Species Coexistence and Periodicity in Host-Host-Pathogen Models

Models for the transmission of an infectious disease in one and two host populations with and without self-regulation are analyzed. Many unusual behaviors such as multiple positive equilibria and periodic solutions occur in previous models that use the mass-action (density-dependent) incidence. In contrast, the models formulated using the frequency-dependent (standard) incidence have the behavior of a classic endemic model, since below the threshold, the disease dies out, and above the threshold, the disease persists and the infectious fractions approach an endemic equilibrium. The results given here reinforce previous examples in which there are major differences in behavior between models using mass-action and frequency-dependent incidences.     

Generalized additive models for cancer mapping with incomplete covariates

Maps depicting cancer incidence rates have become useful tools in public health research, giving valuable information about the spatial variation in rates of disease. Typically, these maps are generated using count data aggregated over areas such as counties or census blocks. However, with the proliferation of geographic information systems and related databases, it is becoming easier to obtain exact spatial locations for the cancer cases and suitable control subjects. The use of such point data allows us to adjust for individual-level covariates, such as age and smoking status, when estimating the spatial variation in disease risk. Unfortunately, such covariate information is often subject to missingness. We propose a method for mapping cancer risk when covariates are not completely observed. We model these data using a logistic generalized additive model. Estimates of the linear and non-linear effects are obtained using a mixed effects model representation. We develop an EM algorithm to account for missing data and the random effects. Since the expectation step involves an intractable integral, we estimate the E-step with a Laplace approximation. This framework provides a general method for handling missing covariate values when fitting generalized additive models. We illustrate our method through an analysis of cancer incidence data from Cape Cod, Massachusetts. These analyses demonstrate that standard complete-case methods can yield biased estimates of the spatial variation of cancer risk.     

一个具有时滞的媒介传播流行病模型中的Hopt分支（英文）

文章研究的是一个具有时滞的媒介传播流行病模型.假定长期的发病率是双线性大规模行动的方式,确定了疾病是否流行的阈值R_0.当R_0≤1时,得到无病平衡点是全局稳定的,即疾病消失;当R_0〉1时,得到地方病平衡点.在具有时滞的微分模型中,时滞与载体转变成传染源的孵化期有关。我们研究了时滞对平衡点稳定性的影响,研究表明,在从寄生源到载体的传播过程中,时滞可以破坏动力系统并且得到了Hopt分支的周期解.     

Testing for concordant equilibria between population samples

A substantial body of theory has been developed to assess the effect of evolutionary forces on the distribution of genotypes, both single and multilocus, within populations. One area where the potential for application of this theory has not been fully appreciated concerns the extent to which population samples differ. Within populations, the divergence of genotype or haplotype frequencies from that expected under Hardy-Weinberg (HW) or linkage equilibrium can be measured as disequilibria coefficients. To assess population samples for concordant equilibria, an analytical framework for comparing disequilibria coefficients between populations is necessary. Here we present log-linear models to evaluate such hypotheses. These models have broad utility ranging from conventional population genetics to genetic epidemiology. We demonstrate the use of these log-linear models (1) as a test for genetic association with disease and (2) as a test for different levels of linkage disequilibria between human populations.     

Dispersal, disease and life-history evolution

Discrete-time susceptible-infective-susceptible (S-I-S) disease transmission models can exhibit bistability (alternative stable equilibria) over a wide range of parameter values. We illustrate the richness generated by such 'simple' non-linear systems in the study of two patch epidemic models with disease-enhanced or disease-suppressed dispersal. Dispersal between patches can have a profound impact on local patch disease dynamics. In fact, dispersal between patches may give rise to bistability in parameter regimes without bistability in single patch models.     

