Minimal and adaptive numerical strategy for critical resource planning in a pandemic

Current epidemiological models can in principle model the temporal evolution of a pandemic. However, any such model will rely on parameters that are unknown, which in practice are estimated using stochastic and poorly measured quantities. As a result, an early prediction of the long-term evolution of a pandemic will quickly lose relevance, while a late model will be too late to be useful for disaster management. Unless a model is designed to be adaptive, it is bound either to lose relevance over time, or lose trust and thus not have a second chance for retraining. We propose a strategy for estimating the number of infections and the number of deaths, that does away with time-series modeling, and instead makes use of a “phase portrait approach.” We demonstrate that, with this approach, there is a universality to the evolution of the disease across countries, that can then be used to make reliable predictions. These same models can also be used to plan the requirements for critical resources during the pandemic. The approach is designed for simplicity of interpretation, and adaptivity over time. Using our model, we predict the number of infections and deaths in Italy and New York State, based on an adaptive algorithm which uses early available data, and show that our predictions closely match the actual outcomes. We also carry out a similar exercise for India, where in addition to projecting the number of infections and deaths, we also project the expected range of critical resource requirements for hospitalizations in a location.

Figure 1

Time series has fluctuations. Countrywise time series data for (a) daily new infections ($dI/dt$) and (b) cumulative infections ($I$). Since the chance of infection and incubation time are stochastic, and data is recorded on a daily basis, $I$ fluctuates. These fluctuations may be country dependent and constrained by other factors such as the frequency of testing, especially in the early days of the pandemic. To obtain a better evolution we instead focus on the ($dI/dt$) versus ($I$) which removes systematic errors.

Figure 2

Universal trend in infections and deaths. The rates at which the number of infections ($I$) and number of deaths ($D$) change are explored using the COVID-19 data from many countries. (a) $dI/dt$ versus $I$ (b) $dD/dt$ versus $D$ show comparable behavior across these countries which can be exploited in our predictions. The data from a 3-day average was used since the number of daily new infections fluctuates significantly. When fitting the data to Eq. (2), the functional form is assumed to be of the form $log\left(y\right)=log\left(x\right)+c$. The values for $c$ in (a) and (b) were $(-0.37\pm 0.03)$ and ($-0.28\pm 0.01$), respectively.

Figure 3

Using the $I\text{−}D$ plot for estimating the number of infections. These plots depict the universal relation between $I$ and $D$, barring a corrective factor. This uncertainty factor that was calculated from the COVID-19 infection data of South Korea as a reference can be used for estimating infections when the number of deaths are known. $\beta =\frac{\text{corrected}\text{infections}}{\text{reported}\text{infections}}$ was assumed to be 1 for South Korea. At any time in the evolution of the pandemic shown on this $I$ versus $D$ graph, using the known number of deaths in that country, the number of infections was rescaled to match with the infections in South Korea and this factor was used as $\beta$. The figure illustrates the procedure used for $\beta$ when deaths are slightly lower than 10, equal to 10 and 20.

Figure 4

Adaptive predictions for Italy. An Illustration of the adaptive predictions for (a) the number of deaths and (b) the number of infections due to COVID-19 in Italy. In both these cases, the top panel uses the the data from week 1, and predictions for weeks 2–4. At the end of week 2, by taking note of the reported data, the predictions for weeks 3–5 are updated. The reported number of deaths and infections are also shown. In comparison, the prediction of the number of deaths was more accurate. It should be noted that number of deaths also gives one a better estimation of the critical resources required. This could potentially be because the infections grow exponentially and the testing resources grow linearly, thus increasing the gap in the reported infections.

Figure 5

Adaptive predictions for New York State. The adaptive predictions were performed for the data from New York sState, just as described in Fig. 4.

Figure 6

Adaptive predictions for India. The adaptive predictions were performed for the data from India, just as described in Figs. 4 and 5.