Asymmetric noise-induced large fluctuations in coupled systems.

Networks of interacting, communicating subsystems are common in many fields, from ecology, biology, and epidemiology to engineering and robotics. In the presence of noise and uncertainty, interactions between the individual components can lead to unexpected complex system-wide behaviors. In this paper, we consider a generic model of two weakly coupled dynamical systems, and we show how noise in one part of the system is transmitted through the coupling interface. Working synergistically with the coupling, the noise on one system drives a large fluctuation in the other, even when there is no noise in the second system. Moreover, the large fluctuation happens while the first system exhibits only small random oscillations. Uncertainty effects are quantified by showing how characteristic time scales of noise-induced switching scale as a function of the coupling between the two coupled parts of the experiment. In addition, our results show that the probability of switching in the noise-free system scales inversely as the square of reduced noise intensity amplitude, rendering the virtual probability of switching an extremely rare event. Our results showing the interplay between transmitted noise and coupling are also confirmed through simulations, which agree quite well with analytic theory.

Equivalent probability density moments determine equivalent epidemics in an SIRS model with temporary immunity.

In an SIRS compartment model for a disease we consider the effect of different probability distributions for remaining immune. We show that to first approximation the first three moments of the corresponding probability densities are sufficient to well describe oscillatory solutions corresponding to recurrent epidemics. Specifically, increasing the fraction who lose immunity, increasing the mean immunity time, and decreasing the heterogeneity of the population all favor the onset of epidemics and increase their severity. We consider six different distributions, some symmetric about their mean and some asymmetric, and show that by tuning their parameters such that they have equivalent moments that they all exhibit equivalent dynamical behavior.

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.

Tight swimming trunks to prevent post scrotal surgery: an experimental justification.

PURPOSE: To conduct a study to measure the pressure effects of the different scrotal supports applied on a simulated expanding scrotal hematoma. MATERIALS AND METHODS: We created a model of an expanding hematoma with simultaneous pressure recording using a urodynamics system. Pressures were recorded independently first without application of any support. Then, three types of scrotal supports were tested, including Euron Net Knickers, scrotal suspensory bandage, and tight swimming trunks brand Speedo® brief and shorts. Subsequent pressures were recorded using the model created, which was applied inside the supports worn by two male volunteers A and B. RESULTS: Without any external compression, the pressure inside the simulated expanding hematoma "balloon" reached a maximum of 15 cmH2O. The pressures measured whilst wearing "Netelast knickers" in both subjects A and B reached a maximum of 15 cmH2O suggesting that this garment exerted no measurable compression. The suspensory scrotal support was then tested in both subjects. As the balloon started to fill with saline, the simulated hematoma pushed the scrotal support forward resulting in falling of the balloon outside the scrotal support. Subsequently, Speedo® briefs and shorts were tested. With Speedo® briefs, maximum filling pressures of 49 cmH2O and 40 cmH2O were reached in subjects A and B, respectively. When using Speedo® shorts, however, maximum pressures of 55 cmH2O in subject A and 54 cmH2O in subject B were reached at the end of the balloon filling to 300 mL of saline. CONCLUSION: The use of tight swimming trunks (Speedo®) has led to satisfactory results in the prevention of hematoma post scrotal surgery.

Synchronous versus asynchronous oscillations for antigenically varying Plasmodium falciparum with host immune response.

We consider a deterministic intra-host model for Plasmodium falciparum (Pf) malaria infection, which accounts for antigenic variation between n clonal variants of PfEMP1 and the corresponding host immune response (IR). Specifically, the model separates the IR into two components, specific and cross-reactive, respectively, in order to demonstrate that the latter can be a mechanism for the sequential appearance of variants observed in actual Pf infections. We show that a strong variant-specific IR relative to the cross-reactive IR favours the asynchronous oscillations (sequential dominance) over the synchronous oscillations in a number of ways. The decay rate of asynchronous oscillations is smaller than that for the synchronous oscillations, allowing for the parasite to survive longer. With the introduction of a delay in the stimulation of the IR, we show that only a small delay is necessary to cause persistent asynchronous oscillations and that a strong variant-specific IR increases the amplitude of the asynchronous oscillations.

Effect of state-dependent delay on a weakly damped nonlinear oscillator.

