Extinction of an epidemic or a species is a rare event that occurs due to a large, rare stochasti... more 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. Stochastic dynamical systems and Lyapunov exponents • Optimal path to extinction 1 Introduction Control and eradication of infectious diseases are among the most important goals for improving public health. Although the global eradication of a disease (e.g. smallpox
Bulletin of the American Physical Society, Mar 17, 2017
Mixed reality framework for collective motion patterns of swarms with delay coupling 1 KLEMENTYNA... more Mixed reality framework for collective motion patterns of swarms with delay coupling 1 KLEMENTYNA SZWAYKOWSKA, IRA SCHWARTZ, Naval Research Lab -The formation of coherent patterns in swarms of interacting self-propelled autonomous agents is an important subject for many applications within the field of distributed robotic systems. However, there are significant logistical challenges associated with testing fully distributed systems in real-world settings. In this paper, we provide a rigorous theoretical justification for the use of mixed-reality experiments as a stepping stone to fully physical testing of distributed robotic systems. We also model and experimentally realize a mixed-reality largescale swarm of delay-coupled agents. Our analyses, assuming agents communicating over an Erdos-Renyi network, demonstrate the existence of stable coherent patterns that can be achieved only with delay coupling and that are robust to decreasing network connectivity and heterogeneity in agent dynamics. We show how the bifurcation structure for emergence of different patterns changes with heterogeneity in agent acceleration capabilities and limited connectivity in the network as a function of coupling strength and delay. Our results are verified through simulation as well as preliminary experimental results of delay-induced pattern formation in a mixed-reality swarm.
Bulletin of the American Physical Society, Mar 23, 2005
The serotypes interact by antibody-dependent enhancement (ADE), in which infection with a single ... more 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 serious illness accompanied by greater infectivity. It has been observed from serotype data that outbreaks of the four serotypes occur asynchronously (Nisalak et al., Am. J. Trop. Med. Hyg. 68: 192). We developed a compartmental model and did bifurcation analysis for multiple serotypes with ADE. Both autonomous and seasonally driven versions were studied. For sufficiently small ADE, we find that the number of infectives of each serotype synchronizes, with outbreaks occurring in phase. However, when the ADE increases past a threshold, the system becomes chaotic, and infectives of each serotype desynchronize.
In some physical and biological swarms, agents effectively move and interact along curved surface... more In some physical and biological swarms, agents effectively move and interact along curved surfaces. The associated constraints and symmetries can affect collective-motion patterns, but little is known about pattern stability in the presence of surface curvature. To make progress, we construct a general model for self-propelled swarms moving on surfaces using Lagrangian mechanics. We find that the combination of self-propulsion, friction, mutual attraction, and surface curvature produce milling patterns where each agent in a swarm oscillates on a limit cycle, with different agents splayed along the cycle such that the swarm's center-of-mass remains stationary. In general, such patterns loose stability when mutual attraction is insufficient to overcome the constraint of curvature, and we uncover two broad classes of stationary milling-state bifurcations. In the first, a spatially periodic mode undergoes a Hopf bifurcation as curvature is increased which results in unstable spatiotemporal oscillations. This generic bifurcation is analyzed for the sphere and demonstrated numerically for several surfaces. In the second, a saddle-node-of-periodic-orbits occurs in which stable and unstable milling states collide and annihilate. The latter is analyzed for milling states on cylindrical surfaces. Our results contribute to the general understanding of swarm pattern-formation and stability in the presence of surface curvature, and may aid in designing robotic swarms that can be controlled to move over complex surfaces. I.
We consider the stochastic patterns of a system of communicating, or coupled, self-propelled part... more 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.
International Journal of Bifurcation and Chaos, Jun 1, 1993
A new class of "toy models" for subaqueous bedform formation are proposed and examined. These mod... more A new class of "toy models" for subaqueous bedform formation are proposed and examined. These models all show a similar mechanism of wavelength selection via bedform unification, and they may have applications to bedform stratigraphy. The models are also useful for exploring general issues of pattern formation and complexity in stochastically driven far from equilibrium systems.
We study rare events in networks with both internal and external noise, and develop a general for... more We study rare events in networks with both internal and external noise, and develop a general formalism for analyzing rare events that combines pair-quenched techniques and largedeviation theory. The probability distribution, shape, and time scale of rare events are considered in detail for extinction in the Susceptible-Infected-Susceptible model as an illustration. We find that when both types of noise are present, there is a cross-over region as the network size is increased, where the probability exponent for large deviations no longer increases linearly with the network size. We demonstrate that the form of the cross-over depends on whether the endemic state is localized near the epidemic threshold or not.
