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Lora Billings

Montclair State University, Mathematical Sciences, Faculty Member

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California Academy of Sciences

The University of the West Indies at St. Augustine Trinidad and Tobago

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Papers by Lora Billings

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

by Lora Billings and Ira Schwartz

Proceedings of the National Academy of Sciences

Antibody-dependent enhancement (ADE), a phenomenon in which viral replication is increased rather... more 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 leve...

Transition to Chaos in Continuous-Time Random Dynamical Systems

Physical Review Letters, 2002

We consider situations where, in a continuous-time dynamical system, a nonchaotic attractor coexi... more 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

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

by Lora Billings and Erik Bollt

Physical review letters, Jan 10, 2002

A general way to classify stochastic chaos is presented and applied to population dynamics models... more 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.

Switching Exponent Scaling near Bifurcation Points for Non-Gaussian Noise

Physical Review Letters, 2010

We study noise-induced switching of a system close to bifurcation parameter values where the numb... more 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.

Instabilities in multiserotype disease models with antibody-dependent enhancement

This paper investigates the complex dynamics induced by antibody-dependent enhancement (ADE) in m... more 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. r

Stochastic bifurcation in a driven laser system: experiment and theory

by Lora Billings, Erik Bollt, and D. Morgan

Physical review. E, Statistical, nonlinear, and soft matter physics, 2004

We analyze the effects of stochastic perturbations in a physical example occurring as a higher-di... more 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 approxima...

<title>Stochastic epidemic outbreaks: why epidemics are like lasers</title>

by Lora Billings, Thomas Carr, and Erik Bollt

Noise in Complex Systems and Stochastic Dynamics II, 2004

Many diseases, such childhood diseases, dengue fever, and West Nile virus, appear to oscillate ra... more Many diseases, such childhood diseases, dengue fever, and West Nile virus, appear to oscillate randomly as a function of seasonal environmental or social changes. Such oscillations appear to have a chaotic bursting character, although it is still uncertain how much is due to random fluctuations. Such bursting in the presence of noise is also observed in driven lasers. In this talk, I will show how noise can excite random outbreaks in simple models of seasonally driven outbreaks, as well as lasers. The models for both population dynamics will be shown to share the same class of underlying topology, which plays a major role in the cause of observed stochastic bursting.

Noise-induced switching and extinction in systems with delay

Physical Review E, 2015

We consider the rates of noise-induced switching between the stable states of dissipative dynamic... more 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.

3. Chemical and Biological Sensing: Modeling and Analysis from the Real World

Bioterrorism: Mathematical Modeling Applications in Homeland Security, 2003

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

Physical Review E - PHYS REV E, 2003

Results are reported concerning the transition to chaos in random dynamical systems. In particula... more 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 Lyapu...

Dynamic and geometric analysis of short time series: a new comparative approach to cell-based biosensors

Physics Letters A, 2001

This Letter describes two approaches for sensing changes in spiking cells when only a limited amo... more This Letter describes two approaches for sensing changes in spiking cells when only a limited amount of spike data is available. The first method detects changes in dynamically constructed local expansion rates, and the other uses spike area distributions. These two methods were tested on time series from cultured neurons. Over-sampled data was generated from experiments on single cells before and after being treated with a small concentration of channel blocker, but relatively few spikes were recorded. In the spontaneously spiking cells, the local expansion rates showed a sensitivity that correlated with the channel concentration level, while the driven cells showed no such correlation. Spike area distributions 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. Published by Elsevier Science B.V. (I.B. Schwartz). 0375-9601/01/$ -see front matter Published by Elsevier Science B.V. PII: S 0 3 7 5 -9 6 0 1 ( 0 1 ) 0 0 4 0 0 -5

Lyapunov Exponents, Singularities, and a Riddling Bifurcation

Physical Review Letters, 1997

There are few examples in dynamical systems theory which lend themselves to exact computations of... more There are few examples in dynamical systems theory which lend themselves to exact computations of macroscopic variables of interest. One such variable is the Lyapunov exponent which measures the average attraction of an invariant set. This article presents .a family of noninvertible transformations of the plane for which such computations are possible. This model sheds additional insight into the notion of what it can mean for an attracting invariant set to have a riddled basin of attraction.

Chaotic desynchronization of multistrain diseases

Physical Review E, 2005

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.

Thermally activated switching in the presence of non-Gaussian noise

Physical Review E, 2008

We study the effect of a non-Gaussian noise on interstate switching activated primarily by Gaussi... more 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. The explicit 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.

