Analysis of the Trojan Y-Chromosome eradication strategy for an invasive species

The Trojan Y-Chromosome (TYC) strategy, an autocidal genetic biocontrol method, has been proposed to eliminate invasive alien species. In this work, we analyze the dynamical system model of the TYC strategy, with the aim of studying the viability of the TYC eradication and control strategy of an invasive species. In particular, because the constant introduction of sex-reversed trojan females for all time is not possible in practice, there arises the question: What happens if this injection is stopped after some time? Can the invasive species recover? To answer that question, we perform a rigorous bifurcation analysis and study the basin of attraction of the recovery state and the extinction state in both the full model and a certain reduced model. In particular, we find a theoretical condition for the eradication strategy to work. Additionally, the consideration of an Allee effect and the possibility of a Turing instability are also studied in this work. Our results show that: (1) with the inclusion of an Allee effect, the number of the invasive females is not required to be very low when the introduction of the sex-reversed trojan females is stopped, and the remaining Trojan Y-Chromosome population is sufficient to induce extinction of the invasive females; (2) incorporating diffusive spatial spread does not produce a Turing instability, which would have suggested that the TYC eradication strategy might be only partially effective, leaving a patchy distribution of the invasive species.     

A stochastic model for transmission,extinction and outbreak of Escherichia coli O157:H7 in cattle as affected by ambient temperature and cleaning practices

Many infectious agents transmitting through a contaminated environment are able to persist in the environment depending on the temperature and sanitation determined rates of their replication and clearance, respectively. There is a need to elucidate the effect of these factors on the infection transmission dynamics in terms of infection outbreaks and extinction while accounting for the random nature of the process. Also, it is important to distinguish between the true and apparent extinction, where the former means pathogen extinction in both the host and the environment while the latter means extinction only in the host population. This study proposes a stochastic-differential equation model as an approximation to a Markov jump process model, using Escherichia coli O157:H7 in cattle as a model system. In the model, the host population infection dynamics are described using the standard susceptible-infected-susceptible framework, and the E. coli O157:H7 population in the environment is represented by an additional variable. The backward Kolmogorov equations that determine the probability distribution and the expectation of the first passage time are provided in a general setting. The outbreak and apparent extinction of infection are investigated by numerically solving the Kolmogorov equations for the probability density function of the associated process and the expectation of the associated stopping time. The results provide insight into E. coli O157:H7 transmission and apparent extinction, and suggest ways for controlling the spread of infection in a cattle herd. Specifically, this study highlights the importance of ambient temperature and sanitation, especially during summer.     

A neutral model of edge effects

In this paper a spatially implicit neutral model for explaining the edge effects between habitats is proposed. To analyze this model we use two different approaches: a discrete approach that is based on the Master equation for a one step jump process and a continuous approach based on the approximation of the discrete jump process with the Kolmogorov-Fokker-Planck forward and backward equations. The discrete and continuous approaches are applied to analyze the species abundance distributions and the time to species extinction. Moreover, with the aid of the continuous approach a realistic classification of the behavior of species in local communities is developed. The species abundance dynamics at the edge between two distinct habitats is compared with those located in the homogeneous interior habitats using species abundance distributions and the first time to species extinction. We show that the structure of the links between local community and the metacommunity plays an important role on species persistence. Specifically, species at the edge between two distinct metacommunities have higher extinction rate than those in the interior habitats connected only to one metacommunity. Moreover, the same species might be persistent in the homogeneous interior habitat, but its probability of extinction from the edge local community could be very high.     

CENTRAL MOMENTS AND PROBABILITY DISTRIBUTION OF COLLESS'S COEFFICIENT OF TREE IMBALANCE

The great increase in the number of phylogenetic studies of a wide variety of organisms in recent decades has focused considerable attention on the balance of phylogenetic trees—the degree to which sister clades within a tree tend to be of equal size—for at least two reasons: (1) the degree of balance of a tree may affect the accuracy of estimates of it; (2) the degree of balance, or imbalance, of a tree may reveal something about the macroevolutionary processes that produced it. In particular, variation among lineages in rates of speciation or extinction is expected to produce trees that are less balanced than those that result from phylogenetic evolution in which each extant species of a group has the same probability of speciation or extinction. Several coefficients for measuring the balance or imbalance of phylogenetic trees have been proposed. I focused on Colless's coefficient of imbalance (7) for its mathematical tractability and ease of interpretation. Earlier work on this statistic produced exact methods only for calculating the expected value. In those studies, the variance and confidence limits, which are necessary for testing the departure of observed values of I from the expected, were estimated by Monte Carlo simulation. I developed recursion equations that allow exact calculation of the mean, variance, skewness, and complete probability distribution of I for two different probability-generating models for bifurcating tree shapes. The Equal-Rates Markov (ERM) model assumes that trees grow by the random speciation and extinction of extant species, with all species that are extant at a given time having the same probability of speciation or extinction. The Equal Probability (EP) model assumes that all possible labeled trees for a given number of terminal taxa have the same probability of occurring. Examples illustrate how these theoretically derived probabilities and parameters may be used to test whether the evolution of a monophyletic group or set of monophyletic groups has proceeded according to a Markov model with equal rates of speciation and extinction among species, that is, whether there has been significant variation among lineages in expected rates of speciation or extinction.     