Bayesian analysis of a time series of counts with covariates: an application to the control of an infectious disease

This paper presents a Bayesian analysis of a time series of counts to assess its dependence on an explanatory variable. The time series represented is the incidence of the infectious disease ESBL-producing Klebsiella pneumoniae in an Australian hospital and the explanatory variable is the number of grams of antibiotic (third generation) cephalosporin used during that time. We demonstrate that there is a statistically significant relationship between disease occurrence and use of the antibiotic, lagged by three months. The model used is a parameter-driven model in the form of a generalized linear mixed model. Comparison of models is made in terms of mean square error.     

Avoidable errors in the modelling of outbreaks of emerging pathogens, with special reference to Ebola

As an emergent infectious disease outbreak unfolds, public health response is reliant on information on key epidemiological quantities, such as transmission potential and serial interval. Increasingly, transmission models fit to incidence data are used to estimate these parameters and guide policy. Some widely used modelling practices lead to potentially large errors in parameter estimates and, consequently, errors in model-based forecasts. Even more worryingly, in such situations, confidence in parameter estimates and forecasts can itself be far overestimated, leading to the potential for large errors that mask their own presence. Fortunately, straightforward and computationally inexpensive alternatives exist that avoid these problems. Here, we first use a simulation study to demonstrate potential pitfalls of the standard practice of fitting deterministic models to cumulative incidence data. Next, we demonstrate an alternative based on stochastic models fit to raw data from an early phase of 2014 West Africa Ebola virus disease outbreak. We show not only that bias is thereby reduced, but that uncertainty in estimates and forecasts is better quantified and that, critically, lack of model fit is more readily diagnosed. We conclude with a short list of principles to guide the modelling response to future infectious disease outbreaks.     

Computing the heteroclinic bifurcation curves in predator–prey systems with ratio-dependent functional response

Predator-prey models with Michaelis-Menten-Holling type ratio- dependent functional response exhibit very rich and complex dynamical behavior, such as the existence of degenerate equilibria, appearance of limit cycles and heteroclinic loops, and the coexistence of two attractive equilibria. In this paper, we study heteroclinic bifurcations of such a predator-prey model. We first calculate the higher order Melnikov functions by transforming the model into a Hamiltonian system and then provide an algorithm for computing higher order approximations of the heteroclinic bifurcation curves.     

Dynamic dissociating homo-oligomers and the control of protein function

Homo-oligomeric protein assemblies are known to participate in dynamic association/disassociation equilibria under native conditions, thus creating an equilibrium of assembly states. Such quaternary structure equilibria may be influenced in a physiologically significant manner either by covalent modification or by the non-covalent binding of ligands. This review follows the evolution of ideas about homo-oligomeric equilibria through the 20th and into the 21st centuries and the relationship of these equilibria to allosteric regulation by the non-covalent binding of ligands. A dynamic quaternary structure equilibria is described where the dissociated state can have alternate conformations that cannot reassociate to the original multimer; the alternate conformations dictate assembly to functionally distinct alternate multimers of finite stoichiometry. The functional distinction between different assemblies provides a mechanism for allostery. The requirement for dissociation distinguishes this morpheein model of allosteric regulation from the classical MWC concerted and KNF sequential models. These models are described alongside earlier dissociating allosteric models. The identification of proteins that exist as an equilibrium of diverse native quaternary structure assemblies has the potential to define new targets for allosteric modulation with significant consequences for further understanding and/or controlling protein structure and function. Thus, a rationale for identifying proteins that may use the morpheein model of allostery is presented and a selection of proteins for which published data suggests this mechanism may be operative are listed.     

Stability analysis and optimal vaccination of an SIR epidemic model

Almost all mathematical models of diseases start from the same basic premise: the population can be subdivided into a set of distinct classes dependent upon experience with respect to the relevant disease. Most of these models classify individuals as either a susceptible individual S, infected individual I or recovered individual R. This is called the susceptible-infected-recovered (SIR) model. In this paper, we describe an SIR epidemic model with three components; S, I and R. We describe our study of stability analysis theory to find the equilibria for the model. Next in order to achieve control of the disease, we consider a control problem relative to the SIR model. A percentage of the susceptible populations is vaccinated in this model. We show that an optimal control exists for the control problem and describe numerical simulations using the Runge-Kutta fourth order procedure. Finally, we describe a real example showing the efficiency of this optimal control.     

