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In this work, we extend the Markov models describing the susceptible-infectedsusceptible (SIS) epidemics over undirected networks to take into account the virus minimum incubation period and the minimum recovery period of an infected individual. We represent both periods as time delays in the states of the extended model. However, due to the addition of time delays, the process loses its Markovain property. We use the generalized semi-Markov theory to introduce both the incubation and recovery delays to the probabilistic dynamical models. Hence, in this paper, we propose a time-delay version of the two principal models of the SIS epidemics over undirected networks: the exact 2 𝑁 −state model and the approximated 𝑁−intertwined model. Finally, using Lyapunov analysis, we give sufficient conditions that guarantee the global exponential stability of the time-delay 𝑁−intertwined model.

In this work, we consider the estimation of the thermodynamic properties (Pressure, Enthalpy, Temperature) and mass flow rates along the pipes of a concentric heat exchanger tube in which we have CO 2 as the working refrigerant. The transport phenomena are modeled using Navier-Stokes equations in 1D for both hot and cold sides, where we consider single and two phase flows. The estimation is done with a PDE observer that uses measurements taken at the tube boundaries to construct the required profiles along the tubes. The convergence of this observer is proved using Lyapunov analysis and the theoretical results are illustrated by numerical simulations.

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