Optimal adaptive testing for epidemic control: combining molecular and serology tests

TL;DR

The paper tackles the problem of containing an epidemic under limited testing by deriving an optimal adaptive testing policy within a baseline SIR framework that splits infections into undetected and detected classes. It shows that, under a fixed transmission rate, the optimal strategy follows a three-phase path: no testing until the undetected-infection level hits a cap $i_{ ext{max}}$, a time-varying testing rate $ heta(t)$ to hold $i_u$ at the threshold, and zero testing after herd immunity is reached; this yields the fastest feasible path to epidemic extinction while meeting the constraint. A key insight is that molecular tests alone cannot identify the epidemic state when $eta(t)$ varies, so the authors advocate baseline serology testing to estimate past infections and improve observability, enabling effective adaptive testing via an extended Kalman filter in stochastic simulations. Extensions to a two-threshold setting and to stochastic dynamics demonstrate robustness of the core principle and substantial cost savings (up to ~60% relative to constant testing strategies). The work thus provides a practical, theory-backed pathway for deploying adaptive testing in future pandemics, highlighting the value of combining serology-based state estimation with targeted molecular testing.

Abstract

The COVID-19 crisis highlighted the importance of non-medical interventions, such as testing and isolation of infected individuals, in the control of epidemics. Here, we show how to minimize testing needs while maintaining the number of infected individuals below a desired threshold. We find that the optimal policy is adaptive, with testing rates that depend on the epidemic state. Additionally, we show that such epidemic state is difficult to infer with molecular tests alone, which are highly sensitive but have a short detectability window. Instead, we propose the use of baseline serology testing, which is less sensitive but detects past infections, for the purpose of state estimation. Validation of such combined testing approach with a stochastic model of epidemics shows significant cost savings compared to non-adaptive testing strategies that are the current standard for COVID-19.

Figures (10)

