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Rule-based models have been successfully used to represent different aspects of the COVID-19 pandemic, including age, testing, hospitalisation, lockdowns, immunity, infectivity, behaviour, mobility and vaccination of individuals. These rule-based approaches are motivated by chemical reaction rules which are traditionally solved numerically with the standard Gillespie algorithm proposed in the context of molecular dynamics. When applying reaction system type of approaches to epidemiology, generalisations of the Gillespie algorithm are required due to the time-dependency of the problems. In this article, we present different generalisations of the standard Gillespie algorithm which address discrete subtypes (e.g., incorporating the age structure of the population), time-discrete updates (e.g., incorporating daily imposed change of rates for lockdowns) and deterministic delays (e.g., given waiting time until a specific change in types such as release from isolation occurs). These algorithms are complemented by relevant examples in the context of the COVID-19 pandemic and numerical results.

Motivated by chemical reaction rules, we introduce a rule-based epidemiological framework for the systematic mathematical modelling of future pandemics. Here we stress that we do not have a specific model in mind, but a whole collection of models which can be transformed into each other, or represent different aspects of a pandemic, and these aspects can change during the course of the emergency, as happened during the Covid-19 pandemic. As conditions for outbreaks in the modern world change on different time-scales, some rapidly, epidemiology has few 'laws', besides perhaps the fundamental infection process described by Kermack-McKendrick. Each single of our variety of models, called framework, is based on a mathematical formulation that we call a rule-based system. They have several advantages, for example that they can be both interpreted stochastically and deterministically, without changing the model structure. Rule-based systems should be easier to communicate to non-specialists, when compared to differential equations. Due to their combinatorial nature, the rule-based model framework we propose is ideal for systematic mathematical modelling, systematic links to statistics, data analysis in general and also machine learning leading to artificial intelligence., Comment: 67 pages, 23 figures

The energy spectrum of cosmic Hydrogen and Helium nuclei has been measured, below the so-called "knee", by using a hybrid experiment with a wide field-of-view Cherenkov telescope and the Resistive Plate Chamber (RPC) array of the ARGO-YBJ experiment at 4300 m above sea level. The Hydrogen and Helium nuclei have been well separated from other cosmic ray components by using a multi-parameter technique. A highly uniform energy resolution of about 25% is achieved throughout the whole energy range (100 TeV - 700 TeV). The observed energy spectrum is compatible with a single power law with index gamma=-2.63+/-0.06., Comment: To be published on Chinese Physics C

The ARGO-YBJ detector is an extensive air shower array that has been used to monitor the northern $\gamma$-ray sky at energies above 0.3 TeV from 2007 November to 2013 January. In this paper, we present the results of a sky survey in the declination band from $-10^{\circ}$ to $70^{\circ}$, using data recorded over the past five years. With an integrated sensitivity ranging from 0.24 to $\sim$1 Crab units depending on the declination, six sources have been detected with a statistical significance greater than 5 standard deviations. Several excesses are also reported as potential $\gamma$-ray emitters. The features of each source are presented and discussed. Additionally, $95\%$ confidence level upper limits of the flux from the investigated sky region are shown. Specific upper limits for 663 GeV $\gamma$-ray AGNs inside the ARGO-YBJ field of view are reported. The effect of the absorption of $\gamma$-rays due to the interaction with extragalactic background light is estimated., Comment: 30 pages, 13 figures, Accepted for publication by ApJ

We report the observation of a very high energy \gamma-ray source, whose position is coincident with HESS J1841-055. This source has been observed for 4.5 years by the ARGO-YBJ experiment from November 2007 to July 2012. Its emission is detected with a statistical significance of 5.3 standard deviations. Parameterizing the source shape with a two-dimensional Gaussian function we estimate an extension \sigma=(0.40(+0.32,-0.22}) degree, consistent with the HESS measurement. The observed energy spectrum is dN/dE =(9.0-+1.6) x 10^{-13}(E/5 TeV)^{-2.32-+0.23} photons cm^{-2} s^{-1} TeV^{-1}, in the energy range 0.9-50 TeV. The integral \gamma-ray flux above 1 TeV is 1.3-+0.4 Crab units, which is 3.2-+1.0 times the flux derived by HESS. The differences in the flux determination between HESS and ARGO-YBJ, and possible counterparts at other wavelengths are discussed., Comment: 17 pages, 4 figures, have been accepted for publication in ApJ

Local bifurcation theory typically deals with the response of a degenerate but isolated equilibrium state or periodic orbit of a dynamical system to perturbations controlled by one or more independent parameters, and characteristically uses tools from singularity theory. There are many situations, however, in which the equilibrium state or periodic orbit is not isolated but belongs to a manifold $S$ of such states, typically as a result of continuous symmetries in the problem. In this case the bifurcation analysis requires a combination of local and global methods, and is most tractable in the case of normal nondegeneracy, that is when the degeneracy is only along $S$ itself and the system is nondegenerate in directions normal to $S$. In this paper we consider the consequences of relaxing normal nondegeneracy, which can generically occur within 1-parameter families of such systems. We pay particular attention to the simplest but important case where $\dim S=1$ and where the normal degeneracy occurs with corank 1. Our main focus is on uniform degeneracy along $S$, although we also consider aspects of the branching structure for solutions when the degeneracy varies at different places on $S$. The tools are those of singularity theory adapted to global topology of $S$, which allow us to explain the bifurcation geometry in natural way. In particular, we extend and give a clear geometric setting for earlier analytical results of Hale and Taboas.

