### Delay differential equations

Organizer: Daniel Alejandro Melchor, dmelchor@ipicyt.edu.mx

In order to model some phenomena in the physical world, we use systems that depend on the state of them in the past, we study their dynamics via delay differential equations.In this sessión we intend to bring together leading experts, to show recent advances, and explore current directions of research, on the qualitative and quantitative properties of delay differential equations and its applications.

Confirmed speakers

Mikhail Basin, Facultad de Ciencias Físico Matemáticas, Universidad Autónoma de Nuevo León.
Cruz Vargas de León, Escuela Superior de Medicina, Instituto Politécnico Nacional
Daniel Melchor Aguilar, División de Matemáticas Aplicadas, Instituto Potosino de Investigación Científica y Tecnológica.
Anatoli Ivanov, Pennsylvania State University

### Abstracts

Existence of a first integral and stability of delayed virus dynamics models with absorption effect

Cruz Vargas de León
Escuela Superior de Medicina, Instituto Politécnico Nacional, Mexico
In this talk, we will consider some systems of delay differential equations (DDEs) modelling the viral dynamics in a cell culture. These models incorporate the loss of viral particles due to the absorption into target cells. Two discrete or distributed delays are incorporated into the model, which describe (i) the lag between the time of the virus contacts the uninfected cell and the time of the cell becomes actively infected and (ii) the time necessary for the newly produced virions to become mature and then infectious particles, respectively. We derived a first integral or conserved quantity, and we proved the use of experimental data in order to test the conservation law. The systems have nonhyperbolic equilibrium points, and the conditions for their stability are obtained by using a Lyapunov functional. We complemented these theoretical results with some numerical simulations.

Mean-Square Filtering for Linear Systems with State and Observation Delays

Michael Basin
The mean-square filtering problem for linear systems with state delay over linear observations is treated proceeding from the general expression for the stochastic Ito differential of the estimate and the error variance. As a result, the estimate equation similar to the traditional Kalman-Bucy one is derived; however, it is impossible to obtain a system of the filtering equations that is closed with respect to the only two variables, the estimate and the error variance, as in the Kalman-Bucy filter. The resulting system of equations for determining the error variance consists of a set of equations, whose number is specified by the ratio between the current filtering horizon and the delay value in the state equation and increases as the filtering horizon tends to infinity. In the example, performance of the designed filter for linear systems with state delay is verified against the Kalman-Bucy filter available for linear systems without delays and two versions of the extended Kalman-Bucy filter for time-delay systems.
The mean-square filtering problem is then presented for linear systems with state and observation delays. The estimate equation similar to the traditional Kalman-Bucy one is derived, and the system of equations for determining the filter gain matrix consists of an infinite set of equations. It is demonstrated that a finite set of the filtering equations can be obtained in case of commensurable delays. In the example, the designed filter is compared to the traditional Kalman-Bucy filter.

Contributions on the stability of functional difference equations

Daniel Melchor
División de Matemáticas Aplicadas, IPICYT, Mexico
We consider some classes of functional difference equations (FDE) with discrete and distributed delays which play a fundamental role in several stability problems of delay differential equations. One of the fields where such FDE arises is in the stability analysis of the difference operator of some neutral delay differential equations (NDDE). We show that the asymptotic stability problem of these FDE cannot be solved by using the existing results for NDDE and present some new results for investigating its asymptotic stability. We firstly give an appropriate definition of the fundamental matrix that allows the description of piecewise continuous solutions of the FDE. Then, we obtain the variation-of-constants formula for nonhomogeneous equations and use it for deriving asymptotic stability results of perturbed equations involving general linear and/or nonlinear perturbations terms with discrete and distributed delays.

Periodic solutions and global stability in multi-dimensional cyclic feedback systems with delay

Anatoli Ivanov
Pennsylvania State University

Schedule

Thursday
15:30-16:30 Cruz Vargas de León
16:30-17:30 Michael Basin

Friday
12:00-12:30 Daniel Melchor
12:30-13:00 Anatoli Ivanov