To trace or not to trace: analytical insights from network-based contact-tracing models

TL;DR

A unified framework yields rigorous and tractable threshold conditions for contact tracing dynamics on networks, extending the applicability of pairwise models beyond the fast-tracing regime and providing new insight into the interplay between disease progression, partial treatment compliance, and higher-order tracing processes.

Abstract

Contact tracing is one of the most important control measures deployed during epidemics. Relying on the identification of contacts of known infected individuals, it necessitates a network perspective. Although pairwise models have been used extensively to study contact tracing, their analysis typically depends on a decoupling assumption-most commonly that contact tracing operates on a much faster timescale than disease transmission. Furthermore, contact tracing models often assume that all infected individuals become contact tracing-triggering, which is unrealistic given partial compliance to treatment. We relax both of these restrictive assumptions and provide a full analytical characterisation of the epidemic threshold in the pairwise mean-field model. Our analysis uses a fast-variables approach that captures the rapid early stabilisation of key network quantities. Inspired by mechanisms from social adoption dynamics, we introduce triplewise contact tracing in which an infected individual can be traced not only through direct contact with a single tracing-triggering neighbor (pairwise tracing), but also indirectly when connected to two tracing-triggering nodes simultaneously. For pure pairwise and pure triplewise contact tracing, we derive analytical expressions for critical contact tracing levels and demonstrate that when many infected individuals bypass treatment, the epidemic can become uncontrollable. When both contact tracing mechanisms operate together, we map out their combined contribution and relative impact on epidemic control. This unified framework yields rigorous and tractable threshold conditions for contact tracing dynamics on networks, extending the applicability of pairwise models beyond the fast-tracing regime and providing new insight into the interplay between disease progression, partial treatment compliance, and higher-order tracing processes.

Figures (7)

• Figure 1: Schematic description of the epidemic model. Individuals exist in four states: susceptible $S$, infected $I$, treated (tracing) $T$, and recovered $R$. A susceptible individual becomes infected at rate $\tau$, conditional on an $[SI]$ contact. An infected individual either recovers naturally at rate $h$ or seeks treatment at rate $g$, thereby moving to the treated state and initiating contact tracing. Treated individuals exit $T$ at rate $a$ and move to $R$. In addition, infected individuals move to $T$ through contact tracing at rate $c_p$ conditional on an $[IT]$ contact, and at rate $c_t$ conditional on a $[TIT]$ triple.
• Figure 2: The fast variables reach a transient equilibrium much sooner compared to the time scale of epidemic evolution. a) Time evolution of $[I]$ and $[T]$. b) Time evolution of the pairs $[SI]$, $[II]$, and $[IT]$. c) Time evolution of the fast variables $x$, $y$, and $z$, computed from the full system in Eq. \ref{['eq:system']} (dotted line) or from the fast-variables system in Eq. \ref{['eq:fast_var_sys']} (solid line). Parameter values: $n = 5$, $\tau = 1.1$, $a = 1$, $q = 0.6$, $c_p = 10$, and $c_t = 0$ (pure pairwise). Initial conditions: $[I](t=0)=[T](t=0) = 10^{-5}$.
• Figure 3: The threshold diverges as a function of tracing coverage and slowness of CT. a) Critical level of pairwise CT as a function of the tracing coverage $q$, for three values of the transmission rate: $\tau = 2, 7, 20$. The dashed vertical lines represent $q_{\text{min}}^p$. Parameter value: $a = 0.1$. b) Ratio between the pure pairwise critical level of CT computed with the method of fast variables ($c_p^*$) and the Eames and Keeling threshold (EK), as a function of the celerity of CT, through $\Lambda$. We consider the same three values of transmission rate as in panel a, and $c_t = 0$. c) Ratio between the pure triplewise critical level of CT computed with the method of fast variables ($c_t^*$) and the pure triplewise critical level obtained in the adjusted fast tracing limit, as a function of the celerity of CT, through $\Lambda$. We consider the same three values of transmission rate as in panel a and $c_p = 0$. Parameter values for panels b and c: $q = 1$, and $a = 1$. Parameter values common across the three panels: $n=3$.
• Figure 4: The social weight of CT increases dramatically for dense networks, as we move away from the fast tracing limit. Contour lines of the (analytical) social weight $(c_p^*/a)/((c_p^*/a) + 1)$ in the following plane: network density $n$ vs probability that infection passes across a contact $\tau/(\tau + 1)$. a) Fast variables result when $\Lambda = 1$. b) Fast variables result when $\Lambda = 100$. c) Eames and Keeling's (EK) fast tracing result (dashed black lines) superimposed on the fast variables result when $\Lambda = 2000$. Parameter values: $c_t = 0$, $a=1$, and $q=1$.
• Figure 5: Triplewise CT is more stringent and less efficient than pairwise CT. a) Analytical critical level of CT as a function of the tracing coverage $q$ for both pairwise (dotted, orange) and triplewise (dashed, blue) contact tracing. The vertical solid lines indicate the minimum compliance $q_{\text{min}}$. Parameter values: $\tau = 5$, $a = 0.1$, and two choices of the network density, $n=3$ (full opacity) and $n=9$ (partial transparency). b) Heatmap and contour lines of the final epidemic size $R_{\infty}$ in the following plane: $c_p$ vs $c_t$. The dotted orange and dashed blue lines are the analytical critical levels of contact tracing for pure pairwise and pure triplewise CT, respectively. Parameter values: $n = 5$, $\tau = 0.7$, $a = 1$, $q = 0.6$. Initial conditions: $[I](0) = [T](0) =5 \cdot 10^{-8}$.