Dynamical epidemic suppression using stochastic prediction and control

by Lora Billings and Erik Bollt

Physical Review E, 2004

We consider the effects of noise on a model of epidemic outbreaks, where the outbreaks appear ran... more

$FIG. 1: This is an uncontrolled time series of the fraction of infectives J for the MSI model under period 2 and large (LA) period 3 amplitude oscillations of the underlying bi-stable deterministic$

$G. 2: (a) An uncontrolled time series of infective fraction as a function of time. (b) Constan$

A manifold independent approach to understanding transport in stochastic dynamical systems

by Lora Billings and Erik Bollt

Physica D: Nonlinear Phenomena, 2002

We develop a new collection of tools aimed at studying stochastically perturbed dynamical systems... more We develop a new collection of tools aimed at studying stochastically perturbed dynamical systems. Specifically, in the setting of bi-stability, that is a two-attractor system, it has previously been numerically observed that a small noise volume is sufficient to destroy would be zero-noise case barriers in the phase space (pseudo-barriers), thus creating a pre-heteroclinic tangency chaos-like behavior. The stochastic dynamical system has a corresponding Frobenius-Perron operator with a stochastic kernel, which describes how densities of initial conditions move under the noisy map. Thus in studying the action of the Frobenius-Perron operator, we learn about the transport of the map; we have employed a Galerkin-Ulam-like method to project the Frobenius-Perron operator onto a discrete basis set of characteristic functions to highlight this action localized in specified regions of the phase space. Graph theoretic methods allow us to re-order the resulting finite dimensional Markov operator approximation so as to highlight the regions of the original phase space which are particularly active pseudo-barriers of the stochastic dynamics. Our toolbox allows us to find: (1) regions of high activity of transport, (2) flux across pseudo-barriers, and also (3) expected time of escape from pseudo-basins. Some of these quantities are also possible via the manifold dependent stochastic Melnikov method, but Melnikov only applies to a very special class of models for which the unperturbed homoclinic orbit is available. Our methods are unique in that they can essentially be considered as a "black-box" of tools which can be applied to a wide range of stochastic dynamical systems in the absence of a priori knowledge of manifold structures. We use here a model of childhood diseases to showcase our methods. Our tools will allow us to make specific observations of: (1) loss of reducibility between basins with increasing noise, (2) identification in the phase space of active regions of stochastic transport, (3) stochastic flux which essentially completes the heteroclinic tangle.

Global manifold control in a driven laser: sustaining chaos and regular dynamics

Physica D: Nonlinear Phenomena, 2004

We present experimental and numerical evidence of a multi-frequency phase control able to preserv... more We present experimental and numerical evidence of a multi-frequency phase control able to preserve periodic behavior within a chaotic window as well as to re-excite chaotic behavior when it is destroyed by the presence of a mitigating unstable periodic orbit created in the presence of the multi-frequency drive. The mitigating saddle controlling the global behavior is identified and the controlling manifolds approximated.

Bi-instability and the global role of unstable resonant orbits in a driven laser

Physica D: Nonlinear Phenomena, 2000

Driven class-B lasers are devices which possess quadratic nonlinearities and are known to exhibit... more Driven class-B lasers are devices which possess quadratic nonlinearities and are known to exhibit chaotic behavior. We describe the onset of global heteroclinic connections which give rise to chaotic saddles. These form the precursor topology which creates both localized homoclinic chaos, as well as global mixed-mode heteroclinic chaos. To locate the relevant periodic orbits creating the precursor topology, approximate maps are derived using matched asymptotic expansions and subharmonic Melnikov theory. Locating the relevant unstable fixed points of the maps provides an organizing framework to understand the global dynamics and chaos exhibited by the laser.

Disease persistence in epidemiological models: The interplay between vaccination and migration

Mathematical Biosciences, 2012

We consider the interplay of vaccination and migration rates on disease persistence in epidemiolo... more We consider the interplay of vaccination and migration rates on disease persistence in epidemiological systems. We show that short-term and long-term migration can inhibit disease persistence. As a result, we show how migration changes how vaccination rates should be chosen to maintain herd immunity. In a system of coupled SIR models, we analyze how disease eradication depends explicitly on vaccine distribution and migration connectivity. The analysis suggests potentially novel vaccination policies that underscore the importance of optimal placement of finite resources.

