Spatial spread of COVID-19 outbreak in Italy using multiscale kinetic transport equations with uncertainty

Abstract

<abstract><p>In this paper we introduce a space-dependent multiscale model to describe the spatial spread of an infectious disease under uncertain data with particular interest in simulating the onset of the COVID-19 epidemic in Italy. While virus transmission is ruled by a SEIAR type compartmental model, within our approach the population is given by a sum of commuters moving on a extra-urban scale and non commuters interacting only on the smaller urban scale. A transport dynamics of the commuter population at large spatial scales, based on kinetic equations, is coupled with a diffusion model for non commuters at the urban scale. Thanks to a suitable scaling limit, the kinetic transport model used to describe the dynamics of commuters, within a given urban area coincides with the diffusion equations that characterize the movement of non-commuting individuals. Because of the high uncertainty in the data reported in the early phase of the epidemic, the presence of random inputs in both the initial data and the epidemic parameters is included in the model. A robust numerical method is designed to deal with the presence of multiple scales and the uncertainty quantification process. In our simulations, we considered a realistic geographical domain, describing the Lombardy region, in which the size of the cities, the number of infected individuals, the average number of daily commuters moving from one city to another, and the epidemic aspects are taken into account through a calibration of the model parameters based on the actual available data. The results show that the model is able to describe correctly the main features of the spatial expansion of the first wave of COVID-19 in northern Italy.</p></abstract>

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

“…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.…”

“…Following [18,21], it is possible to structure a 1D network considering that the nodes of the network identify locations of interest such as municipalities, provinces or, in a wider scale, regions or nations, while the arcs, enclosing the 1D spatial dynamics, represent the paths linking each location to the others. In this configuration, nodes are active since the epidemic state of each one evolves in time influenced by the mobility of the commuting individuals, moving from the other locations included in the network, always considering a part of the population composed by non-commuting individuals which remain at the origin node.…”

“…In order to prescribe the proper coupling between nodes and arcs, it is necessary to impose appropriate transmission conditions at each arc-node interface, which ensure the conservation of total density (population) in the network and of fluxes at the interface and further solving the Riemann problem at each interface employing Riemann Invariants. The complete description of the implementation of transmission conditions at nodes is presented in [21] for a SIR-type spatial model and in [18] for a SEIAR-type transport model.…”

“…For the definition of the incidence function regarding asymptomatic people 𝐹 𝐴 (•, 𝐴 𝑇 ), we consider (84). When introducing the same definition of flux (89) for the additional compartments, 𝐽 𝐸 and 𝐽 𝐴 , integrating system (93) in 𝑣, we get the set of equations for the macroscopic densities [18]. Moreover, defining also 𝐷 𝐸 = 1 2 𝜆 2 𝐸 𝜏 𝐸 and 𝐷 𝐴 = 1 2 𝜆 2 𝐴 𝜏 𝐴 and considering the same procedure discussed in Section 4.2.2, we recover SEIAR system in the diffusive regime for the commuting individuals [18] coupled with (94) for the non-commuting counterpart.…”

“…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.…”