No abstract

166 countries/regions, including cases of human-to-human transmission around the world. The proportions of this epidemics is probably one of the largest challenges faced by our interconnected modern societies. According to the current epidemiological reports, the large basic reproduction number, R0 ∼ 2.3, number of secondary cases produced by an infected individual in a population of susceptible individuals, as well as an asymptomatic period (up to 14 days) in which infectious individuals are undetectable without further analysis, pave the way for a major crisis of the national health capacity systems. Recent scientific reports have pointed out that the detected cases of COVID19 at young ages is strikingly short and that lethality is concentrated at large ages. Here we adapt a Microscopic Markov Chain Approach (MMCA) metapopulation mobility model to capture the spread of COVID-19. We propose a model that stratifies the population by ages, and account for the different incidences of the disease at each strata. The model is used to predict the incidence of the epidemics in a spatial population through time, permitting investigation of control measures. The model is applied to the current epidemic in Spain, using the estimates of the epidemiological parameters and the mobility and demographic census data of the national institute of statistics (INE). The results indicate that the peak of incidence will happen in the first half of April 2020 in absence of mobility restrictions. These results can be refined with improved estimates of epidemiological parameters, and can be adapted to precise mobility restrictions at the level of municipalities. The current estimates largely compromises the Spanish health capacity system, in particular that for intensive care units, from the end of March. However, the model allows for the scrutiny of containment measures that can be used for health authorities to forecast with accuracy their impact in prevalence of COVID-19. Here we show by testing different epidemic containment scenarios that we urge to enforce total lockdown to avoid a massive collapse of the Spanish national health system.

, reaction processes are understood as contagions within each subpopulation (patch), while diffusion represents the mobility of individuals between patches. Recently, the characteristics of human mobility 7 , such as its recurrent nature, have been proven crucial to understand the phase transition to endemic epidemic states 8,9 . Here, by developing a framework able to cope with the elementary epidemic processes, the spatial distribution of populations and the commuting mobility patterns, we discover three different critical regimes of the epidemic incidence as a function of these parameters. Interestingly, we reveal a regime of the reaction-diffussion process in which, counter-intuitively, mobility is detrimental to the spread of disease. We analytically determine the precise conditions for the emergence of any of the three possible critical regimes in real and synthetic networks.Epidemic processes in complex networks have attracted the attention of physicists during the last two decades 6 . Several outstanding results have been the consequence of a mathematical analysis that borrows ideas from other physical processes. In particular, epidemic spread in networks can be thought of as reactiondiffusion processes, referring to the change of the concentration of two or more types of element: local reactions in which the elements are transformed into each other, and diffusion that causes the substances to spread out over the available space. In epidemiology, the elements in play are the subjects (humans or animals), characterized by their states in the evolution of the sickness (for example, susceptible, infected, recovered and so on). In complex networks, the reaction phase corresponds to the infections produced by the local interaction of subjects within a subpopulation (node), and the diffusion phase corresponds to their mobility through the network according to the connections (links) between nodes.This approach to epidemic spread using reaction-diffusion processes, usually referred to as metapopulation models, has been largely studied in network science [10][11][12][13][14] ; however, several challenges remain open 15,16 . The most representative of these challenges, from a physicist's perspective, is to complement large-scale agent-based simulations [17][18][19] , by deriving models amenable to mathematical analysis 20 that capture the influence of human behaviour 21 and the existence of complex social structures.Our proposal to fill this gap is to formulate a general 'microscopic' Markovian model describing the metapopulation reaction-diffusion dynamics. We start by analysing, at the individual level, the probabilities of infection in the scope of the susceptibleinfected-susceptible (SIS) epidemic model 12 . We denote as λ and μ the infection and recovery probabilities respectively. This way, a susceptible (S) individual is infected with probability λ when interacting with an infected (I) subject. In turn, infected (I) individuals become susceptible (S) again with probability μ. Note that if there is no recov...

We propose a theoretical framework for the study of spreading processes in structured metapopulations, with heterogeneous agents, subjected to different recurrent mobility patterns. We propose to represent the heterogeneity in the composition of the metapopulations as layers in a multiplex network, where nodes would correspond to geographical areas and layers account for the mobility patterns of agents of the same class. We analyze classical epidemic models within this framework and obtain an excellent agreement with extensive Monte Carlo simulations. This agreement allows us to derive analytical expressions of the epidemic threshold and to face the challenge of characterizing a real multiplex metapopulation, the city of Medellín in Colombia, where different recurrent mobility patterns are observed depending on the socioeconomic class of the agents. Our framework allows us to unveil the geographical location of those patches that trigger the epidemic state at the critical point. A careful exploration reveals that social mixing between classes and mobility crucially determines these critical patches and, more importantly, it can produce abrupt changes of the critical properties of the epidemic onset.

This expression for R is an extremely useful tool to design containment policies that are able to suppress the epidemics. We applied our epidemic model for the case of Spain, successfully forecasting both the observed incidence in each region and the overload of the health system. The expression for R allowed us to determine the precise reduction of mobility κ 0 needed to bend the curve of epidemic incidence, which turned out to be κ 0 ∼ 0.7.This value, for the case of Spain, translates to a total lockdown with the exception of the mobility associated to essential services, a policy that was finally enforced on March 28.