• Figure 1: The deterministic epidemiological models used for the derivation of optimal testing policies (A) as discussed in Section \ref{['sec:opt']} and for state estimation (B) as discussed in Section \ref{['sec:state_est']}. Individuals in the population are divided in compartments according to their epidemiological state. In (A), Susceptible individuals (S) are infected via contact with infected-undetected individuals ($\textup{I}_\textup{u}$), which either transition to the infected-detected ($\textup{I}_\textup{d}$) compartment if they are tested positive for infection, or to the Recovered (R) compartment if they recover from the infection before being tested positive. Infected-detected individuals also recover from the infection. In (B), infected-undetected individuals ($\textup{I}_\textup{u}$) can transition to the infected-detected state by being tested positive via either adaptive molecular testing or baseline testing. Infected-undetected individuals can also transition to the recovered-undetected ($\textup{R}_\textup{u}$) compartment, if they recover from the infection before being tested positive or displaying symptoms. With serology baseline testing, recovered-undetected individuals can transition to the recovered-detected compartment ($\textup{R}_\textup{d}$) if they are tested positive for past infection. Infected-detected individuals transition directly to the recovered-detected compartment when they recover from the infection. Transitions between compartments are indicated along with the corresponding rates, orange arrows indicate transitions due to serology testing.
• Figure 2: Illustration of the optimal testing policy for problem \ref{['eq_reduced_problem']}. The green and light-blue curves are, respectively, the fractions of susceptible and infected-undetected individuals in the population. The fraction of infected-undetected $i_u$ is kept below the constraint $i_{\max}$ (black, dashed line) at all times by the optimal adaptive testing policy (red, dashed curve), which is equal to zero until $i_u$ reaches $i_{\max}$, is then equal to $(\beta s(t)-\gamma)/\eta$ (yellow curve) until herd immunity is reached (i.e., when $s(t)=\gamma/\beta$), and is equal to zero afterwards. For illustration purposes, we set $i_{\max}=0.1$ and $\eta=1$, the other parameters are as in Table \ref{['table:parameters']}.
• Figure 3: Schematic of the proposed procedure. Measurements of detected-infected and detected-recovered individuals ($i_d(t),r_d(t)$) as defined in Eq. \ref{['eq:SIR_estimation']} obtained via baseline testing with rate $\theta_B$ are used to estimate the aggregate epidemic state ($\hat{s}(t), \hat{i}_u(t)$), which is then used to compute the optimal adaptive testing rate ($\hat{\theta}^\dagger(t)$) according to the results of Theorem \ref{['optimal']} (the hat symbol denotes the fact that the optimal testing rate is evaluated as a function of the estimated state). The objective of adaptive testing is to contain the fraction of infected-undetected individuals below the desired threshold $i_{\max}$.
• Figure 4: Optimal testing strategy for the extended problem \ref{['eq_full_problem']}. The green and blue curves are, respectively, the fractions of susceptible and infected individuals in the population, respectively. The black, dashed line represents the constraint on the total fraction of infected individuals, $i=i_u+i_d\leq i_{\max}$. The optimal testing policy (red, dashed curve) is equal to zero at first, switches to its maximum value $\theta_{\max}$ at time $t_A$, such that $i_u+i_d$ reaches the constraint $i_{\max}$ with zero derivative at time $t_B$. Between times $t_B$ and $t_C$, the optimal testing policy is equal to $(\beta (s(t)-i_u(t)) - \gamma-\kappa)/\eta$ (yellow curve), which keeps $i_u+i_d=i_{\max}$. At time $t_C$, the optimal testing policy switches back to $\theta_{\max}$ until time $t_D$, after which it is equal to zero. The times $t_C$ and $t_D$ are such that, after $t_D$, the total fraction of infected individuals grows initially, reaching the constraint $i_{\max}$ tangentially (inset), and then decreases to zero. The switching times can be computed analytically, given the initial condition (as discussed in the Supplementary Materials). For illustration purposes, we set $i_{\max}=0.1$ and $\eta=1$, while the other parameters are as in Table \ref{['table:parameters']}.
• Figure 5: Control of stochastic trajectories by using a receding horizon version of the optimal testing policy $\theta^*$ in combination with baseline serology testing with rate $\theta_B=1/(14 \textup{\ days})$. Panel A shows the mean number of infected-undetected individuals $\langle I_u(t) \rangle$ (blue, thick curves), the mean total number of infected individuals $\langle I(t) \rangle$ (orange, thick curves) across $500$ realizations and $I_u(t)$, $I(t)$ in five, randomly selected stochastic trajectories (solid, thin lines). Note that capital letters denote absolute numbers instead of fractions of individuals. Colored bands are $95\%$ empirical confidence intervals and the thick, yellow lines show the value of $I_{\max}$. Panel B shows the mean molecular time-varying testing rate in the simulations (thick, green curve) and its $95\%$ confidence interval. The dotted, black line shows $(\beta (\langle S \rangle-\langle I_u\rangle)/N-\gamma-\kappa)/\eta$, which is the functional form of the optimal adaptive testing rate $\theta^{*}(t)$ for the deterministic SIR model in the interval [$t_B,t_C$] (Eq. \ref{['optimalBC']}). The black, dashed line shows the maximum testing rate $\theta_{\max}$. Panel C shows the number of infected-undetected individuals $I_u(t)$ (blue curve) and the total number of infected individuals $I=I_u(t)+I_d(t)$ (orange curve) in a single realization. The estimated number of infected-undetected individuals $\hat{I}_u(t)$ and estimated total number of infected individuals $\hat{I}(t)$ are shown with black, dashed curves. The 95$\%$ confidence intervals for $\hat{I}_u(t)$ and $\hat{I}(t)$ computed using the predicted variance estimated according to the extended Kalman filter are shown as colored bands. The infected threshold value $I_{\max}$ is shown as a yellow line. Panel D shows a zoom of the initial phases of the epidemic highlighting the accuracy of the Kalman filter estimates. Model parameters and initial conditions are as in the Materials and Methods Table \ref{['table:parameters']} and Section \ref{['sec:parameters']}.

Theorems & Definitions (3)

• proof
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