Map of propagation risk of COVID-19 by local contact in PORTUGAL


The spread of COVID-19 is posing many challenges to our social and health systems. One of them is predicting and quantifying the emergence of new cases that follow local contagion at a national scale. By local contagion we refer to those cases of individuals that are infected with the virus but for which the source of infection is unknown, that is, the person affected has not reported any travel to affected areas or any direct contacts with other infected individuals.

One of the major difficulties we are currently facing is the early detection of cases, which is essential for confinement and medical treatment. A crucial feature of this virus is that it presents an asymptomatic phase, where the infected individual is infectious but presents mild or no symptoms. This period is considerably long, and can last up to 14 days according to the present reported data. The current lack of measures to detect infection cases in their early stages in addition to the existence of this asymptomatic phase influences to a large extent the spreading of the epidemics and the implementation of effective control measures.

In this website we show the results of the estimated epidemic risk in Portugal, at a municipal level, as obtained by a model of epidemic spreading which takes into account the recurrent mobility patterns (commuting) among municipalities. This data has been provided by the company NOS.

Our model currently includes the epidemiological data reported up to the present moment for the virus responsible of the COVID-19 (now officially named SARS-CoV-2) and demographical and mobility data for each municipality in Portugal. This model can also be directly applied to other countries and regions for which such mobility and demographics data is available.


Risk map

The risk map generated with our model produces an indicator for each municipality of the fraction of the population that is estimated to have contracted the infection of SARS-CoV-2 through local contact. Those areas shaded in gray correspond to areas for which there is no data available.

The numerical data corresponding to the map can be downloaded here in Excel format and here in PDF format.


A mathematical model for the spatiotemporal epidemic spreading of COVID19
medRxiv 2020.03.21.20040022
The full description of the model is available for download here.

The model we are using is a new version of a family of epidemiological models in discrete time, that has been specifically modified to represent the particular spreading dynamics of SARS-COV-2, the virus causing COVID-19.

The objective of the model is to estimate the risk for every municipality in Portugal, taking into account the following factors: (I) The dynamics of the transmission of SARS-COV-2, and (II) the recurrent mobility flows in Portugal, and (III) the demography of the Portuguese population.

With respect to modeling the transmission of the virus, we use a compartmental model, meaning that we divide the population according to their infection status. These compartments are:

  • Susceptible: an individual that has not been infected yet but is susceptible to infection.
  • Exposed: an individual who is infected but not yet infectious, because he/she still is in an incubation phase.
  • Asymptomatic (or displaying mild symptoms): an individual who is infected and infectious, but does not show clear symptoms of COVID-19.
  • Infected: in our model, an infected individual is infectious and does show clear symptoms compatible with COVID-19, making its detection easier than those in the Asymptomatic compartment.
  • Recovered/removed: an individual who has been infected at a moment in time, but who in the current moment, is not infectious anymore. This could happen because the individual has recovered from the infection and developed immunity, or because the patient has died.

The transitions among the previous compartments are regulated with the specific parameters of the model (transmission probability, recovery probability...etc), and have been obtained from the scientific literature published up to the present moment.

Regarding mobility, we have used data made available by the company NOS, corresponding to the mobility patterns of mobile phone users within a municipality and between all pairs of municipalities in Portugal.


  • The model does not predict the imported international cases (those individuals that have been infected outside the country and then traveled to Portugal).
  • The model is working with epidemiological parameters as reported up to the present moment in the medical and scientific literature, but can be changed upon new discoveries.
  • The model assumes that the mobility data as reported by NOS do not change. Our estimation would substantially differ if mobility restrictions were imposed.


  • The model allows tuning the epidemiological parameters as soon as new epidemiological studies report them.
  • The model allows to study the influence of the asymptomatic period and its associated infectivity.
  • Taking into account the epidemiological factors, the mobility and demography data, we can estimate the map of risk of new cases, which allows us to anticipate the spreading of the virus through asymptomatic individuals.
  • Massive restriction mobility (quarantine policies) can easily be introduced in the model, allowing us to obtain a new estimate of the risk under these new measures. This would allow policy makers and authorities to obtain an estimation of the efficacy of such measures.

