### Session S25 - New Methods and Emerging Applications in Dynamics, Networks, and Control

## Talks

Monday, July 12, 16:15 ~ 16:45 UTC-3

## Optimal control of diseases in prison populations through screening policies of new inmates

### Victor Riquelme

#### Universidad de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

We study an optimal control problem of a communicable disease in a prison population. To control the spread of the disease inside a prison, we consider an active case-finding strategy, consisting of screening a proportion of new inmates at the entry point, followed by a treatment depending on the results of this procedure. The control variable consists then in the coverage of the screening applied to new inmates. The disease dynamics is modeled by an SIS (susceptible-infected-susceptible) model, typically used to represent diseases that do not confer immunity after infection. We determine the optimal strategy that minimizes a combination between the cost of the screening/treatment at the entrance and the cost of maintaining infected individuals inside the prison, in a given time horizon. Using the Pontryagin Maximum Principle and Hamilton-Jacobi-Bellman equation, among other tools, we provide the complete synthesis of an optimal feedback control, consisting of a bang-bang strategy with at most two switching times and no singular arc trajectory, characterizing different profiles depending on model parameters.

Joint work with Pedro Gajardo (Universidad Técnica Federico Santa María, Chile) and Diego Vicencio (Universidad Técnica Federico Santa María, Chile).

Monday, July 12, 16:50 ~ 17:20 UTC-3

## Extension of the Solution Set of the Bounded Finite-Time Stabilization of the Prey-Predator Model via Controllability Function

### Abdon Choque-Rivero

#### Universidad Michoacana de San Nicolás de Hidalgo, México - This email address is being protected from spambots. You need JavaScript enabled to view it.

For the prey-predator model, an extended set of bounded finite-time stabilizing positional controls is given. We use Korobov’s controllability function $\Theta(x)$ which is a Lyapunov-type function. Previously, the controllability function was the unique solution of certain implicit equation on $\Theta$. In the present talk, we consider the case when there are between one and three solutions of the mentioned implicit equation. This occurs for initial positions belonging to certain domain of the phase space.

Monday, July 12, 17:25 ~ 17:45 UTC-3

## Nematic liquid crystals: well posedness, optical solitons and control

### Constanza Sánchez Fernández de la Vega

#### IMAS-CONICET y DM, Facultad de Ciencias Exactas y Naturales, UBA, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk we present results on well-posedness, decay, soliton solutions and control of the coupled nonlinear Schrödinger (NLS) equation \[\partial_{z}u= \frac{1}{2} \mathrm{i} \nabla^{2} u+ \mathrm{i} \gamma (\sin^2(\psi+\theta_0)-\sin^2(\theta_0)) u, \] \[ \nu \nabla^{2} \psi= \frac{1}{2}E_0^2\sin(2\theta_0)-\frac{1}{2}(E_0^2+|u|^{2})\sin(2(\psi+\theta_0)), \] where $u$ and $\psi$ depend on the ``optical axis'' coordinate $z \in \mathbb{R}$, and the ``transverse coordinates'' $(x, y) \in \mathbb{R}^2$. Also $\nabla^{2} = \partial_x^2 + \partial_y^2$ is the Laplacian in the transverse directions, $E_0$, $\nu$ and $\gamma $ are positive constants, and $\theta_0$ is a constant satisfying $\theta_0 \in (\pi/4, \pi/2)$. The model arises in the study of optical beam propagation in nematic liquid crystals, and in particular a set of experiments by Assanto and collaborators [3], [4], [5]. The complex field $u$ represents the electric field amplitude of a linearly polarized laser beam that propagates through a nematic liquid crystal along the optical axis $z$. The elliptic equation describes the effects of the beam electric field on the local orientation (director field) of the nematic liquid crystal and has an important regularizing effect, seen experimentally and understood theoretically in related models. The ``director field'' $\psi + \theta_0$ is a field of angles that describe the macroscopic orientation of the nematic liquid crystal molecules. The laser beam causes an additional deviation $\psi$ in the orientation of the liquid crystal molecules.

In [1] we show a ``saturation'' effect consistent with a bound $\theta_0 + \psi < \pi/2 $ on the total angle, implying that the molecular orientation can not be perpendicular to the optical axis. This seems to be a sharp bound on the saturation of the nonlinearity. In particular it is more precise than the bound obtained in [2] and follows from a more general model that has no small size assumptions for $\theta_0 - \pi/4$ and $\psi$.

Finally, we show some recent results concerning an optimal control problem where the external electric field varying in time is the control.