We consider a weakly damped nonlinear oscillator with state-dependent delay, which has applications in models for lasers, epidemics, and microparasites. More generally, the delay-differential equations considered are a predator-prey system where the delayed term is linear and represents the proliferation of the predator. We determine the critical value of the delay that causes the steady state to become unstable to periodic oscillations via a Hopf bifurcation. Using asymptotic averaging, we determine how the system's behavior is influenced by the functional form of the state-dependent delay. Specifically, we determine whether the branch of periodic solutions will be either sub- or supercritical as well as an accurate estimation of the amplitude. Finally, we choose a few examples of state-dependent delay to test our analytical results by comparing them to numerical continuation.

Oscillations in an intra-host model of Plasmodium Falciparum malaria due to cross-reactive immune response.

We consider an intra-host model of malaria that allows for antigenic variation within a single species. More specifically, the host's immune response is compartmentalized into reactions to major and minor epitopes. We investigate the conditions that lead to transient oscillations, which correspond to recurrent clinical episodes of the diseases, and how a small delay in the activation of the immune response can lead to persistent oscillations. We find that the efficacies of the immune responses to the major and minor epitopes, defined in terms of rate constants, play a crucial role in determining when there will be transient oscillations. The delay necessary to excite persistent oscillations, the time duration between disease episodes and their severity are also expressed in terms of the immune response efficacies. In addition, we describe how the severity and duration of the oscillations depend upon the parasite propagation rates and the immune response efficacies.

An SIR epidemic model with partial temporary immunity modeled with delay.

The SIR epidemic model for disease dynamics considers recovered individuals to be permanently immune, while the SIS epidemic model considers recovered individuals to be immediately resusceptible. We study the case of temporary immunity in an SIR-based model with delayed coupling between the susceptible and removed classes, which results in a coupled set of delay differential equations. We find conditions for which the endemic steady state becomes unstable to periodic outbreaks. We then use analytical and numerical bifurcation analysis to describe how the severity and period of the outbreaks depend on the model parameters.

Coupling-induced resonance in two mutually and asymmetrically coupled oscillators.

We consider two mutually coupled oscillators, where we have independent control over the magnitude, sign, and delay of the coupling signal. For appropriate tuning of the coupling constants, there is a coupling-induced resonance where the amplitude becomes large. We investigate the role of nonlinear dissipation and amplitude-dependent frequency correction on the coupling resonance. With delayed coupling, we track the deformation of the resonant bifurcation equation through imperfect bifurcations and the generation of isolas, which generate intervals of multistability between oscillations of different amplitudes. The resonance should be observable in coupled-oscillator systems, where the amplitude remains an important dynamical variable, but not in limit-cycle oscillators modeled by phase-only descriptions.

Scaling behavior of laser population dynamics with time-delayed coupling: theory and experiment.

We study the influence of asymmetric coupling strengths on the onset of light intensity oscillations in an experimental system consisting of two semiconductor lasers cross coupled optoelectronically with a time delay. We discover a scaling law that relates the amplitudes of oscillations and the coupling strengths. These observations are in agreement with a theoretical model. These results could be applicable to the population dynamics of other systems, such as the spread of disease in human populations coupled by migration.

Open-loop sustained chaos and control: a manifold approach.

We present a general method for preserving chaos in nonchaotic parameter regimes as well as preserving periodic behavior in chaotic regimes using a multifrequency phase control. The systems considered are nonlinear systems driven at a base frequency. Multifrequency phase control is defined as the addition of small subharmonic amplitude modulation coupled with a phase shift. By implementing multifrequency control, stable and unstable manifold intersections in postcrisis regimes may be manipulated to sustain chaos as well as to sustain periodic behavior. The theory and a preliminary experiment are demonstrated for a CO2 driven laser.

Tracking controlled chaos: Theoretical foundations and applications.

Tracking controlled states over a large range of accessible parameters is a process which allows for the experimental continuation of unstable states in both chaotic and non-chaotic parameter regions of interest. In algorithmic form, tracking allows experimentalists to examine many of the unstable states responsible for much of the observed nonlinear dynamic phenomena. Here we present a theoretical foundation for tracking controlled states from both dynamical systems as well as control theoretic viewpoints. The theory is constructive and shows explicitly how to track a curve of unstable states as a parameter is changed. Applications of the theory to various forms of control currently used in dynamical system experiments are discussed. Examples from both numerical and physical experiments are given to illustrate the wide range of tracking applications. (c) 1997 American Institute of Physics.