Achieving maximum speedup in multi-level acceleration for massive coronavirus testing

Abstract

It is well and widely known that sample pooling could provide an effective and efficient way for fast coronavirus testing among massive asymptomatic individuals. The method of multi-level acceleration for asymptomatic COVID-19 screening has been introduced, and for one and two levels, the optimal group sizes have been obtained. However, there are still multiple challenges. First, it is not clear how to find the optimal group sizes for three or more levels. Second, there is lack of closed-form expressions for the optimal group sizes for two or more levels. Third, it is not clear how to determine the optimal number of levels. And last, it is not known what the maximum achievable speedup is. The motivation of this paper is to address all the above challenges. The optimization of a hierarchical pooling strategy includes its number of levels and the group size of each level. In this paper, based on multi-variable optimization and Taylor approximation, we are able to derive closed-form expressions for the optimal number of levels $d^{\ast }=\ln (1/\ln (1/q_0\left)\right)-1$, the optimal group sizes $m_1^{\ast }=e^{d^{\ast }}=1/\left(e{p}_0\right)$, $m_2^{\ast }=e^{d^{\ast }-1}=1/\left(e^2{p}_0\right)$,…, $m_{d^{\ast }}^{\ast }=e=1/\left(e^{d^{\ast }}{p}_0\right)$, and the maximum possible speedup of a hierarchical pooling strategy of $1/(e{p}_0\ln (1/p_0\left)\right)$, where $p_0$ is the fraction of infected people. The above speedup is nearly a linear function of the reciprocal of $p_0$, in the sense that it is asymptotically greater than any sub-linear function $(1/p_0{)}^{1-ϵ}$ of the reciprocal of $p_0$ for any small $ϵ>0$. Using the results in this paper, we can quickly and easily predict the performance of an optimal hierarchical pooling strategy. For instance, if the fraction of infected people is 0.0001, an 8-level hierarchical pooling strategy can achieve speedup of nearly 400.

Keywords:

• Asymptomatic screening
• group test
• hierarchical pooling strategy
• multi-level acceleration
• optimization
• speedup

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of Bo Yang was supported in part by the Scientific Research Key Project of Hunan Education Department [grant number 19A077].