Inferring bifurcation diagrams with transformers.

The construction of bifurcation diagrams is an essential component of understanding nonlinear dynamical systems. The task can be challenging when one knows the equations of the dynamical system and becomes much more difficult if only the underlying data associated with the system are available. In this work, we present a transformer-based method to directly estimate the bifurcation diagram using only noisy data associated with an arbitrary dynamical system. By splitting a bifurcation diagram into segments at bifurcation points, the transformer is trained to simultaneously predict how many segments are present and to minimize the loss with respect to the predicted position, shape, and asymptotic stability of each predicted segment. The trained model is shown, both quantitatively and qualitatively, to reliably estimate the structure of the bifurcation diagram for arbitrarily generated one- and two-dimensional systems experiencing a codimension-one bifurcation with as few as 30 trajectories. We show that the method is robust to noise in both the state variable and the system parameter.

Controlling epidemic extinction using early warning signals.

As the recent COVID-19 pandemic has shown us, there is a critical need to develop new approaches to monitoring the outbreak and spread of infectious disease. Improvements in monitoring will enable a timely implementation of control measures, including vaccine and quarantine, to stem the spread of disease. One such approach involves the use of early warning signals to detect when critical transitions are about to occur. Although the early detection of a stochastic transition is difficult to predict using the generic indicators of early warning signals theory, the changes detected by the indicators do tell us that some type of transition is taking place. This observation will serve as the foundation of the method described in the article. We consider a susceptible-infectious-susceptible epidemic model with reproduction number R 0 > 1 so that the deterministic endemic equilibrium is stable. Stochastically, realizations will fluctuate around this equilibrium for a very long time until, as a rare event, the noise will induce a transition from the endemic state to the extinct state. In this article, we describe how metric-based indicators from early warning signals theory can be used to monitor the state of the system. By measuring the autocorrelation, return rate, skewness, and variance of the time series, it is possible to determine when the system is in a weakened state. By applying a control that emulates vaccine/quarantine when the system is in this weakened state, we can cause the disease to go extinct earlier than it otherwise would without control. We also demonstrate that applying a control at the wrong time (when the system is in a non-weakened, highly resilient state) can lead to a longer extinction time than if no control had been applied. This feature underlines the importance of determining the system's state of resilience before attempting to affect its behavior through control measures.

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.

Model Integration in Computational Biology: The Role of Reproducibility, Credibility and Utility.

During the COVID-19 pandemic, mathematical modeling of disease transmission has become a cornerstone of key state decisions. To advance the state-of-the-art host viral modeling to handle future pandemics, many scientists working on related issues assembled to discuss the topics. These discussions exposed the reproducibility crisis that leads to inability to reuse and integrate models. This document summarizes these discussions, presents difficulties, and mentions existing efforts towards future solutions that will allow future model utility and integration. We argue that without addressing these challenges, scientists will have diminished ability to build, disseminate, and implement high-impact multi-scale modeling that is needed to understand the health crises we face.

Knowledge-based learning of nonlinear dynamics and chaos.

Extracting predictive models from nonlinear systems is a central task in scientific machine learning. One key problem is the reconciliation between modern data-driven approaches and first principles. Despite rapid advances in machine learning techniques, embedding domain knowledge into data-driven models remains a challenge. In this work, we present a universal learning framework for extracting predictive models from nonlinear systems based on observations. Our framework can readily incorporate first principle knowledge because it naturally models nonlinear systems as continuous-time systems. This both improves the extracted models' extrapolation power and reduces the amount of data needed for training. In addition, our framework has the advantages of robustness to observational noise and applicability to irregularly sampled data. We demonstrate the effectiveness of our scheme by learning predictive models for a wide variety of systems including a stiff Van der Pol oscillator, the Lorenz system, and the Kuramoto-Sivashinsky equation. For the Lorenz system, different types of domain knowledge are incorporated to demonstrate the strength of knowledge embedding in data-driven system identification.

Using control to shape stochastic escape and switching dynamics.

We present a strategy to control the mean stochastic switching times of general dynamical systems with multiple equilibrium states subject to Gaussian white noise. The control can either enhance or abate the probability of escape from the deterministic region of attraction of a stable equilibrium in the presence of external noise. We synthesize a feedback control strategy that actively changes the system's mean stochastic switching behavior based on the system's distance to the boundary of the attracting region. With the proposed controller, we are able to achieve a desired mean switching time, even when the strength of noise in the system is not known. The control method is analytically validated using a one-dimensional system, and its effectiveness is numerically demonstrated for a set of dynamical systems of practical importance.

Modeling Agrilus planipennis F. (Coleoptera: Buprestidae) Spread in New Jersey.

Pests and disease have become an increasingly common issue as globalized trade brings non-native species into unfamiliar systems. Emerald ash borer (Agrilus planipennis), is an Asiatic species of boring beetle currently devastating the native population of ash (Fraxinus) trees in the northern forests of the United States, with 85 million trees having already succumbed across much of the Midwest. We have developed a reaction-diffusion partial differential equation model to predict the spread of emerald ash borer over a heterogeneous 2-D landscape, with the initial ash tree distribution given by data from the Forest Inventory and Analysis. As expected, the model predictions show that emerald ash borer consumes ash which causes the local ash population to decline, while emerald ash borer spreads outward to other areas. Once the local ash population begins to decline emerald ash borer also declines due to the loss of available habitat. Our model's strength lies with its focus on the county scale and its linkage between emerald ash borer population growth and ash density. This enables one to make accurate predictions regarding emerald ash borer spread which allows one to consider various methods of control as well as to accurately study the economic effects of emerald ash borer spread.

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.

A Framework for Inferring Unobserved Multistrain Epidemic Subpopulations Using Synchronization Dynamics.

