Characterizing outbreak vulnerability in a stochastic SIS model with an external disease reservoir

In this article, we take a mathematical approach to the study of population-level disease spread, performing a quantitative and qualitative investigation of an SISκ model which is a susceptible-infectious-susceptible ( SIS ) model with exposure to an external disease reservoir. The external reservoir is non-dynamic, and exposure from the external reservoir is assumed to be proportional to the size of the susceptible population. The full stochastic system is modelled using a master equation formalism. A constant population size assumption allows us to solve for the stationary probability distribution, which is then used to investigate the predicted disease prevalence under a variety of conditions. By using this approach, we quantify outbreak vulnerability by performing the sensitivity analysis of disease prevalence to changing population characteristics. In addition, the shape of the probability density function is used to understand where, in parameter space, there is a transition from disease free, to disease present, and to a disease endemic system state. Finally, we use Kullback–Leibler divergence to compare our semi-analytical results for the SISκ model with more complex susceptible-infectious-recovered ( SIR ) and susceptible-exposed-infectious-recovered ( SEIR ) models.

Extinction pathways and outbreak vulnerability in a stochastic Ebola model

A zoonotic disease is a disease that can be passed from animals to humans. Zoonotic viruses may adapt to a human host eventually becoming endemic in humans, but before doing so punctuated outbreaks of the zoonotic virus may be observed. The Ebola virus disease (EVD) is an example of such a disease. The animal population in which the disease agent is able to reproduce in sufficient number to be able to transmit to a susceptible human host is called a reservoir. There is little work devoted to understanding stochastic population dynamics in the presence of a reservoir, specifically the phenomena of disease extinction and reintroduction. Here, we build a stochastic EVD model and explicitly consider the impacts of an animal reservoir on the disease persistence. Our modelling approach enables the analysis of invasion and fade-out dynamics, including the efficacy of possible intervention strategies. We investigate outbreak vulnerability and the probability of local extinction and quantify the effective basic reproduction number. We also consider the effects of dynamic population size. Our results provide an improved understanding of outbreak and extinction dynamics in zoonotic diseases, such as EVD.

Computing the optimal path in stochastic dynamical systems

In stochastic systems, one is often interested in finding the optimal path that maximizes the probability of escape from a metastable state or of switching between metastable states. Even for simple systems, it may be impossible to find an analytic form of the optimal path, and in high-dimensional systems, this is almost always the case. In this article, we formulate a constructive methodology that is used to compute the optimal path numerically. The method utilizes finite-time Lyapunov exponents, statistical selection criteria, and a Newton-based iterative minimizing scheme. The method is applied to four examples. The first example is a two-dimensional system that describes a single population with internal noise. This model has an analytical solution for the optimal path. The numerical solution found using our computational method agrees well with the analytical result. The second example is a more complicated four-dimensional system where our numerical method must be used to find the optimal path. The third example, although a seemingly simple two-dimensional system, demonstrates the success of our method in finding the optimal path where other numerical methods are known to fail. In the fourth example, the optimal path lies in six-dimensional space and demonstrates the power of our method in computing paths in higher-dimensional spaces.

Herbivory and Stoichiometric Feedbacks to Primary Production

Established theory addresses the idea that herbivory can have positive feedbacks on nutrient flow to plants. Positive feedbacks likely emerge from a greater availability of organic carbon that primes the soil by supporting nutrient turnover through consumer and especially microbially-mediated metabolism in the detrital pool. We developed an entirely novel stoichiometric model that demonstrates the mechanism of a positive feedback. In particular, we show that sloppy or partial feeding by herbivores increases detrital carbon and nitrogen allowing for greater nitrogen mineralization and nutritive feedback to plants. The model consists of differential equations coupling flows among pools of: plants, herbivores, detrital carbon and nitrogen, and inorganic nitrogen. We test the effects of different levels of herbivore grazing completion and of the stoichiometric quality (carbon to nitrogen ratio, C:N) of the host plant. Our model analyses show that partial feeding and plant C:N interact because when herbivores are sloppy and plant biomass is diverted to the detrital pool, more mineral nitrogen is available to plants because of the stoichiometric difference between the organisms in the detrital pool and the herbivore. This model helps to identify how herbivory may feedback positively on primary production, and it mechanistically connects direct and indirect feedbacks from soil to plant production.

