Lessons from being challenged by COVID-19

Abstract

We present results of different approaches to model the evolution of the COVID-19 epidemic in Argentina, with a special focus on the megacity conformed by the city of Buenos Aires and its metropolitan area, including a total of 41 districts with over 13 million inhabitants. We first highlight the relevance of interpreting the early stage of the epidemic in light of incoming infectious travelers from abroad. Next, we critically evaluate certain proposed solutions to contain the epidemic based on instantaneous modifications of the reproductive number. Finally, we build increasingly complex and realistic models, ranging from simple homogeneous models used to estimate local reproduction numbers, to fully coupled inhomogeneous (deterministic or stochastic) models incorporating mobility estimates from cell phone location data. The models are capable of producing forecasts highly consistent with the official number of cases with minimal parameter fitting and fine-tuning. We discuss the strengths and limitations of the proposed models, focusing on the validity of different necessary first approximations, and caution future modeling efforts to exercise great care in the interpretation of long-term forecasts, and in the adoption of non-pharmaceutical interventions backed by numerical simulations.

Keywords: COVID-19; Compartmental models; Mathematical epidemiology; Mobility; Nonlinear dynamics.

Fig. 1

Top: Evolution of the number of reported infectious travelers arriving to Argentina since the first official case. Dots and crosses correspond to the accumulated (minus recoveries) and new per-day cases, respectively, and the solid and dashed lines represent the best fits to the data. Bottom: Number of accumulated cases in the first 25 days, and the best fit obtained using a forced deterministic SEIR model.

Fig. 2

Simulation of a deterministic SIR model for a population of 10⁶ individuals. The infection rate parameter is periodically modulated $({\beta }_0=0.3\times {10}^{-6},$$A=0.2\times {10}^{-6}),$ with $\gamma =0.12$ and $\omega =2\pi /10$. This corresponds to variations of the basic reproduction number R₀ ∈ [0.8, 4.16] with a period of 10 days.

Fig. 3

Fifty simulations of the stochastic SEIR model for a set of communes inside the city of Buenos Aires (CABA). The top panels display the simulations for the entire city, and for one of the 15 communes. The solutions of the continuous model are displayed, in both cases, with dashed lines. A detail of the simulation at early times is shown in the bottom panels, also for the whole city (left) and for one of the communes (right). The average of the simulations (thick dark lines) is shown together with the solution of a homogeneous deterministic model (dashed lines).

Fig. 4

Top: Evolution of the number of infectious individuals in an ensemble of realizations of a stochastic SEIR model with a population of 50 individuals. Bottom: Time to extinction of the focus as a function of the size of the population involved, assuming the focus is perfectly isolated.

Fig. 5

Estimated transmission rate β(t) for the whole nation (Nation wide) and for the province of Buenos Aires (PBA). The dashed line corresponds to the values obtained from a least square fit with the deterministic model using a moving window of 10 days. The blue line corresponds to the smooth estimation at early and late times, while the blue shading indicates a window of confidence of 95% for β(t) during the lockdown.

Fig. 6

Accumulated cases in the entire country (Nation wide) and in the province of Buenos Aires (PBA, right), obtained from the SEJIHR model using β(t), with 95% confidence intervals (blue shading) and compared with the official data.

Fig. 7

Left: Normalized inter-regional mobility computed during the 2nd of March. Regions are ordered according to their community membership detected with the Louvain algorithm. The three white diagonal squares indicate major communities corresponding to the southern, centre-northern and western part of the metropolitan area. Right: Topological representation of the major communities using the Yifan Hu Multilevel layout algorithm as implemented in Gephi.

Fig. 8

Left: Average local mobility from the 1st of March to the 12th of April. Right: Average inter-regional mobility (computed as the average of all rows in C_i,j) from the 1st of March to the 12th of April. In both panels the grey vertical line indicates the start of the lockdown (19th March).

Fig. 9

Number of accumulated cases in five districts of the Buenos Aires metropolitan region, and in the city of Buenos Aires (CABA). Infectious travelers were reported in the early stages of the outbreak for all these districts. Red dots indicate the official number of cases, while the blue curves indicate results from the deterministic coupled model.

Fig. 10

Number of accumulated cases in four districts of the Buenos Aires metropolitan region that reported no infectious travelers arriving from countries with reported COVID-19 cases. Red dots indicate the official number of cases, while the blue solid lines indicate results from the deterministic coupled model.

Fig. 11

Number of cases in the city of Buenos Aires (center, enclosed by black lines) and the 40 districts of the metropolitan area (in different shades). The white region corresponds to a river. From top to bottom, days 5, 20, 40, and 60 since the first reported case. The size of blue circles indicates the number of official cases, while the size of violet circles indicates the number of cases according to the deterministic coupled model.

Fig. 12

Number of accumulated cases in four districts of the metropolitan region and Buenos Aires city (CABA), with forecasts for an increase in mobility recovering 30%, 60%, and 100% of the values before the lockdown, starting on the last day with available data.

Fig. 13

Number of accumulated cases in five districts of the Buenos Aires metropolitan region, plus the city of Buenos Aires (CABA). Infectious travelers were reported in the early stages of the outbreak for all these districts. Red dots indicate the official number of cases, while blue curves indicate individual realizations of the stochastic model. The thick blue line indicates the average across all realizations.

Fig. 14

Number of accumulated cases in four districts of the Buenos Aires metropolitan region, for districts that reported no infectious travelers arriving from countries with reported COVID-19 cases. Red dots indicate the official number of cases, while blue curves indicate individual realizations of the stochastic model. The thick blue line indicates the average across all realizations.