Time-continuous dynamical systems defined on graphs are often used to model complex systems with many interacting components in a non-spatial context. In the reverse sense attaching meaningful dynamics to given 'interaction diagrams' is a central bottleneck problem in many application areas, especially in cell biology where various such diagrams with different conventions describing molecular regulation are presently in use. In most situations these diagrams can only be interpreted by the use of both discrete and continuous variables during the modelling process, corresponding to both deterministic and stochastic hybrid dynamics. The conventions in genetics are well-known, and therefore we use this field for illustration purposes. In [25] and [26] the authors showed that with the help of a multi-scale analysis stochastic systems with both continuous variables and finite state spaces can be approximated by dynamical systems whose leading order time evolution is given by a combination of ordinary differential equations (ODEs) and Markov chains. The leading order term in these dynamical systems is called average dynamics and turns out to be an adequate concept to analyse a class of simplified hybrid systems. Once the dynamics is defifined the mutual interaction of both ODEs and Markov chains can be analysed through the (reverse) introduction of the so called Interaction Graph, a concept originally invented for time-continuous dynamical systems, see [5]. Here we transfer this graph concept to the average dynamics, which itself is introduced as an heuristic tool to construct models of reaction or contact networks. The graphical concepts introduced form the basis for any subsequent study of the qualitative properties of hybrid models in terms of connectivity and (feedback) loop formation., Comment: To appear in Advances in Complex Systems

In most natural sciences there is currently the insight that it is necessary to bridge gaps between different processes which can be observed on different scales. This is especially true in the field of chemical reactions where the abilities to form bonds between different types of atoms and molecules create much of the properties we experience in our everyday life, especially in all biological activity. There are essentially two types of processes related to biochemical reaction networks, the interactions among molecules and interactions involving their conformational changes, so in a sense, their internal state. The first type of processes can be conveniently approximated by the so-called mass-action kinetics, but this is not necessarily so for the second kind where molecular states do not define any kind of density or concentration. In this paper we demonstrate the necessity to study reaction networks in a stochastic formulation for which we can construct a coherent approximation in terms of specific space-time scales and the number of particles. The continuum limit procedure naturally creates equations of Fokker-Planck type where the evolution of the concentration occurs on a slower time scale when compared to the evolution of the conformational changes, for example triggered by binding or unbinding events with other (typically smaller) molecules. We apply the asymptotic theory to derive the effective, i.e. macroscopic dynamics of the biochemical reaction system. The theory can also be applied to other processes where entities can be described by finitely many internal states, with changes of states occuring by arrival of other entities described by a birth-death process.

In this paper we consider deterministic limits of molecular stochastic systems with finite and infinite degrees of freedom. The method to obtain the deterministic vector field is based on the continuum limit of such microscopic systems which has been derived in [11]. With the aid of the theory we finally develop a new approach for molecular systems that describe typical enzyme kinetics or other interactions between molecular machines like genetic elements and smaller communicating molecules. In contrast to the literature on enzyme kinetics the resulting deterministic functional responses are not derived by time-scale arguments on the macroscopic level, but are a result of time scaling transition rates on the discrete microscopic level. We present several examples of common functional responses found in the literature, like Michaelis-Menten and Hill equation. We finally give examples of more complex but typical macro-molecular machinery.

The paper analyses stochastic systems describing reacting molecular systems with a combination of two types of state spaces, a finite-dimensional, and an infinite dimenional part. As a typical situation consider the interaction of larger macro-molecules, finite and small in numbers per cell (like protein complexes), with smaller, very abundant molecules, for example metabolites. We study the construction of the continuum approximation of the associated Master Equation (ME) by using the Trotter approximation [27]. The continuum limit shows regimes where the finite degrees of freedom evolve faster than the infinite ones. Then we develop a rigourous asymptotic adiabatic theory upon the condition that the jump process arising from the finite degrees of freedom of the Markov Chain (MC, typically describing conformational changes of the macro-molecules) occurs with large frequency. In a second part of this work, the theory is applied to derive typical enzyme kinetics in an alternative way and interpretation within this framework., Comment: Corrected typos

It is shown that the characteristic vector field associated to a first order PDE has the same form of an infinitesimal generator of an odd-symplectic transformation with contact Hamiltonian the given PDE. It is considered under which condition such PDE has a characteristic vector field commuting with a generator of an odd-symplectic action., Comment: 8 pages, no figure