Noise-induced switching and extinction in systems with delay

We consider the rates of noise-induced switching between the stable states of dissipative dynamical systems with delay and also the rates of noise-induced extinction, where such systems model population dynamics. We study a class of systems where the evolution depends on the dynamical variables at a preceding time with a fixed time delay, which we call hard delay. For weak noise, the rates of interattractor switching and extinction are exponentially small. Finding these rates to logarithmic accuracy is reduced to variational problems. The solutions of the variational problems give the most probable paths followed in switching or extinction. We show that the equations for the most probable paths are acausal and formulate the appropriate boundary conditions. Explicit results are obtained for small delay compared to the relaxation rate. We also develop a direct variational method to find the rates. We find that the analytical results agree well with the numerical simulations for both switching and extinction rates.

Intervention-based stochastic disease eradication

Disease control is of paramount importance in public health, with infectious disease extinction as the ultimate goal. Although diseases may go extinct due to random loss of effective contacts where the infection is transmitted to new susceptible individuals, the time to extinction in the absence of control may be prohibitively long. Intervention controls are typically defined on a deterministic schedule. In reality, however, such policies are administered as a random process, while still possessing a mean period. Here, we consider the effect of randomly distributed intervention as disease control on large finite populations. We show explicitly how intervention control, based on mean period and treatment fraction, modulates the average extinction times as a function of population size and rate of infection spread. In particular, our results show an exponential improvement in extinction times even though the controls are implemented using a random Poisson distribution. Finally, we discover those parameter regimes where random treatment yields an exponential improvement in extinction times over the application of strictly periodic intervention. The implication of our results is discussed in light of the availability of limited resources for control.

Set-based corral control in stochastic dynamical systems: making almost invariant sets more invariant

We consider the problem of stochastic prediction and control in a time-dependent stochastic environment, such as the ocean, where escape from an almost invariant region occurs due to random fluctuations. We determine high-probability control-actuation sets by computing regions of uncertainty, almost invariant sets, and Lagrangian coherent structures. The combination of geometric and probabilistic methods allows us to design regions of control, which provide an increase in loitering time while minimizing the amount of control actuation. We show how the loitering time in almost invariant sets scales exponentially with respect to the control actuation, causing an exponential increase in loitering times with only small changes in actuation force. The result is that the control actuation makes almost invariant sets more invariant.

Effective Vaccination Policies

We present a framework for modeling the spread of pathogens throughout a population and generating policies that minimize the impact of those pathogens on the population. This framework is used to study the spread of human viruses between cities via airplane travel. It combines agent-based simulation, mathematical analysis, and an Evolutionary Algorithm (EA) optimizer. The goal of this study is to develop tools that determine the optimal distribution of a vaccine supply in the model. Using plausible benchmark vaccine allocation policies of uniform and proportional distribution, we compared their effectiveness to policies found by the EA. We then designed and tested a new, more effective policy which increased the importance of vaccinating smaller cities that are flown to more often. This "importance factor" was validated using U.S. influenza data from the last four years.

Switching exponent scaling near bifurcation points for non-Gaussian noise

We study noise-induced switching of a system close to bifurcation parameter values where the number of stable states changes. For non-Gaussian noise, the switching exponent, which gives the logarithm of the switching rate, displays a non-power-law dependence on the distance to the bifurcation point. This dependence is found for Poisson noise. Even weak additional Gaussian noise dominates switching sufficiently close to the bifurcation point, leading to a crossover in the behavior of the switching exponent.

