Selective sweeps in SARS-CoV-2 variant competition.

The main mathematical result in this paper is that change of variables in the ordinary differential equation (ODE) for the competition of two infections in a Susceptible-Infected-Removed (SIR) model shows that the fraction of cases due to the new variant satisfies the logistic differential equation, which models selective sweeps. Fitting the logistic to data from the Global Initiative on Sharing All Influenza Data (GISAID) shows that this correctly predicts the rapid turnover from one dominant variant to another. In addition, our fitting gives sensible estimates of the increase in infectivity. These arguments are applicable to any epidemic modeled by SIR equations.

Viscosity Flow Curves of Agar and the Bounded Ripening Growth Model of the Gelation Onset.

The gelation kinetics of agar aqueous solutions were studied by means of the viscosity flow curves using a coaxial Couette cylinder viscometer. The viscosity curves show an unusual sigmoidal trend or an exponential decay to a viscous steady state. An original theory of gelation kinetics was developed considering the coarsening of increasingly larger and more stable clusters due to Ostwald ripening and the breakup of clusters that were too large due to the instability of rotating large particles induced by the shear rate. The developed Bounded Ripening Growth model takes into account the trend of the viscosity curves by means of an autocatalytic process with negative feedback on aggregation according to the logistic kinetic equation, in which the constants k1(γ) and k-(ν) are governed by the surface tension and shear rate, respectively. A dimensionless equation based on the difference between the Weber number and the ratio of the inverse kinetic constant to forward constant, accounts for the behavior of the dispersed phase in equilibrium conditions or far from the hydrostatic equilibrium.

Nonlinear diffusion in multi-patch logistic model.

We examine a multi-patch model of a population connected by nonlinear asymmetrical migration, where the population grows logistically on each patch. Utilizing the theory of cooperative differential systems, we prove the global stability of the model. In cases of perfect mixing, where migration rates approach infinity, the total population follows a logistic law with a carrying capacity that is distinct from the sum of carrying capacities and is influenced by migration terms. Furthermore, we establish conditions under which fragmentation and nonlinear asymmetrical migration can lead to a total equilibrium population that is either greater or smaller than the sum of carrying capacities. Finally, for the two-patch model, we classify the model parameter space to determine if nonlinear dispersal is beneficial or detrimental to the sum of two carrying capacities.

Reliable and efficient parameter estimation using approximate continuum limit descriptions of stochastic models.

Stochastic individual-based mathematical models are attractive for modelling biological phenomena because they naturally capture the stochasticity and variability that is often evident in biological data. Such models also allow us to track the motion of individuals within the population of interest. Unfortunately, capturing this microscopic detail means that simulation and parameter inference can become computationally expensive. One approach for overcoming this computational limitation is to coarse-grain the stochastic model to provide an approximate continuum model that can be solved using far less computational effort. However, coarse-grained continuum models can be biased or inaccurate, particularly for certain parameter regimes. In this work, we combine stochastic and continuum mathematical models in the context of lattice-based models of two-dimensional cell biology experiments by demonstrating how to simulate two commonly used experiments: cell proliferation assays and barrier assays. Our approach involves building a simple statistical model of the discrepancy between the expensive stochastic model and the associated computationally inexpensive coarse-grained continuum model. We form this statistical model based on a limited number of expensive stochastic model evaluations at design points sampled from a user-chosen distribution, corresponding to a computer experiment design problem. With straightforward design point selection schemes, we show that using the statistical model of the discrepancy in tandem with the computationally inexpensive continuum model allows us to carry out prediction and inference while correcting for biases and inaccuracies due to the continuum approximation. We demonstrate this approach by simulating a proliferation assay, where the continuum limit model is the well-known logistic ordinary differential equation, as well as a barrier assay where the continuum limit model is closely related to the well-known Fisher-KPP partial differential equation. We construct an approximate likelihood function for parameter inference, both with and without discrepancy correction terms. Using maximum likelihood estimation, we provide point estimates of the unknown parameters, and use the profile likelihood to characterise the uncertainty in these estimates and form approximate confidence intervals. For the range of inference problems considered, working with the continuum limit model alone leads to biased parameter estimation and confidence intervals with poor coverage. In contrast, incorporating correction terms arising from the statistical model of the model discrepancy allows us to recover the parameters accurately with minimal computational overhead. The main tradeoff is that the associated confidence intervals are typically broader, reflecting the additional uncertainty introduced by the approximation process. All algorithms required to replicate the results in this work are written in the open source Julia language and are available at GitHub.

