  We introduce a new probabilistic model to estimate the real spread of the
novel SARS-CoV-2 virus along regions or countries. Our model simulates the
behavior of each individual in a population according to a probabilistic model
through an inverse problem; we estimate the real number of recovered and
infected people using mortality records. In addition, the model is dynamic in
the sense that it takes into account the policy measures introduced when we
solve the inverse problem. The results obtained in Spain have particular
practical relevance: the number of infected individuals can be $17$ times
higher than the data provided by the Spanish government on April $26$ $th$ in
the worst-case scenario. Assuming that the number of fatalities reflected in
the statistics is correct, $9.8$ percent of the population may be contaminated
or have already been recovered from the virus in Madrid, one of the most
affected regions in Spain. However, if we assume that the number of fatalities
is twice as high as the official numbers, the number of infections could have
reached $19.5\%$. In Galicia, one of the regions where the effect has been the
least, the number of infections does not reach $2.5 \%$ . Based on our
findings, we can: i) estimate the risk of a new outbreak before Autumn if we
lift the quarantine; ii) may know the degree of immunization of the population
in each region; and iii) forecast or simulate the effect of the policies to be
introduced in the future based on the number of infected or recovered
individuals in the population.
