Dynamics of SIR model with heterogeneous response to intervention policy.

In classical epidemic theory, behavior is assumed to be stationary. In recent years, epidemic models have been extended to include behaviors that transition in response to the current state of the epidemic. However, it is widely known that human behavior can exhibit strong history-dependence as a consequence of learned experiences. This history-dependence is similar to hysteresis phenomena that have been well-studied in control theory. To illustrate the importance of history-dependence for epidemic theory, we study dynamics of a variant of the SIRS model where individuals exhibit lazy-switch responses to prevalence dynamics, based on the Preisach hysteresis operator. The resulting model can possess a continuum of endemic equilibrium states characterized by different proportions of susceptible, infected and recovered populations. We consider how the limit point of the epidemic trajectory and the infection peak along this trajectory depend on the degree of heterogeneity of the response. Our approach supports the argument that public health responses during the emergence of a new disease can have fundamental long-term consequences for subsequent management efforts.

Dynamics of SIR model with vaccination and heterogeneous behavioral response of individuals modeled by the Preisach operator.

We study global dynamics of an SIR model with vaccination, where we assume that individuals respond differently to dynamics of the epidemic. Their heterogeneous response is modeled by the Preisach hysteresis operator. We present a condition for the global stability of the infection-free equilibrium state. If this condition does not hold true, the model has a connected set of endemic equilibrium states characterized by different proportion of infected and immune individuals. In this case, we show that every trajectory converges either to an endemic equilibrium or to a periodic orbit. Under additional natural assumptions, the periodic attractor is excluded, and we guarantee the convergence of each trajectory to an endemic equilibrium state. The global stability analysis uses a family of Lyapunov functions corresponding to the family of branches of the hysteresis operator.

Global dynamics of SIR model with switched transmission rate.

We propose a new epidemiological model, based on the classical SIR model, taking additionally into account a switching prevention strategy. The model has two distinct thresholds that determine the beginning and the end of an intervention and two different transmission rates. We study the global dynamics of the proposed two-dimensional model.

Mixed global dynamics of forced vibro-impact oscillator with Coulomb friction.

This paper revisits a well-known model of forced vibro-impact oscillator with Amonton-Coulomb friction. In the vast majority of the existing studies, this model included also viscous friction, and its global dynamics in the state space is governed by periodic, quasiperiodic, or chaotic attractors. We demonstrate that removal of the viscous friction leads to qualitative modification of the global dynamics, namely, the state space is divided into the regions with "regular" attraction to the aforementioned special solutions and the regions with profoundly Hamiltonian dynamics. The latter regions contain structures typical for forced Hamiltonian systems: stability islands, extended nonattractive chaotic regions, etc. We prove that such local Hamiltonian behavior should occur for phase trajectories with nonvanishing velocity. Stability analysis for the periodic orbits confirms the above statement. It is demonstrated that similar mixed global dynamics can be observed in a broader class of models.

TASEP modelling provides a parsimonious explanation for the ability of a single uORF to derepress translation during the integrated stress response.

Translation initiation is the rate-limiting step of protein synthesis that is downregulated during the Integrated Stress Response (ISR). Previously, we demonstrated that most human mRNAs that are resistant to this inhibition possess translated upstream open reading frames (uORFs), and that in some cases a single uORF is sufficient for the resistance. Here we developed a computational model of Initiation Complexes Interference with Elongating Ribosomes (ICIER) to gain insight into the mechanism. We explored the relationship between the flux of scanning ribosomes upstream and downstream of a single uORF depending on uORF features. Paradoxically, our analysis predicts that reducing ribosome flux upstream of certain uORFs increases initiation downstream. The model supports the derepression of downstream translation as a general mechanism of uORF-mediated stress resistance. It predicts that stress resistance can be achieved with long slowly decoded uORFs that do not favor translation reinitiation and that start with initiators of low leakiness.

Memory and adaptive behavior in population dynamics: anti-predator behavior as a case study.

