Elementary time-delay dynamics of COVID-19 disease

Jose Menendez

The time-delay model predicts correctly that the ratio will be very close to unity. However, in the case of China the actual curve is only in qualitative agreement with the data. This implies that the approximation of a single convalescence time is too crude. In principle, one should use a distribution of convalescence times, but one can obtain a reasonable good agreement with the available data using two convalescence times and , so that the second equation in (2) becomes (5) where . The fit with the two convalescence times is shown as a solid line in Figure 1 , and the corresponding parameters appear in Table I . It is apparent that the agreement is much improved. A two-convalescence time fit for South-Korea also leads to a small but noticeably improvement. The meaning of the new fit parameters, however, is not obvious. According to the Report of the WHO-China Joint Mission on Coronavirus Disease 2019 [9], about 80% of the patients experience a mild disease and recover faster. However, it seems impossible to obtain a good fit using = 0.80 if . This may indicate that the documented cases include a smaller fraction of mild cases, which would go mostly undetected. On the other hand, this finding may simply indicate that the two-step survival function implied by Eq. (5) still too naïve. The model presented here, although extremely simplified, could be useful to health authorities by predicting the total number of cases as well as the peak number of infected people. The fit values could be improved on a daily basis as more data become available. As an example, Fig. 3 shows fits for Italy and Spain, which have not yet reached tpeak but show clear signs of a flattening in the number of cases curve. For other countries, the model does not yet provide a meaningful fit. In the case of the US, the growth as of March 24, 2020 is faster than exponential, perhaps as a result of improved testing. Iran, on the other hand, is a notable case because is quite linear, a behavior that the model here or any SEIR model fail to reproduce. In summary, an elementary model of COVID-19 dynamics has been presented that produces a much better fit of existing data than standard SEIR models. The model is based on time-delay differential equations that take into account the convalescence time . It fits very well existing data for China and S. Korea, and it can be used to make predictions for several countries that have not yet reached the peak of the infected curve.