Maximal total population of species in a diffusive logistic model.

In this paper, we investigate the maximization of the total population of a single species which is governed by a stationary diffusive logistic equation with a fixed amount of resources. For large diffusivity, qualitative properties of the maximizers like symmetry will be addressed. Our results are in line with previous findings which assert that for large diffusion, concentrated resources are favorable for maximizing the total population. Then, an optimality condition for the maximizer is derived based upon rearrangement theory. We develop an efficient numerical algorithm applicable to domains with different geometries in order to compute the maximizer. It is established that the algorithm is convergent. Our numerical simulations give a real insight into the qualitative properties of the maximizer and also lead us to some conjectures about the maximizer.

Spatial ecology, optimal control and game theoretical fishing problems.

Of paramount importance in both ecological systems and economic policies are the problems of harvesting of natural resources. A paradigmatic situation where this question is raised is that of fishing strategies. Indeed, overfishing is a well-known problem in the management of live-stocks, as being too greedy may lead to an overall dramatic depletion of the population we are harvesting. A closely related topic is that of Nash equilibria in the context of fishing policies. Namely, two players being in competition for the same pool of resources, is it possible for them to find an equilibrium situation? The goal of this paper is to provide a detailed analysis of these two queries (i.e optimal fishing strategies for single-player models and study of Nash equilibria for multiple players games) by using a basic yet instructive mathematical model, the logistic-diffusive equation. In this framework, the underlying model simply reads [Formula: see text] where K accounts for natural resources, [Formula: see text] for the density of the population that is being harvested and [Formula: see text] encodes either the single player fishing strategy or, when dealing with Nash equilibria, a combination of the fishing strategies of both players. This article consists of two main parts. The first one gives a very fine characterisation of the optimisers for the single-player game where one aims at solving [Formula: see text], under [Formula: see text] and [Formula: see text] constraints on the fishing strategies [Formula: see text]. In particular, we show that, depending on the value of these constraints, this optimal control problem may behave like a convex or, conversely, concave problem. We also provide a detailed analysis of the large diffusivity limit of this problem. In the case where two players are involved, we rather write [Formula: see text] as [Formula: see text] where [Formula: see text], the fishing strategy of the i-th player, also satisfies [Formula: see text] and [Formula: see text] constraints. Defining [Formula: see text] we aim at finding a Nash equilibrium. We prove the existence of Nash equilibria in several different regimes and investigate several related qualitative queries, for instance providing examples of the well-known tragedy of commons. Our study is completed by a variety of numerical simulations that illustrate our results and allow us to formulate open questions and conjectures.

Gompertz model in COVID-19 spreading simulation.

The paper reports on application of the Gompertz model to describe the growth dynamics of COVID-19 cases during the first wave of the pandemic in different countries. Modeling has been performed for 23 countries: Australia, Austria, Belgium, Brazil, Great Britain, Germany, Denmark, Ireland, Spain, Italy, Canada, China, the Netherlands, Norway, Serbia, Turkey, France, Czech Republic, Switzerland, South Korea, USA, Mexico, and Japan. The model parameters are determined by regression analysis based on official World Health Organization data available for these countries. The comparison of the predictions given by the Gompertz model and the simple logistic model (i.e., Verhulst model) is performed allowing to conclude on the higher accuracy of the Gompertz model.

Effect of different fish feeds on water quality and growth of crucian carp (Carassius carassius) in the presence and absence of prometryn.

