A comparison of a deterministic and stochastic model for Hepatitis C with an isolation stage

We formulate a deterministic epidemic model for the spread of Hepatitis C containing an acute, chronic and isolation class and analyse the effects of the isolation class on the transmission dynamics of the disease. We calculate the basic reproduction number R₀ and show that for R₀≤1, the disease-free equilibrium is globally asymptotically stable. In addition, it is shown that for a special case when R₀>1, the endemic equilibrium is locally asymptotically stable. Furthermore, an analogous stochastic epidemic model for Hepatitis C is formulated using a continuous time Markov chain. Numerical simulations are used to estimate the mean, variance and probability distributions of the discrete random variables and these are compared to the steady-state solutions of the deterministic model. Finally, the expected time to disease extinction is estimated for the stochastic model and the impact of isolation on the time to extinction is explored.     

Missing Data and the Mixtures of Discrete and Continuous Random Variables

In many practical applications we deal with a problem of estimation of a density function of a vector x some components of which are discrete, while the remaining ones are continuous. Among many models that can be used in this case the most useful are the location model and the kernel model. The problem arises when the observed data contain missing values i.e. on some individuals some of the variables have not been observed with no particular pattern of missingness. An application of the EM algorithm will allow us to estimate the parameters of the location model from incomplete data. The method is described in Section 2. In Section 3 some suggestions how to deal with incompleteness when the kernel model is used are made. Finally, Section 4 contains an example.     

Relationships in a simple harmonic mean two-sex fertility model

A continuous time two-sex stable population model that does not recognize age is examined under the “harmonic mean” consistency condition of equation (2). Solutions for the stable population intrinsic growth rate r and the sex composition 5 are presented in equations (5) and (6). The process of stabilization is examined, and it is shown that, given two basic constraints, any initial population sex composition will eventually converge to the stable population value. An algebraic solution for the discrete case where the sex ratio at birth is unity is presented and used to describe the trajectory to stability of several hypothetical populations. A closed form algebraic expression for the trajectory to stability is presented for the continuous model in the special case of no mortality.     

Reanalysis of Models and an Improved Model of Biomass Size Spectra

Our reanalysis aimed at understanding the regularity in empirical biomass size spectra (BSS) suggests that the construction of BSS depends of the size interval and size scales used and different definitions of BSS in literature are therefore very different. Existing empirical models of BSS can be fitted perfectly to the observed data, but the biological basis of the fitted parameters is not explained and comparison and interpretation of the findings is therefore difficult. Parameters of mechanistic models of BSS have a biological background and are interpretable. Discrete mechanistic models based on Lindeman's trophic chain theory assume a constant ratio of size (or body mass) in two adjacent trophic levels. However, this biomass ratio is not comparable with that in two adjacent (logarithmic) size intervals in the measured biomass size spectra. The continuous model by Thiebaux and Dickie (1992) is based on the discrete model by Boudreau et al. (1991). We show how the validity of the transformation of a discrete form into a continous form depends on the size ranges of prey and predator population. The model by Platt and Denman (1977) does not represent a continuous formulation due to the use of normalized biomass defined in logarithmic size intervals. We suggest to eliminate the use of trophic levels and normalized biomass. On the basis of the reanalysis we formulate and improved continuous model based on the model by Silvert and Platt (1978). The model is based on Eulerian strategy which appears more adequate for the problem than the previously used Lagrangian strategy. The model appears to be able to demonstrate the regularity in observed BSS.     

Capacity optimization and scheduling of a multiproduct manufacturing facility for biotech products

A general mathematical framework has been proposed in this work for scheduling of a multiproduct and multipurpose facility involving manufacturing of biotech products. The specific problem involves several batch operations occurring in multiple units involving fixed processing time, unlimited storage policy, transition times, shared units, and deterministic and fixed data in the given time horizon. The different batch operations are modeled using state‐task network representation. Two different mathematical formulations are proposed based on discrete‐ and continuous‐time representations leading to a mixed‐integer linear programming model which is solved using General Algebraic Modeling System software. A case study based on a real facility is presented to illustrate the potential and applicability of the proposed models. The continuous‐time model required less number of events and has a smaller problem size compared to the discrete‐time model. © 2014 American Institute of Chemical Engineers Biotechnol. Prog., 30:1221–1230, 2014     

An individual-based model for competingDrosophila populations

An individual-based model forDrosophila is formulated, based on competition amongst larvae consuming the same batch of food. The predictions of the model are supported by data for single speciesDrosophila populations reared in the laboratory. The model is used to build a simple discrete model for the dynamics ofDrosophila populations that are kept over a number of generations. The dynamics of a single species is shown to give either a stable equilibrium or fluctuations which can be periodic or chaotic. When the dynamics of a species in the absence of the other is periodic or chaotic, we found coexistence or two alternative states, on neither of which the species can coexist.     