References:

[1] Borgna, Juan Pablo and Panayotaros, Panayotis and Rial, Diego and Sánchez de la Vega, Constanza. Optical solitons in nematic liquid crystals: Arbitrary deviation angle model. Physica D: Nonlinear Phenomena 408, 2020.

[2] Borgna, Juan Pablo and Panayotaros, Panayotis and Rial, Diego and Sánchez de la Vega, Constanza. Optical solitons in nematic liquid crystals: model with saturation effects. Nonlinearity 31 (4), 2018.

[3] Conti, C. and Peccianti, M. and Assanto, G. Route to nonlocality and observation of accessible solitons. Physical review letters 91 (7), 2003.

[4] Peccianti, M. and Assanto, G. Nematicons. Physics Reports 516 (4), 2012.

[5] Peccianti, M. and De Rossi, A. and Assanto, G. and De Luca, A. and Umeton, C. and Khoo, I. C. Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells. Applied Physics Letters 77 (1), 2000.

Joint work with Juan Pablo Borgna (CONICET - CEDEMA, Universidad Nacional de San Martin, Buenos Aires, Argentina), Panayotis Panayotaros (Departamento de Matemáticas y Mecánica, IIMAS, Universidad Nacional Autónoma de México, Cd. México, México) and Diego Rial (IMAS - CONICET and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, UBA, CABA, Argentina).

Monday, July 12, 18:00 ~ 18:30 UTC-3

## Conviviendo con coronavirus

### Juan Aparicio

#### INENCO, Universidad Nacional de Salta, Conicet., Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Estudiamos aspectos de la dinámica de Covid-19 en ciudades de porte mediano (del orden de 100 mil habitantes). Para ello desarrollamos un modelo computacional basado en individuos. Como ejemplos consideramos la distribución de habitantes por casa observada en distintas ciudades de Argentina.

Modelamos también distintas herramientas de control como seguimiento y aislamiento de casos y contactos, cierres parciales de trabajos y cierres de escuelas.

Las señales utilizadas para activar o desactivar cada una de ellas se toman del nivel de ocupación de camas de terapia intensiva disponibles.

Finalmente se discuten extensiones y limitaciones de nuestro trabajo.

Joint work with Mario Ignacio Simoy (Instituto Multidisciplinario sobre Ecosistemas y Desarrollo Sustentable, Universidad Nacional del Centro de la Provincia de Buenos Aires, Tandil, Argentina.).

Monday, July 12, 18:35 ~ 19:05 UTC-3

## Simple models for complex realities: The case of COVID-19.

### Fernando Córdova Lepe

#### Universidad Católica del Maule, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

In the differential models construction processes, as instruments for the decision-making, there is a need for a detailed structuring of the reference system, for example, incorporating a large number of variables, specific functional relationships, and precision in the parameters. However, these are supported by more elementary models that capture essential general patterns of the phenomenon; it is the role that the SEIR model has played for the epidemiological analysis of Covid-19. In its version with constant parameters, this base model presents strong resistance to its generalizations to attempts to break the unimodal behavior of the epidemic curve. The main objective of the presentation is to propose a new basic model or "toy model," which is still a generalization to the classic SEIR, that incorporating the concepts of "reaction rate" and "restitution rate" associated with the rate transmission, overcomes the limitation by incorporating the possibility of multiple oscillatory modes that geometrically correlate with the existing data that the pandemic has left for various countries or territories. Also, some related mathematical challenges and other epidemiological considerations are detailed.

Córdova-Lepe, F.; Robledo, G.; Vergaño-Salazar, J.G. (2020) Mitigation effort performance index and bounds for inflection points of the epidemic curve. Supplies to fight COVID-19. Submitted.

Cabrera, M; Córdova-Lepe, F; Gutiérrez-Jara, JP; Vogt-Geisse, K. (2020) An SIR-type epidemiological model that integrates social distancing as a dynamic law based on point prevalence and socio-behavioral factors. Submitted.

Córdova-Lepe, F; Vogt-Geisse, K. (2021) A dynamic law for the contagion rate in SIR-type models. To be submitted.

