# How long is the curve ?

A calculator for the time required to reach herd immunity at a sustainable rate for healthcare.

Numbers of days till herd immunity calculator

A key aspect of the current COVID-19 pandemic has been the rapid overload of Intensive Care Units (ICUs) in countries and regions where the epidemic was not quickly controlled because many patients infected with SARS-CoV-2 develop severe respiratory complications. The relatively low mortality outside the high-risk group, the absence of a vaccine and of a proven antiviral treatment, and the huge socioeconomic impact of the pandemic (which will itself cause significant mortality, morbidity, psychological distress, and economic suffering) suggest that strategies aiming at achieving herd immunity, even at the cost of significant mortality, should be evaluated. Once herd immunity has been achieved, high-risk individuals are protected. By contrast, strategies based on containment are inherently fragile as temporary failure to maintain the basic reproduction number $$R_0$$ below 1 leads to the flare-up of the disease.

Our application allows computing the period for reaching herd immunity taking uncertainty into account. The first parameter is the $$R_0$$ of COVID-19 in a population without containment measures. This allows calculating the fraction of the population $$1-1/R_0$$ needed to reach herd immunity. Given that $$R_0$$ is uncertain, the application takes a range $$[R_0^{-}, R_0^{+}]$$ as input. This allows calculating a range of herd immunity thresholds $$[H^-=1-1/R_0^-;H^+=1-1/R_0^+]$$ (in percent of the population) and considers these values as an error measurement: $$H^{-}=H-\varepsilon_H;H^{+}=H+\varepsilon_H$$.

Next to the herd immunity target, we also need the ICU bed capacity in beds per 100,000 inhabitants. This is not the baseline ICU capacity in the population considered, but the ICU capacity fully dedicated to COVID-19 patients over the course of multiple months. This should take into account the ICU bed capacity that can be additionally deployed in a given population. This ICU bed capacity is input as an error range $$[B^{-}=B-\varepsilon_B;B^{+}=B+\varepsilon_B]$$ (in beds per 100,000 inhabitants). The next variable is the average stay duration at ICU as this determines the average number of daily admissions because the number of ICU COVID-19 patients needs to stay constant. This variable is also input as a range $$[L^{-}=L-\varepsilon_L; L^{+}=L+\varepsilon_L]$$ (in days).

Next, we need the fraction of the population that will require ICU care following SARS-CoV-2, which is the most uncertain factor. The fraction of the general population that recovers from SARS-CoV-2 infection asymptomatically or with only mild symptoms is poorly characterized at this point because testing efforts have focused on symptomatic cases and/or people who have been in contact with infectious patients. Systematic serological surveys are needed to reduce the uncertainty for this factor. This fraction is also input as a range $$[C^{-}=C-\varepsilon_C; C^{+}=C+\varepsilon_C]$$ (in percent of the population).

The average number of days till herd immunity can then be computed as $$P=\frac{H.L.C}{B}\times100000$$ The error range can be calculated using error analysis:

$$\varepsilon_P=P.\sqrt{(\varepsilon_H/H)^2+(\varepsilon_L/L)^2+(\varepsilon_C/C)^2+(\varepsilon_B/B)^2}$$

and a range can be returned as

$$[P^-=P-\varepsilon_P; P^+=P+\varepsilon_P]$$.

## Some results per countries

In the table below we present resulting days required to achieve herd immunity in the population of several countries. For all, we considered an average ICU stay duration of 10 days, as well as a proportion of infected population requiring ICU of 1%. $$R_0$$ was set to 2.5.

Country Population ICU beds Days till herd immunity
 United Sates 328.2 millions 29.4 198 days United Kingdom 64.7 millions 6.6 883 days Belgium 11.2 millions 15.9 366 days Netherlands 16.9 millions 6.4 911 days Italy 60.8 millions 12.5 466 days France 66.3 millions 11.6 502 days Germany 81.1 millions 33.9 172 days Spain 46.4 millions 9.7 601 days

## Heterogeneous confinement simulator

As the calculator above suggest, the time scale required to reach herd immunity at a manageable pace for ICU capacity by keeping a $$R_0$$ close to 1 is prohibitive. In the following simulator, we simulate another strategy that would consist in isolating the risk-group from the non-risk group. Importantly, distinct lockdown measures for both groups would result in a $$R_{0,risk}$$ for the risk population and a $$R_{0,non-risk}$$ for the non-risk population. We further consider a $$R_{0,cross}$$ that model the force of infection across both groups. The strategy would then consist in letting the non-risk population getting infected at a faster pace in order to reach herd immunity in the global population. Once, the total number of recovered people is higher than the ratio $$1-1/R_0$$, the lockdown is eased on both populations. The day at which this general lifting can be tuned with the merge-diff parameter.

The blue line represents the number of patients requiring ICU at a specific time. While the red line gives the ICU capacity limit in % of the general population. $$S_{init}$$, $$R_{init}$$ and $$I_{init}$$ represent respectively the number of Susceptible, Recovered and Infected individuals. We invite you to play with the simulation to try to find a solution that might lead to herd immunity in short amount of time without overloading ICU capacity. Importantly, this seems unfeasible if there is the slight cross contamination between risk groups, which seems unevitable.

Who we are  We are Yves Moreau group, part of STADIUS at KU Leuven University in Leuven, Belgium. Our group expertise includes Bio-Informatics, Machine Learning, Human Genetics, AI for Healthcare and Drug Discovery.

For futher information, please mail us at : moreau[AT]esat[DOT]kuleuven[DOT]be.