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A key factor in the transmission of infectious diseases is the structure of disease transmitting contacts. In the context of the current COVID-19 pandemic and with some data based on the Hungarian population we develop a theoretical epidemic model (susceptible-infected-removed, SIR) on a multilayer network. The layers include the Hungarian household structure, with population divided into children, adults and elderly, as well as schools and workplaces, some spatial embedding and community transmission due to sharing communal spaces, service and public spaces. We investigate the sensitivity of the model (via the time evolution and final size of the epidemic) to the different contact layers and we map out the relation between peak prevalence and final epidemic size. When compared to the classic compartmental model and for the same final epidemic size, we find that epidemics on multilayer network lead to higher peak prevalence meaning that the risk of overwhelming the health care system is higher. Based on our model we found that keeping cliques/bubbles in school as isolated as possible has a major effect while closing workplaces had a mild effect as long as workplaces are of relatively small size.

Modelling epidemics on networks represents an important departure from classical compartmental models which assume random mixing. However, the resulting models are high-dimensional and their analysis is often out of reach. It turns out that mean-field models, low-dimensional systems of differential equations, whose variables are carefully chosen expected quantities from the exact model provide a good approximation and incorporate explicitly some network properties. Despite the emergence of such mean-field models, there has been limited work on investigating whether these can be used for inference purposes. In this paper, we consider network-based mean-field models and explore the problem of parameter identifiability when observations about an epidemic are available. Making use of the analytical tractability of most network-based mean-field models, e.g. explicit analytical expressions for leading eigenvalue and final epidemic size, we set up the parameter identifiability problem as finding the solution or solutions of a system of coupled equations. More precisely, subject to observing/measuring growth rate and final epidemic size, we seek to identify parameter values leading to these measurements. We are particularly concerned with disentangling transmission rate from the network density. To do this, we give a condition for practical identifiability and we find that except for the simplest model, parameters cannot be uniquely determined, that is, they are practically unidentifiable. This means that there exist multiple solutions (a manifold of infinite measure) which give rise to model output that is close to the data. Identifying, formalising and analytically describing this problem should lead to a better appreciation of the complexity involved in fitting models with many parameters to data.

The global behaviour of the compact pairwise approximation of SIS epidemic propagation on networks is studied. It is shown that the system can be reduced to two equations enabling us to carry out a detailed study of the dynamic properties of the solutions. It is proved that transcritical bifurcation occurs in the system at \begin{document}$ \tau = \tau _c = \frac{\gamma n}{\langle n^{2}\rangle-n} $\end{document} , where \begin{document}$ \tau $\end{document} and \begin{document}$ \gamma $\end{document} are infection and recovery rates, respectively, \begin{document}$ n $\end{document} is the average degree of the network and \begin{document}$ \langle n^{2}\rangle $\end{document} is the second moment of the degree distribution. For subcritical values of \begin{document}$ \tau $\end{document} the disease-free steady state is stable, while for supercritical values a unique stable endemic equilibrium appears. We also prove that for subcritical values of \begin{document}$ \tau $\end{document} the disease-free steady state is globally stable under certain assumptions on the graph that cover a wide class of networks.

SummaryLockdowns have been a key infection control measure for many countries during the COVID-19 pandemic. In England’s first lockdown, children of single parent households (SPHs) were permitted to move between parental homes. By the second lockdown, SPH support bubbles between households were also permitted, enabling larger within-household networks. We investigated the combined impact of these approaches on household transmission dynamics, to inform policymaking for control and support mechanisms in a respiratory pandemic context.This network modelling study applied percolation theory to a base model of SPHs constructed with population survey estimates of SPH family size. To explore putative impact, varying estimates were applied regarding extent of bubbling and proportion of Different-parentage SPHs (DSPHs) (in which children do not share both the same parents). Results indicate that the formation of giant components (in which Covid-19 household transmission accelerates) are more contingent on DSPHs than on formation of bubbles between SPHs; and that bubbling with another SPH will accelerate giant component formation where one or both are DSPHs. Public health guidance should include supportive measures that mitigate the increased transmission risk afforded by support bubbling among DSPHs. Future network, mathematical and epidemiological studies should examine both independent and combined impact of policies.

