#2 – What does this model predict ?

To go from the diagram to the model predictions, you need to define :

• the initial conditions, i.e. how individuals in the population are distributed between the different health states at the beginning of the simulation (at t , the first time step);
• the values of model parameters.

In the absence of accurate information, as is often the case at the beginning of an epidemic, it is assumed that the epidemic starts with the arrival of a (single) infected individual in a fully susceptible population. Here we consider the arrival of an asymptomatic individual (A) in a population of constant size N. As a reminder (see article #1), individuals in the population are grouped by health status according to whether they are susceptible (S), asymptomatic (A), symptomatic (I), recovered (R), or dead (M). Initially S(t ) = N-1, A(t ) = 1, I(t ) = 0, R(t ) = 0 et M(t ) = 0.

The values of the parameters were chosen to be consistent with the available knowledge on the propagation of SARS-CoV-2, so that on March 16 (with t  = January 1^st) there were 148 cumulative deaths (the death toll in France on that date). Of course, this rudimentary model is not intended to be used in the current situation. The figures are mainly there to illustrate our point. Some parameter values may even seem surprising, such as the transmission rate of I's here lower than that of A's. It should not be forgotten, however, that some of the I's see their contacts decrease when they are ill !

Parameter values used to obtain a cumulative number of 148 deaths as of March 16,
the epidemic starting following the arrival of individual A in the susceptible population on January 1^st.

Under the assumptions we listed in article #1, and for the initial conditions and parameter values used, the model will predict the headcount by health status over time, as well as the flows between states. It is therefore possible to construct multiple outputs (which the model will calculate for us) depending on what we are interested in. Thus, the epidemic curve provides the date of the epidemic peak, the proportion of individuals affected simultaneously (i.e. prevalence, 20% of the population at the date of the epidemic peak here), the duration of the epidemic (80-100 days here), or the date of reaching a threshold of cumulative number of cases or deaths.

Epidemic dynamics predicted by the SAIRM model described in article #1,
whose parameters are calibrated as shown in the table above, and with 
as initial conditions 1 asymptomatic individual introduced into a susceptible population,
without any measures to control the epidemic. The grey vertical dotted line indicates March 16.

Let us now zoom in on the period from January 1st (t=0) to March 16 (t=76). We see very little variation in the number of individuals by health status during this period, which marks the beginning of the epidemic in France. We can nevertheless look at the number of new daily cases (also called incidence of infection, i.e. the sum of the flows from state S to states A and I) and the number of deaths (numbers in state M for the cumulative number of deaths; daily flow from state I to state M for the number of new deaths per day).

Evolution of the cumulative number of deaths since the introduction of the virus into the population up to March 16.

Article#3 will discuss the predictive quality of models.