Reaction-Diffusion Systems as Complex Networks

The spatially distributed reaction networks are indispensable for the understanding of many important phenomena concerning the development of organisms, coordinated cell behavior, and pattern formation. The purpose of this brief discussion paper is to point out some open problems in the theory of PDE and compartmental ODE models of balanced reaction-diffusion networks.Comment: A discussion paper for the 1st IFAC Workshop on Control of Systems Governed by Partial Differential Equation

Bayesian inference for indirectly observed stochastic processes, applications to epidemic modelling

Stochastic processes are mathematical objects that offer a probabilistic representation of how some quantities evolve in time. In this thesis we focus on estimating the trajectory and parameters of dynamical systems in cases where only indirect observations of the driving stochastic process are available. We have first explored means to use weekly recorded numbers of cases of Influenza to capture how the frequency and nature of contacts made with infected individuals evolved in time. The latter was modelled with diffusions and can be used to quantify the impact of varying drivers of epidemics as holidays, climate, or prevention interventions. Following this idea, we have estimated how the frequency of condom use has evolved during the intervention of the Gates Foundation against HIV in India. In this setting, the available estimates of the proportion of individuals infected with HIV were not only indirect but also very scarce observations, leading to specific difficulties. At last, we developed a methodology for fractional Brownian motions (fBM), here a fractional stochastic volatility model, indirectly observed through market prices. The intractability of the likelihood function, requiring augmentation of the parameter space with the diffusion path, is ubiquitous in this thesis. We aimed for inference methods robust to refinements in time discretisations, made necessary to enforce accuracy of Euler schemes. The particle Marginal Metropolis Hastings (PMMH) algorithm exhibits this mesh free property. We propose the use of fast approximate filters as a pre-exploration tool to estimate the shape of the target density, for a quicker and more robust adaptation phase of the asymptotically exact algorithm. The fBM problem could not be treated with the PMMH, which required an alternative methodology based on reparameterisation and advanced Hamiltonian Monte Carlo techniques on the diffusion pathspace, that would also be applicable in the Markovian setting

Physics of interdependent dynamical processes.

La emergencia de fenómenos colectivos a escalas macroscópicas no observados en escalas microscópicas cuestiona la validez de las teorías reduccionistas. Para explicar estos fenómenos se necesitan enfoques sistémicos que den cuenta de los patrones de interacción no triviales existentes entre los constituyentes de los sistemas sociales, biológicos o económicos, lo que ha dado lugar al nacimiento de la disciplina conocida como ciencia de los sistemas complejos. Una vía habitual para caracterizar los sistemas complejos ha sido la búsqueda de la conexión entre la estructura de interacciones y el comportamiento colectivo observado en sistemas reales mediante el estudio individual de dinámicas aisladas. No obstante, los sistemas complejos no son inmutables y se encuentran constantemente intercambiando información mediante estímulos internos y externos. Esta tesis se centra en la adaptación de modelos sobre diferentes dinámicas en el campo de los sistemas complejos para caracterizar el impacto de este flujo de información, ya sea entre escalas microscópicas y macroscópicas de un mismo sistema o mediante la existencia de interdependencias entre procesos dinámicos que se propagan de forma simultánea.La primera parte de la tesis aborda el estudio dinámicas acopladas en redes de contacto estáticas. Adaptando los modelos compartimentales introducidos en el siglo XX a la naturaleza de cada dinámica, caracterizamos cuatro problemas diferentes: la propagación de patógenos que interactúan, cuya coexistencia puede ser beneficiosa o perjudicial para su evolución, el control de brotes epidémicos con el uso del rastreo de contactos digital, la aparición de movimientos sociales desencadenados por pequeñas minorías sociales bien coordinadas y la competencia entre honestidad y la corrupción en las sociedades modernas. En todas estas dinámicas, encontramos que el flujo de información cambia las propiedades críticas del sistema así como algunas de las conclusiones extraídas sobre el papel de la estructura de contactos al estudiar cada dinámica de forma individual.La segunda parte de la tesis se centra en el impacto de la movilidad recurrente en la propagación de epidemias en entornos urbanos. Derivamos un modelo sencillo que permite incorporar fácilmente la distribución de la población en las ciudades reales y sus patrones habituales de desplazamiento sin ninguna pérdida de información. Demostramos que los efectos de las políticas de contención basadas en la reducción de la movilidad no son universales y dependen en gran medida de las características estructurales de las ciudades y los parámetros epidemiológicos del virus circulante en la población. En particular, descubrimos y caracterizamos un nuevo fenómeno, el detrimento epidémico, que refleja el efecto beneficioso de la movilidad en algunos escenarios para contener un brote epidémico. Por último, exploramos tres casos de estudio reales, mostrando que nuestro modelo permite capturar algunos de los mecanismos que han convertido a los núcleos urbanos en importantes focos de contagio en recientes epidemias y que el modelo desarrollado puede servir como base para desarrollar marcos teóricos más realistas que reproducen la evolución de distintas enfermedades como la COVID-19 o el dengue.<br /

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...