The epidemic threshold is probably the most studied quantity in the modelling of epidemics on networks. For a large class of networks and dynamics, it is well studied and understood. However, it is less so for clustered networks where theoretical results are mostly limited to idealised networks. In this paper we focus on a class of models known as pairwise models where, to our knowledge, no analytical result for the epidemic threshold exists. We show that by exploiting the presence of fast variables and using some standard techniques from perturbation theory we are able to obtain the epidemic threshold analytically. We validate this new threshold by comparing it to the threshold based on the numerical solution of the full system. The agreement is found to be excellent over a wide range of values of the clustering coefficient, transmission rate and average degree of the network. Interestingly, we find that the analytical form of the threshold depends on the choice of closure, highlighting the importance of model selection when dealing with real-world epidemics. Nevertheless, we expect that our method will extend to other systems in which fast variables are present.

The dynamical behaviour of continuous time recurrent neural network models is studied with emphasis on global stability of a unique equilibrium. First we show in a unified context two Liapunov functions that were introduced in the nineties by Hopfield, Grossberg, Matsouka and Forti. Then we introduce a class of networks for which the model becomes a special cooperative system with a unique globally stable steady state. Finally, we show that periodic orbits may occur when the sufficient conditions for the existence of Liapunov functions are violated.

Networks are widely used to model the contact structure within a population and in the resulting models of disease spread. While networks provide a high degree of realism, the analysis of the exact model is out of reach and even numerical methods fail for modest network size. Hence, mean-field models (e.g. pairwise) focusing on describing the evolution of some summary statistics from the exact model gained a lot of traction over the last few decades. In this paper we revisit the problem of deriving triple closures for pairwise models and we investigate in detail the assumptions behind some of the well-known closures as well as their validity. Using a top-down approach we start at the level of the entire graph and work down to the level of triples and combine this with information around nodes and pairs. We use our approach to derive many of the existing closures and propose new ones and theoretically connect the two well-studied models of multinomial link and Poisson link selection. The theoretical work is backed up by numerical examples to highlight where the commonly used assumptions may fail and provide some recommendations for how to choose the most appropriate closure when using graphs with no or modest degree heterogeneity.

Final epidemic size relations play a central role in mathematical epidemiology. These can be written in the form of an implicit equation which is not analytically solvable in most of the cases. While final size relations were derived for several complex models, including multiple infective stages and models in which the durations of stages are arbitrarily distributed, the solvability of those implicit equations have been less studied. In this paper the SIR homogeneous mean-field and pairwise models and the heterogeneous mean-field model are studied. It is proved that the implicit equation for the final epidemic size has a unique solution, and that through writing the implicit equation as a fixed point equation in a suitable form, the iteration of the fixed point equation converges to the unique solution. The Markovian SIR epidemic model on finite networks is also studied by using the generation-based approach. Explicit analytic formulas are derived for the final size distribution for line and star graphs of arbitrary size. Iterative formulas for the final size distribution enable us to study the accuracy of mean-field approximations for the complete graph.

Pairwise models are widely used to model epidemic spread on networks. This includes the modelling of susceptible-infected-removed (SIR) epidemics on regular networks and extensions to SIS dynamics and contact tracing on more exotic networks exhibiting degree heterogeneity, directed and/or weighted links and clustering. However, extra features of the disease dynamics or of the network lead to an increase in system size and analytical tractability becomes problematic. Various “closures” can keep the system tractable. Focusing on SIR epidemics on regular but clustered networks, we show that even for the most complex closure we can determine the epidemic threshold as an asymptotic expansion in terms of the clustering coefficient. We do this by exploiting the presence of a system of fast variables, specified by the correlation structure of the epidemic, whose steady state determines the epidemic threshold. While we do not find the steady state analytically, we create an elegant asymptotic expansion of it. We validate this new threshold by comparing it to the numerical solution of the full system and find excellent agreement over a wide range of values of the clustering coefficient, transmission rate and average degree of the network. The technique carries over to pairwise models with other closures [1], and we note that the epidemic threshold will be model dependent. This emphasises the importance of model choice when dealing with realistic outbreaks.