Monday, July 12, 19:10 ~ 19:40 UTC-3

## Modeling COVID-19 Transmission Dynamics and Control Strategies in Close-Contact facilities

### Anuj Mubayi

#### Illinois State University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

The COVID-19 pandemic has caused havoc in all walks of our life and for each and every sub-populations. Since SARS-COV-2, a novel coronavirus, has been identified in 2019, local and national public health departments have tried different kinds of control policies to constrain its outbreaks with varied success. Cumulative global deaths related to COVID-19 have been continuously on rise, especially due to considerable mortality among the vulnerable populations such as residents of long-term care facilities. In this talk, I will discuss our recent study, where we developed and analyzed mathematical models that captured infection dynamics in a close-contact place and estimated the transmissibility of the SARS-CoV-2 virus via distinct mechanisms in the presence of unprecedented control measures. The aim of the study was to identify key factors associated with SARS-CoV-2 infection and outbreaks among staff, residents and visitors in long-term care facilities. Using reported data from long-term care facilities in USA and optimal control methods, we investigated the influence of the different control measures on the size of COVID-19 outbreak in these locations. Identified optimal data-driven strategies have highlighted pressure points of the disease system in the close-contacts places. There is not a one size fits all approach to any aspect of COVID interventions, though, rapid adoption of certain measures could support efforts to protect this vulnerable group of society from future waves of SARS-CoV-2 infection.

Joint work with Aditi Ghosh (Texas A&M University-Commerce, USA),, Doménica N. Garzón (Yachay Tech University, Ecuador), and Padmanabhan Seshaiyer (George Mason University, USA).

Monday, July 12, 19:50 ~ 20:20 UTC-3

## COVID-19 epidemic scenarios based on observed key superdispersion events

### Jorge Velasco-Hernandez

#### UNAM, Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.

Key high transmission dates for the year 2020 are used to create scenarios to model the evolution of the COVID-19 pandemic in several states of Mexico for 2021. These scenarios are obtained through the estimation of a time-dependent contact rate, where the main assumption is that the dynamic of the disease is heavily determined by the mobility and social activity of the population during holidays and other important calendar dates. First, changes in the effective contact rate on predetermined dates of 2020 are estimated. Then, using the instantaneous reproduction number to characterize the status of the epidemic ($R_t\approx 1$, $R_t>1$ or $R_t<1$) this information is used to propose different scenarios for the number of cases and deaths for 2021. The main assumption is that the effective contact rate during 2021 will maintain a similar trend to that observed during 2020 on key calendar dates. All other conditions are assumed to remain constant in the time scale of the projections. The objective is to generate a range of scenarios that could be useful to evaluate the possible evolution of the epidemic and its likely impact on incidence and mortality.

Joint work with Mario Santana Cibrian, Adrán Acuña Zegarra, Carlos E. Rodriguez Hernández-Vela and Ramsés Mena.

Monday, July 12, 20:25 ~ 20:55 UTC-3

## Monotonicity properties arising in a simple model of Wolbachia invasion for wild mosquito populations

### Diego Vicencio

#### Universidad Tecnica Federico Santa Maria, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this presentation, we propose a simplified bi-dimensional Wolbachia infestation model in a population of Aedes aegypti mosquitoes, preserving the main features associated with the biology of this species that can be found in higher dimensional models.

Using tools borrowed from monotone dynamical system theory, in the proposed model, we prove the existence of an invariant threshold manifold that allows us to provide practical recommendations for performing single and periodic releases of Wolbachia-carrying mosquitoes, seeking the eventual elimination of wild insects that are capable of transmitting infections to humans.

We illustrate these findings with numerical simulations using parameter values corresponding to the wMelPop strain of Wolbachia that is considered the best virus blocker but induces fitness loss in its carriers.

Joint work with Pedro Gajardo (Universidad Tecnica Federico Santa Maria, Chile) and Olga Vasilieva (Universidad del Valle, Colombia).

Tuesday, July 13, 16:00 ~ 16:30 UTC-3

## Impact of vector control and the eradication of diseased plants in the Diaphorina citri-HLB pathosystem

### Lilian Sofia Sepulveda-Salcedo

#### Universidad Autónoma de Occidente, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.

This work is within the framework of the research project Models and mathematical methods for the control of Diaphorina citri. In this project, we hope to develop new tools for the management of the Huanglongbing disease (HLB) of citrus in Colombia, through the use of mathematical tools such as dynamic modeling of populations in time and space, control theory and complex network techniques.

In order to contribute to the evaluation, development and design of effective and efficient strategies for the management of Diaphorina citri and the HLB disease in the conditions of Colombian citrus, the work proposal that we present is aligned with the recommendations of the Plan Strategic of the Colombian Agricultural Sector (PECTIA 2017-2027) and wants to evaluate the impact of vector control and the eradication of diseased plants in the effective management of the disease and in reducing economic losses.

On the one hand, we adapt the method for computing sustainable thresholds for controlled cooperative models introduce for Barrios-Rivera, E, et al. for phatosytem Diaphorina citri-HLB. Taking account of uncertainty in the transmission of the bacteria from the vector to the plant, we want to make the characterization of set of thresholds that refers to constraints that can be sustained when the eradication of diseased plants is implemented as a control of HLB.

