Thomas W. Carr

Modeling of cancer virotherapy with recombinant measles viruses.

The Edmonston vaccine strain of measles virus has potent and selective activity against a wide range of tumors. Tumor cells infected by this virus or genetically modified strains express viral proteins that allow them to fuse with neighboring cells to form syncytia that ultimately die. Moreover, infected cells may produce new virus particles that proceed to infect additional tumor cells. We present a model of tumor and virus interactions based on established biology and with proper accounting of the free virus population. The range of model parameters is estimated by fitting to available experimental data. The stability of equilibrium states corresponding to complete tumor eradication, therapy failure and partial tumor reduction is discussed. We use numerical simulations to explore conditions for which the model predicts successful therapy and tumor eradication. The model exhibits damped, as well as stable oscillations in a range of parameter values. These oscillatory states are organized by a Hopf bifurcation.

Dynamics of multiple myeloma tumor therapy with a recombinant measles virus

Replication-competent viruses are being tested as tumor therapy agents. The fundamental premise of this therapy is the selective infection of the tumor cell population with the amplification of the virus. Spread of the virus in the tumor ultimately should lead to eradication of the cancer. Tumor virotherapy is unlike any other form of cancer therapy as the outcome depends on the dynamics that emerge from the interaction between the virus and tumor cell populations both of which change in time. We explore these interactions using a model that captures the salient biological features of this system in combination with in vivo data. Our results show that various therapeutic outcomes are possible ranging from tumor eradication to oscillatory behavior. Data from in vivo studies support these conclusions and validate our modeling approach. Such realistic models can be used to understand experimental observations, explore alternative therapeutic scenarios and develop techniques to optimize therapy.

An SIR epidemic model with partial temporary immunity modeled with delay

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

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

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.

Dimensionless rate equations and simple conditions for self-pulsing in laser diodes

Rate equations modeling self-pulsating laser diodes have been investigated numerically by various groups. In the paper, we formulate dimensionless equations; which unifies these independent studies. The low values of the decay rates of the carriers motivate analytical approximations for the domain of self-pulsation. These approximations highlight the effect of some physical processes (diffusion of the carriers, radiative recombination rate) which are important for self-pulsating diode lasers.

Oscillations in an Intra-host Model of Plasmodium Falciparum Malaria Due to Cross-reactive Immune Response

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

Near-threshold bursting is delayed by a slow passage near a limit point

In a general model for square-wave bursting oscillations, we examine the fast transition between the slowly varying quiescent and active phases. In this type of bursting, the transition occurs at a saddle-node (SN) bifurcation point of the fast-variable subsystem when the slow variable is taken to be the bifurcation parameter. A critical case occurs when the SN bifurcation point is also a steady solution of the full bursting system. In this case near the bursting threshold, the transition suffers a large delay. We propose a first investigation of this critical case that has been noted accidentally but never explored. We present an asymptoticanalysis local to the SN point of the fast subsystem and quantitatively describe the slow passage near the SN point underlying the transition delay. Our analysis reveals that bursting solutions showing the longest delays and, correspondingly, the bursting threshold appear near but not exactly at the SN point, as is commonly assumed.

Tracking controlled chaos: Theoretical foundations and applications.

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

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

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

Delayed‐Mutual Coupling Dynamics of Lasers: Scaling Laws and Resonances

We consider a model for two lasers that are mutually coupled optoelectronically by modulating the pump of one laser with the intensity deviations of the other. Signal propagation time in the optoelectronic loop causes a significant delay leading to the onset of oscillatory output. Multiscale perturbation methods are used to describe the amplitude and period of oscillations as a function of the coupling strength and delay time. For weak coupling the oscillations have the laser’s relaxation period, and the amplitude varies as the one‐fourth power of the parameter deviations from the bifurcation point. For order‐one coupling strength the period is determined as multiples of the delay time, and the amplitude varies with a square‐root power law. Because we allow for independent control of the individual coupling constants, for certain parameter values there is an atypical amplitude‐resonance phenomena. Finally, our theoretical results are consistent with recent experimental observations when the inclusion of a...