Deterministic limit of a class of continuous time Markov chains is considered based purely on differential equation techniques. Starting from the linear system of master equations, ordinary differential equations for the moments and a partial differential equation, called Fokker–Planck equation, for the distribution is derived. Introducing closures at the level of the second and third moments, mean-field approximations are introduced. The accuracy of the mean-field approximations and the Fokker–Planck equation is investigated by using two differential equation-based and an operator semigroup-based approach.

Several network processes exhibiting transcritical bifurcation have globally stable steady states. Their dynamical behaviour is captured by a simple property of the right hand side of the corresponding system of ODEs. Based on this property, a class of dynamical systems is introduced, for which the local stability of the trivial steady state determines the global behaviour of the system. It is shown that this condition is satisfied by three network models, namely the individual-based and degree-based ODE approximations of SIS epidemic propagation on networks and the Hopfield model with non-negative weights. The general result enables us to describe the global behaviour of these systems that was not available for the first and third models and was proved in a significantly more complicated way for the second.

In this paper we provide the derivation of a super compact pairwise model with only 4 equations in the context of describing susceptible-infected-susceptible (SIS) epidemic dynamics on heterogenous networks. The super compact model is based on a new closure relation that involves not only the average degree but also the second and third moments of the degree distribution. Its derivation uses an a priori approximation of the degree distribution of susceptible nodes in terms of the degree distribution of the network. The new closure gives excellent agreement with heterogeneous pairwise models that contain significantly more differential equations.

In this paper we make the first steps to bridge the gap between classic control theory and modern, network-based epidemic models. In particular, we apply nonlinear model predictive control (NMPC) to a pairwise ODE model which we use to model a susceptible-infectious-susceptible (SIS) epidemic on non-trivial contact structures. While classic control of epidemics concentrates on aspects such as vaccination, quarantine and fast diagnosis, our novel setup allows us to deliver control by altering the contact network within the population. Moreover, the ideal outcome of control is to eradicate the disease while keeping the network well connected. The paper gives a thorough and detailed numerical investigation of the impact and interaction of system and control parameters on the controllability of the system. The analysis reveals, that for certain set parameters it is possible to identify critical control bounds above which the system is controllable. We foresee, that our approach can be extended to even more realistic or simulation-based models with the aim to apply these to real-world situations., Comment: 20 pages, 8 figures

In this paper we use comparison theorems from classical ODE theory in order to rigorously show that the N-Intertwined Mean-Field Approximation (NIMFA) model provides an upper estimate on the exact stochastic process. The proof of the results relies on the observation that the epidemic process is negatively correlated (in the sense that the probability of an edge being in the susceptible-infected state is smaller than the product of the probabilities of the nodes being in the susceptible and infected states, respectively), which we also prove rigorously. The results in the paper hold for arbitrary weighted and directed networks. We cast the results in a more general framework where alternative closures, other than that assuming the independence of nodes connected by an edge, are possible and provide a succinct summary of the stability analysis of the resulting more general mean-field models., Comment: 18 pages

In this paper, an SIS (susceptible-infected-susceptible)-type epidemic propagation is studied on a special class of 3-regular graphs, called modified cycle graphs. The modified cycle graph is constructed from a cycle graph with N nodes by connecting node i to the node i + d in a way that every node has exactly three links. Monte-Carlo simulations show that the propagation process depends on the value of d in a non-monotone way. A new theoretical model is developed to explain this phenomenon. This reveals a new relation between the spreading process and the average path length in the graph.

