Control of rare events in reaction and population systems by deterministically imposed transitions.

We consider control of reaction and population systems by imposing transitions between states with different numbers of particles or individuals. The transitions take place at predetermined instants of time. Even where they are significantly less frequent than spontaneous transitions, they can exponentially strongly modify the rates of rare events, including switching between metastable states or population extinction. We also study optimal control of rare events. Specifically, we are interested in the optimal control of disease extinction for a limited vaccine supply. A comparison is made with control of rare events by modulating the rates of elementary transitions rather than imposing transitions deterministically. It is found that, unexpectedly, for the same mean control parameters, controlling the transitions rates can be more efficient.

Noise and controllability: suppression of controllability in large quantum systems.

A closed quantum system is defined as completely controllable if an arbitrary unitary transformation can be executed using the available controls. In practice, control fields are a source of unavoidable noise. Can one design control fields such that the effect of noise is negligible on the timescale of the transformation? Complete controllability in practice requires that the effect of noise can be suppressed for an arbitrary transformation. The present study considers a paradigm of control, where the Lie-algebraic structure of the control Hamiltonian is fixed, while the size of the system increases, determined by the dimension of the Hilbert space representation of the algebra. We show that for large quantum systems, generic noise in the controls dominates for a typical class of target transformation; i.e., complete controllability is destroyed by the noise.

Speeding up disease extinction with a limited amount of vaccine.

We consider optimal vaccination protocol where the vaccine is in short supply. In this case, the endemic state remains dynamically stable; disease extinction happens at random and requires a large fluctuation, which can come from the intrinsic randomness of the population dynamics. We show that vaccination can exponentially increase the disease extinction rate. For a time-periodic vaccination with fixed average rate, the optimal vaccination protocol is model independent and presents a sequence of short pulses. The effect can be resonantly enhanced if the vaccination pulse period coincides with the characteristic period of the disease dynamics or its multiples. This resonant effect is illustrated using a simple epidemic model. The analysis is based on the theory of fluctuation-induced population extinction in periodically modulated systems that we develop. If the system is strongly modulated (for example, by seasonal variations) and vaccination has the same period, the vaccination pulses must be properly synchronized; a wrong vaccination phase can impede disease extinction.

Time-resolved extinction rates of stochastic populations.

Extinction of a long-lived isolated stochastic population can be described as an exponentially slow decay of quasistationary probability distribution of the population size. We address extinction of a population in a two-population system in the case when the population turnover-renewal and removal--is much slower than all other processes. In this case there is a time-scale separation in the system which enables one to introduce a short-time quasistationary extinction rate W1 and a long-time quasistationary extinction rate W2, and to develop a time-dependent theory of the transition between the two rates. It is shown that W1 and W2 coincide with the extinction rates when the population turnover is absent and present, but very slow, respectively. The exponentially large disparity between the two rates reflects fragility of the extinction rate in the population dynamics without turnover.

Spectrum of an oscillator with jumping frequency and the interference of partial susceptibilities.

We study an underdamped oscillator with random frequency jumps. We describe the oscillator spectrum in terms of coupled susceptibilities for different-frequency states. Depending on the parameters, the spectrum has a fine structure or displays a single asymmetric peak. For nanomechanical resonators with a fluctuating number of attached molecules, it is found in a simple analytical form. The results bear on dephasing in various types of systems with jumping frequency.

Extinction rate fragility in population dynamics.

Population extinction is of central interest for population dynamics. It may occur from a large rare fluctuation. We find that, in contrast to related large-fluctuation effects like noise-induced interstate switching, quite generally extinction rates in multipopulation systems display fragility, where the height of the effective barrier to be overcome in the fluctuation depends on the system parameters nonanalytically. We show that one of the best-known models of epidemiology, the susceptible-infectious-susceptible model, is fragile to total population fluctuations.

Multiphase control of a nonlinear lattice.

Large amplitude, multiphase excitations of the periodic Toda lattice (n-gap solutions) are created and controlled by small forcing. The approach uses passage through an ensemble of resonances and subsequent multiphase self-locking of the system with adiabatic wave-like perturbations. The synchronization of each phase in the excited lattice proceeds from the weakly nonlinear stage, where the problem can be reduced to that for a number of independent, driven, one-degree-of-freedom oscillatory systems. Due to this separability, the phase locking at this stage is robust, provided the amplitude of the corresponding forcing component exceeds a threshold, which scales as 3/4 power of the corresponding frequency chirp rate. The adiabatic synchronization continues into a fully nonlinear stage, as the driven lattice self-adjusts its state to remain in a persisting and stable multifrequency resonance with the driving perturbation. Thus, a complete control of the n-gap state becomes possible by slow variation of external parameters.