Stochastic models of tumor growth and the probability of elimination by cytotoxic cells

The probability of tumor extinction due to the action of cytotoxic cell populations is investigated by several one dimensional stochastic models of the population growth and elimination processes of a tumor. The several models are made necessary by the nonlinearity of the processes and the different parameter ranges explored. The deterministic form of the model is where γ₀, k′₆ and k ₁ are positive constants. The parameter of most import is which determines the stability of the T = 0 equilibrium. With an initial tumor size of one, a (linear) branching process is used to estimate the extinction probability. However, in the case λ = 0 when the linearization of the deterministic model gives no information (T = 0 is actually unstable) the branching model is unsatisfactory. This makes necessary the utilization of a density-dependent branching process to approximate the population. Through scaling a diffusion limit is reached which enables one to again compute the probability of extinction. For populations away from one a sequence of density-dependent jump Markov processes are approximated by a sequence of diffusion processes. In limiting cases, the estimates of extinction correspond to that computed from the original branching process. Table 1 summarizes the results.     

Reversibility and the age of an allele II. Two-allele models,with selection and mutation

A Markov process with absorbing boundaries may be made recurrent by returning the process to the interior whenever a boundary is reached. The age of such a process may be defined as the length of time since the last return event. Examples drawn from two-allele genetic models are discussed, in which reversibility of the return process means that the age of an allele, whose present frequency in the population is known, has the same probability distribution as its future extinction time. Some discrete models are not reversible, yet if approximated by diffusion processes, the (approximate) age distribution is the same as the future extinction time distribution. Various results in the literature are unified by this viewpoint.     

Alternative Predictions for Optimal Dispersal in Response to Local Catastrophic Mortality

What dispersal strategy should be employed by an organism in response to local catastrophic mortality? Here we contrast predictions from an analytical solution derived from an ESS model which optimizes competitive ability (Comins et al., 1980) with those from a stochastic, branching process model (Karlson and Taylor, 1992) which maximizes survivorship of a clonal lineage. The optimal dispersal fraction varies directly with the probability of local extinction in the ESS model, yet varies inversely with this probability over much of the parameter space in the latter model. In order to conform more closely with the assumptions of the ESS model, we have modified the branching process model to have a random, Poisson-distributed number of offspring and compared the predictions of these models. Both models invoke dispersal as an escape from local extinction and predict mixed dispersal strategies over a wide range of conditions. However, increasing local catastrophic mortality favors more dispersal in the ESS model, but it can be so severe in the branching process model that no dispersal strategy is adaptive. In this model, the predicted optimal proportion of dispersed offspring is highest at low to intermediate levels of catastrophic mortality depending on the total number of offspring produced. We suggest that this observed discrepancy is sufficiently large to warrant empirical tests of these qualitatively different predictions.     

Analysis of the Trojan Y chromosome model for eradication of invasive species in a dendritic riverine system

The use of Trojan Y chromosomes has been proposed as a genetic strategy for the eradication of invasive species. The strategy is particularly relevant to invasive fish species that have XY sex determination system and are amenable to sex-reversal. In this paper we study the dynamics of an invasive fish population occupying a dendritic domain in which Trojan individuals bearing multiple Y chromosomes have been released as a means of eradication. We demonstrate the existence of a bounded absorbing set that represents extinction of the invasive species irrespective of the dendritic configuration. The method of analysis used to obtain global estimates could be applied to other population problems and other geometries.     