We extrapolate from the exact master equations of epidemic dynamics on fully connected graphs to non-fully connected by keeping the size of the state space N + 1, where N is the number of nodes in the graph. This gives rise to a system of approximate ODEs (ordinary differential equations) where the challenge is to compute/approximate analytically the transmission rates. We show that this is possible for graphs with arbitrary degree distributions built according to the configuration model. Numerical tests confirm that: (a) the agreement of the approximate ODEs system with simulation is excellent and (b) that the approach remains valid for clustered graphs with the analytical calculations of the transmission rates still pending. The marked reduction in state space gives good results, and where the transmission rates can be analytically approximated, the model provides a strong alternative approximate model that agrees well with simulation. Given that the transmission rates encompass information both about the dynamics and graph properties, the specific shape of the curve, defined by the transmission rate versus the number of infected nodes, can provide a new and different measure of network structure, and the model could serve as a link between inferring network structure from prevalence or incidence data.

The bifurcations in a three-variable ODE model describing the oxygen reduction reaction on a platinum surface is studied. The investigation is motivated by the fact that this reaction plays an important role in fuel cells. The goal of this paper is to determine the dynamical behaviour of the ODE system, with emphasis on the number and type of the stationary points, and to find the possible bifurcations. It is shown that a non-trivial steady state can appear through a transcritical bifurcation, or a stable and an unstable steady state can arise as a result of saddle-node bifurcation. The saddle-node bifurcation curve is determined by using the parametric representation method, and this enables us to determine numerically the parameter domain where bistability occurs, which is important from the chemical point of view.

For a model of gasless combustion with heat loss, we use geometric singular perturbation theory to show existence of traveling combustion fronts. We show that the fronts are nonlinearly stable in an appropriate sense if an Evans function criterion, which can be verified numerically, is satisfied. For a solid reactant and exothermicity parameter that is not too large, we verify numerically that the criterion is satisfied.

For many diseases (e.g., sexually transmitted infections, STIs), most individuals are aware of the potential risks of becoming infected, but choose not to take action ('respond') despite the information that aims to raise awareness and to increases the responsiveness or alertness of the population. We propose a simple mathematical model that accounts for the diffusion of health information disseminated as a result of the presence of a disease and an 'active' host population that can respond to it by taking measures to avoid infection or if infected by seeking treatment early. In this model, we assume that the whole population is potentially aware of the risk but only a certain proportion chooses to respond appropriately by trying to limit their probability of becoming infectious or seeking treatment early. The model also incorporates a level of responsiveness that decays over time. We show that if the dissemination of information is fast enough, infection can be eradicated. When this is not possible, information transmission has an important effect in reducing the prevalence of the infection. We derive the full characterisation of the global behaviour of the model, and we show that the parameter space can be divided into three parts according to the global attractor of the system which is one of the two disease-free steady states or the endemic equilibrium.

In this paper, the rigorous linking of exact stochastic models to mean-field approximations is studied. Using a continuous-time Markov chain, we start from the exact formulation of a simple epidemic model on a certain class of networks, including completely connected and regular random graphs, and rigorously derive the well-known mean-field approximation that is usually justified based on biological hypotheses. We propose a unifying framework that incorporates and discusses the details of two existing proofs and we put forward a new ordinary differential equation (ODE)-based proof. The more well-known proof is based on a first-order partial differential equation approximation, while the other, more technical one, uses Martingale and Semigroup theory. We present the main steps of both proofs to investigate their applicability in different modelling contexts and to make these ideas more accessible to a broader group of applied researchers. The main result of the paper is a new ODE-based proof that may serve as a building block to prove similar convergence results for more complex networks. The new proof is based on deriving a countable system of ODEs for the moments of a distribution of interest and proving a perturbation theorem for this infinite system.