In parallel, we also want to evaluate the effect of uncertainty in the proportion of the infected vector with phytophathogen Candidatus Liberibacter asiaticus causing of HLB, on the optimal solution of the problem that aims to maximize production and minimize the costs of applying a control strategy that combines vector control and eradication of diseased plants in a citric crop.

Joint work with Edwin Barrios-Riveras (Universidad del Valle, Colombia), Lumey Pérez-Artiles (Corporación Colombiana De Investigación Agropecuaria - AGROSAVIA, Colombia) and Diego Vicencio-Morales (Universidad Técnica Federico Santa María, Chile).

Tuesday, July 13, 16:35 ~ 17:05 UTC-3

## Carryover of a saddle-node bifurcation after transforming a parameter into a variable

### Gustavo Carrero

#### Athabasca University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this presentation, we introduce and study the carryover of a saddle-node bifurcation, a concept that describes how a saddle-node bifurcation of a dynamical system is carried over into an extended dynamical system obtained by transforming one of the parameters of the original system into a variable. We show the conditions needed to guarantee the carryover of a saddle-node bifurcation, provide a graphical methodology with a two-parameter bifurcation diagram to verify that such conditions are met, and illustrate the results with examples.

Joint work with Gerda de Vries (University of Alberta, Canada) and Carlos Contreras (University of Alberta, Canada).

Tuesday, July 13, 17:10 ~ 17:40 UTC-3

## Complex Dynamics of the Interactions Between Nucleus in the Basal Ganglia

### Julián Hurtado

#### Universidad Autónoma de Occidente, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.

The oscillatory nature of basal ganglia activity is highly linked with some movement disorders. This talk shows a study of oscillations in a reduced neural circuit model of the subthalamic nucleus (STN)-external globus pallidus (GPe) loop. The role of their interactions has been mainly discussed but mostly in the context of a dysfunction, as in Parkinson's disease, and not of its function.

Understanding the interaction between the basal ganglia with other brain areas is essential for finding new treatments for disorders affecting the neural systems supporting motor and cognitive behaviors. Reaching a comprehensive computational model of BG will allow a strengthening of the theoretical neurobiology. There will be important contributions to different fields as computer science and applied mathematics, robotics, and machine learning, as well.

The studied model is a mean firing rate mathematical model of a coupled pair of STN and GPe populations based on the GEN model (Chakravarthy, 2018). The goal of this study is to investigate how changes in network structure can lead to different dynamical modes such as steady state, oscillatory, and bistable behavior. Specifically, we perform a bifurcation analysis of a network of two single artificial neurons to investigate different dynamical modes with respect to changes in inputs and interconnection strengths.

Joint work with David Fernando Ramírez Moreno (Universidad Autónoma de Occidente).

Tuesday, July 13, 18:20 ~ 18:50 UTC-3

## Response to perturbations as a built-in feature in a mathematical model for paced finger tapping

### Rodrigo Laje

#### University of Quilmes, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

Paced finger tapping is one of the simplest tasks to study sensorimotor synchronization. The subject is instructed to tap in synchrony with a periodic sequence of brief tones, and the time difference (called asynchrony) between each response and the corresponding stimulus is recorded. Despite its simplicity, this task helps to unveil interesting features of the underlying neural system and the error correction mechanism responsible for synchronization. Perturbation experiments are usually performed to probe the subject's response, for example in the form of a ``step change'', i.e.\ an unexpected change in tempo. The asynchrony is the usual observable in such experiments and it is chosen as the main variable in many mathematical models that attempt to describe the phenomenon. In this work we show that although asynchrony can be perfectly described in operational terms, it is not well defined as a model variable when tempo perturbations are considered. We introduce an alternative variable and a mathematical model that intrinsically takes into account the perturbation, and make theoretical predictions about the response to novel perturbations based on the geometrical organization of the trajectories in phase space. Our proposal is relevant to understand interpersonal synchronization and the synchronization to non-periodic stimuli.

Joint work with Claudia R. González (University of Quilmes).

Tuesday, July 13, 18:55 ~ 19:25 UTC-3

## Ideas of maximal entropy to understand biodiversity and genomics composition of microbial communities

### Alejandro Maass

#### Universidad de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

The configuration of microbial biodiversity in relation to its environmental conditions is an open problem in ecology. In this talk, using massive data from the TARA_Ocean expeditions we explore principles of maximal entropy that could explain configurations of biodiversity, but also of some important genomics signals.

Joint work with Ignacio Arroyo (Universidad de Chile y Pontificia Universidad Católica), Pablo Marquet (Pontificia Universidad Católica), Ricardo Palma (Universidad de Chile), Dante Travisany (Universidad de Chile-INRIA Chile).