On real-valued SDE and nonnegative-valued SDE population models with demographic variability

Population dynamics with demographic variability is frequently studied using discrete random variables with continuous-time Markov chain (CTMC) models. An approximation of a CTMC model using continuous random variables can be derived in a straightforward manner by applying standard methods based on the reaction rates in the CTMC model. This leads to a system of Itô stochastic differential equations (SDEs) which generally have the form \( d \mathbf {y} = \varvec{\mu } \, dt + G \, d \mathbf {W},\) where \(\mathbf {y}\) is the population vector of random variables, \(\varvec{\mu }\) is the drift vector, and G is the diffusion matrix. In some problems, the derived SDE model may not have real-valued or nonnegative solutions for all time. For such problems, the SDE model may be declared infeasible. In this investigation, new systems of SDEs are derived with real-valued solutions and with nonnegative solutions. To derive real-valued SDE models, reaction rates are assumed to be nonnegative for all time with negative reaction rates assigned probability zero. This biologically realistic assumption leads to the derivation of real-valued SDE population models. However, small but negative values may still arise for a real-valued SDE model. This is due to the magnitudes of certain problem-dependent diffusion coefficients when population sizes are near zero. A slight modification of the diffusion coefficients when population sizes are near zero ensures that a real-valued SDE model has a nonnegative solution, yet maintains the integrity of the SDE model when sizes are not near zero. Several population dynamic problems are examined to illustrate the methodology.

     

A comparison of the Trojan Y Chromosome and daughterless carp eradication strategies

Two autocidal genetic biocontrol methods have been proposed as a means to eliminate invasive fish by changing the sex ratio of the population: the Trojan Y Chromosome (TYC) strategy and the Daughterless Carp (DC) strategy. Both strategies were modeled using ordinary differential equations that allow the kinetics of female decline to be assessed under identical modeling conditions. When compared directly in an ordinary differential equation (ODE) model, the TYC strategy was found to result in female extinction more rapidly than a DC strategy (in each of three models tested in which the Daughterless autocidal fish contained an aromatase inhibitor gene in either two or eight copies). The TYC strategy additionally required the introduction of fewer autocidal fish to the target population to achieve local extinction of females as compared to the DC approach. The results suggest that the relatively lower efficiency of female reduction associated with the DC approach is a consequence of a greater capacity to produce females and also a reduced capacity to produce males as compared to the TYC system.     

Probability distribution of molecular evolutionary trees: A new method of phylogenetic inference

A new method is presented for inferring evolutionary trees using nucleotide sequence data. The birth-death process is used as a model of speciation and extinction to specify the prior distribution of phylogenies and branching times. Nucleotide substitution is modeled by a continuous-time Markov process. Parameters of the branching model and the substitution model are estimated by maximum likelihood. The posterior probabilities of different phylogenies are calculated and the phylogeny with the highest posterior probability is chosen as the best estimate of the evolutionary relationship among species. We refer to this as the maximum posterior probability (MAP) tree. The posterior probability provides a natural measure of the reliability of the estimated phylogeny. Two example data sets are analyzed to infer the phylogenetic relationship of human, chimpanzee, gorilla, and orangutan. The best trees estimated by the new method are the same as those from the maximum likelihood analysis of separate topologies, but the posterior probabilities are quite different from the bootstrap proportions. The results of the method are found to be insensitive to changes in the rate parameter of the branching process. Correspondence to: Z. Yang     

Extinction models for cancer stem cell therapy

Cells with stem cell-like properties are now viewed as initiating and sustaining many cancers. This suggests that cancer can be cured by driving these cancer stem cells to extinction. The problem with this strategy is that ordinary stem cells are apt to be killed in the process. This paper sets bounds on the killing differential (difference between death rates of cancer stem cells and normal stem cells) that must exist for the survival of an adequate number of normal stem cells. Our main tools are birth-death Markov chains in continuous time. In this framework, we investigate the extinction times of cancer stem cells and normal stem cells. Application of extreme value theory from mathematical statistics yields an accurate asymptotic distribution and corresponding moments for both extinction times. We compare these distributions for the two cell populations as a function of the killing rates. Perhaps a more telling comparison involves the number of normal stem cells N_H at the extinction time of the cancer stem cells. Conditioning on the asymptotic time to extinction of the cancer stem cells allows us to calculate the asymptotic mean and variance of N_H. The full distribution of N_H can be retrieved by the finite Fourier transform and, in some parameter regimes, by an eigenfunction expansion. Finally, we discuss the impact of quiescence (the resting state) on stem cell dynamics. Quiescence can act as a sanctuary for cancer stem cells and imperils the proposed therapy. We approach the complication of quiescence via multitype branching process models and stochastic simulation. Improvements to the τ-leaping method of stochastic simulation make it a versatile tool in this context. We conclude that the proposed therapy must target quiescent cancer stem cells as well as actively dividing cancer stem cells. The current cancer models demonstrate the virtue of attacking the same quantitative questions from a variety of modeling, mathematical, and computational perspectives.     