Few data are available regarding comprehensive or quantitative assessment of fish feed considering both the environmental and feeding impacts. Aiming to fill the gap, an experimental study to investigate the effects of three fish feeds on concentrations of nutrients and crucian carp (Carassius carassius) growth was conducted in laboratory aquariums in the presence and absence of prometryn. Results showed that weight gain rates of crucian carp treated with Tong Wei (TW) feed were 106.3% and 2.0% higher than that of Zhong Shan (ZS) and Zhong Liang (ZL) feeds, a possible explanation was that the quality of protein in TW feed was highest as evidenced by the protein efficiency ratios. Meanwhile, TW feed posed relatively lighter effects on water qualities (between ZL and ZS). Prometryn significantly inhibited the growth of crucian carp and thus affected concentrations of nutrients in water indirectly. The relationships between weight gain rates of fish and concentrations of nutrients in water (R2 = 0.929-0.990) were developed. In sum, this study suggested that it is realizable to obtain better fish growth performance with lesser degrading effects on water qualities by producing and selecting appropriate feed regardless of prometryn existence, and the developed equations could be used as a basis for future studies.

Microwave pasteurization of apple juice: Modeling the inactivation of Escherichia coli O157:H7 and Salmonella Typhimurium at 80-90 °C.

Although due to their acidity some fruit juices are considered safe, several outbreaks have been reported. For processing fruit juices, microwave heating offers advantages such as shorter come-up time, faster and uniform heating, and energy efficiency. Thus, it could be a beneficial alternative to conventional pasteurization. The objective of this study was to study the inactivation kinetics of Escherichia coli O157:H7 and Salmonella Typhimurium under microwave pasteurization at temperatures between 80 and 90 °C, i.e., at conditions that are employed in conventional pasteurization. Inoculated juices were treated at different power levels (600 W, 720 W) and treatment times (5s, 10s, 15s, 20s, 25s). Time-temperature profiles were obtained by fiber-optic sensors in contact with the samples allowing continuous data collection. The log-logistic and Arrhenius equations were used to account for the influence of the temperature history; thus, resulting in two different modeling approaches that were compared in terms of their prediction abilities. Survival kinetics including non-isothermal conditions were described by a non-linear ordinary differential equation that was numerically solved by the Runge-Kutta method (ode45 in MATLAB ®). The lsqcurvefit function (MATLAB®) was employed to estimate the corresponding survival parameters, which were obtained from freshly made apple juice, whereas the prediction ability of these parameters was evaluated on commercial apple juices. Results indicated that inactivation increased with power level, temperature, and treatment time reaching a microbial reduction up to 7 Log10 cycles. The study is relevant to the food industry because it provides a quantitative tool to predict survival characteristics of pathogens at other non-isothermal processing conditions.

Logistic equation and COVID-19.

The generalized logistic equation is used to interpret the COVID-19 epidemic data in several countries: Austria, Switzerland, the Netherlands, Italy, Turkey and South Korea. The model coefficients are calculated: the growth rate and the expected number of infected people, as well as the exponent indexes in the generalized logistic equation. It is shown that the dependence of the number of the infected people on time is well described on average by the logistic curve (within the framework of a simple or generalized logistic equation) with a determination coefficient exceeding 0.8. At the same time, the dependence of the number of the infected people per day on time has a very uneven character and can be described very roughly by the logistic curve. To describe it, it is necessary to take into account the dependence of the model coefficients on time or on the total number of cases. Variations, for example, of the growth rate can reach 60%. The variability spectra of the coefficients have characteristic peaks at periods of several days, which corresponds to the observed serial intervals. The use of the stochastic logistic equation is proposed to estimate the number of probable peaks in the coronavirus incidence.

Interpreting, analysing and modelling COVID-19 mortality data.