Accurate noise projection for reduced stochastic epidemic models

We consider a stochastic susceptible-exposed-infected-recovered (SEIR) epidemiological model. Through the use of a normal form coordinate transform, we are able to analytically derive the stochastic center manifold along with the associated, reduced set of stochastic evolution equations. The transformation correctly projects both the dynamics and the noise onto the center manifold. Therefore, the solution of this reduced stochastic dynamical system yields excellent agreement, both in amplitude and phase, with the solution of the original stochastic system for a temporal scale that is orders of magnitude longer than the typical relaxation time. This new method allows for improved time series prediction of the number of infectious cases when modeling the spread of disease in a population. Numerical solutions of the fluctuations of the SEIR model are considered in the infinite population limit using a Langevin equation approach, as well as in a finite population simulated as a Markov process.

Thermally activated switching in the presence of non-Gaussian noise

We study the effect of a non-Gaussian noise on interstate switching activated primarily by Gaussian noise. Even weak non-Gaussian noise can strongly change the switching rate. The effect is determined by all moments of the noise distribution. It is expressed in a closed form in terms of the noise characteristic functional. The analytical results are compared with the results of simulations for an overdamped system driven by white Gaussian noise and a Poisson noise. Switching induced by a purely Poisson noise is also discussed.

Identifying almost invariant sets in stochastic dynamical systems

We consider the approximation of fluctuation induced almost invariant sets arising from stochastic dynamical systems. The dynamical evolution of densities is derived from the stochastic Frobenius-Perron operator. Given a stochastic kernel with a known distribution, approximate almost invariant sets are found by translating the problem into an eigenvalue problem derived from reversible Markov processes. Analytic and computational examples of the methods are used to illustrate the technique, and are shown to reveal the probability transport between almost invariant sets in nonlinear stochastic systems. Both small and large noise cases are considered.

Vaccinations in disease models with antibody-dependent enhancement

This paper examines the effects of single-strain vaccine campaigns on the dynamics of an epidemic multistrain model with antibody-dependent enhancement (ADE). ADE is a disease spreading process causing individuals with their secondary infection to be more infectious than during their first infection by a different strain. We follow the two-strain ADE model described in Cummings et al. [D.A.T. Cummings, Doctoral Thesis, Johns Hopkins University, 2004] and Schwartz et al. [I.B. Schwartz, L.B. Shaw, D.A.T. Cummings, L. Billings, M. McCrary, D. Burke, Chaotic desynchronization of multi-strain diseases, Phys. Rev. E, 72:art. no. 066201, 2005]. After describing the model and its steady state solutions, we modify it to include vaccine campaigns and explore if there exists vaccination rates that can eradicate one or more strains of a virus with ADE.

Using dimension reduction to improve outbreak predictability of multistrain diseases

Multistrain diseases have multiple distinct coexisting serotypes (strains). For some diseases, such as dengue fever, the serotypes interact by antibody-dependent enhancement (ADE), in which infection with a single serotype is asymptomatic, but contact with a second serotype leads to higher viral load and greater infectivity. We present and analyze a dynamic compartmental model for multiple serotypes exhibiting ADE. Using center manifold techniques, we show how the dynamics rapidly collapses to a lower dimensional system. Using the constructed reduced model, we can explain previously observed synchrony between certain classes of primary and secondary infectives (Schwartz et al. in Phys Rev E 72:066201, 2005). Additionally, we show numerically that the center manifold equations apply even to noisy systems. Both deterministic and stochastic versions of the model enable prediction of asymptomatic individuals that are difficult to track during an epidemic. We also show how this technique may be applicable to other multistrain disease models, such as those with cross-immunity.

Instabilities in multiserotype disease models with antibody-dependent enhancement

This paper investigates the complex dynamics induced by antibody-dependent enhancement (ADE) in multiserotype disease models. ADE is the increase in viral growth rate in the presence of immunity due to a previous infection of a different serotype. The increased viral growth rate is thought to increase the infectivity of the secondary infectious class. In our models, ADE induces the onset of oscillations without external forcing. The oscillations in the infectious classes represent outbreaks of the disease. In this paper, we derive approximations of the ADE parameter needed to induce oscillations and analyze the associated bifurcations that separate the types of oscillations. We then investigate the stability of these dynamics by adding stochastic perturbations to the model. We also present a preliminary analysis of the effect of a single serotype vaccination in the model.