In this paper a random link activation–deletion (RLAD) model is proposed that gives rise to a stochastically evolving network. This dynamic network is then coupled to a simple susceptible-infectious-suceptible ( SIS ) dynamics on the network, and the resulting spectrum of model behaviour is explored via simulation and a novel pairwise model for dynamic networks. First, the dynamic network model is systematically analysed by considering link-type independent and dependent network dynamics coupled with globally constrained link creation. This is done rigorously with some analytical results and we highlight where such analysis can be performed and how these simpler models provide a benchmark to test and validate full simulations. The pairwise model is used to study the interplay between SIS -type dynamics on the network and link-type-dependent activation–deletion. Assumptions of the pairwise model are identified and their implications interpreted in a way that complements our current understanding. Furthermore, we also discuss how the strong assumptions of the closure relations can lead to disagreement between the simulation and pairwise model. Unlike on a static network, the resulting spectrum of behaviour is more complex with the prevalence of infections exhibiting not only a single steady state, but also bistability and oscillations.

The bifurcations in a four-variable ODE model of an SIS type epidemic on an adaptive network are studied. The model describes the propagation of the epidemic on a network where links (or edges) of different type (i.e. SI;II and SS ) can be activated or deleted according to a simple rule consisting of random link activation and deletion. In the case when II links cannot be neither deleted nor created it is proved that the system can have at most three steady states with the trivial, disease-free steady state being one of them. It is shown that a stable endemic steady state can appear through a transcritical bifurcation, or a stable and an unstable endemic steady state arise as a result of saddle-node bifurcation. Moreover, at the endemic steady state a Hopf bifurcation may occur giving rise to stable oscillation. The bifurcation curves in the parameter space are determined analytically using the parametric representation method. For certain parameter regimes or bifurcation types, analytical results based on the ODE model show good agreement when compared to results based on individual- based network simulations. When agreement between the two modelling approaches holds, the ODE-based model provides a faster and more reliable tool that can be used to explore full spectrum of model behaviour.

Motivated by industrial applications, we investigate the so-called Dynamic Matrix Control (DMC) strategy for single-input single-output linear continuous-time time-invariant systems. DMC is a type of Model Predictive Control based on the step response model of the process. We show that if the process is governed by a one-dimensional stable dynamical system, then the method drives the output of the sampled system into the desired setpoint as time goes to infinity, that is, the system is asymptotically output controllable with DMC. For two-dimensional systems, sufficient condition on the asymptotic output controllability is given. Mathematics subject classification (2010): Dynamic Matrix Control, step response, asymptotic output controllability, discrete-time Hurwitz matrix.

Many if not all models of disease transmission on networks can be linked to the exact state-based Markovian formulation. However the large number of equations for any system of realistic size limits their applicability to small populations. As a result, most modelling work relies on simulation and pairwise models. In this paper, for a simple SIS dynamics on an arbitrary network, we formalise the link between a well known pairwise model and the exact Markovian formulation. This involves the rigorous derivation of the exact ODE model at the level of pairs in terms of the expected number of pairs and triples. The exact system is then closed using two different closures, one well established and one that has been recently proposed. A new interpretation of both closures is presented, which explains several of their previously observed properties. The closed dynamical systems are solved numerically and the results are compared to output from individual-based stochastic simulations. This is done for a range of networks with the same average degree and clustering coefficient but generated using different algorithms. It is shown that the ability of the pairwise system to accurately model an epidemic is fundamentally dependent on the underlying large-scale network structure. We show that the existing pairwise models are a good fit for certain types of network but have to be used with caution as higher-order network structures may compromise their effectiveness.

The maximal operators for Cesàro or (C,α) and Riesz summability with respect to Walsh–Fourier series are investigated as mappings between dyadic Hardy and Lebesgue spaces. It is well known that they are bounded from Hp to Lp for all 1/(α+1)

A two variable model describing the circadian fluctuation of two proteins (PER and TIM) in cells is considered. The original model was set up by Leloup and Goldbeter [1998], the present form was developed by Tyson et al. [1999]. Periodic solutions with 24-h period were investigated in those papers. Here the possible phase portraits and bifurcations are studied in detail. The saddle-node and Hopf bifurcation curves are determined in the plane of two parameters by using the parametric representation method [Simon et al., 1999]. It is shown that there are four cases according to their mutual position (as the remaining parameters of the system are varied). Using these curves the number and type of the stationary points are determined in all four cases. The global bifurcation diagram, which is a system of bifurcation curves that divide the given parameter plane into regions according to topological equivalence of global phase portraits, is also determined. Finally, the so-called constant period curves are computed numerically. These curves consist of those parameter pairs on the parameter plane for which the system has a limit cycle with a given period. It turns out that in a wide range of parameters the system has limit cycles with period close to 24-h.

