Partitioning of energy in highly polydisperse granular gases

A highly polydisperse granular gas is modeled by a continuous distribution of particle sizes, a, giving rise to a corresponding continuous temperature profile, T(a), which we compute approximately, generalizing previous results for binary or multicomponent mixtures. If the system is driven, it evolves towards a stationary temperature profile, which is discussed for several driving mechanisms in dependence on the variance of the size distribution. For a uniform distribution of sizes, the stationary temperature profile is nonuniform with either hot small particles (constant force driving) or hot large particles (constant velocity or constant energy driving). Polydispersity always gives rise to non-Gaussian velocity distributions. Depending on the driving mechanism the tails can be either overpopulated or underpopulated as compared to the molecular gas. The deviations are mainly due to small particles. In the case of free cooling the decay rate depends continuously on particle size, while all partial temperatures decay according to Haff's law. The analytical results are supported by event driven simulations for a large, but discrete number of species.Comment: 10 pages; 5 figure

Vaccination strategies when vaccines are scarce: on conflicts between reducing the burden and avoiding the evolution of escape mutants

When vaccine supply is limited but population immunization urgent, theallocation of the available doses needs to be carefully considered. Oneaspect of dose allocation is the time interval between the first and thesecond injections in two-dose vaccines. By stretching this interval, more indi-viduals can be vaccinated with the first dose more quickly, which can bebeneficial in reducing case numbers, provided a single dose is sufficientlyeffective. On the other hand, there has been concern that intermediatelevels of immunity in partially vaccinated individuals may favour theevolution of vaccine escape mutants. In that case, a large fraction of half-vaccinated individuals would pose a risk—but only if they encounter thevirus. This raises the question whether there is a conflict between reducingthe burden and the risk of vaccine escape evolution or not. We develop anSIR-type model to assess the population-level effects of the timing of thesecond dose. Trade-offs can occur both if vaccine escape evolution is morelikely or if it is less likely in half-vaccinated than in unvaccinated individ-uals. Their presence or absence depends on the efficacies for susceptibilityand transmissibility elicited by a single dose

Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations

Using spatial domain techniques developed by the authors and Myunghyun Oh in the context of parabolic conservation laws, we establish under a natural set of spectral stability conditions nonlinear asymptotic stability with decay at Gaussian rate of spatially periodic traveling-waves of systems of reaction diffusion equations. In the case that wave-speed is identically zero for all periodic solutions, we recover and slightly sharpen a well-known result of Schneider obtained by renormalization/Bloch transform techniques; by the same arguments, we are able to treat the open case of nonzero wave-speeds to which Schneider's renormalization techniques do not appear to appl

The Covid-19 pandemic: basic insights from basic mathematical models

Mathematical models for the spread of infectious diseases have a long history. From the start of the Covid-19 pandemic, there was a huge public interest in applying such models, since they help to understand general features of epidemic spread and support the assessment of possible mitigation measures – and their later relaxation. We describe and discuss some well-established mathematical models for epidemic spread, starting from the susceptible-infected-recovered (SIR) model and branching processes and discussing insights from network-based models. During the Covid-19 pandemic, such classical models have also been extended to include many additional aspects that affect epidemic spread, such as mobility patterns or testing possibilities. However, such complex models are increasingly difficult to assess from the outside. In a situation where their predictions can directly affect the lives of millions of people, this can become a severe problem. We argue that simple mathematical models have huge merits and can explain many of the key features of more complex models, such as the importance of heterogeneity in disease transmission. For example, basic models allow inferring whether super-spreading, where very few infected individuals cause the vast majority of secondary cases, should be the rule or the exception – with wide-ranging consequences for the possible success of mitigation measures. In addition, these basic models are simple enough to be understood and implemented without expert knowledge in theoretical epidemiology or computer science. Thus, they offer a level of transparency that can be important for a society to accept mitigation measures

Fixation dynamics of beneficial alleles in prokaryotic polyploid chromosomes and plasmids

Theoretical population genetics has been mostly developed for sexually reproducing diploid and for monoploid (haploid) organisms, focusing on eukaryotes. The evolution of bacteria and archaea is often studied by models for the allele dynamics in monoploid populations. However, many prokaryotic organisms harbor multicopy replicons—chromosomes and plasmids—and theory for the allele dynamics in populations of polyploid prokaryotes remains lacking. Here, we present a population genetics model for replicons with multiple copies in the cell. Using this model, we characterize the fixation process of a dominant beneficial mutation at 2 levels: the phenotype and the genotype. Our results show that depending on the mode of replication and segregation, the fixation of the mutant phenotype may precede genotypic fixation by many generations; we term this time interval the heterozygosity window. We furthermore derive concise analytical expressions for the occurrence and length of the heterozygosity window, showing that it emerges if the copy number is high and selection strong. Within the heterozygosity window, the population is phenotypically adapted, while both alleles persist in the population. Replicon ploidy thus allows for the maintenance of genetic variation following phenotypic adaptation and consequently for reversibility in adaptation to fluctuating environmental conditions

Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation

In this paper we consider the spectral and nonlinear stability of periodic traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In particular, we resolve the long-standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized (integrable) perturbations. Our analysis is based upon detailed estimates of the linearized solution operator, which are complicated by the fact that the (necessarily essential) spectrum of the associated linearization intersects the imaginary axis at the origin. We carry out a numerical Evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves, in close agreement with previous numerical studies of Frisch-She-Thual, Bar-Nepomnyashchy, Chang-Demekhin-Kopelevich, and others carried out by other techniques. We also compare predictions of the associated Whitham modulation equations, which formally describe the dynamics of weak large scale perturbations of a periodic wave train, with numerical time evolution studies, demonstrating their effectiveness at a practical level. For the reader's convenience, we include in an appendix the corresponding treatment of the Swift-Hohenberg equation, a nonconservative counterpart of the generalized Kuramoto-Sivashinsky equation for which the nonlinear stability analysis is considerably simpler, together with numerical Evans function analyses extending spectral stability analyses of Mielke and Schneider.Comment: 78 pages, 11 figure