Global dispersal reduces local diversity

Metapopulation models and stepping-stone models in genetics are based on very different underlying dispersal structures, yet it can be difficult to distinguish the behaviour of the two kinds of models. We demonstrate a striking qualitative difference in the equilibrium behaviour possible with these two kinds of dispersal. If, in a local patch, there are multiple stable equilibria (and consequently an unstable equilibrium), we demonstrate that, for the spatial system with a metapopulation structure, at equilibrium every patch has to be near one of the stable equilibria. This contrasts with the clinal structure possible with a stepping-stone or continuous space model; thus the result can be used to deduce qualitative information about the form of dispersal from observations of allele frequencies.     

Multiplicity of stable states in freshwater systems

It is shown with the use of minimal models that several ecological relationships in freshwater systems potentially give rise to the existence of alternative equilibria over a certain range of nutrient values. The existence of alternative stable states has some implications for the management of such systems. An important consequence is that signs of eutrophication are only apparent after the occurrence of changes that are very difficult to reverse. Reduction of the nutrient level as a measure to restore such systems gives poor results, but biomanipulation as an additional measure can have significant effects, provided that the nutrient level has been reduced enough to allow the existence of a stable alternative clear water equilibrium.     

Coevolution to the edge of chaos: coupled fitness landscapes, poised states, and coevolutionary avalanches

We introduce a broadened framework to study aspects of coevolution based on the NK class of statistical models of rugged fitness landscapes. In these models the fitness contribution of each of N genes in a genotype depends epistatically on K other genes. Increasing epistatic interactions increases the rugged multipeaked character of the fitness landscape. Coevolution is thought of, at the lowest level, as a coupling of landscapes such that adaptive moves by one player deform the landscapes of its immediate partners. In these models we are able to tune the ruggedness of landscapes, how richly intercoupled any two landscapes are, and how many other players interact with each player. All these properties profoundly alter the character of the coevolutionary dynamics. In particular, these parameters govern how readily coevolving ecosystems achieve Nash equilibria, how stable to perturbations such equilibria are, and the sustained mean fitness of coevolving partners. In turn, this raises the possibility that an evolutionary metadynamics due to natural selection may sculpt landscapes and their couplings to achieve coevolutionary systems able to coadapt well. The results suggest that sustained fitness is optimized when landscape ruggedness relative to couplings between landscapes is tuned such that Nash equilibria just tenuously form across the ecosystem. In this poised state, coevolutionary avalanches appear to propagate on all length scales in a power law distribution. Such avalanches may be related to the distribution of small and large extinction events in the record.     

Modeling the Underlying Dynamics of the Spread of Crime

The spread of crime is a complex, dynamic process that calls for a systems level approach. Here, we build and analyze a series of dynamical systems models of the spread of crime, imprisonment and recidivism, using only abstract transition parameters. To find the general patterns among these parameters—patterns that are independent of the underlying particulars—we compute analytic expressions for the equilibria and for the tipping points between high-crime and low-crime equilibria in these models. We use these expressions to examine, in particular, the effects of longer prison terms and of increased incarceration rates on the prevalence of crime, with a follow-up analysis on the effects of a Three-Strike Policy.     

Maintenance of polygenic variation through mutation-selection balance: bifurcation analysis of a biallelic model

Biallelic models which ignore linkage disequilibrium have been used to study variability maintained by mutation in the presence of Gaussian stabilizing selection. Recent work of Barton (1986) showed that these models have stable equilibria at which the mean phenotype differed from the optimum, and that the variability maintained at such equilibria would be higher than at the symmetric equilibria calculated by Bulmer (1980) and others. Here I determine the bifurcation structure of this model, and confirm and extend Barton's results. The form of the bifurcations gives information about the domains of attraction of various equilibria, and shows why the nonsymmetric equilibria may not be observed. The techniques may prove useful in the analysis of other population genetic models.     

