Hyperbolic compartmental models for epidemic spread on networks with uncertain data: Application to the emergence of COVID-19 in Italy

Abstract

The importance of spatial networks in the spread of an epidemic is an essential aspect in modeling the dynamics of an infectious disease. Additionally, any realistic data-driven model must take into account the large uncertainty in the values reported by official sources such as the amount of infectious individuals. In this paper, we address the above aspects through a hyperbolic compartmental model on networks, in which nodes identify locations of interest such as cities or regions, and arcs represent the ensemble of main mobility paths. The model describes the spatial movement and interactions of a population partitioned, from an epidemiological point of view, on the basis of an extended compartmental structure and divided into commuters, moving on a suburban scale, and non-commuters, acting on an urban scale. Through a diffusive rescaling, the model allows us to recover classical diffusion equations related to commuting dynamics. The numerical solution of the resulting multiscale hyperbolic system with uncertainty is then tackled using a stochastic collocation approach in combination with a finite volume Implicit–Explicit (IMEX) method. The ability of the model to correctly describe the spatial heterogeneity underlying the spread of an epidemic in a realistic city network is confirmed with a study of the outbreak of COVID-19 in Italy and its spread in the Lombardy Region.

Modeling Effects of Spatial Heterogeneities and Layered Exposure Interventions on the Spread of COVID-19 across New Jersey

“…Thomas et al [63] used a networkbased model to demonstrate that heterogeneity in population distribution can have large impacts on local pandemic timing and severity, based on simulations for 19 US cities. It should be mentioned that, although PDE-based approaches were not included in the CDC ensemble of models [43], they provide systematic ways to account for the effects of spatial heterogeneities on epidemic dynamics, and such approaches have been utilized in studies of the spread of COVID-19 in Arizona, US [64] and the Lombardy Region of Italy [65][66][67][68]. However, there has been a lack of modeling work focusing on evaluating the effectiveness of interventions that reduce exposures at local regions, while also exploring how key variables (e.g., population density, compliance with wearing face masks, prior exposures to various stressors, age stratification, comorbidities in the population, etc.)…”

Kinetic modelling of epidemic dynamics: social contacts, control with uncertain data, and multiscale spatial dynamics

“…Therefore, the closure of the kinetic system (19) around a Gamma-type equilibrium of social contacts leads then to the system of six equations ( 20)- (22) for the pairs of mass fractions 𝐽 (𝑡) and local mean values 𝑚 𝐽 (𝑡), 𝐽 ∈ {𝑆, 𝐼, 𝑅}. In the following, we refer to the coupled systems (20) and (22) as the social SIR model (S-SIR).…”

“…A limiting case of system (22) is obtained by letting the parameter 𝜇 → +∞, which corresponds to push the variance to zero (absence of heterogeneity). In this case, if the whole population starts with a common number of daily contacts, say w, it is immediate to show that the number of contacts remains fixed in time, thus reducing system (20) to a classical SIR model with contact rate β w2 . Hence this classical epidemiological model is contained in (20)-( 22) and corresponds to consider the case of a population that, regardless of the presence of the epidemic, maintains the same fixed number of daily contacts.…”

“…To account for more complex compartmental models capable of better analyzing the evolution of specific infectious diseases, we consider extending the simple SIR compartmentalization by taking into account two additional population compartments, 𝐸 and 𝐴, resulting in a SEIAR model [18,20]. Subjects in the 𝐸 compartment are the exposed, hence infected but not yet infectious, being in the latent period.…”

“…These problems will be addressed both in the case of dynamics on networks connecting different cities [21], and in the case of completely two-dimensional dynamics at regional level [23]. Applications of these models to the early stages of the COVID-19 pandemic in Italy will also be illustrated [18,20]. Some open issues and future developments are also discussed at the end of this review along with detailed references to the data sources used in the simulations.…”