Dynamic discrete model of flashing light effect in photosynthesis of microalgae

For a photobioreactor for mass-culturing microalgae, it is known that flashing light effect enhances the efficiency of photosynthesis. A dynamic model for photosynthesis was developed to elucidate this effect. A particular feature of the model is that discrete RuBP particles circulate in the Calvin cycle and their speeds in the cycle are determined by the amount of ATP generated in the photon reception process. This can realise the light saturation under continuous light and the flashing light effect under fluctuating illumination. Laboratory experiments were conducted to obtain model parameters by curve-fitting for Chaetoceros calcitrans. The present model demonstrates the light flashing effect moderately well and elucidates its mechanism reasonably.     

Optimal resource allocation in perennial plants: A continuous-time model

A continuous-time model, similar to W. M. Schaffer's (1983, Amer. Nat. 121, 418–431), of growth and reproduction for a perennial herb with discrete growing seasons is considered. Assuming that metabolic rates of reproductive and storage structures are equal, it was possible, through the reduction of the continuous model to a discrete one, to find the optimal allocations to the vegetative, reproductive, and reserve structures. The main feature of the optimal strategy is the existence of an optimal reserve size. The allocation to vegetative structures is, every growing season, the allocation which maximizes the total of reproductive and reserve structures at the end of the season. The relative allocation between reserve and reproductive structures is given, when reproductive success is a linear function of investment, by the fastest growth to the optimal size: no reproduction until the optimal size is reached, and, afterwards, allocation to reproduction of everything beyond what is needed to maintain size R*. Asymptotic growth to the equilibrium and cycles are possible, when reproductive success is a nonlinear function of investment (A. Pugliese, 1988, in “Biomathematics and Related Computational Problems” (L. M. Ricciardi, Ed.), Reidel, Dordrecht, to appear). It has therefore been possible to solve the “general life history problem” ( Schaffer, 1983) when growth is in general a concave function of body size. In the Discussion discrete and continuous-time models are compared; if the real dynamics is described by a continuous model of the type analyzed here, life history predictions made by analyzing the system with a discrete model are upheld.     

A simulation of anthropogenic Columbian mammoth (Mammuthus columbi) extinction

The cause of the extinction of the Columbian mammoth (Mammuthus columbi) and other species of megafauna during the end of the Pleistocene epoch is an ongoing debate. In this study, we used mathematical modelling to test the overkill hypothesis first proposed by Martin in 1973. The overkill hypothesis claims that early humans migrating from Asia through Beringia and into North America hunted the majority of the continent’s megafauna to extinction. Previous research has been conducted on the overkill hypothesis for the Columbian mammoth using a continuous differential equations model. We improved on this work by developing a computationally more efficient and more realistic discrete stochastic model. Most model parameters were obtained directly from the literature; migration parameters were informed by the literature and calibrated for the model. Our results provide evidence in support of the overkill hypothesis.     

Investigation of Combining Plant Genotypic Values and Molecular Marker Information for Constructing Core Subsets

In the present study, a strategy was proposed for constructing plant core subsets by clusters based on the combination of continuous data for genotypic values and discrete data for molecular marker InformaUon. A mixed linear model approach was used to predict genotyplc values for eliminating the environment effect. The ＂mixed genetic distance＂ was designed to solve the difficult problem of combining continuous and discrete data to construct a core subset by cluster. Four commonly used genetic distances for continuous data （Euclidean distance, standardized Euclidean distance, city block distance, and Mahalanobls distance） were used to assess the validity of the conUnuous data part of the mixed genetic distance; three commonly used genetic distances for discrete data （cosine distance, correlaUon distance, and Jaccard distance） were used to assess the validity of the discrete data part of the mixed genetic distance, A rice germplasm group with eight quantitative traits and information for 60 molecular markers was used to evaluate the validity of the new strategy. The results suggest that the validity of both parts of the mixed geneUc distance are equal to or higher than the common geneUc distance. The core subset constructed on the basis of a combination of data for genotyplc values and molecular marker information was more representative than that constructed on the basis of data from genotypic values or molecular marker informaUon alone. Moreover, the strategy of using combined data was able to treat dominant marker informaUon and could combine any other continuous data and discrete data together to perform cluster to construct a plant core subset.     

