Condition for Global Stability for a SEIR Model Incorporating Exogenous Reinfection and Primary Infection Mechanisms

A mathematical model incorporating exogenous reinfection and primary progression infection processes is proposed. Global stability is examined using the geometric approach which involves the generalization of Poincare-Bendixson criterion for systems of n -ordinary differential equations. Analytical results show that for a Susceptible-Exposed-Infective-Recovered (SEIR) model incorporating exogenous reinfection and primary progression infection mechanisms, an additional condition is required to fulfill the Bendixson criterion for global stability. That is, the model is globally asymptotically stable whenever a parameter accounting for exogenous reinfection is less than the ratio of background mortality to effective contact rate. Numerical simulations are also presented to support theoretical findings.

Nonlinear, Phase-Based Oscillator to Generate and Assist Periodic Motions

Oscillatory behavior is important for tasks, such as walking and running. We are developing methods for wearable robotics to add energy to enhance or vary the oscillatory behavior based on the system's phase angle. We define a nonlinear oscillator using a forcing function based on the sine and cosine of the system's phase angle that can modulate the amplitude and frequency of oscillation. This method is based on the state of the system and does not use off-line trajectory planning. The behavior of a limit cycle is shown using the Poincaré–Bendixson criterion. Linear and rotational models are simulated using our phase controller. The method is implemented and tested to control a pendulum.

Nonlinear Phase-Based Oscillator to Generate and Assist Periodic Motions

Oscillatory behavior is important for tasks such as walking and running. We are developing methods to add energy to enhance or vary the oscillatory behavior based on the system’s phase angle. We define a nonlinear oscillator using a forcing function based on the sine and cosine of the system’s phase angle that can modulate the amplitude and frequency of oscillation. The stability of the system is proved using the Poincaré-Bendixson criterion. Linear and rotational mechanical systems are simulated using our phase controller. The method is implemented and tested to control a pendulum. Lastly, we propose how to assist hip motion during walking using the phase-based forcing function.

HOPF BIFURCATION ANALYSIS IN A MACKEY–GLASS SYSTEM

The dynamics of a Mackey–Glass equation with delay are investigated. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Global existence of periodic solutions are established using a global Hopf bifurcation result due to Wu [1998] and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney [1994].