Conceptual and statistical problems with the DEC+J model of founder‐event speciation and its comparison with DEC via model selection

Phylogenetic studies of geographic range evolution are increasingly using statistical model selection methods to choose among variants of the dispersal‐extinction‐cladogenesis (DEC) model, especially between DEC and DEC+J, a variant that emphasizes “jump dispersal,” or founder‐event speciation, as a type of cladogenetic range inheritance scenario. Unfortunately, DEC+J is a poor model of founder‐event speciation, and statistical comparisons of its likelihood with DEC are inappropriate. DEC and DEC+J share a conceptual flaw: cladogenetic events of range inheritance at ancestral nodes, unlike anagenetic events of dispersal and local extinction along branches, are not modelled as being probabilistic with respect to time. Ignoring this probability factor artificially inflates the contribution of cladogenetic events to the likelihood, and leads to underestimates of anagenetic, time‐dependent range evolution. The flaw is exacerbated in DEC+J because not only is jump dispersal allowed, expanding the set of cladogenetic events, its probability relative to non‐jump events is assigned a free parameter, j, that when maximized precludes the possibility of non‐jump events at ancestral nodes. DEC+J thus parameterizes the mode of speciation, but like DEC, it does not parameterize the rate of speciation. This inconsistency has undesirable consequences, such as a greater tendency towards degenerate inferences in which the data are explained entirely by cladogenetic events (at which point branch lengths become irrelevant, with estimated anagenetic rates of 0). Inferences with DEC+J can in some cases depart dramatically from intuition, e.g. when highly unparsimonious numbers of jump dispersal events are required solely because j is maximized. Statistical comparison with DEC is inappropriate because a higher DEC+J likelihood does not reflect a more close approximation of the “true” model of range evolution, which surely must include time‐dependent processes; instead, it is simply due to more weight being allocated (via j) to jump dispersal events whose time‐dependent probabilities are ignored. In testing hypotheses about the geographic mode of speciation, jump dispersal can and should instead be modelled using existing frameworks for state‐dependent lineage diversification in continuous time, taking appropriate cautions against Type I errors associated with such methods. For simple inference of ancestral ranges on a fixed phylogeny, a DEC‐based model may be defensible if statistical model selection is not used to justify the choice, and it is understood that inferences about cladogenetic range inheritance lack any relation to time, normally a fundamental axis of evolutionary models.     

Coalescent process with fluctuating population size and its effective size

We consider a Wright-Fisher model whose population size is a finite Markov chain. We introduce a sequence of two-dimensional discrete time Markov chains whose components describe the coalescent process and the fluctuation of population size. For the limiting process of the sequence of Markov chains, the relationship of the expectation of coalescence time to the harmonic and the arithmetic means of population sizes is shown, and the Laplace transform of the distribution of coalescence time is calculated. We define the coalescence effective population size (cEPS) by the expectation of coalescence time. We show that cEPS is strictly larger (resp. smaller) than the harmonic (resp. arithmetic) mean. As the population size fluctuates more quickly (resp. slowly), cEPS is closer to the harmonic (resp. arithmetic) mean. For the case of a two-valued Markov chain, we show the explicit expression of cEPS and its dependency on the sample size.     

Bayesian Analysis of Growth Curves Using Mixed Models Defined by Stochastic Differential Equations

Summary Growth curve data consist of repeated measurements of a continuous growth process over time in a population of individuals. These data are classically analyzed by nonlinear mixed models. However, the standard growth functions used in this context prescribe monotone increasing growth and can fail to model unexpected changes in growth rates. We propose to model these variations using stochastic differential equations (SDEs) that are deduced from the standard deterministic growth function by adding random variations to the growth dynamics. A Bayesian inference of the parameters of these SDE mixed models is developed. In the case when the SDE has an explicit solution, we describe an easily implemented Gibbs algorithm. When the conditional distribution of the diffusion process has no explicit form, we propose to approximate it using the Euler–Maruyama scheme. Finally, we suggest validating the SDE approach via criteria based on the predictive posterior distribution. We illustrate the efficiency of our method using the Gompertz function to model data on chicken growth, the modeling being improved by the SDE approach.     

