Parametric dependence in model epidemics. I: contact-related parameters

One of the interesting properties of nonlinear dynamical systems is that arbitrarily small changes in parameter values can induce qualitative changes in behavior. The changes are called bifurcations, and they are typically visualized by plotting asymptotic dynamics against a parameter. In some cases, the resulting bifurcation diagram is unique: irrespective of initial conditions, the same dynamical sequence obtains. In other cases, initial conditions do matter, and there are coexisting sequences. Here we study an epidemiological model in which multiple bifurcation sequences yield to a single sequence in response to varying a second parameter. We call this simplification the emergence of unique parametric dependence (UPD) and discuss how it relates to the model's overall response to parameters. In so doing, we tie together a number of threads that have been developing since the mid-1980s. These include period-doubling; subharmonic resonance, attractor merging and subduction and the evolution of strange invariant sets. The present paper focuses on contact related parameters. A follow-up paper, to be published in this journal, will consider the effects of non-contact related parameters.     

Multiparametric bifurcations of an epidemiological model with strong Allee effect

In this paper we completely study bifurcations of an epidemic model with five parameters introduced by Hilker et al. (Am Nat 173:72–88, 2009), which describes the joint interplay of a strong Allee effect and infectious diseases in a single population. Existence of multiple positive equilibria and all kinds of bifurcation are examined as well as related dynamical behavior. It is shown that the model undergoes a series of bifurcations such as saddle-node bifurcation, pitchfork bifurcation, Bogdanov–Takens bifurcation, degenerate Hopf bifurcation of codimension two and degenerate elliptic type Bogdanov–Takens bifurcation of codimension three. Respective bifurcation surfaces in five-dimensional parameter spaces and related dynamical behavior are obtained. These theoretical conclusions confirm their numerical simulations and conjectures by Hilker et al., and reveal some new bifurcation phenomena which are not observed in Hilker et al. (Am Nat 173:72–88, 2009). The rich and complicated dynamics exhibit that the model is very sensitive to parameter perturbations, which has important implications for disease control of endangered species.     

Dynamical behavior of epidemiological models with nonlinear incidence rates

Epidemiological models with nonlinear incidence rates I ^pS^qshow a much wider range of dynamical behaviors than do those with bilinear incidence rates IS. These behaviors are determined mainly by p and , and secondarily by q. For such models, there may exist multiple attractive basins in phase space; thus whether or not the disease will eventually die out may depend not only upon the parameters, but also upon the initial conditions. In some cases, periodic solutions may appear by Hopf bifurcation at critical parameter values.     

Parametric dependence in model epidemics. II: Non-contact rate-related parameters

In a previous paper, we discussed the bifurcation structure of SEIR equations subject to seasonality. There, the focus was on parameters that affect transmission: the mean contact rate, β₀, and the magnitude of seasonality, ? _B . Using numerical continuation and brute force simulation, we characterized a global pattern of parametric dependence in terms of subharmonic resonances and period-doublings of the annual cycle. In the present paper, we extend this analysis and consider the effects of varying non-contact-related parameters: periods of latency, infection and immunity, and rates of mortality and reproduction, which, following the usual practice, are assumed to be equal. The emergence of several new forms of dynamical complexity notwithstanding, the pattern previously reported is preserved. More precisely, the principal effect of varying non-contact related parameters is to displace bifurcation curves in the β₀?? _B parameter plane and to expand or contract the regions of resonance and period-doubling they delimit. Implications of this observation with respect to modeling real-world epidemics are considered.     

Transient oscillations induced by delayed growth response in the chemostat

In this paper, in order to try to account for the transient oscillations observed in chemostat experiments, we consider a model of single species growth in a chemostat that involves delayed growth response. The time delay models the lag involved in the nutrient conversion process. Both monotone response functions and nonmonotone response functions are considered. The nonmonotone response function models the inhibitory effects of growth response of certain nutrients when concentrations are too high. By applying local and global Hopf bifurcation theorems, we prove that the model has unstable periodic solutions that bifurcate from unstable nonnegative equilibria as the parameter measuring the delay passes through certain critical values and that these local periodic solutions can persist, even if the delay parameter moves far from the critical (local) bifurcation values.When there are two positive equilibria, then positive periodic solutions can exist. When there is a unique positive equilibrium, the model does not have positive periodic oscillations and the unique positive equilibrium is globally asymptotically stable. However, the model can have periodic solutions that change sign. Although these solutions are not biologically meaningful, provided the initial data starts close enough to the unstable manifold of one of these periodic solutions they may still help to account for the transient oscillations that have been frequently observed in chemostat experiments. Numerical simulations are provided to illustrate that the model has varying degrees of transient oscillatory behaviour that can be controlled by the choice of the initial data.Mathematics Subject Classification: 34D20, 34K20, 92D25Research was partially supported by NSERC of Canada.This work was partly done while this author was a postdoc at McMaster.     