We present results on the mortality statistics of the COVID-19 epidemic in a number of countries. Our data analysis suggests classifying countries in five groups, (1) Western countries, (2) East Block, (3) developed Southeast Asian countries, (4) Northern Hemisphere developing countries and (5) Southern Hemisphere countries. Comparing the number of deaths per million inhabitants, a pattern emerges in which the Western countries exhibit the largest mortality rate. Furthermore, comparing the running cumulative death tolls as the same level of outbreak progress in different countries reveals several subgroups within the Western countries and further emphasises the difference between the five groups. Analysing the relationship between deaths per million and life expectancy in different countries, taken as a proxy of the preponderance of elderly people in the population, a main reason behind the relatively more severe COVID-19 epidemic in the Western countries is found to be their larger population of elderly people, with exceptions such as Norway and Japan, for which other factors seem to dominate. Our comparison between countries at the same level of outbreak progress allows us to identify and quantify a measure of efficiency of the level of stringency of confinement measures. We find that increasing the stringency from 20 to 60 decreases the death count by about 50 lives per million in a time window of 20  days. Finally, we perform logistic equation analyses of deaths as a means of tracking the dynamics of outbreaks in the "first wave" and estimating the associated ultimate mortality, using four different models to identify model error and robustness of results. This quantitative analysis allows us to assess the outbreak progress in different countries, differentiating between those that are at a quite advanced stage and close to the end of the epidemic from those that are still in the middle of it. This raises many questions in terms of organisation, preparedness, governance structure and so on.

Generating visually coherent encrypted images with reversible data hiding in wavelet domain by fusing chaos and pairing function.

For secure transmission of digital images, existing cryptographic algorithms transform coherent visual information into a noise-like appearance prompting an adversary of the presence of a possible cipher. This paper proposes an algorithm that produces a visually coherent and meaningful cipher image. The proposed algorithm consists of a permutation-substitution subroutine to obtain a partial cipher. The Arnold-3D map does the permutation, and a delayed logistic map performs the substitution in this subroutine. The hiding of this partial cipher is done in the reference image using an integer wavelet transform. The pixels of the partial cipher are embedded in the four sub-bands of the decomposed reference image as 4 to 1-pixel encoding using Cantor-like pairing function. In addition to the lossless encryption scheme, the integer nature of all the sub-bands in the wavelet decomposition and the invertible pairing function facilitates the perfect reconstruction of the reference image. One of the significant novelty of this work lies in the subtle use of simple pairing functions, which prohibits the unnecessary increase in the size of the cipher, thereby reducing the storage and transmission costs.

The study on the early warning period of varicella outbreaks based on logistic differential equation model.

Chickenpox is a common acute and highly contagious disease in childhood; moreover, there is currently no targeted treatment. Carrying out an early warning on chickenpox plays an important role in taking targeted measures in advance as well as preventing the outbreak of the disease. In recent years, the infectious disease dynamic model has been widely used in the research of various infectious diseases. The logistic differential equation model can well demonstrate the epidemic characteristics of epidemic outbreaks, gives the point at which the early epidemic rate changes from slow to fast. Therefore, our study aims to use the logistic differential equation model to explore the epidemic characteristics and early-warning time of varicella. Meanwhile, the data of varicella cases were collected from first week of 2008 to 52nd week of 2017 in Changsha. Finally, our study found that the logistic model can be well fitted with varicella data, besides the model illustrated that there are two peaks of varicella at each year in Changsha City. One is the peak in summer-autumn corresponding to the 8th-38th week; the other is in winter-spring corresponding to the time from the 38th to the seventh week next year. The 'epidemic acceleration week' average value of summer-autumn and winter-spring are about the 16th week (ranging from the 15th to 17th week) and 45th week (ranging from the 44th to 47th week), respectively. What is more, taking warning measures during the acceleration week, the preventive effect will be delayed; thus, we recommend intervene during recommended warning weeks which are the 15th and 44th weeks instead.

Predicting Patient-Specific Radiotherapy Protocols Based on Mathematical Model Choice for Proliferation Saturation Index.

Radiation is commonly used in cancer treatment. Over 50% of all cancer patients will undergo radiotherapy (RT) as part of cancer care. Scientific advances in RT have primarily focused on the physical characteristics of treatment including beam quality and delivery. Only recently have inroads been made into utilizing tumor biology and radiobiology to design more appropriate RT protocols. Tumors are composites of proliferating and growth-arrested cells, and overall response depends on their respective proportions at irradiation. Prokopiou et al. (Radiat Oncol 10:159, 2015) developed the concept of the proliferation saturation index (PSI) to augment the clinical decision process associated with RT. This framework is based on the application of the logistic equation to pre-treatment imaging data in order to estimate a patient-specific tumor carrying capacity, which is then used to recommend a specific RT protocol. It is unclear, however, how dependent clinical recommendations are on the underlying tumor growth law. We discuss a PSI framework with a generalized logistic equation that can capture kinetics of different well-known growth laws including logistic and Gompertzian growth. Estimation of model parameters on the basis of clinical data revealed that the generalized logistic model can describe data equally well for a wide range of the generalized logistic exponent value. Clinical recommendations based on the calculated PSI, however, are strongly dependent on the specific growth law assumed. Our analysis suggests that the PSI framework may best be utilized in clinical practice when the underlying tumor growth law is known, or when sufficiently many tumor growth models suggest similar fractionation protocols.