Magnetically Interesting Coordination Complexes Based on Macrocyclic Ligands

The synthesis and study of select 3d and/or 4f coordination complexes prepared from crown ether and Schiff-base dual compartmental macrocycles are described herein, working towards the discovery and study of new families of macrocyclic-based single molecule magnets (SMMs). Chapter 1 introduces the general theory of magnetism, molecular magnetism and SMMs and provides the reader with a brief overview of the relevant coordination chemistry of the two families of macrocycles. In Chapter 2, two 15-crown-5 complexes [Ln(NO3)3(OH2)2(MeOH)], (where Ln(III) = Tb (I) and Dy (II)) have been prepared and characterized. X-ray diffraction studies reveal the two complexes crystallize as 1-D chains. Variable temperature ac magnetic susceptibility studies reveal that (II) is an SMM with two effective energy barriers, Ueff = 26 K (18 cm−1); τ0 = 4.10 × 10−7 s and Ueff = 41 K (29 cm−1); τ0 = 1.35 × 10−8 s, whereas ab initio studies suggest that the observation of slow relaxation of magnetization in the Tb complex (I) is hindered by the presence of rapid quantum tunneling mechanisms (QTM). Solid state photoluminescence measurements reveal the two complexes have well-resolved f–f transitions, where a Gaussian fit of the fine structure of the highest-energy emission band for the Dy(III) complex allows the Stark splitting of the ground state to be determined. In Chapter 3, select Ln(III) complexes with benzo and dibenzo 15-crown-5 macrocycles were synthesized and characterized. Reaction of Dy(III) together with benzo 15-crown-5 afforded a unique [Dy(OH2)8]3+ complex (III), where the hydrated Dy(III) cation is fully encapsulated within a supramolecular cage formed by three benzo 15-crown-5 macrocycles. Interestingly, the close to perfect square antiprismatic geometry of the 4f ion enhances its axial anisotropy, which suppresses quantum tunnelling mechanisms (QTM) in the ground and first excited states, resulting in the observation of SMM behavior in zero dc field. For this system the magnetic data were further supported by solid-state photoluminescence and ab initio studies, The introduction of a second benzene ring into the organic framework of the macrocycle increases its rigidity, where on coordination to Dy(III), affords the partially encapsulated complex (IV), which displays slow relaxation of magnetisation, consistent with SMM properties. In Chapter 4, the coordination chemistry of a dual compartmental Schiff-base macrocycle H2L3 containing O3O2 and N3O2 cavities was explored together with select 3d and 4f ions. In the first part of this chapter, the coordination chemistry of H2L3 with 3d metal ions is presented, where in the presence of NaOH, the Na(I) ions reside in the O3O2 cavity and the 3d ions occupy the second N3O2 cavity. Three coordination complexes containing Cu(II), Zn(II), and Mn(II) ions were prepared and characterized. The Cu(II), and Zn(II) complexes are monomeric with molecular formulae [CuNa(L3b)ClCH3OH]‧6H2O (V) and [ZnNa(L3b)(CH3COO)(CH3OH)]‧H2O (VI) respectively, while the Mn(II) complex crystallizes as a trimer with stoichiometry [Mn3Na2(L3)2(CH3COO)4]·5.75CH3OH·0.5H2O (VII). For complexes (V) and (VI), nucleophilic addition of the NH of the N3O2 cavity to the carbon atom of the adjacent imine results in a contraction of the N3O2 cavity and the formation of a five-membered imidazoline ring to afford the modified ligand L3b.The magnetic properties of (V) and (VII) are also reported. In the second part of this chapter, coordination of the macrocycle to select 4f ions in the absence of any base afforded the mononuclear complexes [Dy(H2L3)(H2O)2(CH3OH)2]Cl3·CH3OH, (VIII), and [Ln(H2L3)(H2O)3(CH3OH)] Cl3, where Ln(III) = Tb (IX), Er (X), and Gd (XI), in which the Ln(III) ion is coordinated in the O3O2 cavity. Magneto-structural studies on these complexes reveal that the Dy complex has a slightly different structure than the other three complexes, however all four 4f ions crystallize with square antiprismatic geometries, where only the Dy(III) complex (VIII) displays SMM properties