Dynamics of epidemic spreading on connected graphs

The emergence of spatial patterns for compartmental reaction kinetics coupled by two bulk diffusing species with comparable diffusivities

Originating from the pioneering study of Alan Turing, the bifurcation analysis predicting spatial pattern formation from a spatially uniform state for diffusing morphogens or chemical species that interact through nonlinear reactions is a central problem in many chemical and biological systems. From a mathematical viewpoint, one key challenge with this theory for two component systems is that stable spatial patterns can typically only occur from a spatially uniform state when a slowly diffusing 'activator' species reacts with a much faster diffusing 'inhibitor' species. However, from a modelling perspective, this large diffusivity ratio requirement for pattern formation is often unrealistic in biological settings since different molecules tend to diffuse with similar rates in extracellular spaces. As a result, one key long-standing question is how to robustly obtain pattern formation in the biologically realistic case where the time scales for diffusion of the interacting species are comparable. For a coupled one-dimensional bulk-compartment theoretical model, we investigate the emergence of spatial patterns for the scenario where two bulk diffusing species with comparable diffusivities are coupled to nonlinear reactions that occur only in localized 'compartments', such as on the boundaries of a one-dimensional domain. The exchange between the bulk medium and the spatially localized compartments is modelled by a Robin boundary condition with certain binding rates. As regulated by these binding rates, we show for various specific nonlinearities that our one-dimensional coupled PDE-ODE model admits symmetry-breaking bifurcations, leading to linearly stable asymmetric steady-state patterns, even when the bulk diffusing species have equal diffusivities. Depending on the form of the nonlinear kinetics, oscillatory instabilities can also be triggered. Moreover, the analysis is extended to treat a periodic chain of compartments. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

A Graph-Based Framework for Multiscale Modeling of Physiological Transport

Trillions of chemical reactions occur in the human body every second, where the generated products are not only consumed locally but also transported to various locations in a systematic manner to sustain homeostasis. Current solutions to model these biological phenomena are restricted in computability and scalability due to the use of continuum approaches in which it is practically impossible to encapsulate the complexity of the physiological processes occurring at diverse scales. Here, we present a discrete modeling framework defined on an interacting graph that offers the flexibility to model multiscale systems by translating the physical space into a metamodel. We discretize the graph-based metamodel into functional units composed of well-mixed volumes with vascular and cellular subdomains; the operators defined over these volumes define the transport dynamics. We predict glucose drift governed by advective-dispersive transport in the vascular subdomains of an islet vasculature and cross-validate the flow and concentration fields with finite-element-based COMSOL simulations. Vascular and cellular subdomains are coupled to model the nutrient exchange occurring in response to the gradient arising out of reaction and perfusion dynamics. The application of our framework for modeling biologically relevant test systems shows how our approach can assimilate both multi-omics data from in vitro-in vivo studies and vascular topology from imaging studies for examining the structure-function relationship of complex vasculatures. The framework can advance simulation of whole-body networks at user-defined levels and is expected to find major use in personalized medicine and drug discovery.

Spreading Properties for SIR Models on Homogeneous Trees

We consider an epidemic model of SIR type set on a homogeneous tree and investigate the spreading properties of the epidemic as a function of the degree of the tree, the intrinsic basic reproduction number and the strength of the interactions between the populations of infected individuals at each node. When the degree is one, the homogeneous tree is nothing but the standard lattice on the integers and our model reduces to a SIR model with discrete diffusion for which the spreading properties are very similar to the continuous case. On the other hand, when the degree is larger than two, we observe some new features in the spreading properties. Most notably, there exists a critical value of the strength of interactions above which spreading of the epidemic in the tree is no longer possible.

Multi-generational SIR modeling: Determination of parameters, epidemiological forecasting and age-dependent vaccination policies

We use an age-dependent SIR system of equations to model the evolution of the COVID-19. Parameters that measure the amount of interaction in different locations (home, work, school, other) are approximated from in-sample data using a random optimization scheme, and indicate changes in social distancing along the course of the pandemic. That allows the estimation of the time evolution of classical and age-dependent reproduction numbers. With those parameters we predict the disease dynamics, and compare our results with out-of-sample data from the City of Rio de Janeiro. Finally, we provide a numerical investigation regarding age-based vaccination policies, shedding some light on whether is preferable to vaccinate those at most risk (the elderly) or those who spread the disease the most (the youngest). There is no clear upshot, as the results depend on the age of those immunized, contagious parameters, vaccination schedules and efficiency.