Mesa-type patterns in the one-dimensional Brusselator and their stability

The Brusselator is a generic reaction-diffusion model for a tri-molecular chemical reaction. We consider the case when the input and output reactions are slow. In this limit, we show the existence of $K$-periodic, spatially bi-stable structures, \emph{mesas}, and study their stability. Using singular perturbation techniques, we find a threshold for the stability of $K$ mesas. This threshold occurs in the regime where the exponentially small tails of the localized structures start to interact. By comparing our results with Turing analysis, we show that in the generic case, a Turing instability is followed by a slow coarsening process whereby logarithmically many mesas are annihilated before the system reaches a steady equilibrium state. We also study a ``breather''-type instability of a mesa, which occurs due to a Hopf bifurcation. Full numerical simulations are shown to confirm the analytical results.Comment: to appear, Physica

Theoretical basis to measure the impact of short-lasting control of an infectious disease on the epidemic peak

Background. While many pandemic preparedness plans have promoted disease control effort to lower and delay an epidemic peak, analytical methods for determining the required control effort and making statistical inferences have yet to be sought. As a first step to address this issue, we present a theoretical basis on which to assess the impact of an early intervention on the epidemic peak, employing a simple epidemic model. Methods. We focus on estimating the impact of an early control effort (e.g. unsuccessful containment), assuming that the transmission rate abruptly increases when control is discontinued. We provide analytical expressions for magnitude and time of the epidemic peak, employing approximate logistic and logarithmic-form solutions for the latter. Empirical influenza data (H1N1-2009) in Japan are analyzed to estimate the effect of the summer holiday period in lowering and delaying the peak in 2009. Results. Our model estimates that the epidemic peak of the 2009 pandemic was delayed for 21 days due to summer holiday. Decline in peak appears to be a nonlinear function of control-associated reduction in the reproduction number. Peak delay is shown to critically depend on the fraction of initially immune individuals. Conclusions. The proposed modeling approaches offer methodological avenues to assess empirical data and to objectively estimate required control effort to lower and delay an epidemic peak. Analytical findings support a critical need to conduct population-wide serological survey as a prior requirement for estimating the time of peak. © 2011 Omori and Nishiura; licensee BioMed Central Ltd.published_or_final_versio

Stability of cluster solutions in a cooperative consumer chain model

This is the author's accepted manuscript. The final published article is available from the link below. Copyright @ Springer-Verlag Berlin Heidelberg 2012.We study a cooperative consumer chain model which consists of one producer and two consumers. It is an extension of the Schnakenberg model suggested in Gierer and Meinhardt [Kybernetik (Berlin), 12:30-39, 1972] and Schnakenberg (J Theor Biol, 81:389-400, 1979) for which there is only one producer and one consumer. In this consumer chain model there is a middle component which plays a hybrid role: it acts both as consumer and as producer. It is assumed that the producer diffuses much faster than the first consumer and the first consumer much faster than the second consumer. The system also serves as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir. In the small diffusion limit we construct cluster solutions in an interval which have the following properties: The spatial profile of the third component is a spike. The profile for the middle component is that of two partial spikes connected by a thin transition layer. The first component in leading order is given by a Green's function. In this profile multiple scales are involved: The spikes for the middle component are on the small scale, the spike for the third on the very small scale, the width of the transition layer for the middle component is between the small and the very small scale. The first component acts on the large scale. To the best of our knowledge, this type of spiky pattern has never before been studied rigorously. It is shown that, if the feedrates are small enough, there exist two such patterns which differ by their amplitudes.We also study the stability properties of these cluster solutions. We use a rigorous analysis to investigate the linearized operator around cluster solutions which is based on nonlocal eigenvalue problems and rigorous asymptotic analysis. The following result is established: If the time-relaxation constants are small enough, one cluster solution is stable and the other one is unstable. The instability arises through large eigenvalues of order O(1). Further, there are small eigenvalues of order o(1) which do not cause any instabilities. Our approach requires some new ideas: (i) The analysis of the large eigenvalues of order O(1) leads to a novel system of nonlocal eigenvalue problems with inhomogeneous Robin boundary conditions whose stability properties have been investigated rigorously. (ii) The analysis of the small eigenvalues of order o(1) needs a careful study of the interaction of two small length scales and is based on a suitable inner/outer expansion with rigorous error analysis. It is found that the order of these small eigenvalues is given by the smallest diffusion constant ε22.RGC of Hong Kon