Vaccinations in disease models with antibody-dependent enhancement

Mathematical Biosciences, 2008

This paper examines the effects of single-strain vaccine campaigns on the dynamics of an epidemic... more 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.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.

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

by Lora Billings and Ira Schwartz

Proceedings of the National Academy of Sciences

Antibody-dependent enhancement (ADE), a phenomenon in which viral replication is increased rather... more 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 leve...

Transition to Chaos in Continuous-Time Random Dynamical Systems

Physical Review Letters, 2002

We consider situations where, in a continuous-time dynamical system, a nonchaotic attractor coexi... more 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

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

by Lora Billings and Erik Bollt

Physical review letters, Jan 10, 2002

A general way to classify stochastic chaos is presented and applied to population dynamics models... more 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.

Switching Exponent Scaling near Bifurcation Points for Non-Gaussian Noise

Physical Review Letters, 2010

We study noise-induced switching of a system close to bifurcation parameter values where the numb... more 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.

Instabilities in multiserotype disease models with antibody-dependent enhancement

This paper investigates the complex dynamics induced by antibody-dependent enhancement (ADE) in m... more 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. r

Stochastic bifurcation in a driven laser system: experiment and theory

by Lora Billings, Erik Bollt, and D. Morgan

Physical review. E, Statistical, nonlinear, and soft matter physics, 2004

We analyze the effects of stochastic perturbations in a physical example occurring as a higher-di... more 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 approxima...

<title>Stochastic epidemic outbreaks: why epidemics are like lasers</title>

by Lora Billings, Thomas Carr, and Erik Bollt

Noise in Complex Systems and Stochastic Dynamics II, 2004

Many diseases, such childhood diseases, dengue fever, and West Nile virus, appear to oscillate ra... more Many diseases, such childhood diseases, dengue fever, and West Nile virus, appear to oscillate randomly as a function of seasonal environmental or social changes. Such oscillations appear to have a chaotic bursting character, although it is still uncertain how much is due to random fluctuations. Such bursting in the presence of noise is also observed in driven lasers. In this talk, I will show how noise can excite random outbreaks in simple models of seasonally driven outbreaks, as well as lasers. The models for both population dynamics will be shown to share the same class of underlying topology, which plays a major role in the cause of observed stochastic bursting.

Noise-induced switching and extinction in systems with delay

Physical Review E, 2015

We consider the rates of noise-induced switching between the stable states of dissipative dynamic... more 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.

3. Chemical and Biological Sensing: Modeling and Analysis from the Real World

Bioterrorism: Mathematical Modeling Applications in Homeland Security, 2003

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

Physical Review E - PHYS REV E, 2003

Results are reported concerning the transition to chaos in random dynamical systems. In particula... more 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 Lyapu...

Dynamic and geometric analysis of short time series: a new comparative approach to cell-based biosensors

Physics Letters A, 2001

This Letter describes two approaches for sensing changes in spiking cells when only a limited amo... more This Letter describes two approaches for sensing changes in spiking cells when only a limited amount of spike data is available. The first method detects changes in dynamically constructed local expansion rates, and the other uses spike area distributions. These two methods were tested on time series from cultured neurons. Over-sampled data was generated from experiments on single cells before and after being treated with a small concentration of channel blocker, but relatively few spikes were recorded. In the spontaneously spiking cells, the local expansion rates showed a sensitivity that correlated with the channel concentration level, while the driven cells showed no such correlation. Spike area distributions 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. Published by Elsevier Science B.V. (I.B. Schwartz). 0375-9601/01/$ -see front matter Published by Elsevier Science B.V. PII: S 0 3 7 5 -9 6 0 1 ( 0 1 ) 0 0 4 0 0 -5

Lyapunov Exponents, Singularities, and a Riddling Bifurcation

Physical Review Letters, 1997

There are few examples in dynamical systems theory which lend themselves to exact computations of... more There are few examples in dynamical systems theory which lend themselves to exact computations of macroscopic variables of interest. One such variable is the Lyapunov exponent which measures the average attraction of an invariant set. This article presents .a family of noninvertible transformations of the plane for which such computations are possible. This model sheds additional insight into the notion of what it can mean for an attracting invariant set to have a riddled basin of attraction.

Chaotic desynchronization of multistrain diseases

Physical Review E, 2005

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.

Thermally activated switching in the presence of non-Gaussian noise

Physical Review E, 2008

We study the effect of a non-Gaussian noise on interstate switching activated primarily by Gaussi... more 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. The explicit 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.