A discrete mathematical model of unlabelled granulocyte kinetics

The equations used in formulating the continuous model of granulocyte kinetics developed by O'Fallon et al. (1971) were analyzed to see if they could be altered to simulate a feedback mechanism operating on the production and development of granulocytes. After extensive study and modification of the continuous model, it was found that a discrete model based on a Leslie matrix procedure was more effective for simulating the feedback system. This discrete model was used to show experimentally, from a mathematical view point, that a feedback mechanism of some kind must be operating on the production and development of granulocytes. Further, the discrete model was subjected to preliminary tests (simultaneous and cascading feedback) to demonstrate that it has the capability of responding to feedback control.This work was completed while the first author was at North Carolina State University at Raleigh, Department of Statistics, Biomathematics Division, part of it under grant number 5T1 GM 678-15, National Institutes of Health, and part of it under grant number 5 F32 CA05964-02 from the National Cancer Institute     

Cumulative Damage Survival Models for the Randomized Complete Block Design

A continuous time discrete state cumulative damage process {X(t), t ≥ 0} is considered, based on a non‐homogeneous Poisson hit‐count process and discrete distribution of damage per hit, which can be negative binomial, Neyman type A, Polya‐Aeppli or Lagrangian Poisson. Intensity functions considered for the Poisson process comprise a flexible three‐parameter family. The survival function is S(t) = P(X(t) ≤ L) where L is fixed. Individual variation is accounted for within the construction for the initial damage distribution {P(X(0) = x) | x = 0, 1, …,}. This distribution has an essential cut‐off before x = L and the distribution of L – X(0) may be considered a tolerance distribution. A multivariate extension appropriate for the randomized complete block design is developed by constructing dependence in the initial damage distributions. Our multivariate model is applied (via maximum likelihood) to litter‐matched tumorigenesis data for rats. The litter effect accounts for 5.9 percent of the variance of the individual effect. Cumulative damage hazard functions are compared to nonparametric hazard functions and to hazard functions obtained from the PVF‐Weibull frailty model. The cumulative damage model has greater dimensionality for interpretation compared to other models, owing principally to the intensity function part of the model.     

Did we miss some evidence of chaos in laboratory insect populations?

The collection and documentation of the experimental evidence of chaos in biological populations have always been elusive. We were puzzled by the observation that the most frequently computed demographic parameter for laboratory insect populations, the population intrinsic rate of increase (r _m), seems too small to induce chaotic dynamics. For example, when it is directly utilized as an approximation to the parameter (a) of the one-parameter discrete logistic model, the parameter seems well out of the chaotic range. In a recent reanalysis of our early laboratory demographic data of 1800 Russian wheat aphids (RWA), we discovered that a proper measure unit conversion should be performed to make the link between r _m and the discrete logistic model. We think that this conversion issue may have been ignored historically in biological literature since we are not aware of any uses of the r _m in the discussion of chaos. It should be noted that r _m is different from the r in discrete logistic model (e.g., May in Nature 261:459–467, 1976). Since extensive demographic data purported to estimate r _m have been accumulated in literature on population growth, the finding revealed with our RWA experiment data can easily be verified with the published data in literature. We near-arbitrarily surveyed 10 studies (containing 37 datasets of r _m) published in the literature and archived in JSTOR or BioOne databases to test the conversion, and the results confirmed our finding. The finding should significantly expand the evidence base of chaos in laboratory populations of insects and possibly microorganisms, too.     

Modeling animals' behavioral response by Markov chain models for capture-recapture experiments

A bivariate Markov chain approach that includes both enduring (long-term) and ephemeral (short-term) behavioral effects in models for capture-recapture experiments is proposed. The capture history of each animal is modeled as a Markov chain with a bivariate state space with states determined by the capture status (capture/noncapture) and marking status (marked/unmarked). In this framework, a conditional-likelihood method is used to estimate the population size and the transition probabilities. The classical behavioral model that assumes only an enduring behavioral effect is included as a special case of the bivariate Markovian model. Another special case that assumes only an ephemeral behavioral effect reduces to a univariate Markov chain based on capture/noncapture status. The model with the ephemeral behavioral effect is extended to incorporate time effects; in this model, in contrast to extensions of the classical behavioral model, all parameters are identifiable. A data set is analyzed to illustrate the use of the Markovian models in interpreting animals' behavioral response. Simulation results are reported to examine the performance of the estimators.     

Circadian rhythms and photic entrainment of swimming activity in cave-dwelling fish Astyanax mexicanus (Actinopterygii: Characidae), from El Sotano La Tinaja,San Luis Potosi,Mexico

Circadian regulation has a profound adaptive meaning in timing the best performance of biological functions in a cyclic niche. However, in cave-dwelling animals (troglobitic), a lack of photic cyclic environment may represent a disadvantage for persistence of circadian rhythms. There are different populations of cave-dwelling fish Astyanax mexicanus in caves of the Sierra El Abra, Mexico, with different evolutive history. In the present work, we report that fish collected from El Sótano la Tinaja show circadian rhythms of swimming activity in laboratory conditions. Rhythms observed in some of the organisms entrain to either continuous light–dark cycles or discrete skeleton photoperiods tested. Our results indicate that circadian rhythm of swimming activity and their ability to entrain in discrete and continuous photoperiods persist in some organisms that might represent one of the oldest populations of cave-dwelling A. mexicanus in the Sierra El Abra.     