A new method is proposed to infer unobserved epidemic subpopulations by exploiting the synchronization properties of multistrain epidemic models. A model for dengue fever is driven by simulated data from secondary infective populations. Primary infective populations in the driven system synchronize to the correct values from the driver system. Most hospital cases of dengue are secondary infections, so this method provides a way to deduce unobserved primary infection levels. We derive center manifold equations that relate the driven system to the driver system and thus motivate the use of synchronization to predict unobserved primary infectives. Synchronization stability between primary and secondary infections is demonstrated through numerical measurements of conditional Lyapunov exponents and through time series simulations.

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.

Analysis and control of pre-extinction dynamics in stochastic populations.

We consider a stochastic population model, where the intrinsic or demographic noise causes cycling between states before the population eventually goes extinct. A master equation approach coupled with a (Wentzel-Kramers-Brillouin) WKB approximation is used to construct the optimal path to extinction. In addition, a probabilistic argument is used to understand the pre-extinction dynamics and approximate the mean time to extinction. Analytical results agree well with numerical Monte Carlo simulations. A control method is implemented to decrease the mean time to extinction. Analytical results quantify the effectiveness of the control and agree well with numerical simulations.

Predicting unobserved exposures from seasonal epidemic data.

We consider a stochastic Susceptible-Exposed-Infected-Recovered (SEIR) epidemiological model with a contact rate that fluctuates seasonally. Through the use of a nonlinear, stochastic projection, we are able to analytically determine the lower dimensional manifold on which the deterministic and stochastic dynamics correctly interact. Our method produces a low dimensional stochastic model that captures the same timing of disease outbreak and the same amplitude and phase of recurrent behavior seen in the high dimensional model. Given seasonal epidemic data consisting of the number of infectious individuals, our method enables a data-based model prediction of the number of unobserved exposed individuals over very long times.

Coherent Pattern Prediction in Swarms of Delay-Coupled Agents.

We consider a general swarm model of self-propelling agents interacting through a pairwise potential in the presence of noise and communication time delay. Previous work has shown that a communication time delay in the swarm induces a pattern bifurcation that depends on the size of the coupling amplitude. We extend these results by completely unfolding the bifurcation structure of the mean field approximation. Our analysis reveals a direct correspondence between the different dynamical behaviors found in different regions of the coupling-time delay plane with the different classes of simulated coherent swarm patterns. We derive the spatiotemporal scales of the swarm structures, as well as demonstrate how the complicated interplay of coupling strength, time delay, noise intensity, and choice of initial conditions can affect the swarm. In particular, our studies show that for sufficiently large values of the coupling strength and/or the time delay, there is a noise intensity threshold that forces a transition of the swarm from a misaligned state into an aligned state. We show that this alignment transition exhibits hysteresis when the noise intensity is taken to be time dependent.

Noise, Bifurcations, and Modeling of Interacting Particle Systems.

We consider the stochastic patterns of a system of communicating, or coupled, self-propelled particles in the presence of noise and communication time delay. For sufficiently large environmental noise, there exists a transition between a translating state and a rotating state with stationary center of mass. Time delayed communication creates a bifurcation pattern dependent on the coupling amplitude between particles. Using a mean field model in the large number limit, we show how the complete bifurcation unfolds in the presence of communication delay and coupling amplitude. Relative to the center of mass, the patterns can then be described as transitions between translation, rotation about a stationary point, or a rotating swarm, where the center of mass undergoes a Hopf bifurcation from steady state to a limit cycle. Examples of some of the stochastic patterns will be given for large numbers of particles.

Converging towards the optimal path to extinction.

Extinction appears ubiquitously in many fields, including chemical reactions, population biology, evolution and epidemiology. Even though extinction as a random process is a rare event, its occurrence is observed in large finite populations. Extinction occurs when fluctuations owing to random transitions act as an effective force that drives one or more components or species to vanish. Although there are many random paths to an extinct state, there is an optimal path that maximizes the probability to extinction. In this paper, we show that the optimal path is associated with the dynamical systems idea of having maximum sensitive dependence to initial conditions. Using the equivalence between the sensitive dependence and the path to extinction, we show that the dynamical systems picture of extinction evolves naturally towards the optimal path in several stochastic models of epidemics.

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.

Maximal sensitive dependence and the optimal path to epidemic extinction.

Extinction of an epidemic or a species is a rare event that occurs due to a large, rare stochastic fluctuation. Although the extinction process is dynamically unstable, it follows an optimal path that maximizes the probability of extinction. We show that the optimal path is also directly related to the finite-time Lyapunov exponents of the underlying dynamical system in that the optimal path displays maximum sensitivity to initial conditions. We consider several stochastic epidemic models, and examine the extinction process in a dynamical systems framework. Using the dynamics of the finite-time Lyapunov exponents as a constructive tool, we demonstrate that the dynamical systems viewpoint of extinction evolves naturally toward the optimal path.

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.

Delay-induced instabilities in self-propelling swarms.

We consider a general model of self-propelling particles interacting through a pairwise attractive force in the presence of noise and communication time delay. Previous work by Erdmann [Phys. Rev. E 71, 051904 (2005)] has shown that a large enough noise intensity will cause a translating swarm of individuals to transition to a rotating swarm with a stationary center of mass. We show that with the addition of a time delay, the model possesses a transition that depends on the size of the coupling amplitude. This transition is independent of the initial swarm state (traveling or rotating) and is characterized by the alignment of all of the individuals along with a swarm oscillation. By considering the mean field equations without noise, we show that the time-delay-induced transition is associated with a Hopf bifurcation. The analytical result yields good agreement with numerical computations of the value of the coupling parameter at the Hopf point.