Existence and Stability of a Spike in the Central Component for a Consumer Chain Model

We study a three-component consumer chain model which is based on Schnakenberg type kinetics. In this model there is one consumer feeding on the producer and a second consumer feeding on the first consumer. This means that the first consumer (central component) plays a hybrid role: it acts both as consumer and producer. The model is an extension of the Schnakenberg model suggested in \cite{gm,schn1} for which there is only one producer and one consumer. It is assumed that both the producer and second consumer diffuse much faster than the central component. We construct single spike solutions on an interval for which the profile of the first consumer is that of a spike. The profiles of the producer and the second consumer only vary on a much larger spatial scale due to faster diffusion of these components. It is shown that there exist two different single spike solutions if the feed rates are small enough: a large-amplitude and a small-amplitude spike. We study the stability properties of these solutions in terms of the system parameters. We use a rigorous analysis for the linearized operator around single spike solutions based on nonlocal eigenvalue problems. The following result is established: If the time-relaxation constants for both producer and second consumer vanish, the large-amplitude spike solution is stable and the small-amplitude spike solution is unstable. We also derive results on the stability of solutions when these two time-relaxation constants are small. We show a new effect: if the time-relaxation constant of the second consumer is very small, the large-amplitude spike solution becomes unstable. To the best of our knowledge this phenomenon has not been observed before for the stability of spike patterns. It seems that this behavior is not possible for two-component reaction-diffusion systems but that at least three components are required. Our main motivation to study this system is mathematical since the novel interaction of a spike in the central component with two other components results in new types of conditions for the existence and stability of a spike. This model is realistic if several assumptions are made: (i) cooperation of consumers is prevalent in the system, (ii) the producer and the second consumer diffuse much faster than the first consumer, and (iii) there is practically an unlimited pool of producer. The first assumption has been proven to be correct in many types of consumer groups or populations, the second assumption occurs if the central component has a much smaller mobility than the other two, the third assumption is realistic if the consumers do not feel the impact of the limited amount of producer due to its large quantity. This chain model plays a role in population biology, where consumer and producer are often called predator and prey. This system can also be used as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir

SO(10) GUT and Quark-Lepton Mass Matrices

The phenomenological model that all quark and lepton mass matrices have the same zero texture, namely their (1,1), (1,3) and (3,1) components are zeros, is discussed in the context of SO(10) Grand Unified Theories (GUTs). The mass matrices of type I for quarks are consistent with the experimental data in the quark sector. For the lepton sector, consistent fitting to the data of neutrino oscillation experiments force us to use the mass matrix for the charged leptons which is slightly deviated from type I. Given quark masses and charged lepton masses, the model includes 19 free parameters, whereas the SO(10) GUTs gives 16 constrained equations. Changing the remaining three parameters freely, we can fit all the entries of the CKM quark mixing matrix and the MNS lepton mixing matrix, and three neutrino masses consistently with the present experimental data.Comment: 32pp, REV TeX, 12 EPS Figure

The Time Required to Estimate the Case Fatality Ratio of Influenza Using Only the Tip of an Iceberg: Joint Estimation of the Virulence and the Transmission Potential

Estimating the case fatality ratio (CFR) of a novel strain of influenza virus during the early stage of the pandemic is one of key epidemiological tasks to be conducted as rapid research response. Past experience during the epidemics of severe acute respiratory syndrome (SARS) and influenza A (H1N1-2009) posed several technical challenges in estimating the CFR in real time. The present study aimed to develop a simple method to estimate the CFR based on readily available datasets, that is, confirmed cases and deaths, while addressing some of the known technical issues. To assess the reliability and validity of the proposed method, we examined the minimum length of time required for the assigned CFR to be included within the 95% confidence intervals and for the estimated CFR to be below a prespecified cut-off value by means of Monte Carlo simulations. Overall, the smaller the transmission potential was, the longer it took to compare the estimated CFR against the cut-off value. If policymaking and public health response have to be made based on the CFR estimate derived from the proposed method and readily available data, it should be noted that the successful estimation may take longer than a few months