Tuesday, July 13, 19:30 ~ 20:00 UTC-3

## An alternative approach to overcome the ''odd number limitation'' of Pyragas stabilizability problem.

### Verónica Estela Pastor

#### Universidad de Buenos Aires, Facultad de Ingeniería, Departamento de Matemática, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

The problem of Pyragas stabilizability was stated in [1] as follows. It is proposed to stabilize the nonlinear system given by: \[ \dot x = f(x) \ \ \ \ \ (1) \] in one of its (unknown) unstable equilibrium points by adding the feedback control: \[ u(t)= K (x(t- \tau)-x(t)) \ \ \ \ \ (2) \] where the real constant matrix $K$ and the real number $\tau$ are the control parameters.

This method presents an essential constraint known as the ''odd number limitation''. Namely, if the jacobian matrix of the system evaluated on the equilibrium point has an odd number of positive eigenvalues or a zero eigenvalue, stabilization is not achieved for any value of the control parameters ([2]).

To overcome this drawback, different methods have been designed by introducing a non-stationary feedback control. In particular, in [3], the constant gain $K$ of (2) is replaced by a periodic $K(t)$, defined by some adequate constants. The choice of these constants is based on analytical arguments but a complete characterization of the available set of stability parameters is not determined.

As an alternative, we propose the following scheme: \[ u(t)= K(t) (x(t- 2\tau)-x(t- \tau)) \ \ \ \ \ (3) \] where the periodic gain yields to an oscillatory type control.

This proposal keeps the non-invasive feature of its antecedents and it is based on the methodology developed in [4] for the one dimensional case. Its efficiency for any hyperbolic equilibrium point is proved and a full description of the set of stability parameters is deduced.

References:

$[1]$ K. Pyragas, Control of chaos via extended delay feedback, Phys. Lett. A 206 (1995).

$[2]$ H. Kokame, K. Hirata, K. Konishi, T. Mori, Difference feedback can stabilize uncer-tain steady states, IEEE Trans. Autom. Control 46 (2001).

$[3]$ G.A. Leonov, M.M. Shumafov, Pyragas stabilizability of unstable equilibria by non-stationary time-delayed feedback, Autom. Remote Control 6 (2018).

$[4]$ V. E. Pastor, G. A. González, Oscillating delayed feedback control schemes for stabilizing equilibrium points, Heliyon 5 (2019).

Joint work with Graciela Adriana González (Universidad de Buenos Aires. Facultad de Ingeniería. Departamento de Matemática y CONICET, Argentina).

Tuesday, July 13, 20:05 ~ 20:35 UTC-3

## Modeling Energy Markets: Advances in Networks and PWS Dynamical Systems

### Gerard Olivar-Tost

#### Universidad de Aysén, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

Energy markets can be thought in the framework of general economic models for the supply-and-demand when investments are required. We improve a well-known model from System Dynamics, and we translate it to the mathematical equations in order to be precise at the simulation level. The model is shown as a system of piecewise-smooth differential equations.

Piecewise smooth and hybrid dynamical systems have been increasingly used in Engineer- ing and Applied Sciences. More recently, these systems appeared also in Economics and Social Science, mainly in Sustainability Development, Bioeconomics and new knowledge areas. Several nonsmooth bifurcations have been reported in the literature. They are the fingerprint of an intrinsic complex system.

When several markets are connected, complex networks (in the dynamics and structure) naturally appear. Finally, stochasticity is introduced in the system in order to model the risk aversion of investment agents. This is done through Markov chains. This combination of deterministic paths and stochasticity leads to the so-called Piecewise-Deterministic Markov Processes.

Joint work with Johnny Valencia-Calvo (Universidad de Aysén, Chile).

Tuesday, July 20, 18:30 ~ 19:00 UTC-3

## Synchronized path following for LTI systems and closed paths

### Christopher Nielsen

#### University of Waterloo , Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we examine a synchronized path following control design problem for a collection of homogenous, minimum phase, linear time-invariant systems assigned closed curves in their output space. In the first part of the talk, we define a \emph{path following normal form} for LTI systems. We isolate the role that the zero dynamics play in determining the feasibility of using the path following normal form for control design. In the second part of the talk, we leverage recent results from the literature on synchronizing LTI systems to develop synchronizing controllers. The main result is a distributed feedback law that drives all the agents to their respective paths and ensures path invariance while simultaneously synchronizing their positions along the path. Laboratory results are presented to illustrate the effectiveness of the proposed approach.

Joint work with Maxwell Steinfeld (University of Waterloo).