An integral boundary layer equation for film flow over inclined wavy bottoms

We study the flow of an incompressible liquid film down a wavy incline. Applying a Galerkin method with only one ansatz function to the Navier-Stokes equations we derive a second order weighted residual integral boundary layer equation, which in particular may be used to describe eddies in the troughs of the wavy bottom. We present numerical results which show that our model is qualitatively and quantitatively accurate in wide ranges of parameters, and we use the model to study some new phenomena, for instance the occurrence of a short wave instability for laminar flows which does not exist over flat bottom.Comment: 23 pages, 12 figures. We added a new Section "Regularization" in which we additionally apply a Pade-like regularization to the weighted residual integral boundary layer equatio

Sequential antibiotic therapy in the laboratory and in the patient

Laboratory experiments suggest that rapid cycling of antibiotics during the course of treatment could successfully counter resistance evolution. Drugs involving collateral sensitivity could be particularly suitable for such therapies. However, the environmental conditions in vivo differ from those in vitro. One key difference is that drugs can be switched abruptly in the laboratory, while in the patient, pharmacokinetic processes lead to changing antibiotic concentrations including periods of dose overlaps from consecutive administrations. During such overlap phases, drug–drug interactions may affect the evolutionary dynamics. To address the gap between the laboratory and potential clinical applications, we set up two models for comparison—a ‘laboratory model’ and a pharmacokinetic-pharmacodynamic ‘patient model’. The analysis shows that in the laboratory, the most rapid cycling suppresses the bacterial population always at least as well as other regimens. For patient treatment, however, a little slower cycling can sometimes be preferable if the pharmacodynamic curve is steep or if drugs interact antagonistically. When resistance is absent prior to treatment, collateral sensitivity brings no substantial benefit unless the cell division rate is low and drug cycling slow. By contrast, drug–drug interactions strongly influence the treatment efficiency of rapid regimens, demonstrating their importance for the optimal choice of drug pairs

An Archaeological Study of the McPherson Road Extension Project Laredo, Texas

In October 1977, the Commissioners Court of Webb County and the Center for Archaeological Research, The University of Texas at San Antonio, entered into a contract for an archaeological assessment of the McPherson Road Extension Project, Laredo, Texas. The object of this survey was to locate, record and evaluate archaeological or historical resources that might be present within the proposed right-of-way. As a result of the survey, three archaeological sites (41 WB 64, 41 WB 65 and 41 WB 66) were located within the right-of-way, and it was determined that these sites would be affected by road construction. Sites 41 WB 64 and 41 WB 65 were recommended for limited testing to evaluate their potential. Site 41 WB 66 was considered to be of little archaeological significance

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

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