It is known that introducing time delays into the communication network of mobile-agent swarms pr... more It is known that introducing time delays into the communication network of mobile-agent swarms produces coherent rotational patterns, from both theory and experiments. Often such spatiotemporal rotations can be bistable with other swarming patterns, such as milling and flocking. Yet, most known bifurcation results related to delay-coupled swarms rely on inaccurate mean-field techniques. As a consequence, the utility of applying macroscopic theory as a guide for predicting and controlling swarms of mobile robots has been limited. To overcome this limitation, we perform an exact stability analysis of two primary swarming patterns in a general model with time-delayed interactions. By correctly identifying the relevant spatio-temporal modes, we are able to accurately predict unstable oscillations beyond the mean-field dynamics and bistability in large swarms-laying the groundwork for comparisons to robotics experiments.
Motivated by recent epidemic outbreaks, including those of COVID-19, we solve the canonical probl... more Motivated by recent epidemic outbreaks, including those of COVID-19, we solve the canonical problem of calculating the dynamics and likelihood of extensive outbreaks in a population within a large class of stochastic epidemic models with demographic noise, including the Susceptible-Infected-Recovered (SIR) model and its general extensions. In the limit of large populations, we compute the probability distribution for all extensive outbreaks, including those that entail unusually large or small (extreme) proportions of the population infected. Our approach reveals that, unlike other well-known examples of rare events occurring in discrete-state stochastic systems, the statistics of extreme outbreaks emanate from a full continuum of Hamiltonian paths, each satisfying unique boundary conditions with a conserved probability flux.
Without vaccines and treatments, societies must rely on non-pharmaceutical intervention strategie... more Without vaccines and treatments, societies must rely on non-pharmaceutical intervention strategies to control the spread of emerging diseases such as COVID-19. Though complete lockdown is epidemiologically effective, because it eliminates infectious contacts, it comes with significant costs. Several recent studies have suggested that a plausible compromise strategy for minimizing epidemic risk is periodic closure, in which populations oscillate between wide-spread social restrictions and relaxation. However, no underlying theory has been proposed to predict and explain optimal closure periods as a function of epidemiological and social parameters. In this work we develop such an analytical theory for SEIR-like model diseases, showing how characteristic closure periods emerge that minimize the total outbreak, and increase predictably with the reproductive number and incubation periods of a disease-as long as both are within predictable limits. Using our approach we demonstrate a sweet-spot effect in which optimal periodic closure is maximally effective for diseases with similar incubation and recovery periods. Our results compare well to numerical simulations, including in COVID-19 models where infectivity and recovery show significant variation.
During an epidemic individual nodes in a network may adapt their connections to reduce the chance... more During an epidemic individual nodes in a network may adapt their connections to reduce the chance of infection. A common form of adaption is avoidance rewiring, where a non-infected node breaks a connection to an infected neighbor, and forms a new connection to another non-infected node. Here we explore the effects of such adaptivity on stochastic fluctuations in the susceptibleinfected-susceptible model, focusing on the largest fluctuations that result in extinction of infection. Using techniques from large-deviation theory, combined with a measurement of heterogeneity in the susceptible degree distribution at the endemic state, we are able to predict and analyze large fluctuations and extinction in adaptive networks. We find that in the limit of small rewiring there is a sharp exponential reduction in mean extinction times compared to the case of zero-adaption. Furthermore, we find an exponential enhancement in the probability of large fluctuations with increased rewiring rate, even when holding the average number of infected nodes constant.
We present a general theory of stochastic model reduction which is based on a normal form coordin... more We present a general theory of stochastic model reduction which is based on a normal form coordinate transform method of A.J. Roberts. This nonlinear, stochastic projection allows for the deterministic and stochastic dynamics to interact correctly on the lower-dimensional manifold so that the dynamics predicted by the reduced, stochastic system agrees well with the dynamics predicted by the original, high-dimensional stochastic system. The method may be applied to any system with well-separated time scales. In this article, we consider a physical problem that involves a singularly perturbed Duffing oscillator as well as a biological problem that involves the prediction of infectious disease outbreaks.
