Analytical approximation for invasion and endemic thresholds, and the optimal control of epidemics in spatially explicit individual-based models.

Computer simulations of individual-based models are frequently used to compare strategies for the control of epidemics spreading through spatially distributed populations. However, computer simulations can be slow to implement for newly emerging epidemics, delaying rapid exploration of different intervention scenarios, and do not immediately give general insights, for example, to identify the control strategy with a minimal socio-economic cost. Here, we resolve this problem by applying an analytical approximation to a general epidemiological, stochastic, spatially explicit SIR(S) model where the infection is dispersed according to a finite-ranged dispersal kernel. We derive analytical conditions for a pathogen to invade a spatially explicit host population and to become endemic. To derive general insights about the likely impact of optimal control strategies on invasion and persistence: first, we distinguish between 'spatial' and 'non-spatial' control measures, based on their impact on the dispersal kernel; second, we quantify the relative impact of control interventions on the epidemic; third, we consider the relative socio-economic cost of control interventions. Overall, our study shows a trade-off between the two types of control interventions and a vaccination strategy. We identify the optimal strategy to control invading and endemic diseases with minimal socio-economic cost across all possible parameter combinations. We also demonstrate the necessary characteristics of exit strategies from control interventions. The modelling framework presented here can be applied to a wide class of diseases in populations of humans, animals and plants.

A unified framework for analysis of individual-based models in ecology and beyond.

Individual-based models, 'IBMs', describe naturally the dynamics of interacting organisms or social or financial agents. They are considered too complex for mathematical analysis, but computer simulations of them cannot give the general insights required. Here, we resolve this problem with a general mathematical framework for IBMs containing interactions of an unlimited level of complexity, and derive equations that reliably approximate the effects of space and stochasticity. We provide software, specified in an accessible and intuitive graphical way, so any researcher can obtain analytical and simulation results for any particular IBM without algebraic manipulation. We illustrate the framework with examples from movement ecology, conservation biology, and evolutionary ecology. This framework will provide unprecedented insights into a hitherto intractable panoply of complex models across many scientific fields.

Modelling chronotaxicity of cellular energy metabolism to facilitate the identification of altered metabolic states.

Altered cellular energy metabolism is a hallmark of many diseases, one notable example being cancer. Here, we focus on the identification of the transition from healthy to abnormal metabolic states. To do this, we study the dynamics of energy production in a cell. Due to the thermodynamic openness of a living cell, the inability to instantaneously match fluctuating supply and demand in energy metabolism results in nonautonomous time-varying oscillatory dynamics. However, such oscillatory dynamics is often neglected and treated as stochastic. Based on experimental evidence of metabolic oscillations, we show that changes in metabolic state can be described robustly by alterations in the chronotaxicity of the corresponding metabolic oscillations, i.e. the ability of an oscillator to resist external perturbations. We also present a method for the identification of chronotaxicity, applicable to general oscillatory signals and, importantly, apply this to real experimental data. Evidence of chronotaxicity was found in glycolytic oscillations in real yeast cells, verifying that chronotaxicity could be used to study transitions between metabolic states.

Generalized chronotaxic systems: time-dependent oscillatory dynamics stable under continuous perturbation.

Chronotaxic systems represent deterministic nonautonomous oscillatory systems which are capable of resisting continuous external perturbations while having a complex time-dependent dynamics. Until their recent introduction in Phys. Rev. Lett. 111, 024101 (2013) chronotaxic systems had often been treated as stochastic, inappropriately, and the deterministic component had been ignored. While the previous work addressed the case of the decoupled amplitude and phase dynamics, in this paper we develop a generalized theory of chronotaxic systems where such decoupling is not required. The theory presented is based on the concept of a time-dependent point attractor or a driven steady state and on the contraction theory of dynamical systems. This simplifies the analysis of chronotaxic systems and makes possible the identification of chronotaxic systems with time-varying parameters. All types of chronotaxic dynamics are classified and their properties are discussed using the nonautonomous Poincaré oscillator as an example. We demonstrate that these types differ in their transient dynamics towards a driven steady state and according to their response to external perturbations. Various possible realizations of chronotaxic systems are discussed, including systems with temporal chronotaxicity and interacting chronotaxic systems.

Inverse approach to chronotaxic systems for single-variable time series.

Following the development of a new class of self-sustained oscillators with a time-varying but stable frequency, the inverse approach to these systems is now formulated. We show how observed data arranged in a single-variable time series can be used to recognize such systems. This approach makes use of time-frequency domain information using the wavelet transform as well as the recently developed method of Bayesian-based inference. In addition, a set of methods, named phase fluctuation analysis, is introduced to detect the defining properties of the new class of systems by directly analyzing the statistics of the observed perturbations.We apply these methods to numerical examples but also elaborate further on the cardiac system.

Chronotaxic systems with separable amplitude and phase dynamics.

Until recently, deterministic nonautonomous oscillatory systems with stable amplitudes and time-varying frequencies were not recognized as such and have often been mistreated as stochastic. These systems, named chronotaxic, were introduced in Phys. Rev. Lett. 111, 024101 (2013). In contrast to conventional limit cycle models of self-sustained oscillators, these systems posses a time-dependent point attractor or steady state. This allows oscillations with time-varying frequencies to resist perturbations, a phenomenon which is ubiquitous in living systems. In this work a detailed theory of chronotaxic systems is presented, specifically in the case of separable amplitude and phase dynamics. The theory is extended by the introduction of chronotaxic amplitude dynamics. The wide applicability of chronotaxic systems to a range of fields from biological and condensed matter systems to robotics and control theory is discussed.

Chronotaxic systems: a new class of self-sustained nonautonomous oscillators.

Nonautonomous oscillatory systems with stable amplitudes and time-varying frequencies have often been treated as stochastic, inappropriately. We therefore formulate them as a new class and discuss how they generate complex behavior. We show how to extract the underlying dynamics, and we demonstrate that it is simple and deterministic, thus paving the way for a diversity of new systems to be recognized as deterministic. They include complex and nonautonomous oscillatory systems in nature, both individually and in ensembles and networks.