Dynamical epidemic suppression using stochastic prediction and control

by Lora Billings and Erik Bollt

Physical Review E, 2004

We consider the effects of noise on a model of epidemic outbreaks, where the outbreaks appear ran... more

$FIG. 1: This is an uncontrolled time series of the fraction of infectives J for the MSI model under period 2 and large (LA) period 3 amplitude oscillations of the underlying bi-stable deterministic$

$G. 2: (a) An uncontrolled time series of infective fraction as a function of time. (b) Constan$

A manifold independent approach to understanding transport in stochastic dynamical systems

by Lora Billings and Erik Bollt

Physica D: Nonlinear Phenomena, 2002

We develop a new collection of tools aimed at studying stochastically perturbed dynamical systems... more We develop a new collection of tools aimed at studying stochastically perturbed dynamical systems. Specifically, in the setting of bi-stability, that is a two-attractor system, it has previously been numerically observed that a small noise volume is sufficient to destroy would be zero-noise case barriers in the phase space (pseudo-barriers), thus creating a pre-heteroclinic tangency chaos-like behavior. The stochastic dynamical system has a corresponding Frobenius-Perron operator with a stochastic kernel, which describes how densities of initial conditions move under the noisy map. Thus in studying the action of the Frobenius-Perron operator, we learn about the transport of the map; we have employed a Galerkin-Ulam-like method to project the Frobenius-Perron operator onto a discrete basis set of characteristic functions to highlight this action localized in specified regions of the phase space. Graph theoretic methods allow us to re-order the resulting finite dimensional Markov operator approximation so as to highlight the regions of the original phase space which are particularly active pseudo-barriers of the stochastic dynamics. Our toolbox allows us to find: (1) regions of high activity of transport, (2) flux across pseudo-barriers, and also (3) expected time of escape from pseudo-basins. Some of these quantities are also possible via the manifold dependent stochastic Melnikov method, but Melnikov only applies to a very special class of models for which the unperturbed homoclinic orbit is available. Our methods are unique in that they can essentially be considered as a "black-box" of tools which can be applied to a wide range of stochastic dynamical systems in the absence of a priori knowledge of manifold structures. We use here a model of childhood diseases to showcase our methods. Our tools will allow us to make specific observations of: (1) loss of reducibility between basins with increasing noise, (2) identification in the phase space of active regions of stochastic transport, (3) stochastic flux which essentially completes the heteroclinic tangle.

Global manifold control in a driven laser: sustaining chaos and regular dynamics

Physica D: Nonlinear Phenomena, 2004

We present experimental and numerical evidence of a multi-frequency phase control able to preserv... more We present experimental and numerical evidence of a multi-frequency phase control able to preserve periodic behavior within a chaotic window as well as to re-excite chaotic behavior when it is destroyed by the presence of a mitigating unstable periodic orbit created in the presence of the multi-frequency drive. The mitigating saddle controlling the global behavior is identified and the controlling manifolds approximated.

Bi-instability and the global role of unstable resonant orbits in a driven laser

Physica D: Nonlinear Phenomena, 2000

Driven class-B lasers are devices which possess quadratic nonlinearities and are known to exhibit... more Driven class-B lasers are devices which possess quadratic nonlinearities and are known to exhibit chaotic behavior. We describe the onset of global heteroclinic connections which give rise to chaotic saddles. These form the precursor topology which creates both localized homoclinic chaos, as well as global mixed-mode heteroclinic chaos. To locate the relevant periodic orbits creating the precursor topology, approximate maps are derived using matched asymptotic expansions and subharmonic Melnikov theory. Locating the relevant unstable fixed points of the maps provides an organizing framework to understand the global dynamics and chaos exhibited by the laser.

Disease persistence in epidemiological models: The interplay between vaccination and migration

Mathematical Biosciences, 2012

We consider the interplay of vaccination and migration rates on disease persistence in epidemiolo... more We consider the interplay of vaccination and migration rates on disease persistence in epidemiological systems. We show that short-term and long-term migration can inhibit disease persistence. As a result, we show how migration changes how vaccination rates should be chosen to maintain herd immunity. In a system of coupled SIR models, we analyze how disease eradication depends explicitly on vaccine distribution and migration connectivity. The analysis suggests potentially novel vaccination policies that underscore the importance of optimal placement of finite resources.

Vaccinations in disease models with antibody-dependent enhancement

Mathematical Biosciences, 2008

This paper examines the effects of single-strain vaccine campaigns on the dynamics of an epidemic... more 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.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.