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Sideward contact tracing in an epidemic model with mixing groups

Dongni Zhang, Martina Favero

TL;DR

The paper develops a macro-branching-process framework to analyze sideward contact tracing in an epidemic with mixing-group transmission. By aggregating siblings infected in the same mixing event into independent macro-individuals, it derives the effective macro reproduction number $R_e$ (shown to equal the effective individual reproduction number $R^{(ind)}_e$) and an extinction-probability characterization, establishing a clear threshold for major outbreaks. Numerical results reveal that increasing the diagnosis fraction $\delta/(\delta+\gamma)$ or the tracing probability $p$ reduces $R_e$, with larger mean event sizes enhancing the effectiveness of sideward tracing, though unlimited gathering sizes can limit impact. The approach provides analytic tools for evaluating sideward tracing, and it suggests avenues for extending the model to integrate forward and backward tracing, delays, and population heterogeneity to better inform public health interventions.

Abstract

We consider a stochastic epidemic model with sideward contact tracing. We assume that infection is driven by interactions within mixing events (gatherings of two or more individuals). Once an infective is diagnosed, each individual who was infected at the same event as the diagnosed individual is contact traced with some given probability. Assuming few initial infectives in a large population, the early phase of the epidemic is approximated by a branching process with sibling dependencies. To address the challenges given by the dependencies, we consider sibling groups (individuals who become infected at the same event) as macro-individuals and define a macro-branching process. This allows us to derive an expression for the effective macro-reproduction number which corresponds to the effective individual reproduction number and represents a threshold for the behaviour of the epidemic. Through numerical examples, we show how the reproduction number varies with the distribution of the mixing event size, the mean size, the rate of diagnosis and the tracing probability.

Sideward contact tracing in an epidemic model with mixing groups

TL;DR

The paper develops a macro-branching-process framework to analyze sideward contact tracing in an epidemic with mixing-group transmission. By aggregating siblings infected in the same mixing event into independent macro-individuals, it derives the effective macro reproduction number (shown to equal the effective individual reproduction number ) and an extinction-probability characterization, establishing a clear threshold for major outbreaks. Numerical results reveal that increasing the diagnosis fraction or the tracing probability reduces , with larger mean event sizes enhancing the effectiveness of sideward tracing, though unlimited gathering sizes can limit impact. The approach provides analytic tools for evaluating sideward tracing, and it suggests avenues for extending the model to integrate forward and backward tracing, delays, and population heterogeneity to better inform public health interventions.

Abstract

We consider a stochastic epidemic model with sideward contact tracing. We assume that infection is driven by interactions within mixing events (gatherings of two or more individuals). Once an infective is diagnosed, each individual who was infected at the same event as the diagnosed individual is contact traced with some given probability. Assuming few initial infectives in a large population, the early phase of the epidemic is approximated by a branching process with sibling dependencies. To address the challenges given by the dependencies, we consider sibling groups (individuals who become infected at the same event) as macro-individuals and define a macro-branching process. This allows us to derive an expression for the effective macro-reproduction number which corresponds to the effective individual reproduction number and represents a threshold for the behaviour of the epidemic. Through numerical examples, we show how the reproduction number varies with the distribution of the mixing event size, the mean size, the rate of diagnosis and the tracing probability.
Paper Structure (14 sections, 3 theorems, 38 equations, 5 figures, 3 tables)

This paper contains 14 sections, 3 theorems, 38 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model
  3. The epidemic model with mixing groups
  4. Sideward contact tracing
  5. Early epidemic approximation
  6. The macro branching process
  7. Main results
  8. Effective macro reproduction number
  9. Effective individual reproduction number
  10. Extinction probability
  11. Numerical results
  12. Conclusion and discussion
  13. Growth rate of the macro process $\mathcal{M}$
  14. Additional numerical results

Key Result

Theorem 5.1

The macro reproduction number $R_{e}$ for the macro branching process $\mathcal{M}$ is given by where $m_{i0}$ is given by Equation (eq:m_k0).

Figures (5)

  • Figure 4.1: Illustration of an individual process $\mathcal{B}_{CT}$ and its corresponding macro process $\mathcal{M}$. On the left, circles represent individuals grouped into sibling groups, enclosed within dashed rectangles. The macro process on the right simplifies the individual process by aggregating sibling groups into macro-individuals, represented as squares labeled by the respective sibling group sizes.
  • Figure 6.1: Heatmaps of the reproduction number $R_{e}$ as function of $\delta/(\delta+\gamma)$ in $[0,0.99]$ and $p$ in $[0,1]$. The size of mixing event $C$ follows a geometric distribution with mean $\mu_{C}=20$, $\gamma=1/7$ and $\beta= 3/266$. In the left panel: $\pi_{c}= 2/c$ for $c \geq 2$, while $\pi_{c}\equiv 0.05$ on the right.
  • Figure 6.2: Plot of the relative reduction $r$ as a function of the tracing probability $p$ for different mean event size $\mu_{C}=5, 10$ and $20$. We fix $\delta=\gamma=1/7$ and in the upper panel, the infection probability $\pi_{c}= {2}/{c}$ for all $c$; in the lower panel, $\pi_{c} \equiv \pi$ is chosen specifically for each distribution of $C$ to result in the same average number of infections per event as the $\pi_{c}= {2}/{c}$ case.
  • Figure 6.3: Plot of the reproduction number $R_{e}$ as a function of the tracing probability $p$ for different mean event size $\mu_{C}=5, 10$ and $20$. We fix $\delta=\gamma=1/7$ and $\beta= 3/28$ such that $R_{0}=3$ with $\mu_{C}=5$. In the upper panel, the infection probability $\pi_{c}= {2}/{c}$ for all $c$; in the lower panel, $\pi_{c} \equiv \pi$ is chosen specifically for each distribution of $C$ to result in the same average number of infections per event as the $\pi_{c}= {2}/{c}$ case.
  • Figure B.1: Heatmaps of the reproduction number $R_{e}$ as function of $\delta/(\delta+\gamma)$ in $[0,0.99]$ and $p$ in $[0,1]$, where $\gamma=1/7$ and $\beta= 3/266$. The size of mixing event $C$ follows a logarithmic distribution with mean $\mu_{C}=20$ in \ref{['fig:heatmap_R_log']}, where in the left panel: $\pi_{c}=2/c$ for $c \geq 2$ and $\pi_{c}\equiv 0.03$ on the right. In \ref{['fig:heatmap_R_fix']}, the event size is fixed: $C \equiv 20$ and hence $\pi_{c}\equiv \pi = 0.1$.

Theorems & Definitions (7)

  • Theorem 5.1
  • Proposition 5.2
  • Remark 5.3
  • proof : Proof of Proposition \ref{['pro:ind_reprod_number']}
  • Theorem 5.4
  • Remark 6.1
  • Remark 6.2