Predator–prey system with strong Allee effect in prey

Global bifurcation analysis of a class of general predator–prey models with a strong Allee effect in prey population is given in details. We show the existence of a point-to-point heteroclinic orbit loop, consider the Hopf bifurcation, and prove the existence/uniqueness and the nonexistence of limit cycle for appropriate range of parameters. For a unique parameter value, a threshold curve separates the overexploitation and coexistence (successful invasion of predator) regions of initial conditions. Our rigorous results justify some recent ecological observations, and practical ecological examples are used to demonstrate our theoretical work.     

Dynamics and bifurcations of the adaptive exponential integrate-and-fire model

Recently, several two-dimensional spiking neuron models have been introduced, with the aim of reproducing the diversity of electrophysiological features displayed by real neurons while keeping a simple model, for simulation and analysis purposes. Among these models, the adaptive integrate-and-fire model is physiologically relevant in that its parameters can be easily related to physiological quantities. The interaction of the differential equations with the reset results in a rich and complex dynamical structure. We relate the subthreshold features of the model to the dynamical properties of the differential system and the spike patterns to the properties of a Poincaré map defined by the sequence of spikes. We find a complex bifurcation structure which has a direct interpretation in terms of spike trains. For some parameter values, spike patterns are chaotic.     

Steady-state solutions in mathematical models of atrial cell electrophysiology and their stability

The steady states of the Fenton-Karma, the Courtemanche and the Nygren cell models were studied by determining the fixed points of the dynamical system describing their cell kinetics. The linear stability of the fixed points was investigated, as well as their response to external stimuli. Symbolic calculations were carried out as far as possible in order to prove the existence of these fixed points. In the Fenton-Karma model, a unique stable fixed point was found, namely the resting state. In contrast, the Courtemanche model had an infinite number of fixed points. A bifurcation diagram was constructed by classifying these fixed points according to a conservation law. Initial conditions were identified, for which the dynamical behavior of the cell was auto-oscillatory. In its original formulation, the Nygren model had no fixed point. After having restored charge conservation, the system was found to have an infinite number of fixed points, resulting in a bifurcation diagram similar to that of the Courtemanche model. The approach proposed in this paper assists in the exploration of the high-dimensional parameter space of the cell models and the identification of the conditions leading to spontaneous pacemaker activity.     

Stability and Hopf bifurcation for a five-dimensional virus infection model with Beddington–DeAngelis incidence and three delays

In this paper, the dynamical behaviours for a five-dimensional virus infection model with three delays which describes the interactions of antibody, cytotoxic T-lymphocyte (CTL) immune responses and Beddington–DeAngelis incidence are investigated. The reproduction numbers for virus infection, antibody immune response, CTL immune response, CTL immune competition and antibody immune competition, respectively, are calculated. By using the Lyapunov functionals and linearization method, the threshold conditions on the local and global stability of the equilibria for infection-free, immune-free, antibody response, CTL response and interior, respectively, are established. The existence of Hopf bifurcation with immune delay as a bifurcation parameter is investigated by using the bifurcation theory. Numerical simulations are presented to justify the analytical results.     

Bifurcation analysis of a neural network model

This paper describes the analysis of the well known neural network model by Wilson and Cowan. The neural network is modeled by a system of two ordinary differential equations that describe the evolution of average activities of excitatory and inhibitory populations of neurons. We analyze the dependence of the model's behavior on two parameters. The parameter plane is partitioned into regions of equivalent behavior bounded by bifurcation curves, and the representative phase diagram is constructed for each region. This allows us to describe qualitatively the behavior of the model in each region and to predict changes in the model dynamics as parameters are varied. In particular, we show that for some parameter values the system can exhibit long-period oscillations. A new type of dynamical behavior is also found when the system settles down either to a stationary state or to a limit cycle depending on the initial point.     