The astrobiology primer: an outline of general knowledge--version 1, 2006

The Astrobiology Primer has been created as a reference tool for those who are interested in the interdisciplinary field of astrobiology. The field incorporates many diverse research endeavors, but it is our hope that this slim volume will present the reader with all he or she needs to know to become involved and to understand, at least at a fundamental level, the state of the art. Each section includes a brief overview of a topic and a short list of readable and important literature for those interested in deeper knowledge. Because of the great diversity of material, each section was written by a different author with a different expertise. Contributors, authors, and editors are listed at the beginning, along with a list of those chapters and sections for which they were responsible. We are deeply indebted to the NASA Astrobiology Institute (NAI), in particular to Estelle Dodson, David Morrison, Ed Goolish, Krisstina Wilmoth, and Rose Grymes for their continued enthusiasm and support. The Primer came about in large part because of NAI support for graduate student research, collaboration, and inclusion as well as direct funding. We have entitled the Primer version 1 in hope that it will be only the first in a series, whose future volumes will be produced every 3-5 years. This way we can insure that the Primer keeps up with the current state of research. We hope that it will be a great resource for anyone trying to stay abreast of an ever-changing field.

Chaotic desynchronization of multistrain diseases

Multistrain diseases are diseases that consist of several strains, or serotypes. The serotypes may interact by antibody-dependent enhancement (ADE), in which infection with a single serotype is asymptomatic, but infection with a second serotype leads to serious illness accompanied by greater infectivity. It has been observed from serotype data of dengue hemorrhagic fever that outbreaks of the four serotypes occur asynchronously. Both autonomous and seasonally driven outbreaks were studied in a model containing ADE. For sufficiently small ADE, the number of infectives of each serotype synchronizes, with outbreaks occurring in phase. When the ADE increases past a threshold, the system becomes chaotic, and infectives of each serotype desynchronize. However, certain groupings of the primary and secondary infectives remain synchronized even in the chaotic regime.

Dynamic effects of antibody-dependent enhancement on the fitness of viruses

Antibody-dependent enhancement (ADE), a phenomenon in which viral replication is increased rather than decreased by immune sera, has been observed in vitro for a large number of viruses of public health importance, including flaviviruses, coronaviruses, and retroviruses. The most striking in vivo example of ADE in humans is dengue hemorrhagic fever, a disease in which ADE is thought to increase the severity of clinical manifestations of dengue virus infection by increasing virus replication. We examine the epidemiological impact of ADE on the prevalence and persistence of viral serotypes. Using a dynamical system model of n cocirculating dengue serotypes, we find that ADE may provide a competitive advantage to those serotypes that undergo enhancement compared with those that do not, and that this advantage increases with increasing numbers of cocirculating serotypes. Paradoxically, there are limits to the selective advantage provided by increasing levels of ADE, because greater levels of enhancement induce large amplitude oscillations in incidence of all dengue virus infections, threatening the persistence of both the enhanced and nonenhanced serotypes. Although the models presented here are specifically designed for dengue, our results are applicable to any epidemiological system in which partial immunity increases pathogen replication rates. Our results suggest that enhancement is most advantageous in settings where multiple serotypes circulate and where a large host population is available to support pathogen persistence during the deep troughs of ADE-induced large amplitude oscillations of virus replication.

'Active' filters for upgrading phosphorus removal from pond systems

This paper investigates limestone and iron slag filters as an upgrade option for phosphorus removal from wastewater treatment ponds. A review of 'active' filter technology and the results from laboratory and field research using packed columns of the different media is presented. It is shown that both limestone and iron slag can remove phosphorus but highlights that different types of limestone give markedly different performance. Filter performance appears to be improved by increasing temperature and by the presence of algae, presumably because of its tendency to elevate pH. Performance is related to hydraulic retention time (HRT), but this relationship is not linear, particularly at low HRTs. Importantly for future research, the results from field-testing with pond effluent show significant differences compared to those obtained when using a synthetic feed in the laboratory. For the iron slag filter, higher performance was observed in the field (72% in field vs. 27% in laboratory, at a 12 hour-HRT), while the opposite was observed for the limestone (64% in laboratory vs. 18% in field, at a 12-hour HRT).

