Evaluation of the potential incidence of COVID-19 and effectiveness of contention measures in Spain: a data-driven approach

Alberto Aleta; Yamir Moreno

where P (j|j) is the normalized flow matrix without row and column j, p(j) is the j-column of P with element j removed and δ is a dimensionless parameter that depends on the infection, recovery and mobility rates [14] . This quantity, defined for SIR metapopulation models, gives us the expected time that it would take for the disease to reach each subpopulation of the system, also known as the hitting time. In Figure 3 , we compare the hitting time obtained from stochastic simulations of the SEIR metapopulation model with the theoretical distances derived for the simplified SIR model. We can see that the correlation is nearly perfect, implying that the spreading itself is quite similar in both models. However, we find that the hitting times in the SEIR implementation are at least two times larger than the theoretical ones for the SIR scenario (on its turn, stochastic simulations of the SIR model agree very well with the theoretical expectations for the model, see appendix C). Thus, the addition of the latent state produces a substantial delay on the spreading of the epidemic. This is in line with the fact that the epidemic is thought to have started in mid-November or early December, however, a noticeable number of cases was only reported by early January. Figure 4 shows the expected hitting time for each province when the disease starts from 5 different locations, as well as one case with seeds in multiple places, as obtained from the SEIR metapopulation model. As before, the hitting times might seem long, but this is due to the long latent periods of the disease, in agreement with the evolution of reported cases in mainland China. We also note that Spanish major cities are expected to be affected by the outbreak in no more than 40 days −but often within 20 days− from the initial time in all the situations considered. Indeed, to mimic the situation in the country as of February 28th, 2020, and to make projections about the evolution of the disease into the future, we considered the scenario in which the model is initialized with infected individuals in the capital of each region where there are cases of COVID19: Madrid, Tenerife, Barcelona, Balearic Islands, Zaragoza, Seville and Valencia [15] . The results show that the spreading is much faster (note that the outer circle in the 5 individual provinces represents 60 days while in this case it represents only 40 days) in such a situation.