Estimating the Final Epidemic Size for COVID-19 Outbreak using Improved Epidemiological Models

Rajesh Ranjan

The infection rate predicted by all the above models follow almost a normal distribution with dI dt = a exp(− t−µ σ ) 2 , with µ represents the day of peak cases, σ is spread around this day from the beginning of the epidemic to the end, and a is the constant. Because of the availability of limited data, predicting both -day of the peak (mean) as well as the spread (variance) is challenging for every model. Further, even if the day of the peak is known the decline rate is often not correctly predicted giving an incorrect estimation of the final epidemic size. Because of the symmetric distribution of the curve, the decline rate is predicted as negative of the acceleration. This is often not true as observed from the COVID-19 data of Italy and Spain, where the decline is very slow. In order to illustrate this point, we show the total infection as well as the rate of infection for Italy in Fig. 5 . The predictions by the three models are also shown. All the three fitted curves show underprediction of total confirmed cases from April 8 despite using data till April 11 for modeling. If we look at the infection rate, we note that the decline rate given by Gaussian-like distribution is much higher than actual decline leading to a lower estimate of the epidemic size. In order to account for this difference, we introduce a new parameter η, which changes the variance of this distribution for predictions after the peak. Briefly, we fit a normal distribution after getting predictions from SEIQRDP model, and then adjust the spread to make the decline rate realistic. Hence, the new distribution is given by dI dt = ζa exp(− t−µ ησ ) 2 , where the factor ζ = a 1 /a ensures that number of infections on the peak day remains unchanged. The dash-dot magenta curve shows this new distribution which is much closer to the actual values. The value of η is obtained by fitting the curve to Italy and used for subsequent predictions. This corrected model is designated as SEIQRDP(C). Predictions from this model give the upper limit of the predicted cases. Note that, this adjustment was not needed for fitting data of China as the flattening of curve in China was very fast compared to what is seen in the recent past for European countries (see Fig. 3 ).