The problem with the data

Our model can calculate, starting from a set of initial conditions (i.e. a certain number of detected cases and their exact location), an estimation of how would the epidemic evolve in Portugal. In the first phase of the spreading of SARS-COV-2 in Portugal, the majority of the reported cases were “imported”, that is, individuals that had traveled to other infected countries, got infected there and then traveled to Portugal. These cases cannot be detected by our model because they are external to the transmission dynamics of the virus inside the country. For that reason, in a phase where the imported cases conform the majority of the detected positives in the country, it is crucial to have accurate, reliable data to be able to calculate our estimates.

Frequently Asked Questions

In this map we plot, for each municipality in Portugal, an indicator of the fraction of the population that we estimate might have been infected by COVID-19 by local contact.

A mathematical model is a simplified abstraction of reality in the form of mathematical equations. The equations obtained can be used to understand what phenomena would we observe under certain conditions. The most common example of a mathematical model are meteorology models, used for weather forecasting.

Our model is a new version of a family of models already established in the scientific literature, called compartmental epidemiological models. In this particular model, which has been especially designed for the spreading of COVID-19, we assume that an individual might be in a susceptible state, in the exposed state (infected but not yet infectious), in an infectious asymptomatic state, in an infectious and symptomatic state, or in a recovered or removed state. The transitions between the previous compartments are regulated by the specific parameters for this disease.

The parameters have been obtained from the most recent scientific literature about the epidemiological traits of this new virus causing the disease COVID-19 (now officially termed SARS-CoV-2). The parameters could be modified depending on the results of the ongoing research.

No. A risk indicator of a 0.1% in a municipality means that we estimate that 1 of every 1000 individuals of that municipality might be infected (either in a symptomatic or asymptomatic state). The probability of contracting the disease depends on more factors, like for instance, the number of contacts that one individual makes, the infectivity of the disease, etc. The risk of infection of each municipality that we plot on the map could be interpreted as the “potential health state with respect to COVID-19 of that municipality".

No. That is the process we followed during the first days of infection in Portugal, because most of the detected cases were “imported”, and therefore they had to be introduced in the model by hand. At the present moment, the spreading of the epidemic is in another phase, where most of the infection cases are result of local contagion. These local infections are well captured by our model, meaning our model is able to reproduce them. We are now feeding our model with the initial conditions of the first phase of the epidemics (the number of reported cases in that phase), we let the model evolve and we plot the risk estimate prediction for the following days.

We are using the mobility data from NOS, which accounts for the mobility of the population who travels daily in each municipality. This data reports, not only, the home-to-work mobility and viceversa, bult also include other mobility factors not necessarily work-related.


Principal Investigators

Alex Arenas (Universitat Rovira i Virgili, Tarragona, Spain)

Jesús Gómez-Gardeñes (Universidad de Zaragoza, Zaragoza, Spain)

Participating researchers

Wesley Cota (Universidade Federal de Viçosa, Minas Gerais, Brazil)

Sergio Gómez (Universitat Rovira i Virgili, Tarragona, Spain)

Clara Granell (Universidad de Zaragoza, Zaragoza, Spain)

Joan T. Matamalas (Harvard Medical School, Boston, USA)

David Soriano-Paños (Universidad de Zaragoza, Zaragoza, Spain)

Benjamin Steinegger (Universitat Rovira i Virgili, Tarragona, Spain)


Portugal Section

Nuno Araújo (Centro de Física Teórica e Computacional, Faculdade de Ciências, U Lisboa, Portugal)

Hygor Piaget Melo (Centro de Física Teórica e Computacional, Faculdade de Ciências, U Lisboa, Portugal)

Partners: NOS, Data Science Portuguese Association, and Closer Consulting

Brazil Section

Wesley Cota (Universidade Federal de Viçosa, Minas Gerais, Brazil)

Silvio C. Ferreira (Universidade Federal de Viçosa, Minas Gerais, Brazil)