A large system of ordinary differential equations is approximated by a parabolic partial differential equation with dynamic boundary condition and a different one with Robin boundary condition. Using the theory of differential operators with Wentzell boundary conditions and similar theories, we give estimates on the order of approximation. The theory is demonstrated on a voter model where the Fourier method applied to the PDE is of great advantage., Comment: reviewer requests incorporated, to appear in "Operators and Matrices"

The effects of an electric field on the travelling waves arising in Belousov-Zhabotinsky systems are analysed using the Oregonator to describe the kinetics. The model is reduced to a two-variable version involving the concentrations of HBrO2 and M-ox(3+), the oxidized form of the catalyst, using previously-suggested scalings. The travelling wave equations for this two-variable model are solved numerically for a range of kinetic parameters and the ratio of diffusion coefficients D. An upper bound on the field strength E is found, arising from a saddle-node bifurcation, for the existence of travelling waves. There can also be a lower bound on E for their existence, dependent on the other parameters in the system. The conditions for this termination of a solution at a finite field strength are determined. In other cases, travelling waves exist for all negative field strengths and an asymptotic solution for large \E\ is constructed. This acts as a confirmation of the numerical results and provides further insights into the structure of the wave profiles. Numerical integrations of the corresponding initial-value problem are undertaken. These show wave deceleration and annihilation in positive fields and wave acceleration in negative fields, in line with experimental observations. In cases when there is termination at a finite value of E, wave trains are seen to develop for (negative) field strength less than this value.

The effects of the endothermic decomposition of an inhibitory species W to form a radical scavenger on a laminar, pre-mixed flame supported by an exothermic second-order branching reaction are considered. This work extends a previous study, where the effects of the radical scavenger S were ignored. Two cases are identified, dependent on a parameter β measuring the relative rate of the decomposition of W. These are described by an high-activation-energy asymptotic analysis and through numerical integration of the propagating-flame equations for representative parameter values. For larger values of β the effect of the radical scavenger is to introduce a critical value of the heat-loss parameter α for flame propagation. For smaller values of β, where there is a critical value of α without any S being produced, the effect is to lower this critical value. In both cases the effect of the radical scavenger is to reduce the propagation speed and, if sufficient amounts of S are produced from the decomposition of W, to totally suppress flame propagation, even without any heat loss.

The Walsh system will be investigated in the Kaczmarz rearrangement. In an earlier paper we have shown that the maximal operator of the (C, 1)-means of the Walsh-Kaczmarz-Fourier series is bounded from the dyadic Hardy space Hp into Lp for every ½ < p ≤ 1. In the present work, we extend this result to the (C, α) means when 0 < α ≤ 1 and prove their maximal operator σα : Hp → Lp is bounded for all 1/(α + 1) < p ≤ 1. By known results on interpolation we get from this theorem that σα is of weak type (1,1) and bounded from Lq into Lq if 1 < q ≤ ∞. Moreover, the (C, α) means of an integrable function f converge to f a.e.

The linear stability of the travelling wave solutions of a general reaction-diffusion system is investigated. The spectrum of the corresponding second order differential operator $L$ is studied. The problem is reduced to an asymptotically autonomous first order linear system. The relation between the spectrum of $L$ and the corresponding first order system is dealt with in detail. The first order system is investigated using exponential dichotomies. A self-contained short presentation is shown for the study of the spectrum, with elementary proofs. An algorithm is given for the determination of the exact position of the essential spectrum. The Evans function method for determining the isolated eigenvalues of $L$ is also presented. The theory is illustrated by three examples: a single travelling wave equation, a three variable combustion model and the generalized KdV equation.