Model Construction and Model Examination on the Basis of Simulated Selection

By reason nonlinear relations founded between selection differential and realised selection response we have been made investigations about variants of the genetic-statistical model, which include this nonlinearity. The variations of the model would not only referred to the postulate pattern of the connection between phenotype, genotype and environment but also enclosed the postulate assumption about the distribution of the variates. In an investigated special case the linear model equation P = G ± e was held, however the distributions of P and G were defined over a limited range in one direction. For P we have defined a modified normal distribution and the distribution of the random vector (G, e) non normal regarded with cov (G, e) ≠ 0, By means of a solution set of an integral equation a density function of the random vector (P, G) has been received, in which the expectation of the selection response of the usual genetic-statistical model approximate is included as a special case. The genetical parameters has been derived, which result from changed model. However their representation was only possible partially as an integral function. A subsequent paper informs of the examination this mode! variants, which depend on a parameter of the nonlinearity c.     

Comparison and content of the Wright-Fisher model of random genetic drift, the diffusion approximation, and an intermediate model

We investigate the detailed connection between the Wright-Fisher model of random genetic drift and the diffusion approximation, under the assumption that selection and drift are weak and so cause small changes over a single generation. A representation of the mathematics underlying the Wright-Fisher model is introduced which allows the connection to be made with the corresponding mathematics underlying the diffusion approximation. Two ‘hybrid’ models are also introduced which lie ‘between’ the Wright-Fisher model and the diffusion approximation. In model 1 the relative allele frequency takes discrete values while time is continuous; in model 2 time is discrete and relative allele frequency is continuous. While both hybrid models appear to have a similar status and the same level of plausibility, the different nature of time and frequency in the two models leads to significant mathematical differences. Model 2 is mathematically inconsistent and has to be ruled out as being meaningful. Model 1 is used to clarify the content of Kimura's solution of the diffusion equation, which is shown to have the natural interpretation as describing only those populations where alleles are segregating. By contrast the Wright-Fisher model and the solution of the diffusion equation of McKane and Waxman cover populations of all categories, namely populations where alleles segregate, are lost, or fix.     

Description of a new species and phylogenetic analysis of the subtribe Cynopoecilina,including continuous characters without discretization (Cyprinodontiformes: Rivulidae)

A new species of the subtribe Cynopoecilina is described from the rio Gravataí basin, laguna dos Patos system, southern Brazil. The relationships of the new species among taxa of the subtribe Cynopoecilina is discussed based on two analyses: one using 71 discrete characters and other with the addition of six continuous characters analyzed without discretization. The addition of the continuous characters resulted in the first fully resolved phylogenies for Cynopoecilus and Leptolebias species, not obtained in the analysis including only discrete characters. The new species is assigned to Cynopoecilus as sister group to the remaining species of the genus. A new diagnosis is proposed for Cynopoecilus to accommodate the new species. The resulting phylogeny indicates that the occupation of the grasslands of the Pampa biome by the species of Cynopoecilus occurred along the evolution of the genus and that this event was significant for the diversification of the genus. © 2014 The Linnean Society of London     

Leslie matrix models as “stroboscopic snapshots” of McKendrick PDE models

 High dimensional Leslie matrix models have long been viewed as discretizations of McKendrick PDE models. However, these two fundamental classes of models can be linked in a completely different way. For populations with periodic birth pulses, Leslie models of any dimension can be viewed as “stroboscopic snapshots” (in time) of an associated impulsive McKendrick model; that is, the solution of the discrete model matches the solution of the corresponding continuous model at every discrete time step. In application, McKendrick models of populations with birth pulses can be used to identify the state of the population between the discrete census times of the associated Leslie model. Furthermore, McKendrick models describing populations with near-synchronous birth pulses can be viewed as realistic perturbations of the associated Leslie model. Received: 7 August 1997 / Revised version: 15 January 1998     

The probability of treatment induced drug resistance

We propose a discrete time branching process to model the appearance of drug resistance under treatment. Under our assumptions at every discrete time a pathogen may die with probability 1−p or divide in two with probability p. Each newborn pathogen is drug resistant with probability μ. We start with N drug sensitive pathogens and with no drug resistant pathogens. We declare the treatment successful if all pathogens are eradicated before drug resistance appears. The model predicts that success is possible only if p<1/2. Even in this case the probability of success decreases exponentially with the parameter m=μN. In particular, even with a very potent drug (i.e. p very small) drug resistance is likely if m is large.