Memory allows organisms to forecast the future on the basis of experience, and thus, in some form, is important for the development of flexible adaptive behavior by animal communities. To model memory, we use the concept of hysteresis, which mathematically is described by the Preisach operator. As a case study, we consider anti-predator adaptation in the classic Lotka-Volterra predator-prey model. Despite its simplicity, the model allows us to naturally incorporate essential features of an adaptive system and memory. Our analysis and simulations show that a system with memory can have a continuum of equilibrium states with non-trivial stability properties. The main factor that determines the actual equilibrium state to which a trajectory converges is the maximal number achieved by the population of predator along this trajectory.

Adaptive behaviour and multiple equilibrium states in a predator-prey model.

There is evidence that multiple stable equilibrium states are possible in real-life ecological systems. Phenomenological mathematical models which exhibit such properties can be constructed rather straightforwardly. For instance, for a predator-prey system this result can be achieved through the use of non-monotonic functional response for the predator. However, while formal formulation of such a model is not a problem, the biological justification for such functional responses and models is usually inconclusive. In this note, we explore a conjecture that a multitude of equilibrium states can be caused by an adaptation of animal behaviour to changes of environmental conditions. In order to verify this hypothesis, we consider a simple predator-prey model, which is a straightforward extension of the classic Lotka-Volterra predator-prey model. In this model, we made an intuitively transparent assumption that the prey can change a mode of behaviour in response to the pressure of predation, choosing either "safe" of "risky" (or "business as usual") behaviour. In order to avoid a situation where one of the modes gives an absolute advantage, we introduce the concept of the "cost of a policy" into the model. A simple conceptual two-dimensional predator-prey model, which is minimal with this property, and is not relying on odd functional responses, higher dimensionality or behaviour change for the predator, exhibits two stable co-existing equilibrium states with basins of attraction separated by a separatrix of a saddle point.

Hysteresis can grant fitness in stochastically varying environment.

Although the existence of multiple stable phenotypes of living organisms enables random switching between phenotypes as well as non-random history dependent switching called hysteresis, only random switching has been considered in prior experimental and theoretical models of adaptation to variable environments. This work considers the possibility that hysteresis may also evolve together with random phenotype switching to maximize population growth. In addition to allowing the possibility that switching rates between different phenotypes may depend not only on a continuous environmental input variable, but also on the phenotype itself, the present work considers an opportunity cost of the switching events. This opportunity cost arises as a result of a lag phase experimentally observed after phenotype switching and stochastic behavior of the environmental input. It is shown that stochastic environmental variation results in maximal asymptotic growth rate when organisms display hysteresis for sufficiently slowly varying environmental input. At the same time, sinusoidal input does not cause evolution of memory suggesting that the connection between the lag phase, stochastic environmental variation and evolution of hysteresis is a result of a stochastic resonance type phenomenon.

Chaos in a seasonally perturbed SIR model: avian influenza in a seabird colony as a paradigm.

Seasonality is a complex force in nature that affects multiple processes in wild animal populations. In particular, seasonal variations in demographic processes may considerably affect the persistence of a pathogen in these populations. Furthermore, it has been long observed in computer simulations that under seasonal perturbations, a host-pathogen system can exhibit complex dynamics, including the transition to chaos, as the magnitude of the seasonal perturbation increases. In this paper, we develop a seasonally perturbed Susceptible-Infected-Recovered model of avian influenza in a seabird colony. Numerical simulations of the model give rise to chaotic recurrent epidemics for parameters that reflect the ecology of avian influenza in a seabird population, thereby providing a case study for chaos in a host- pathogen system. We give a computer-assisted exposition of the existence of chaos in the model using methods that are based on the concept of topological hyperbolicity. Our approach elucidates the geometry of the chaos in the phase space of the model, thereby offering a mechanism for the persistence of the infection. Finally, the methods described in this paper may be immediately extended to other infections and hosts, including humans.

Bistable regimes in an optically injected mode-locked laser.

We study experimentally the dynamics of quantum-dot (QD) passively mode-locked semiconductor lasers under external optical injection. The lasers demonstrated multiple dynamical states, with bifurcation boundaries that depended upon the sign of detuning variation. The area of the hysteresis loops grew monotonically at small powers of optical injection and saturated at moderate powers. At high injection levels the hysteresis decreased and eventually disappeared.