Dynamical epidemic suppression using stochastic prediction and control

We consider the effects of noise on a model of epidemic outbreaks, where the outbreaks appear randomly. Using a constructive transition approach that predicts large outbreaks prior to their occurrence, we derive an adaptive control scheme that prevents large outbreaks from occurring. The theory is applicable to a wide range of stochastic processes with underlying deterministic structure.

Stochastic bifurcation in a driven laser system: experiment and theory

We analyze the effects of stochastic perturbations in a physical example occurring as a higher-dimensional dynamical system. The physical model is that of a class- B laser, which is perturbed stochastically with finite noise. The effect of the noise perturbations on the dynamics is shown to change the qualitative nature of the dynamics experimentally from a stochastic periodic attractor to one of chaoslike behavior, or noise-induced chaos. To analyze the qualitative change, we apply the technique of the stochastic Frobenius-Perron operator [L. Billings et al., Phys. Rev. Lett. 88, 234101 (2002)] to a model of the experimental system. Our main result is the identification of a global mechanism to induce chaoslike behavior by adding stochastic perturbations in a realistic model system of an optics experiment. In quantifying the stochastic bifurcation, we have computed a transition matrix describing the probability of transport from one region of phase space to another, which approximates the stochastic Frobenius-Perron operator. This mechanism depends on both the standard deviation of the noise and the global topology of the system. Our result pinpoints regions of stochastic transport whereby topological deterministic dynamics subjected to sufficient noise results in noise-induced chaos in both theory and experiment.

Multi-scale continuum mechanics: from global bifurcations to noise induced high-dimensional chaos

Many mechanical systems consist of continuum mechanical structures, having either linear or nonlinear elasticity or geometry, coupled to nonlinear oscillators. In this paper, we consider the class of linear continua coupled to mechanical pendula. In such mechanical systems, there often exist several natural time scales determined by the physics of the problem. Using a time scale splitting, we analyze a prototypical structural-mechanical system consisting of a planar nonlinear pendulum coupled to a flexible rod made of linear viscoelastic material. In this system both low-dimensional and high-dimensional chaos is observed. The low-dimensional chaos appears in the limit of small coupling between the continua and oscillator, where the natural frequency of the primary mode of the rod is much greater than the natural frequency of the pendulum. In this case, the motion resides on a slow manifold. As the coupling is increased, global motion moves off of the slow manifold and high-dimensional chaos is observed. We present a numerical bifurcation analysis of the resulting system illustrating the mechanism for the onset of high-dimensional chaos. Constrained invariant sets are computed to reveal a process from low-dimensional to high-dimensional transitions. Applications will be to both deterministic and stochastic bifurcations. Practical implications of the bifurcation from low-dimensional to high-dimensional chaos for detection of damage as well as global effects of noise will also be discussed.

Noise-induced unstable dimension variability and transition to chaos in random dynamical systems

Results are reported concerning the transition to chaos in random dynamical systems. In particular, situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle, which can be expected in any periodic window of a nonlinear dynamical system. Under noise, the asymptotic attractor of the system can become chaotic, as characterized by the appearance of a positive Lyapunov exponent. Generic features of the transition include the following: (1) the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions (unstable dimension variability), and this nonhyperbolicity develops as soon as the attractor becomes chaotic; (2) for systems described by differential equations, the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero Lyapunov exponent after the noisy attractor becomes chaotic; and (3) the largest Lyapunov exponent becomes positive from zero in a continuous manner, and its scaling with the variation of the noise amplitude is algebraic. Formulas for the scaling exponent are derived in all dimensions. Numerical support using both low- and high-dimensional systems is provided.

Phase-space transport of stochastic chaos in population dynamics of virus spread

A general way to classify stochastic chaos is presented and applied to population dynamics models. A stochastic dynamical theory is used to develop an algorithmic tool to measure the transport across basin boundaries and predict the most probable regions of transport created by noise. The results of this tool are illustrated on a model of virus spread in a large population, where transport regions reveal how noise completes the necessary manifold intersections for the creation of emerging stochastic chaos.