We model recruitment in adaptive social networks in the presence of birth and death processes. Re... more We model recruitment in adaptive social networks in the presence of birth and death processes. Recruitment is characterized by nodes changing their status to that of the recruiting class as a result of contact with recruiting nodes. Only a susceptible subset of nodes can be recruited. The recruiting individuals may adapt their connections in order to improve recruitment capabilities, thus changing the network structure adaptively. We derive a mean field theory to predict the dependence of the growth threshold of the recruiting class on the adaptation parameter. Furthermore, we investigate the effect of adaptation on the recruitment level, as well as on network topology. The theoretical predictions are compared with direct simulations of the full system. We identify two parameter regimes with qualitatively different bifurcation diagrams depending on whether nodes become susceptible frequently (multiple times in their lifetime) or rarely (much less than once per lifetime).
Multi-strain diseases are diseases that consist of several strains, or serotypes. The serotypes m... more Multi-strain 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.
study the effects of time dependent noise and discrete, randomly distributed time delays on the d... more study the effects of time dependent noise and discrete, randomly distributed time delays on the dynamics of a large coupled system of self-propelling particles. Bifurcation analysis on a mean field approximation of the system reveals that the system possesses patterns with certain universal characteristics that depend on distinguished moments of the time delay distribution. We show both theoretically and numerically that although bifurcations of simple patterns, such as translations, change stability only as a function of the first moment of the time delay distribution, more complex bifurcating patterns depend on all of the moments of the delay distribution. In addition, we show that for sufficiently large values of the coupling strength and/or the mean time delay, there is a noise intensity threshold, dependent on the delay distribution width, 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.
Submitted for the MAR08 Meeting of The American Physical Society Enhancement of epidemic extincti... more Submitted for the MAR08 Meeting of The American Physical Society Enhancement of epidemic extinction by random vaccination 1 IRA SCHWARTZ, Naval Research Laboratory, MARK DYKMAN, Michigan State University-We study the probability of epidemic extinction in large populations. We use the susceptible-infected-susceptible (SIS) model since it forms the foundation of many epidemic processes. Fluctuations in the SIS system have two sources. The major source is the randomness of the "reactions" in which the number of susceptibles and/or infected changes. In addition, we assume that vaccination is done at random, leading to the decrease of the number of susceptibles. The vaccination is modeled by a Poisson process. The probability distribution is found from the master equation, which is solved in the eikonal approximation. It is shown that, even in the absence of vaccination, the logarithm of the extinction rate displays scaling dependence on the parameters. It scales as the square of the distance to the parameter value where the average number of infected vanishes. This is very different from the familiar 3/2 scaling law for saddle-node bifurcations. Finally, we show that even weak vaccination can dramatically increase the extinction probability. The correction to the logarithm of the probability becomes exponential in the vaccination rate when this rate is not too small.
2015 18th International Conference on Information Fusion (Fusion), 2015
Real networks consisting of social contacts do not possess static connections. That is, social co... more Real networks consisting of social contacts do not possess static connections. That is, social connections may be time dependent due to a variety of individual behavioral decisions based on current network links between people. Examples of adaptive networks occur in epidemics, where information about infectious individuals may change the rewiring of healthy people, or in the recruitment of individuals to a cause or fad, where rewiring may optimize recruitment of susceptible individuals. In this talk, we will review some of the dynamical properties of adaptive and random networks, such as bifurcation structure and the size of fluctuations. We will also show how adaptive networks predict novel phenomena as well as yield insight into new controls. Applying a new transition rate approximation that incorporates link dynamics, we extend the theory of large deviations to stochastic network extinction to predict extinction times. In particular, we use the theory to find the most probable pa...
Bulletin of the American Physical Society, Mar 17, 2016
present the development of a stochastic control strategy that leverages the environmental dynamic... more present the development of a stochastic control strategy that leverages the environmental dynamics and uncertainty to navigate in a stochastic fluidic environment. We assume that the domain is composed of the union of a collection of disjoint regions, each bounded by Lagrangian coherent structures (LCSs). We analyze a passive particle's noise-induced transition between adjacent LCS-bounded regions and show how most probable escape trajectories with respect to the transition probability between adjacent LCS-bounded regions can be determined. Additionally, we show how the likelihood of transition can be controlled through minimal actuation. The result is an energy efficient navigation strategy that leverages the inherent uncertainty of the surrounding flow field for controlling sensors in a noisy fluidic environment. We experimentally validate the proposed control strategy and show that the single vehicle control parameter exhibits a predictable exponential scaling with respect to the escape times and is effective even in situations where the structure of the flow is not fully known and control effort is costly.