Two-parameter neutrino mass matrices with two texture zeros

We reanalyse Majorana-neutrino mass matrices M_nu with two texture zeros, by searching for viable hybrid textures in which the non-zero matrix elements of M_nu have simple ratios. Referring to the classification scheme of Frampton, Glashow and Marfatia, we find that the mass matrix denoted by A1 allows the ratios (M_nu)_{mu mu} : (Mnu)_{tau tau} = 1:1 and (M_nu)_{e tau} : (Mnu)_{mu tau} = 1:2. There are analogous ratios for texture A2. With these two hybrid textures, one obtains, for instance, good agreement with the data if one computes the three mixing angles in terms of the experimentally determined mass-squared differences Delta m^2_21 and Delta m^2_31. We could not find viable hybrid textures based on mass matrices different from those of cases A1 and A2.Comment: 10 pages, no figures, minor changes, some references adde

Dynamical effects of interactions and the Tully-Fisher relation for Hickson compact groups

We investigate the properties of the B-band Tully-Fisher (T-F) relation for 25 compact group galaxies, using Vmax derived from 2-D velocity maps. Our main result is that the majority of the Hickson Compact Group galaxies lie on the T-F relation. However, about 20% of the galaxies, including the lowest-mass systems, have higher B luminosities for a given mass, or alternatively, a mass which is too low for their luminosities. We favour a scenario in which outliers have been brightened due to either enhanced star formation or merging. Alternatively, the T-F outliers may have undergone truncation of their dark halo due to interactions. It is possible that in some cases, both effects contribute. The fact that the B-band T-F relation is similar for compact group and field galaxies tells us that these galaxies show common mass-to-size relations and that the halos of compact group galaxies have not been significantly stripped inside R25. We find that 75% of the compact group galaxies studied (22 out of 29) have highly peculiar velocity fields. Nevertheless, a careful choice of inclination, position angle and center, obtained from the velocity field, and an average of the velocities over a large sector of the galaxy enabled the determination of fairly well-behaved rotation curves for the galaxies. However, two of the compact group galaxies which are the most massive members in M51--like pairs, HCG 91a and HCG 96a, have very asymmetric rotation curves, with one arm rising and the other one falling, indicating, most probably, a recent perturbation by the small close companions.Comment: 15 pages, 4 figures, accepted for publication in the Astronomical Journa

Effects of a burst of formation of first-generation stars on the evolution of galaxies

First-generation (Population III) stars in the universe play an important role inearly enrichment of heavy elements in galaxies and intergalactic medium and thus affect the history of galaxies. The physical and chemical properties of primordial gas clouds are significantly different from those of present-day gas clouds observed in the nearby universe because the primordial gas clouds do not contain any heavy elements which are important coolants in the gas. Previous theoretical considerations have suggested that typical masses of the first-generation stars are between several $M_\odot$ and $\approx 10 M_\odot$ although it has been argued that the formation of very massive stars (e.g., $> 100 M_\odot$) is also likely. If stars with several $M_\odot$ are most popular ones at the epoch of galaxy formation, most stars will evolve to hot (e.g., $\gtrsim 10^5$ K), luminous ($\sim 10^4 L_\odot$) stars with gaseous and dusty envelope prior to going to die as white dwarf stars. Although the duration of this phase is short (e.g., $\sim 10^5$ yr), such evolved stars could contribute both to the ionization of gas in galaxies and to the production of a lot of dust grains if the formation of intermediate-mass stars is highly enhanced. We compare gaseous emission-line properties of such nebulae with some interesting high-redshift galaxies such asIRAS F10214+4724 and powerful radio galaxies.Comment: 25 pages, 7 figures, ApJ, in pres

Large θ13ν\theta_{13}^\nu and Unified Description of Quark and Lepton Mixing Matrices

We present a revised version of the so-called "yukawaon model", which was proposed for the purpose of a unified description of the lepton mixing matrix $U_{PMNS}$ and the quark mixing matrix $V_{CKM}$. It is assumed from a phenomenological point of view that the neutrino Dirac mass matrix $M_D$ is given with a somewhat different structure from the charged lepton mass matrix $M_e$, although $M_D=M_e$ was assumed in the previous model. As a result, the revised model predicts a reasonable value $\sin^2 2\theta_{13} \sim 0.07$ with keeping successful results for other parameters in $U_{PMNS}$ as well as $V_{CKM}$ and quark and lepton mass ratios.Comment: 13 pages, 3 figures, version accepted by EPJ