Free boundary models for mosquito range movement driven by climate warming.

As vectors, mosquitoes transmit numerous mosquito-borne diseases. Among the many factors affecting the distribution and density of mosquitoes, climate change and warming have been increasingly recognized as major ones. In this paper, we make use of three diffusive logistic models with free boundary in one space dimension to explore the impact of climate warming on the movement of mosquito range. First, a general model incorporating temperature change with location and time is introduced. In order to gain insights of the model, a simplified version of the model with the change of temperature depending only on location is analyzed theoretically, for which the dynamical behavior is completely determined and presented. The general model can be modified into a more realistic one of seasonal succession type, to take into account of the seasonal changes of mosquito movements during each year, where the general model applies only for the time period of the warm seasons of the year, and during the cold season, the mosquito range is fixed and the population is assumed to be in a hibernating status. For both the general model and the seasonal succession model, our numerical simulations indicate that the long-time dynamical behavior is qualitatively similar to the simplified model, and the effect of climate warming on the movement of mosquitoes can be easily captured. Moreover, our analysis reveals that hibernating enhances the chances of survival and successful spreading of the mosquitoes, but it slows down the spreading speed.

Invasion probabilities, hitting times, and some fluctuation theory for the stochastic logistic process.

We consider excursions for a class of stochastic processes describing a population of discrete individuals experiencing density-limited growth, such that the population has a finite carrying capacity and behaves qualitatively like the classical logistic model Verhulst (Corresp Math Phys 10:113-121, 1838) when the carrying capacity is large. Being discrete and stochastic, however, our population nonetheless goes extinct in finite time. We present results concerning the maximum of the population prior to extinction in the large population limit, from which we obtain establishment probabilities and upper bounds for the process, as well as estimates for the waiting time to establishment and extinction. As a consequence, we show that conditional upon establishment, the stochastic logistic process will with high probability greatly exceed carrying capacity an arbitrary number of times prior to extinction.

Kinetic modeling of bacteriocin-like inhibitory substance secretion by Pediococcus acidilactici Kp10 and its stability in food manufacturing conditions.

This paper deliberates the modelling and validation of bacteriocin-like inhibitory substance (BLIS) secretion by Pediococcus acidilactici Kp10 at different agitation speeds in a stirred tank bioreactor. A range of models namely the re-parameterised logistic, Luedeking-Piret and maintenance energy were assessed to predict the culture performance of the said bacterium. Growth of P. acidilactici Kp10 was enhanced with increased agitation speed up to 600 rpm while BLIS secretion was maximum at 400 rpm but decreased at higher agitation speed. Growth of P. acidilactici aptly subscribed to the re-parameterised logistic model while BLIS secretion and lactose consumption fitted well with the Luedeking-Piret model. The models revealed a relationship between growth of the bacterium and BLIS secretion. Bacterial growth and BLIS secretion were largely affected by the agitation speed of the stirred tank bioreactor which regulated the oxygen transfer to the culture. BLIS secretion by P. acidilactici Kp10 was however enhanced in oxygen-limited culture. The study also assessed BLIS from the perspective of its stability when subjected to factors such as temperature, pH and detergents. Results showed that BLIS produced by this strain was not affected by heat (at 25-100 °C for 20 min and at 121 °C for 15 min), surfactant (Tween 40, 60 and 80 and urea), detergents (up to 1% SDS), organic solvents (50% each of acetone, methanol and ethanol) and stable in a wide range of pH (2-10). The above information are pertinent with reference to commercial applications of this bacterial product in food manufacturing which invariably involve various sterilization processes and subjected to a wide pH range.