Extinction of an epidemic or a species is a rare event that occurs due to a large, rare stochasti... more 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. Stochastic dynamical systems and Lyapunov exponents • Optimal path to extinction 1 Introduction Control and eradication of infectious diseases are among the most important goals for improving public health. Although the global eradication of a disease (e.g. smallpox
Bulletin of the American Physical Society, Mar 17, 2017
Mixed reality framework for collective motion patterns of swarms with delay coupling 1 KLEMENTYNA... more Mixed reality framework for collective motion patterns of swarms with delay coupling 1 KLEMENTYNA SZWAYKOWSKA, IRA SCHWARTZ, Naval Research Lab -The formation of coherent patterns in swarms of interacting self-propelled autonomous agents is an important subject for many applications within the field of distributed robotic systems. However, there are significant logistical challenges associated with testing fully distributed systems in real-world settings. In this paper, we provide a rigorous theoretical justification for the use of mixed-reality experiments as a stepping stone to fully physical testing of distributed robotic systems. We also model and experimentally realize a mixed-reality largescale swarm of delay-coupled agents. Our analyses, assuming agents communicating over an Erdos-Renyi network, demonstrate the existence of stable coherent patterns that can be achieved only with delay coupling and that are robust to decreasing network connectivity and heterogeneity in agent dynamics. We show how the bifurcation structure for emergence of different patterns changes with heterogeneity in agent acceleration capabilities and limited connectivity in the network as a function of coupling strength and delay. Our results are verified through simulation as well as preliminary experimental results of delay-induced pattern formation in a mixed-reality swarm.
Bulletin of the American Physical Society, Mar 23, 2005
The serotypes interact by antibody-dependent enhancement (ADE), in which infection with a single ... more 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 serious illness accompanied by greater infectivity. It has been observed from serotype data that outbreaks of the four serotypes occur asynchronously (Nisalak et al., Am. J. Trop. Med. Hyg. 68: 192). We developed a compartmental model and did bifurcation analysis for multiple serotypes with ADE. Both autonomous and seasonally driven versions were studied. For sufficiently small ADE, we find that the number of infectives of each serotype synchronizes, with outbreaks occurring in phase. However, when the ADE increases past a threshold, the system becomes chaotic, and infectives of each serotype desynchronize.
In some physical and biological swarms, agents effectively move and interact along curved surface... more In some physical and biological swarms, agents effectively move and interact along curved surfaces. The associated constraints and symmetries can affect collective-motion patterns, but little is known about pattern stability in the presence of surface curvature. To make progress, we construct a general model for self-propelled swarms moving on surfaces using Lagrangian mechanics. We find that the combination of self-propulsion, friction, mutual attraction, and surface curvature produce milling patterns where each agent in a swarm oscillates on a limit cycle, with different agents splayed along the cycle such that the swarm's center-of-mass remains stationary. In general, such patterns loose stability when mutual attraction is insufficient to overcome the constraint of curvature, and we uncover two broad classes of stationary milling-state bifurcations. In the first, a spatially periodic mode undergoes a Hopf bifurcation as curvature is increased which results in unstable spatiotemporal oscillations. This generic bifurcation is analyzed for the sphere and demonstrated numerically for several surfaces. In the second, a saddle-node-of-periodic-orbits occurs in which stable and unstable milling states collide and annihilate. The latter is analyzed for milling states on cylindrical surfaces. Our results contribute to the general understanding of swarm pattern-formation and stability in the presence of surface curvature, and may aid in designing robotic swarms that can be controlled to move over complex surfaces. I.
We consider the stochastic patterns of a system of communicating, or coupled, self-propelled part... more 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.
International Journal of Bifurcation and Chaos, Jun 1, 1993
A new class of "toy models" for subaqueous bedform formation are proposed and examined. These mod... more A new class of "toy models" for subaqueous bedform formation are proposed and examined. These models all show a similar mechanism of wavelength selection via bedform unification, and they may have applications to bedform stratigraphy. The models are also useful for exploring general issues of pattern formation and complexity in stochastically driven far from equilibrium systems.
We study rare events in networks with both internal and external noise, and develop a general for... more We study rare events in networks with both internal and external noise, and develop a general formalism for analyzing rare events that combines pair-quenched techniques and largedeviation theory. The probability distribution, shape, and time scale of rare events are considered in detail for extinction in the Susceptible-Infected-Susceptible model as an illustration. We find that when both types of noise are present, there is a cross-over region as the network size is increased, where the probability exponent for large deviations no longer increases linearly with the network size. We demonstrate that the form of the cross-over depends on whether the endemic state is localized near the epidemic threshold or not.