Biomass and biodiversity in species-rich tritrophic communities

We consider a tritrophic system with one basal and one top species and a large number of primary consumers, and derive upper and lower bounds for the total biomass of the middle trophic level. These estimates do not depend on dynamical regime, holding for fixed point, periodic, or chaotic dynamics. We have two kinds of estimates, depending on whether the predator abundance is zero. All these results are uniform in a self-limitation parameter, which regulates prey diversity in the system. For strong self-limitation, diversity is large; for weak self-limitation, it is small. Diversity depends on the variance of species’ parameter values. The larger this variance, the lower the diversity, and vice versa. Moreover, variation in the parameters of the Holling type II functional response changes the bifurcation character, with the equilibrium state with nonzero predator abundance losing stability. If that variation is small then the bifurcation can lead to oscillations (the Hopf bifurcation). Under certain conditions, there exists a supercritical Hopf bifurcation. We then find a connection between diversity and Hopf bifurcations. We also show that the system exhibits top-down regulation and a hump-shaped diversity-productivity curve.We then extend the model by allowing species to experience self-regulation. For this extended model, explicit estimates of prey diversity are obtained. We study the dynamics of this system and find the following. First, diversity and system dynamics crucially depend on variation in species parameters. We show that under certain conditions, the system undergoes a supercritical Hopf bifurcation. We also establish a connection between diversity and Hopf bifurcations. For strong self-limitation, diversity is large and complex dynamics are absent. For weak self-limitation, diversity is small and the equilibrium with non-zero predator abundance is unstable.     

Parameter inference for biochemical systems that undergo a Hopf bifurcation

The increasingly widespread use of parametric mathematical models to describe biological systems means that the ability to infer model parameters is of great importance. In this study, we consider parameter inferability in nonlinear ordinary differential equation models that undergo a bifurcation, focusing on a simple but generic biochemical reaction model. We systematically investigate the shape of the likelihood function for the model's parameters, analyzing the changes that occur as the model undergoes a Hopf bifurcation. We demonstrate that there exists an intrinsic link between inference and the parameters’ impact on the modeled system's dynamical stability, which we hope will motivate further research in this area.     

Modeling the dynamics of natural rotifer populations: Phase-parametric analysis

A model of the dynamics of natural rotifer populations is described as a discrete non-linear map depending on three parameters, which reflect characteristics of the population and environment. Model dynamics and their change by variation of these parameters were investigated by methods of bifurcation theory. A phase-parametric portrait of the model was constructed and domains of population persistence (stable equilibrium, periodic and a-periodic oscillations of population size) as well as population extinction were identified and investigated. The criteria for population persistence and approaches to determining critical parameter values are described. The results identify parameter values that lead to population extinction under various environmental conditions. They further illustrate that the likelihood of extinction can be substantially increased by small changes in environmental quality, which shifts populations into new dynamical regimes.     

Periodic metabolic systems: Oscillations in multiple-loop negative feedback biochemical control networks

Summary For a general multiple loop feedback inhibition system in which the end product can inhibit any or all of the intermediate reactions it is shown that biologically significant behaviour is always confined to a bounded region of reaction space containing a unique equilibrium. By explicit construction of a Liapunov function for the general n dimensional differential equation it is shown that some values of reaction parameters cause the concentration vector to approach the equilibrium asymptotically for all physically realizable initial conditions. As the parameter values change, periodic solutions can appear within the bounded region. Some information about these periodic solutions can be obtained from the Hopf bifurcation theorem. Alternatively, if specific parameter values are known a numerical method can be used to find periodic solutions and determine their stability by locating a zero of the displacement map. The single loop Goodwin oscillator is analysed in detail. The methods are then used to treat an oscillator with two feedback loops and it is found that oscillations are possible even if both Hill coefficients are equal to one.     

Estimating the Stochastic Bifurcation Structure of Cellular Networks

High throughput measurement of gene expression at single-cell resolution, combined with systematic perturbation of environmental or cellular variables, provides information that can be used to generate novel insight into the properties of gene regulatory networks by linking cellular responses to external parameters. In dynamical systems theory, this information is the subject of bifurcation analysis, which establishes how system-level behaviour changes as a function of parameter values within a given deterministic mathematical model. Since cellular networks are inherently noisy, we generalize the traditional bifurcation diagram of deterministic systems theory to stochastic dynamical systems. We demonstrate how statistical methods for density estimation, in particular, mixture density and conditional mixture density estimators, can be employed to establish empirical bifurcation diagrams describing the bistable genetic switch network controlling galactose utilization in yeast Saccharomyces cerevisiae. These approaches allow us to make novel qualitative and quantitative observations about the switching behavior of the galactose network, and provide a framework that might be useful to extract information needed for the development of quantitative network models.     

Fitness landscapes for coupled map lattices

Our goal is to match some dynamical aspects of biological systems with that of networks of coupled logistic maps. With these networks we generate sequences of iterates, convert them to symbol sequences by coarse-graining, and count the number of times combinations of symbols occur. Comparison of this with the number of times these combinations occur in experimental data—a sequence of interbeat intervals for example—is a measure of the fitness of each network to describe the target data. The most fit networks provide a cartoon that suggests a decomposition of the experimental data into a component that may be produced by a simple dynamical subsystem, and a residual component, the result of detailed, particular characteristics of the system that generated the target data. In the space of all network parameters, each point corresponds to a particular network. We construct a fitness landscape when we assign a fitness to each point. Because the parameters are distributed continuously over their ranges, and because fitnesses are estimated numerically, any plot of the landscape involves a finite sample of parameter values. We’ll investigate how the local landscape geometry changes when the array of sample parameters is refined, and use the landscape geometry to explore complex relations between local fitness maxima.     