Global Stability for Delay SIR and SEIR Epidemic Models with Nonlinear Incidence Rate

In this paper, based on SIR and SEIR epidemic models with a general nonlinear incidence rate, we incorporate time delays into the ordinary differential equation models. In particular, we consider two delay differential equation models in which delays are caused (i) by the latency of the infection in a vector, and (ii) by the latent period in an infected host. By constructing suitable Lyapunov functionals and using the Lyapunov–LaSalle invariance principle, we prove the global stability of the endemic equilibrium and the disease-free equilibrium for time delays of any length in each model. Our results show that the global properties of equilibria also only depend on the basic reproductive number and that the latent period in a vector does not affect the stability, but the latent period in an infected host plays a positive role to control disease development.     

Zero‐inflated spatio‐temporal models for disease mapping

In this paper, our aim is to analyze geographical and temporal variability of disease incidence when spatio‐temporal count data have excess zeros. To that end, we consider random effects in zero‐inflated Poisson models to investigate geographical and temporal patterns of disease incidence. Spatio‐temporal models that employ conditionally autoregressive smoothing across the spatial dimension and B‐spline smoothing over the temporal dimension are proposed. The analysis of these complex models is computationally difficult from the frequentist perspective. On the other hand, the advent of the Markov chain Monte Carlo algorithm has made the Bayesian analysis of complex models computationally convenient. Recently developed data cloning method provides a frequentist approach to mixed models that is also computationally convenient. We propose to use data cloning, which yields to maximum likelihood estimation, to conduct frequentist analysis of zero‐inflated spatio‐temporal modeling of disease incidence. One of the advantages of the data cloning approach is that the prediction and corresponding standard errors (or prediction intervals) of smoothing disease incidence over space and time is easily obtained. We illustrate our approach using a real dataset of monthly children asthma visits to hospital in the province of Manitoba, Canada, during the period April 2006 to March 2010. Performance of our approach is also evaluated through a simulation study.     

Testing,tracing and isolation in compartmental models

Existing compartmental mathematical modelling methods for epidemics, such as SEIR models, cannot accurately represent effects of contact tracing. This makes them inappropriate for evaluating testing and contact tracing strategies to contain an outbreak. An alternative used in practice is the application of agent- or individual-based models (ABM). However ABMs are complex, less well-understood and much more computationally expensive. This paper presents a new method for accurately including the effects of Testing, contact-Tracing and Isolation (TTI) strategies in standard compartmental models. We derive our method using a careful probabilistic argument to show how contact tracing at the individual level is reflected in aggregate on the population level. We show that the resultant SEIR-TTI model accurately approximates the behaviour of a mechanistic agent-based model at far less computational cost. The computational efficiency is such that it can be easily and cheaply used for exploratory modelling to quantify the required levels of testing and tracing, alone and with other interventions, to assist adaptive planning for managing disease outbreaks.     

Analysis of SIR epidemic models with nonlinear incidence rate and treatment

This paper deals with the nonlinear dynamics of a susceptible-infectious-recovered (SIR) epidemic model with nonlinear incidence rate, vertical transmission, vaccination for the newborns of susceptible and recovered individuals, and the capacity of treatment. It is assumed that the treatment rate is proportional to the number of infectives when it is below the capacity and constant when the number of infectives reaches the capacity. Under some conditions, it is shown that there exists a backward bifurcation from an endemic equilibrium, which implies that the disease-free equilibrium coexists with an endemic equilibrium. In such a case, reducing the basic reproduction number less than unity is not enough to control and eradicate the disease, extra measures are needed to ensure that the solutions approach the disease-free equilibrium. When the basic reproduction number is greater than unity, the model can have multiple endemic equilibria due to the effect of treatment, vaccination and other parameters. The existence and stability of the endemic equilibria of the model are analyzed and sufficient conditions on the existence and stability of a limit cycle are obtained. Numerical simulations are presented to illustrate the analytical results.