The structure of flame filaments resulting from chaotic mixing within a combustion reaction is considered. The transverse profile of the filaments is investigated numerically and analytically based on a one-dimensional model that represents the effect of stirring as a convergent flow. The dependence of the steady solutions on the Damkohler number and Lewis number is treated in detail. It is found that, below a critical Damkohler number Dacrit, the flame is quenched by the flow. The quenching transition appears as a result of a saddle-node bifurcation where the stable steady filament solution collides with an unstable one. The shape of the steady solutions for the concentration and temperature profiles changes with the Lewis number and the value of Dacrit increases monotonically with the Lewis number. Properties of the solutions are studied analytically in the limit of large Damkohler number and for small and large Lewis number.

An adaptive network model using SIS epidemic propagation with link-type dependent link activation and deletion is considered. Bifurcation analysis of the pairwise ODE approximation and the network-based stochastic simulation is carried out, showing that three typical behaviours may occur; namely, oscillations can be observed besides disease-free or endemic steady states. The oscillatory behaviour in the stochastic simulations is studied using Fourier analysis, as well as through analysing the exact master equations of the stochastic model. A compact pairwise approximation for the dynamic network case is also developed and, for the case of link-type independent rewiring, the outcome of epidemics and changes in network structure are concurrently presented in a single bifurcation diagram. By going beyond simply comparing simulation results to mean-field models, our approach yields deeper insights into the observed phenomena and help better understand and map out the limitations of mean-field models.

We investigate the stability of nontrivial nonnegative stationary solutions of semilinear initial-boundary value problems with convex or concave nonlinearity. In the convex case f(0)⩽0 implies instability, in the concave case f(0)⩾0 implies stability. We also discuss the necessity of the sign condition on f(0).

We prove existence of solutions for parabolic initial value problems $\partial_t u = \Delta u + f(u)$ on $R^N$ , where $f : R \rightarrow R$ is a bounded, but possibly discontinuous function.

The Walsh system will be considered in the Kaczmarz rearrangement. We show that the maximal operator σ* of the (C,1)-means of the Walsh–Kaczmarz–Fourier series is bounded from the dyadic Hardy space Hp into Lp for every 1/2

The so-called Vilenkin systems and the Hardy spaces H p (0 p ≤ 1) with respect to Vilenkin groups will be considered. We investigate certain means of the partial sums of Vilenkin–Fourier series. It will be shown that these means, as operators from H p to L p , are bounded.

The parameter dependence of the solution x of equation f0(x)+u1f1(x)+u2f2(x) = 0 is considered. Our aim is to divide the parameter plane (u1;u2) according to the number of the solutions, that is to construct a bifurcation curve. This curve is given by the singularity set, but in practice it is dicult to depict it, because it is often derived in implicit form. Here we apply the parametric representation method which has the following advantages: (1) the singularity set can be easily constructed as a curve parametrized by x, called D-curve; (2) the solutions belonging to a given parameter pair can be determined by a simple geometric algorithm based on the tangential property; (3) the global bifurcation diagram, that divides the parameter plane according to the number of solutions can be geometrically constructed with the aid of the D-curve. c 1999 Elsevier Science B.V. All rights reserved.