Ralstonia eutropha as a biocatalyst for desulfurization of dibenzothiophene.

The potential of Ralstonia eutropha as a biocatalyst for desulfurization of dibenzothiophene (DBT) was studied in growing and resting cell conditions. The results of both conditions showed that sulfur was removed from DBT which accompanied by the formation of 2-hydroxybiphenyl (2-HBP). In growing cell experiments, glucose was used as an energy supplying substrate in initial concentrations of 55 mM (energy-limited) and 111 mM (energy-sufficient). The growing cell behaviors were quantitatively described using the logistic equation and maintenance concept. The results indicated that 2-HBP production was higher for the energy-sufficient cultures, while the values of the specific growth rate and the maintenance coefficient for these media were lower than those of the energy-limited cultures. Additionally, the kinetic studies showed that the half-saturation constant for the energy-limited cultures was 2 times higher than the energy-sufficient ones where the inhibition constant (0.08 mM) and the maximum specific DBT desulfurization rate (0.002 mmol gcell-1 h-1) were almost constant. By defining desulfurizing capacity (D DBT) including both the biomass concentration and time to reach a particular percentage of DBT conversion, the best condition for desulfurizing cell was determined at 23% gcell L-1 h-1 which corresponded with the resting cells that were harvested at the mid-exponential growth phase.

Dynamical System Modeling to Simulate Donor T Cell Response to Whole Exome Sequencing-Derived Recipient Peptides Demonstrates Different Alloreactivity Potential in HLA-Matched and -Mismatched Donor-Recipient Pairs.

Immune reconstitution kinetics and subsequent clinical outcomes in HLA-matched recipients of allogeneic stem cell transplantation (SCT) are variable and difficult to predict. Considering SCT as a dynamical system may allow sequence differences across the exomes of the transplant donors and recipients to be used to simulate an alloreactive T cell response, which may allow better clinical outcome prediction. To accomplish this, whole exome sequencing was performed on 34 HLA-matched SCT donor-recipient pairs (DRPs) and the nucleotide sequence differences translated to peptides. The binding affinity of the peptides to the relevant HLA in each DRP was determined. The resulting array of peptide-HLA binding affinity values in each patient was considered as an operator modifying a hypothetical T cell repertoire vector, in which each T cell clone proliferates in accordance with the logistic equation of growth. Using an iterating system of matrices, each simulated T cell clone's growth was calculated with the steady-state population being proportional to the magnitude of the binding affinity of the driving HLA-peptide complex. Incorporating competition between T cell clones responding to different HLA-peptide complexes reproduces a number of features of clinically observed T cell clonal repertoire in the simulated repertoire, including sigmoidal growth kinetics of individual T cell clones and overall repertoire, Power Law clonal frequency distribution, increase in repertoire complexity over time with increasing clonal diversity, and alteration of clonal dominance when a different antigen array is encountered, such as in SCT. The simulated, alloreactive T cell repertoire was markedly different in HLA-matched DRPs. The patterns were differentiated by rate of growth and steady-state magnitude of the simulated T cell repertoire and demonstrate a possible correlation with survival. In conclusion, exome wide sequence differences in DRPs may allow simulation of donor alloreactive T cell response to recipient antigens and may provide a quantitative basis for refining donor selection and titration of immunosuppression after SCT.

Production of Hyaluronic Acid by Streptococcus zooepidemicus on Protein Substrates Obtained from Scyliorhinus canicula Discards.

This work investigates the production of hyaluronic acid (H) by Streptococcus equi subsp. zooepidemicus in complex media formulated with peptones obtained from Scyliorhinus canicula viscera by-products. Initially, in batch cultures, the greatest productions were achieved using commercial media (3.03 g/L) followed by peptones from alcalase hydrolyzed viscera (2.32 g/L) and peptones from non-hydrolyzed viscera (2.26 g/L). An increase of between 12% and 15% was found in subsequent fed-batch cultures performed on waste peptones. Such organic nitrogen sources were shown to be an excellent low-cost substrate for microbial H, saving more than 50% of the nutrient costs.