Waning Immunity and the Second Wave: Some Projections for SARS-CoV-2

This paper offers projections of future transmission dynamics for SARS-CoV-2 in an SEIRS model with demographics and waning immunity. In a stylized optimal control setting calibrated to the United States, we show that the disease is endemic in steady state and that its dynamics are characterized by damped oscillations. The magnitude of the oscillations depends on how fast immunity wanes. The optimal social distancing policy both curbs peak prevalence and postpones the infection waves relative to the uncontrolled dynamics. Last, we perform sensitivity analysis with respect to the duration of immunity, the infection fatality rate, and the planning horizon. (JEL I12, I18, J11)

Damped Oscillations – A smartphone approach

The study of the influence of geometric factors on an oscillating pendulum under various damping conditions is reported. Different cross-section areas perpendicular to the motion of the pendulum mass were studied. A smartphone was used as a pendulum and at the same time as a data recorder. Results show that the smartphone is an effective and reliable tool to be used when performing educational activities, and at the same time it presents students with a variety of ways for learning new content and physical concepts. It offers the opportunity to carry out experiments in the classroom, in the laboratory, or at home. In this way, the increment in the cross-section area slightly increases the damping coefficient, and rapidly decreases the oscillation amplitude as time passes. Additionally, the time necessary to decrease the amplitude by half is inversely proportional to the cross-section area of the pendulum. As expected, no significant variation in the period nor the angular frequency were found, due to the air-pendulum drag properties and to the slow pendulum speed.

Damped Oscillations – A smartphone approach

The study of the influence of geometric factors on an oscillating pendulum under various damping conditions is reported. Different cross-section areas perpendicular to the motion of the pendulum mass were studied. A smartphone was used as a pendulum and at the same time as a data recorder. Results show that the smartphone is an effective and reliable tool to be used when performing educational activities, and at the same time it presents students with a variety of ways for learning new content and physical concepts. It offers the opportunity to carry out experiments in the classroom, in the laboratory, or at home. In this way, the increment in the cross-section area slightly increases the damping coefficient, and rapidly decreases the oscillation amplitude as time passes. Additionally, the time necessary to decrease the amplitude by half is inversely proportional to the cross-section area of the pendulum. As expected, no significant variation in the period nor the angular frequency were found, due to the air-pendulum drag properties and to the slow pendulum speed.

Approximating Damped Vibrations of Large Space Structures

It is possible to use data recorded by onboard acceleration sensors to verify mathematical models of large modular space structures in terms of simulating dynamic processes. The paper investigates an approach to approximating damped oscillations caused by dynamic impacts during operation. Initially, we approximate the response of the structure by summing damped harmonics derived from analysing the frequency spectrum of the dynamic process; then we use the Levenberg --- Marquardt algorithm in the parameter space of the harmonic set to find the best match between the real dynamic process and its approximation. We propose a modification of the approach considered which involves employing single harmonics to perform successive approximations of the function of time to be fitted. We show that it is possible to apply the approach proposed to identifying the frequency and dissipative parameters of the structure under consideration. The paper presents the results of testing the approach proposed via artificially generated noisy acceleration functions of time with known parameters, which were reconstructed with a sufficient degree of accuracy. A real-world example provided comprises the results of analysing the ISS accelerometer readings recorded against the background of damped vibrations in its structure that were caused by burns of its attitude control engines

Correction to: Viscosity Measurement by the “Oscillating Drop Method”: The Case of Strongly Damped Oscillations

A correction to this paper has been published: https://doi.org/10.1007/s10765-021-02870-5

Listening to ultrasound from plants reveals xylem vessel anatomy

Plants emit ultrasound pulses under drought stress, which originate in their water-carrying xylem vessels, and can be recorded externally. We demonstrate that these ultrasound pulses consist of superposed damped oscillations at plant-specific frequencies in the range of 10 – 150 kHz, that are correlated to xylem dimensions. We present a method to relate geometrical and viscoelastic properties of xylem vessels with the time- and frequency-domain characteristics of the observed oscillations. We apply the method to ultrasound pulses from drying shoots of three vascular dicot plant species. The extracted parameters are validated with destructive measurements of xylem vessel radii, wall thickness, length of xylem vessel elements, and the elastic modulus of the vascular bundle by optical and scanning cryo-electron microscopy and tensile loading. Our method demonstrates the potential for non-invasive and continuous monitoring of plant vascular anatomy. We foresee applications in high-throughput phenotyping and early detection of vascular wilt diseases.

The Tipping Effect of Delayed Interventions on the Evolution of COVID-19 Incidence

We combine infectious disease transmission and the non-pharmaceutical intervention response to disease incidence into one closed model consisting of two coupled delay differential equations for the incidence rate and the time-dependent reproduction number. The model contains three parameters, the initial reproduction number, the intervention strength, and the response delay. The response is modeled by assuming that the rate of change of the reproduction number is proportional to the negative deviation of the incidence rate from an intervention threshold. This delay dynamical system exhibits damped oscillations in one part of the parameter space, and growing oscillations in another, and these are separated by a surface where the solution is a strictly periodic nonlinear oscillation. For the COVID-19 pandemic, the tipping transition from damped to growing oscillations occurs for response delays of about one week, and suggests that, without vaccination, effective control and mitigation of successive epidemic waves cannot be achieved unless NPIs are implemented in a precautionary manner, rather as a response to the present incidence rate. Vaccination increases the quiet intervals between waves, but with delayed response, future flare-ups can only be prevented by establishing a post-pandemic normal with lower basic reproduction number.

One Game Problem for Oscillatory Systems

The paper concerns the linear differential game of approaching a cylindrical terminal set. We study the case when classic Pontryagin’s condition does not hold. Instead, the modified considerably weaker condition, dealing with the function of time stretching, is used. The latter allows expanding the range of problems susceptible to analytical solution by the way of passing to the game with delayed information. Investigation is carried out in the frames of Pontryagin’s First Direct method that provides hitting the terminal set by a trajectory of the conflict-controlled process at finite instant of time. In so doing, the pursuer’s control, realizing the game goal, is constructed on the basis of the Filippov-Castaing theorem on measurable choice. The outlined scheme is applied to solving the problem of pursuit for two different second-order systems, describing damped oscillations. For this game, we constructed the function of time stretching and deduced conditions on the game parameters, ensuring termination of the game at a finite instant of time, starting from arbitrary initial states and under all admissible controls of the evader. Keywords: differential game, time-variable information delay, Pontryagin’s condition, Aumann’s integral, principle of time stretching, Minkowski’ difference, damped oscillations.