一类具分段常数变量的捕食-食饵系统的Neimark-Sacker分支

讨论了具时滞与分段常数变量的捕食-食饵生态模型的稳定性及Neimark-Sacker分支;通过计算得到连续模型对应的差分模型,基于特征值理论和Schur-Cohn判据得到正平衡态局部渐进稳定的充分条件;以食饵的内禀增长率为分支参数,运用分支理论和中心流形定理分析了Neimark-Sacker分支的存在性与稳定性条件;通过举例和数值模拟验证了理论的正确性。     

Organization of the Euplotes crassus Micronuclear Genome1

Euplotes crassus, like other hypotrichous ciliated protozoa, eliminates most of its micronuclear chromosomal DNA in the process of forming the small linear DNA molecules that comprise the macronuclear genome. By characterizing randomly selected lambda phage clones of E. crassus micronuclear DNA, we have determined the distribution of repetitive and unique sequences and the arrangement of macronuclear genes relative to eliminated DNA. This allows us to compare the E. crassus micronuclear genome organization to that of another distantly related hypotrichous ciliate, Oxytricha nova. The clones from E. crassus segregate into three prevalent classes: those containing primarily eliminated repetitive DNA (Class I); those containing macronuclear genes in addition to repetitive sequences (Class II); and those containing only eliminated unique sequence DNA (Class III). All of the repetitive sequences in these clones belong to the same highly abundant repetitive element family. Our results demonstrate that the sequence organization of the E. crassus and O. nova micronuclear genomes is related in that the macronuclear genes are clustered together in the micronuclear genome and the eliminated unique sequences occur in long stretches that are uninterrupted by repetitive sequences. In both organisms a single repetitive element family comprises the majority of the eliminated interspersed middle repetitive DNA and appears to be preferentially associated with the macronuclear sequence clusters. The similarities in the sequence organization in these two organisms suggest that clustering of macronuclear genes plays a role in the chromosome fragmentation process.     

Global analysis of competition for perfectly substitutable resources with linear response

We study a model of the chemostat with two species competing for two perfectly substitutable resources in the case of linear functional response. Lyapunov methods are used to provide sufficient conditions for the global asymptotic stability of the coexistence equilibrium. Then, using compound matrix techniques, we provide a global analysis in a subset of parameter space. In particular, we show that each solution converges to an equilibrium, even in the case that the coexistence equilibrium is a saddle. Finally, we provide a bifurcation analysis based on the dilution rate. In this context, we are able to provide a geometric interpretation that gives insight into the role of the other parameters in the bifurcation sequence. Funding was provided by the National Science Foundation-funded ADVANCE Institutional Transformation Program at New Mexico State University, fund # NSF0123690. Research partially supported by the Natural Science and Engineering Research Council of Canada.     

Robustness analysis and behavior discrimination in enzymatic reaction networks

Characterizing the behavior and robustness of enzymatic networks with numerous variables and unknown parameter values is a major challenge in biology, especially when some enzymes have counter-intuitive properties or switch-like behavior between activation and inhibition. In this paper, we propose new methodological and tool-supported contributions, based on the intuitive formalism of temporal logic, to express in a rigorous manner arbitrarily complex dynamical properties. Our multi-step analysis allows efficient sampling of the parameter space in order to define feasible regions in which the model exhibits imposed or experimentally observed behaviors. In a first step, an algorithmic methodology involving sensitivity analysis is conducted to determine bifurcation thresholds for a limited number of model parameters or initial conditions. In a second step, this boundary detection is supplemented by a global robustness analysis, based on quasi-Monte Carlo approach that takes into account all model parameters. We apply this method to a well-documented enzymatic reaction network describing collagen proteolysis by matrix metalloproteinase MMP2 and membrane type 1 metalloproteinase (MT1-MMP) in the presence of tissue inhibitor of metalloproteinase TIMP2. For this model, our method provides an extended analysis and quantification of network robustness toward paradoxical TIMP2 switching activity between activation or inhibition of MMP2 production. Further implication of our approach is illustrated by demonstrating and analyzing the possible existence of oscillatory behaviors when considering an extended open configuration of the enzymatic network. Notably, we construct bifurcation diagrams that specify key parameters values controlling the co-existence of stable steady and non-steady oscillatory proteolytic dynamics.