It is known that introducing time delays into the communication network of mobile-agent swarms pr... more It is known that introducing time delays into the communication network of mobile-agent swarms produces coherent rotational patterns, from both theory and experiments. Often such spatiotemporal rotations can be bistable with other swarming patterns, such as milling and flocking. Yet, most known bifurcation results related to delay-coupled swarms rely on inaccurate mean-field techniques. As a consequence, the utility of applying macroscopic theory as a guide for predicting and controlling swarms of mobile robots has been limited. To overcome this limitation, we perform an exact stability analysis of two primary swarming patterns in a general model with time-delayed interactions. By correctly identifying the relevant spatio-temporal modes, we are able to accurately predict unstable oscillations beyond the mean-field dynamics and bistability in large swarms-laying the groundwork for comparisons to robotics experiments.
Motivated by recent epidemic outbreaks, including those of COVID-19, we solve the canonical probl... more Motivated by recent epidemic outbreaks, including those of COVID-19, we solve the canonical problem of calculating the dynamics and likelihood of extensive outbreaks in a population within a large class of stochastic epidemic models with demographic noise, including the Susceptible-Infected-Recovered (SIR) model and its general extensions. In the limit of large populations, we compute the probability distribution for all extensive outbreaks, including those that entail unusually large or small (extreme) proportions of the population infected. Our approach reveals that, unlike other well-known examples of rare events occurring in discrete-state stochastic systems, the statistics of extreme outbreaks emanate from a full continuum of Hamiltonian paths, each satisfying unique boundary conditions with a conserved probability flux.
Without vaccines and treatments, societies must rely on non-pharmaceutical intervention strategie... more Without vaccines and treatments, societies must rely on non-pharmaceutical intervention strategies to control the spread of emerging diseases such as COVID-19. Though complete lockdown is epidemiologically effective, because it eliminates infectious contacts, it comes with significant costs. Several recent studies have suggested that a plausible compromise strategy for minimizing epidemic risk is periodic closure, in which populations oscillate between wide-spread social restrictions and relaxation. However, no underlying theory has been proposed to predict and explain optimal closure periods as a function of epidemiological and social parameters. In this work we develop such an analytical theory for SEIR-like model diseases, showing how characteristic closure periods emerge that minimize the total outbreak, and increase predictably with the reproductive number and incubation periods of a disease-as long as both are within predictable limits. Using our approach we demonstrate a sweet-spot effect in which optimal periodic closure is maximally effective for diseases with similar incubation and recovery periods. Our results compare well to numerical simulations, including in COVID-19 models where infectivity and recovery show significant variation.
During an epidemic individual nodes in a network may adapt their connections to reduce the chance... more During an epidemic individual nodes in a network may adapt their connections to reduce the chance of infection. A common form of adaption is avoidance rewiring, where a non-infected node breaks a connection to an infected neighbor, and forms a new connection to another non-infected node. Here we explore the effects of such adaptivity on stochastic fluctuations in the susceptibleinfected-susceptible model, focusing on the largest fluctuations that result in extinction of infection. Using techniques from large-deviation theory, combined with a measurement of heterogeneity in the susceptible degree distribution at the endemic state, we are able to predict and analyze large fluctuations and extinction in adaptive networks. We find that in the limit of small rewiring there is a sharp exponential reduction in mean extinction times compared to the case of zero-adaption. Furthermore, we find an exponential enhancement in the probability of large fluctuations with increased rewiring rate, even when holding the average number of infected nodes constant.
We present a general theory of stochastic model reduction which is based on a normal form coordin... more We present a general theory of stochastic model reduction which is based on a normal form coordinate transform method of A.J. Roberts. This nonlinear, stochastic projection allows for the deterministic and stochastic dynamics to interact correctly on the lower-dimensional manifold so that the dynamics predicted by the reduced, stochastic system agrees well with the dynamics predicted by the original, high-dimensional stochastic system. The method may be applied to any system with well-separated time scales. In this article, we consider a physical problem that involves a singularly perturbed Duffing oscillator as well as a biological problem that involves the prediction of infectious disease outbreaks.
