The interactive graphs below are based on a simple model for the evolution of an epidemic. The total population is divided in four sub-populations: infected *(I)*, active cases *(A)*, recovered *(R)* and deceased *(D)*. The user can change five input parameters called: infection fatality rate *(IFR),* infectiousness *(IS)*, initial number of infected patients *(InI)*, total population *(P) *and infection-to-death delay time (*T _{D}*).

Our interactive graphs show how the time evolution of these sub-populations is affected by the values digited as input parameters. In the following,

**Fig. 1**plots vs. time the curves of cumulative values for

*I(t), A(t), R(t)*, and

*D(t)*;

**Fig. 2**plots vs. time their daily variations

*δI(t), δA(t), δR(t),*and

*δD(t)*;

**Fig. 3**plots vs. time the case fatality rate

*CFR(t)*defined as the ratio

*D(t)/I(t)*.

When looking at the official data during a real epidemic, only parts of the four subpopulations *I, A, R* and *D* are detected, and all analyses are only made based on detected data. The simulation refers to real cases, whether they are detected or not.

We remark here a few basic observations that can be made already at first sight.

In **Fig. 1**, while the four sub-populations evolutions change according the input parameters, the relationship *I(t) = A(t) + R(t) + D(t) *remains valid at any time*.* *I(t)* converges at the end of the epidemic to a value slightly lower than the total population, corresponding to herd immunity. *A(t)* is the only curve converging back to zero at the end of the epidemic, when all the infected patients *I* become either recovered *R* or deceased *D*. A high infectiousness *IS* increases the peak of active patients *A* and shortens all time scales. A low infectiousness broadens the *A* peak, delaying the end of the epidemic beyond the 200 days period of our simulation. A high initial number *InI *of infected patients shortens all timescales.

In **Fig. 2**, we can observe the same dynamics reflected in the daily changes. Once more, *δI(t) = δA(t) + δR(t) + δD(t).* *δA(t)* is the only quantity considered here that becomes negative: it happens in the second phase of the epidemic, before converging to zero. A high infectiousness sharpens all peaks, while a low infectiousness broadens and delays them all. The peak of daily deaths *δD(t) *follows the peak of daily infections *δI(t) *by the delay time *T _{D.}*

In **Fig. 3**, we observe that *CFR(t)=D(t)/I(t) *is highly time-dependent for any finite *T _{D}* value. This explains why the apparent fatality rate reported in all statistics (deaths/cases) is deceptively low in the early phases of the outbreak and grows in the second and final phases. Only at the end of the epidemics, the convergence

*CFR(t) → IFR*take place. This effect, that is often underestimated, is related to presence of a delay time between

*T*infection and death. It is the same effect that causes the shifts between

_{D}*and*

*δ*D(t)*curves in Fig. 2.*

*δ*I(t)*Comparison between the simulation and real epidemics*

The simulation above is very instructive for didactic purposes but, similarly to the most classical models (e.g. the SIR model), should be taken with great care when simulating data from real epidemics. As mentioned above, the data refer to detected cases, which are only a subset of the real *I, A, R* and *D.* While the error on *D* is often assumed to be small, the error on *I, A* and *R *has been suggested to exceed an order of magnitude for COVID-19 epidemics in many countries.

While it is tempting to apply the simulation to fit only the subsets of detected cases, this approach is not rigorous, because detected and undetected cases are obviously coupled (e.g.: hidden infected patients can infect new patients that will be detected; hidden recoveries contribute to the immunity of the population and decrease new detected infections; and so on). Furthermore, confirmed and hidden cases typically have different fatality rates, because patients that remain hidden are typically those reacting with little or no symptoms to the disease.

Another difference between simple models and reality, is that in the former ones the number of infected patients *I* sooner or later reaches the level a herd immunity, i.e. becomes comparable to the total population. Corrections to such model are needed and have been recently proposed to simulate the effect of collective social distancing or lockdown that prevent the spread of the disease to a large fraction of the population.

For the reasons above, we retained so far from using our simulation to fit real COVID-19 epidemics and only use it here as a tool to improve our understanding of the problem.

For comments or questions please contact us at covid19@spin.cnr.it