Numerical study of COVID-19 spatial-temporal spreading in London

Recent study reported that an aerosolised virus (COVID-19) can survive in the air for a few hours. It is highly possible that people get infected with the disease by breathing and contact with items contaminated by the aerosolised virus. However, the aerosolised virus transmission and trajectories in various meteorological environments remain unclear. This paper has investigated the movement of aerosolised viruses from a high concentration source across a dense urban area. The case study looks at the highly air polluted areas of London: University College Hospital (UCH) and King Cross and St Pancras International Station (KCSPI). We explored the spread and decay of COVID-19 released from the hospital and railway stations with the prescribed meteorological conditions. The study has three key findings: the primary result is that it is possible for the virus to travel from meters up to hundred meters from the source location. The secondary finding shows viruses released into the atmosphere from entry and exit points at KCSPI remain trapped within a small radial distance of<50m. This strengthens the case for the use of face coverings to reduce the infection rate. The final finding shows that there are different levels of risk at various door locations for UCH, depending on which door is used there can be a higher concentration of COVID-19. Although our results are based on London, since the fundamental knowledge processes are the same, our study can be further extended to other locations (especially the highly air polluted areas) in the world.

The earliest confirmed case of COVID-19 was in December 2019 and has since become a global pandemic with over 93,688,066 confirmed cases and 2,005,773 deaths worldwide as of January 14, 2021 (https://www.worldometers.info/coronavirus/).
Existing studies showed that the virus is mostly spread through breathing, coughing and sneezing (Chakraborty and Maity 2020;Howard et al., 2020), and direct contact with unsterilised abiotic surfaces such as plastics and stainless steel (where the virus can remain infectious for 28 days, https://www.bbc.co.uk/news/health-54500673). Social distancing in the range 1 ~ 2.5 meters has been recommended to mitigate this situation (Blocken et al. 2020;Wei and Li, 2015). However, there is evidence that the virus can remain in the air for more prolonged periods of time (remaining infectious for over 3 hours after expulsion, van Doremalen et al. 2020;Setti et al. 2020), travelling further contained within aerosols suspended in the air . Studies have found that similar particles containing viral matter can travel up to ten meters, suggesting that the recommended social distancing may be insufficient (Morawska and Cao, 2020). A higher rate of mortality is also linked with an increased concentration of particulate matter (PM, Wu et al. 2020) and the number of COVID-19 cases has been linked with the level of PM in Italy and France (Kowalski and Konior, 2020).
Prior research on the effect of aerosols in transmitting the virus, suggest measures like better ventilation to keep the aerosols outside of buildings, keeping those within safe (Morawksa and Cao, 2020) but few have considered the effect of aerosols on outdoor transmission over longer distances, partly because there are no simple methods to collect data, and therefore a lack of data to analyse (Carducci et al., 2020).
The aim of this computational study is to tackle the aforementioned issues with the computational fluid dynamics (CFD) model Fluidity, developed by Imperial College. Fluidity is an open source Large Eddy Simulation (LES) model with an advanced adaptive mesh capability (https://github.com/fluidityproject). This study investigates the effect of the exponential decay of the virus and the complexity of the spreading phenomenon: how long can the virus spread for given certain meteorological conditions? We will explore these in two different locations in London: University College Hospital (UCH); King's Cross Station and St Pancras International (KCSPI). This study will explore the spread of this virus from hospitals and railway stations, and the impact of meteorological conditions on a virus. Our aim is to explore the impact of the aerosol transport of the virus under specific meteorological conditions and highlight the importance of treating the pandemic seriously. Although our study is based on London, it can be extended to other locations (especially the highly air polluted areas) in the world since all the fundamental knowledge processes are the same.

