Fig. 1.

Disease dynamics and contact tracing framework.(left) Upon infection, susceptible individuals ($S$) enter a latent exposed stage ($E$) when the disease incubates before infectiousness. Upon becoming infectious, a proportion $p_a$ of the population remain asymptomatic ($A$), while the remainder pass through a presymptomatic stage ($P$) before becoming symptomatic ($I$). Infectious individuals are removed ($R$) through recovery/isolation (rates ${\gamma }_A$ and ${\gamma }_I$), isolation after testing (dotted arrow, rate $\tau$), or quarantining as a consequence of contact tracing (dashed arrows, rate $\alpha (1-\Theta )I+\alpha \Theta I({\mu }_K∕K)$ for $K\in \{E,A,P,I\}$). The total contact tracing rate $\alpha$ depends on the proportion of the population who adopt the contact tracing $u_a$, the testing rate $\tau$, the factor $(1-u_s\left(t\right))$, representing the reduction of transmission rates due to social intervention measures and the notification threshold, $u_n$, representing the minimal exposure required to notify traced contacts. Removal via contact tracing is partitioned into the fraction of contacts who were infected by the tested case $\Theta$, and those who were not $(1-\Theta )$, where $\Theta$ represents the tracing precision and depends on the susceptible proportion $S$ and the notification threshold $u_n$. Contact tracing also causes quarantine of susceptible individuals ($Q$) that had non-infectious contact with an infected case (dashed arrow at rate $\alpha (1-\Theta )I$, return rate $q$). The force of infection, $F$ depends on transmission rates ${\beta }_K$ and infection densities, and social intervention measures. (right) Contact heterogeneity is incorporated via a distribution of exposure levels: contact are encountered at exposure $e$ with density $\rho \left(e\right)$ and result in infection with probability $p_i\left(e\right)$. The notification threshold (illustrated for $u_n=1$) affects the contact tracing rate and precision via the integral $f_c$ (blue and green shaded region) expressing the fraction of contacts notified, and the integral $f_i$ (green shaded region), which captures the proportion of contacts who had infectious contact with the tested case. (Color version online.)