In this paper we study a simple model of a purely excitatory neural network that, by construction, operates at a critical point. This model allows us to consider various markers of criticality and illustrate how they should perform in a finite-size system. By calculating the exact distribution of avalanche sizes we are able to show that, over a limited range of avalanche sizes which we precisely identify, the distribution has scale free properties but is not a power law. This suggests that it would be inappropriate to dismiss a system as not being critical purely based on an inability to rigorously fit a power law distribution as has been recently advocated. In assessing whether a system, especially a finite-size one, is critical it is thus important to consider other possible markers. We illustrate one of these by showing the divergence of susceptibility as the critical point of the system is approached. Finally, we provide evidence that power laws may underlie other observables of the system, that may be more amenable to robust experimental assessment., 33 pages, 10 figures

Low dimensional ODE approximations that capture the main characteristics of SIS-type epidemic propagation along a cycle graph are derived. Three different methods are shown that can accurately predict the expected number of infected nodes in the graph. The first method is based on the derivation of a master equation for the number of infected nodes. This uses the average number of SI edges for a given number of the infected nodes. The second approach is based on the observation that the epidemic spreads along the cycle graph as a front. We introduce a continuous time Markov chain describing the evolution of the front. The third method we apply is the subsystem approximation using the edges as subsystems. Finally, we compare the steady state value of the number of infected nodes obtained in different ways.

The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its Kolmogorov equations, which is a system of linear ODEs that depends on the state space size ($N$) and can be written as $\dot u_N=A_N u_N$. Our results rely on the convergence of the transition matrices $A_N$ to an operator $A$. This convergence also implies that the solutions $u_N$ converge to the solution $u$ of $\dot u=Au$. The limiting ODE can be easily used to derive simpler mean-field-type models such that the moments of the stochastic process will converge uniformly to the solution of appropriately chosen mean-field equations. A bi-product of this method is the proof that the rate of convergence is $\mathcal{O}(1/N)$. In addition, it turns out that the proof holds for cases that are slightly more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a new approach for proving convergence to the mean-field limit is proposed. The starting point in this case is the derivation of a countable system of ordinary differential equations for all the moments. This is followed by the proof of a perturbation theorem for this infinite system, which in turn leads to an estimate for the difference between the moments and the corresponding quantities derived from the solution of the mean-field ODE.

In this paper, we investigate the exact number of positive solutions of the Dirichlet boundary-value problem $u''-u^{-\gamma}+\beta=0$. We will show that the exact number of positive solutions can be 2, 1, 0, depending on the length of the interval and $\gamma$. This solves some open problems posed in [1].

Human immunodeficiency virus (HIV) induces a chronic generalized activation of the immune system, which plays an important role in the pathogenesis of AIDS. This ability of the virus might either be an evolved (adaptive) trait or a coincidental side effect of jumping to a new host species. We argue that selection favours the ability of HIV to induce immune activation at the local sites of infection (e.g. lymph follicles) but not at the systemic level. Immune activation increases the supply of susceptible target cells; however, mutations that increase systemic immune activation benefit all virus variants equally and are therefore selectively neutral. We thus conclude that the generalized immune activation that is probably responsible for pathogenesis is probably not directly under selection.

Necrotic cell death is a common feature in numerous human neurodegenerative disorders. In the nematode Caenorhabditis elegans, gain-of-function mutations in genes that encode specific ion channel subunits such as the degenerins DEG-1 and MEC-4, and the acetylcholine receptor subunit DEG-3 lead to necrotic-like degeneration of a subset of neurons. Neuronal demise caused by ion channel hyperactivity is accompanied by intense degradation of cytoplasmic contents, dramatic membrane infolding and vacuole formation; however, the cellular pathways underlying such processes remain largely unknown. Here we show that the function of three autophagy genes, whose yeast and mammalian orthologs are implicated in cytoplasmic self-degradation, membrane trafficking and the cellular response to starvation, contributes to ion-channel-dependent neurotoxicity in C. elegans. Inactivation of unc-51, bec-1 and lgg-1, the worm counterparts of the yeast autophagy genes Atg1, Atg6 and Atg8 respectively, partially suppresses degeneration of neurons with toxic ion channel variants. We also demonstrate that the TOR-kinase-mediated signaling pathway, a nutrient sensing system that downregulates the autophagy gene cascade, protects neurons from undergoing necrotic cell death, whereas nutrient deprivation promotes necrosis. Our findings reveal a role for autophagy genes in neuronal cell loss in C. elegans.