Governing equations in virus spreading simulations
The three-dimensional (3D) Navier-Stokes equations and generic atmospheric chemical transport (advection-diffusion) equation are utilized for COVID-19 spreading simulations. The 3D Navier-Stokes equations are written as: is the velocity vector, t is the time, x y z ρ is the (assumed uniform) density of the atmosphere, S u represents the source, absorption or the drag forcing term of velocity (e.g. there is an absorption term when the flow passes the trees) and  represents the dynamic viscosity.
The virus transport equation is: where C is the mass concentration of the virus,  is the tensor of turbulent diffusivity, S represents the source term of virus, and D is the decay of virus.
The virus exponential decay formulation (van Doremalen et al, 2020) is given as follows: where 0 C is the initial virus concentration, λ denotes the decay rate. The decay term in Equation (2) can thus expressed as Studies have shown that the decay of airborne viruses is sensitive to the metrological conditions (wind velocity, ambient humidity and temperature (Yang and Marr, 2012;Schuit et al. 2020;Chan et al. 2011). In this work, to ensure stability and suppress spurious oscillations, the control volume (CV) method is used for resolving the virus concentration. To avoid spurious oscillations, the CV-TVD (control volume total variation diminishing) limiter is used to make the solutions total variation diminishing.
For control volume discretization, an explicit scheme is simple but strictly limited by the CFL number, which can be restrictive on adaptive meshes as the minimum mesh size can be very small. Here, we adopt a new time stepping scheme based on traditional Crank-Nicolson scheme because of its robustness, unconditional stability and second-order accurate in time.

Meteorological boundary conditions
The Synthetic-Eddy Method (Pavlidis et al. 2010) is used to set up the inlet boundary condition: are the spatial coordinates, t is the time, x are the velocity components at the inlet along the x, y, z directions, respectively, the mean velocity ( ) in U x is the function of the vertical z coordinate obeying the standard log-law over roughness height z0:

A dynamically adaptive mesh model for Computational Fluid Dynamics (CFD) and virus spreading simulations
Fluidity is a computational fluid dynamics code capable of numerically solving the Navier-Stokes Equation (1) with the large eddy simulation (LES) and accompanying field equations shown in Equation (2)  Once viruses are emitted into the air, the dynamic and transmission processes involve a wide range of spatial scales. An artificial dilution of viruses may lead to a shorter lifetime in existing fixed grid models if the resolution of grids is not high enough. It has been proved (Zheng et al., 2015(Zheng et al., , 2020) that mesh adaptivity is the most efficient and effective approach for resolving multi-scale dynamic processes. The mesh is adapted with respect to the dynamic flow features and virus concentrations in time and space. Using the adaptive mesh, the detailed flow dynamics and the temporal and spatial evolution of COVID-19 viruses during the spreading process can be captured, especially the local turbulent flows around buildings.

Modelling setup • Study area: University College Hospital (UCH) and King's Cross and St Pancras
International Station (KCSPI) on the Euston road in the center of London, is close to significant traffic along Euston road resulting in high pollution levels around that area. The computational area is a rectangular 2 km * 1.5 km domain centered on UCH, and 1000 m high to capture the dynamic part of the atmospheric boundary layer. As shown in Figure   1, the computational domain includes many buildings of different heights and configurations. • Adaptive mesh resolutions: The use of dynamically adaptive meshes optimizes the computational effort to resolve the flow dynamic and virus transport processes over a wide range of spatial scales. In this study, the mesh is dynamically adapted with respect to both the wind velocity field and virus concentration. The a-priori error measure for adapting the mesh is 0.3 m/s for velocity solutions and the relative error measure is 0.01 for virus concentration. The maximum number of nodes is set to be 600,000, which is large enough to ensure the a-priori error to be achieved. To avoid spurious dilution of virus emission, high resolution meshes are located around the sources (see Figure 2 (f)-(g)).
• Boundary and initial conditions: At the inlet boundary, the velocity profile is given by Equation (4)

Results and discussions
In this discussion we will focus on (1) the spatial distribution of viruses released from UCH and KCSPI for given wind field; (2)

Conclusions
Our findings suggest that the aerosolised virus particles can be transmitted a long distance (hundreds of meters) due to the fully developed turbulent flows around the source locations.
For example, around the UCH, there is a strong wind field (~ 2.5 m/s) at the height of 5 m and viruses with the concentration of > 0.2 copies/m 3 can be found within 60 ~ 500 m away from UCH for given meteorological conditions (e.g. wind field). We also notice that the infectious area from the virus released from Door 2 in UCH is larger than the other doors. This suggests that Door 2 in UCH is not a good location for A & E; the entry location in hospital must be chosen carefully. Our study found that the majority of the viruses released from the St Pancras International and King's Cross stations remain trapped within a short radical distance of less than 50 meters and will not affect the people living nearby. In summary, it is suggested that a face cover is needed for personal protection if people travel to public dense places (hospital, train stations etc.). Finally, the impact of urban green (tree, for example) environment on reducing the virus spreading will be further investigated in our future work.

Availability of data:
The data that supports the findings of this study are available within the article.