We model recruitment in adaptive social networks in the presence of birth and death processes. Re... more We model recruitment in adaptive social networks in the presence of birth and death processes. Recruitment is characterized by nodes changing their status to that of the recruiting class as a result of contact with recruiting nodes. Only a susceptible subset of nodes can be recruited. The recruiting individuals may adapt their connections in order to improve recruitment capabilities, thus changing the network structure adaptively. We derive a mean field theory to predict the dependence of the growth threshold of the recruiting class on the adaptation parameter. Furthermore, we investigate the effect of adaptation on the recruitment level, as well as on network topology. The theoretical predictions are compared with direct simulations of the full system. We identify two parameter regimes with qualitatively different bifurcation diagrams depending on whether nodes become susceptible frequently (multiple times in their lifetime) or rarely (much less than once per lifetime).
Multi-strain diseases are diseases that consist of several strains, or serotypes. The serotypes m... more Multi-strain 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.
study the effects of time dependent noise and discrete, randomly distributed time delays on the d... more study the effects of time dependent noise and discrete, randomly distributed time delays on the dynamics of a large coupled system of self-propelling particles. Bifurcation analysis on a mean field approximation of the system reveals that the system possesses patterns with certain universal characteristics that depend on distinguished moments of the time delay distribution. We show both theoretically and numerically that although bifurcations of simple patterns, such as translations, change stability only as a function of the first moment of the time delay distribution, more complex bifurcating patterns depend on all of the moments of the delay distribution. In addition, we show that for sufficiently large values of the coupling strength and/or the mean time delay, there is a noise intensity threshold, dependent on the delay distribution width, 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.
Submitted for the MAR08 Meeting of The American Physical Society Enhancement of epidemic extincti... more Submitted for the MAR08 Meeting of The American Physical Society Enhancement of epidemic extinction by random vaccination 1 IRA SCHWARTZ, Naval Research Laboratory, MARK DYKMAN, Michigan State University-We study the probability of epidemic extinction in large populations. We use the susceptible-infected-susceptible (SIS) model since it forms the foundation of many epidemic processes. Fluctuations in the SIS system have two sources. The major source is the randomness of the "reactions" in which the number of susceptibles and/or infected changes. In addition, we assume that vaccination is done at random, leading to the decrease of the number of susceptibles. The vaccination is modeled by a Poisson process. The probability distribution is found from the master equation, which is solved in the eikonal approximation. It is shown that, even in the absence of vaccination, the logarithm of the extinction rate displays scaling dependence on the parameters. It scales as the square of the distance to the parameter value where the average number of infected vanishes. This is very different from the familiar 3/2 scaling law for saddle-node bifurcations. Finally, we show that even weak vaccination can dramatically increase the extinction probability. The correction to the logarithm of the probability becomes exponential in the vaccination rate when this rate is not too small.
2015 18th International Conference on Information Fusion (Fusion), 2015
Real networks consisting of social contacts do not possess static connections. That is, social co... more Real networks consisting of social contacts do not possess static connections. That is, social connections may be time dependent due to a variety of individual behavioral decisions based on current network links between people. Examples of adaptive networks occur in epidemics, where information about infectious individuals may change the rewiring of healthy people, or in the recruitment of individuals to a cause or fad, where rewiring may optimize recruitment of susceptible individuals. In this talk, we will review some of the dynamical properties of adaptive and random networks, such as bifurcation structure and the size of fluctuations. We will also show how adaptive networks predict novel phenomena as well as yield insight into new controls. Applying a new transition rate approximation that incorporates link dynamics, we extend the theory of large deviations to stochastic network extinction to predict extinction times. In particular, we use the theory to find the most probable pa...
Bulletin of the American Physical Society, Mar 17, 2016
present the development of a stochastic control strategy that leverages the environmental dynamic... more present the development of a stochastic control strategy that leverages the environmental dynamics and uncertainty to navigate in a stochastic fluidic environment. We assume that the domain is composed of the union of a collection of disjoint regions, each bounded by Lagrangian coherent structures (LCSs). We analyze a passive particle's noise-induced transition between adjacent LCS-bounded regions and show how most probable escape trajectories with respect to the transition probability between adjacent LCS-bounded regions can be determined. Additionally, we show how the likelihood of transition can be controlled through minimal actuation. The result is an energy efficient navigation strategy that leverages the inherent uncertainty of the surrounding flow field for controlling sensors in a noisy fluidic environment. We experimentally validate the proposed control strategy and show that the single vehicle control parameter exhibits a predictable exponential scaling with respect to the escape times and is effective even in situations where the structure of the flow is not fully known and control effort is costly.
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Papers by Ira Schwartz