Species assembly in model ecosystems, II: Results of the assembly process

In the companion paper of this set (Capitán and Cuesta, 2010) we have developed a full analytical treatment of the model of species assembly introduced in Capitán et al. (2009). This model is based on the construction of an assembly graph containing all viable configurations of the community, and the definition of a Markov chain whose transitions are the transformations of communities by new species invasions. In the present paper we provide an exhaustive numerical analysis of the model, describing the average time to the recurrent state, the statistics of avalanches, and the dependence of the results on the amount of available resource. Our results are based on the fact that the Markov chain provides an asymptotic probability distribution for the recurrent states, which can be used to obtain averages of observables as well as the time variation of these magnitudes during succession, in an exact manner. Since the absorption times into the recurrent set are found to be comparable to the size of the system, the end state is quickly reached (in units of the invasion time). Thus, the final ecosystem can be regarded as a fluctuating complex system where species are continually replaced by newcomers without ever leaving the set of recurrent patterns. The assembly graph is dominated by pathways in which most invasions are accepted, triggering small extinction avalanches. Through the assembly process, communities become less resilient (e.g., have a higher return time to equilibrium) but become more robust in terms of resistance against new invasions.     

Extinction dynamics in mainland-island metapopulations: an N-patch stochastic model

A generalization of the well-known Levins’ model of metapopulations is studied. The generalization consists of (i) the introduction of immigration from a mainland, and (ii) assuming the dynamics is stochastic, rather than deterministic. A master equation, for the probability that n of the patches are occupied, is derived and the stationary probability P _s (n), together with the mean and higher moments in the stationary state, determined. The time-dependence of the probability distribution is also studied: through a Gaussian approximation for general n when the boundary at n = 0 has little effect, and by calculating P(0, t), the probability that no patches are occupied at time t, by using a linearization procedure. These analytic calculations are supplemented by carrying out numerical solutions of the master equation and simulations of the stochastic process. The various approaches are in very good agreement with each other. This allows us to use the forms for P _s 0) and P(0, t) in the linearization approximation as a basis for calculating the mean time for a metapopulation to become extinct. We give an analytical expression for the mean time to extinction derived within a mean field approach. We devise a simple method to apply our mean field approach even to complex patch networks in realistic model metapopulations. After studying two spatially extended versions of this nonspatial metapopulation model—a lattice metapopulation model and a spatially realistic model—we conclude that our analytical formula for the mean extinction time is generally applicable to those metapopulations which are really endangered, where extinction dynamics dominates over local colonization processes. The time evolution and, in particular, the scope of our analytical results, are studied by comparing these different models with the analytical approach for various values of the parameters: the rates of immigration from the mainland, the rates of colonization and extinction, and the number of patches making up the metapopulation.     

Further contributions to the mathematical biophysics of automobile driving

The discussions of a previous paper (Bull. Math. Biophysics,21, 299–308, 1959) are generalized by considering that the angular direction error made by the driver, as well as the driver's reaction time are not constant but are randomly distributed. Instead of a critical speed, at which the car will jump off the road, we now find that for every speed there is a probability of the car to jump off the road but that this probability is vanishingly small for sufficiently low speeds, yet increases rapidly for high speeds. Thus a more realistic picture of the process of driving is obtained. When the standard deviation of the distribution functions for the angle and the reaction time are very small, the expression obtained here reduces to the expression obtained previously.     

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.     

Novel moment closure approximations in stochastic epidemics

Moment closure approximations are used to provide analytic approximations to non-linear stochastic population models. They often provide insights into model behaviour and help validate simulation results. However, existing closure schemes typically fail in situations where the population distribution is highly skewed or extinctions occur. In this study we address these problems by introducing novel second-and third-order moment closure approximations which we apply to the stochastic SI and SIS epidemic models. In the case of the SI model, which has a highly skewed distribution of infection, we develop a second-order approximation based on the beta-binomial distribution. In addition, a closure approximation based on mixture distribution is developed in order to capture the behaviour of the stochastic SIS model around the threshold between persistence and extinction. This mixture approximation comprises a probability distribution designed to capture the quasi-equilibrium probabilities of the system and a probability mass at 0 which represents the probability of extinction. Two third-order versions of this mixture approximation are considered in which the log-normal and the beta-binomial are used to model the quasi-equilibrium distribution. Comparison with simulation results shows: (1) the beta-binomial approximation is flexible in shape and matches the skewness predicted by simulation as shown by the stochastic SI model and (2) mixture approximations are able to predict transient and extinction behaviour as shown by the stochastic SIS model, in marked contrast with existing approaches. We also apply our mixture approximation to approximate a likehood function and carry out point and interval parameter estimation.