Transition to chaos in continuous-time random dynamical systems

We consider situations where, in a continuous-time dynamical system, a nonchaotic attractor coexists with a nonattracting chaotic saddle, as in a periodic window. Under the influence of noise, chaos can arise. We investigate the fundamental dynamical mechanism responsible for the transition and obtain a general scaling law for the largest Lyapunov exponent. A striking finding is that the topology of the flow is fundamentally disturbed after the onset of noisy chaos, and we point out that such a disturbance is due to changes in the number of unstable eigendirections along a continuous trajectory under the influence of noise.

Exciting chaos with noise: unexpected dynamics in epidemic outbreaks

In this paper, we identify a mechanism for chaos in the presence of noise. In a study of the SEIR model, which predicts epidemic outbreaks in childhood diseases, we show how chaotic dynamics can be attained by adding stochastic perturbations at parameters where chaos does not exist apriori. Data recordings of epidemics in childhood diseases are still argued as deterministic chaos. There also exists noise due to uncertainties in the contact parameters between those who are susceptible and those who are infected, as well as random fluctuations in the population. Although chaos has been found in deterministic models, it only occurs in parameter regions that require a very large population base or other large seasonal forcing. Our work identifies the mechanism whereby chaos can be induced by noise for realistic parameter regions of the deterministic model where it does not naturally occur.

Methods for short time series analysis of cell-based biosensor data

This paper describes two approaches for sensing changes in spiking cells when only a limited amount of spike data is available, i.e., dynamically constructed local expansion rates and spike area distributions. The two methods were tested on time series from cultured neuron cells that exhibit spiking both autonomously and in the presence of periodic stimulation. Our tested hypothesis was that minute concentrations of toxins could affect the local statistics of the dynamics. Short data sets having relatively few spikes were generated from experiments on cells before and after being treated with a small concentration of channel blocker. In spontaneous spiking cells, local expansion rates show a sensitivity that correlates with channel concentration level, while stimulated cells show no such correlation. Spike area distributions on the other hand showed measurable differences between control and treated conditions for both types of spiking, and a much higher degree of sensitivity. Because these methods are based on analysis of short time series analysis, they might provide novel means for cell drug and toxin detection.

Efficient cw end-pumped, end-cooled Nd:YVO(4) diode-pumped laser

Using a novel end-pumped, end-cooled geometry, we demonstrated a Nd:YVO(4) cw laser that produces close to 2 W of single-mode output when pumped by a high-brightness 100-mum-core diode fiber source. An optical-optical conversion efficiency of 49% was obtained. Beam quality was found to be nearly diffraction-limited (M(2) = 1.08) and independent of pumping power.

On noninvertible mappings of the plane: Eruptions

In this paper we are concerned with the dynamics of noninvertible transformations of the plane. Three examples are explored and possibly a new bifurcation, or "eruption," is described. A fundamental role is played by the interactions of fixed points and singular curves. Other critical elements in the phase space include periodic points and an invariant line. The dynamics along the invariant line, in two of the examples, reduces to the one-dimensional Newton's method which is conjugate to a degree two rational map. We also determine, computationally, the characteristic exponents for all of the systems. An unexpected coincidence is that the parameter range where the invariant line becomes neutrally stable, as measured by a zero Lyapunov exponent, coincides with the merging of a periodic point with a point on a singular curve. (c) 1996 American Institute of Physics.

Wound infection rates in dogs and cats after use of cotton muslin or disposable impermeable fabric as barrier material: 720 cases (1983-1989)

A retrospective study involving 720 dogs and cats that underwent a variety of elective surgical procedures was done to compare the effectiveness of reusable cotton barrier materials with that of a commercially available disposable barrier system for prevention of wound infection. The overall wound infection rate, using cotton barrier materials, was 3.1% and for disposable materials, was 4.4%. The difference between groups was not significant.