Epidemiological models and Lyapunov functions

International audienceWe give a survey of results on global stability for deterministic compartmental epidemiological models. Using Lyapunov techniques we revisit a classical result, and give a simple proof. By the same methods we also give a new result on differential susceptibility and infectivity models with mass action and an arbitrary number of compartments. These models encompass the so-called differential infectivity and staged progression models. In the two cases we prove that if the basic reproduction ratio R0 \leq 1, then the disease free equilibrium is globally asymptotically stable. If R0 > 1, there exists an unique endemic equilibrium which is asymptotically stable on the positive orthant

Tracking Control of Redundant Manipulators with Singularity-Free Orientation Representation and Null-Space Compliant Behaviour

This paper presents a suitable solution to control the pose of the end-effector of a redundant robot along a pre-planned trajectory, while addressing an active compliant behaviour in the null-space. The orientation of the robot is expressed through a singularity-free representation form. To accomplish the task, no exteroceptive sensor is needed. While a rigorous stability proof confirms the developed theory, experimental results bolster the performance of the proposed approach

The optimal treatment of an infectious disease with two strains

This paper explores the optimal treatment of an infectious disease in a Susceptible-Infected-Susceptible model, where there are two strains of the disease and one strain is more infectious than the other. The strains are perfectly distinguishable, instantly diagnosed and equally costly in terms of social welfare. Treatment is equally costly and effective for both strains. Eradication is not possible, and there is no superinfection. In this model, we characterise two types of fixed points: coexistence equilibria, where both strains prevail, and boundary equilibria, where one strain is asymptotically eradicated and the other prevails at a positive level. We derive regimes of feasibility that determine which equilibria are feasible for which parameter combinations. Numerically, we show that optimal policy exhibits switch points over time, and that the paths to coexistence equilibria exhibit spirals, suggesting that coexistence equilibria are never the end points of optimal paths

New Results on the Stability of Discrete-Time Systems and Applications to Control Problems

AbstractThe aim of this article is to present some new stability sufficient conditions for discrete-time nonlinear systems. It shows how to use nonnegative semi-definite functions as Lyapunov functions instead of positive definite ones for studying the stability of a given system. Several examples and some applications to control theory are presented to illustrate the various theorems

Epidemiological Models and Lyapunov Functions

We give a survey of results on global stability for deterministic compartmental epidemiological models. Using Lyapunov techniques we revisit a classical result, and give a simple proof. By the same methods we also give a new result on differential susceptibility and infectivity models with mass action and an arbitrary number of compartments. These models encompass the so-called differential infectivity and staged progression models. In the two cases we prove that if the basic reproduction ratio $\mathcal{R}_0$ ≤ 1, then the disease free equilibrium is globally asymptotically stable. If $\mathcal{R}_0$ > 1, there exists an unique endemic equilibrium which is asymptotically stable on the positive orthant