidence rates. There is a symmetry group G acting on parameter space A, such that systems with G-equivalent parameters are isomorphic and map to the same normalized system. Using (Xrep,I) as canonical coordinates, Xrep the replacement number, normalization reduces to parameter space A/G with 5 parameters only. This approach reveals unexpected relations between various models in the literature. Part two of this work will analyze equilibria, stability and backward bifurcation and part three will further reduce the number of essential parameters from 5 to 3.

q-bio.PE↗

arXiv2022

Endemic Oscillations for SARS-CoV-2 Omicron -- A SIRS model analysis

The SIRS model with constant vaccination and immunity waning rates is well known to show a transition from a disease-free to an endemic equilibrium as the basic reproduction number $r_0$ is raised above threshold. It is shown that this model maps to Hethcote's classic endemic model originally published in 1973. In this way one obtains unifying formulas for a whole class of models showing endemic bifurcation. In particular, if the vaccination rate is smaller than the recovery rate and $r_- < r_0 < r_+$ for certain upper and lower bounds $r_\pm$, then trajectories spiral into the endemic equilibrium via damped infection waves. Latest data of the SARS-CoV-2 Omicron variant suggest that according to this simplified model continuous vaccination programs will not be capable to escape the oscillating endemic phase. However, in view of the strong damping factors predicted by the model, in reality these oscillations will certainly be overruled by time-dependent contact behaviors.

q-bio.PE↗

arXiv1999

Integral Theory for Quasi-Hopf Algebras

We generalize the fundamental structure Theorem on Hopf (bi)-modules by Larson and Sweedler to quasi-Hopf algebras H. If H is finite dimensional this proves the existence and uniqueness (up to scalar multiples) of integrals in H. Among other applications we prove a Maschke type Theorem for diagonal crossed products as constructed by the authors.

math.QA↗

arXiv1998

Weak Hopf Algebras and Reducible Jones Inclusions of Depth 2. I: From Crossed products to Jones towers

We apply the theory of finite dimensional weak C^*-Hopf algebras A as developed by G. Böhm, F. Nill and K. Szlachányi to study reducible inclusion triples of von-Neumann algebras N \subset M \subset (M\cros\A). Here M is an A-module algebra, N is the fixed point algebra and \M\cros\A is the crossed product extension. ``Weak'' means that the coproduct Δon A is non-unital, requiring various modifications of the standard definitions for (co-)actions and crossed products. We show that acting with normalized positive and nondegenerate left integrals l\in\A gives rise to faithful conditional expectations E_l: M-->N, where under certain regularity conditions this correspondence is one-to-one. Associated with such left integrals we construct ``Jones projections'' e_l\in\A obeying the Jones relations as an identity in M\cros\A. Finally, we prove that N\subset M always has finite index and depth 2 and that the basic Jones construction is given by the ideal M_1:=M e_l M \subset M\cros\A, where under appropriate conditions M_1 = M\cros\A. In a subsequent paper we will show that converseley any reducible finite index and depth-2 Jones tower of von-Neumann factors (with finite dimensional centers) arises in this way.

math.QA↗

arXiv1998

Axioms for Weak Bialgebras

Let A be a finite dimensional unital associative algebra over a field K, which is also equipped with a coassociative counital coalgebra structure (Δ,\eps). A is called a Weak Bialgebra if the coproduct Δis multiplicative. We do not require Δ(1) = 1 \otimes 1 nor multiplicativity of the counit \eps. Instead, we propose a new set of counit axioms, which are modelled so as to guarantee that \Rep\A becomes a monoidal category with unit object given by the cyclic A-submodule \E := (A --> \eps) \subset \hat A (\hat A denoting the dual weak bialgebra). Under these monoidality axioms \E and \bar\E := (\eps <-- A) become commuting unital subalgebras of \hat A which are trivial if and only if the counit \eps is multiplicative. We also propose axioms for an antipode S such that the category \Rep\A becomes rigid. S is uniquely determined, provided it exists. If a monoidal weak bialgebra A has an antipode S, then its dual \hat A is monoidal if and only if S is a bialgebra anti-homomorphism, in which case S is also invertible. In this way we obtain a definition of weak Hopf algebras which in Appendix A will be shown to be equivalent to the one given independently by G. Böhm and K. Szlachányi. Special examples are given by the face algebras of T. Hayashi and the generalised Kac algebras of T. Yamanouchi.

math.QA↗

arXiv1997

Doubles of Quasi-Quantum Groups

Drinfeld showed that any finite dimensional Hopf algebra \G extends to a quasitriangular Hopf algebra \D(\G), the quantum double of \G. Based on the construction of a so--called diagonal crossed product developed by the authors, we generalize this result to the case of quasi--Hopf algebras \G. As for ordinary Hopf algebras, as a vector space the ``quasi--quantum double'' \D(\G) is isomorphic to the tensor product of \G and its dual \dG. We give explicit formulas for the product, the coproduct, the R--matrix and the antipode on \D(\G) and prove that they fulfill Drinfeld's axioms of a quasitriangular quasi--Hopf algebra. In particular \D(\G) becomes an associative algebra containing \G as a quasi--Hopf subalgebra. On the other hand, \dG \otimes 1 is not a subalgebra of \D(\G) unless the coproduct on \G is strictly coassociative. It is shown that the category of finite dimensional representations of \D(\G) coincides with what has been called the double category of \G--modules by S. Majid [M2]. Thus our construction gives a concrete realization of Majid's abstract definition of quasi--quantum doubles in terms of a Tannaka--Krein--like reconstruction procedure. The whole construction is shown to generalize to weak quasi--Hopf algebras with \D(\G) now being linearly isomorphic to a subspace of \dG \otimes \G.

q-alg↗

arXiv1997

Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry

Given a finite dimensional C^*-Hopf algebra H and its dual H^ we construct the infinite crossed product A=... x H x H^ x H ... and study its superselection sectors in the framework of algebraic quantum field theory. A is the observable algebra of a generalized quantum spin chain with H-order and H^-disorder symmetries, where by a duality transformation the role of order and disorder may also appear interchanged. If H=\CC G is a group algebra then A becomes an ordinary G-spin model. We classify all DHR-sectors of A --- relative to some Haag dual vacuum representation --- and prove that their symmetry is described by the Drinfeld double D(H). To achieve this we construct localized coactions ρ: A \to (A \otimes D(H)) and use a certain compressibility property to prove that they are universal amplimorphisms on A. In this way the double D(H) can be recovered from the observable algebra A as a universal cosymmetrty.

hep-th↗

arXiv1997

Diagonal Crossed Products by Duals of Quasi-Quantum Groups

Let \G be a (weak) quasi-Hopf algebra. Using a two-sided \G-coaction on an algebra \M, we construct what we call the diagonal crossed product as a new associative algebra structure on \M\otimes \dG, where \dG is the dual of \G. This construction is largely motivated by the special case \M = \G, for which we obtain an explicit definition of the quantum double \D(\G) for quasi-Hopf algebras. Applications of our formalism include the field algebra construction of Mack and Schomerus as well as the formulation of Hopf Spin chains or lattice current algebras based on truncated quantum groups at roots of unity. A complete proof that \D(\G) is even a (weak) quasi-triangular quasi-Hopf algebra will be given in a separate paper.

q-alg↗

arXiv1996

On the Structure of Monodromy Algebras and Drinfeld Doubles

We give a review and some new relations on the structure of the monodromy algebra (also called loop algebra) associated with a quasitriangular Hopf algebra H. It is shown that as an algebra it coincides with the so-called braided group constructed by S. Majid on the dual of H. Gauge transformations act on monodromy algebras via the coadjoint action. Applying a result of Majid, the resulting crossed product is isomorphic to the Drinfeld double D(H). Hence, under the so-called factorizability condition given by N. Reshetikhin and M. Semenov-Tian- Shansky, both algebras are isomorphic to the algbraic tensor product H\otimes H. It is indicated that in this way the results of Alekseev et al. on lattice current algebras are consistent with the theory of more general Hopf spin chains given by K. Szlachányi and the author. In the Appendix the multi-loop algebras L_m of Alekseev and Schomerus [AS] are identified with braided tensor products of monodromy algebras in the sense of Majid, which leads to an explanation of the ``bosonization formula'' of [AS] representing L_m as H\otimes\dots\otimes H.

q-alg↗

arXiv1995

Quantum Chains of Hopf Algebras with Order-Disorder Fields and Quantum Double Symmetry

Given a finite dimensional C-*-Hopf algebra H and its dual H^ we construct the infinite crossed product A=... x H x H^ x H x ... and study its representations. A is the observable algebra of a generalized spin model with H-order and H^-disorder symmetries. By pointing out that A possesses a certain compressibility property we can classify all DHR-sectors of A --- relative to some Haag dual vacuum representation --- and prove that their symmetry is described by the Drinfeld double D(H). Complete, irreducible, translation covariant field algebra extensions F > A are shown to be in one-to-one correspondence with cohomology classes of 2-cocycles u in D(H) @ D(H).

hep-th↗

arXiv1994

A Comment on Jones Inclusions with infinite Index

Given an irreducible inclusion of infinite von-Neumann-algebras $\cn \subset \cm$ together with a conditional expectation $ E : \cm \rightarrow \cm $ such that the inclusion has depth 2, we show quite explicitely how $\cn $ can be viewed as the fixed point algebra of $\cm$ w.r.t. an outer action of a compact Kac-algebra acting on $\cm$. This gives an alternative proof, under this special setting of a more general result of M. Enock and R. Nest, [E-N], see also S. Yamagami, [Ya2].

hep-th↗

arXiv1994

Electrically and Magnetically Charged States and Particles in the 2+1-Dimensional Z_N-Higgs Gauge Model

Electrically as well as magnetically charged states are constructed in the 2+1-dimensional Euclidean Z_N-Higgs lattice gauge model, the former following ideas of Fredenhagen and Marcu and the latter using duality transformations on the algebra of observables. The existence of electrically and of magnetically charged particles is also established. With this work we prepare the ground for the constructive study of anyonic statistics of multiparticle scattering states of electrically and magnetically charged particles in this model (work in progress).

hep-th↗