# Causal identification of infectious disease intervention effects
in a clustered population

Xiaoxuan Cai1, Eben Kenah2, and Forrest W. Crawford1,3,4,5
1\. Department of Biostatistics, Yale School of Public Health
2\. Division of Biostatistics, College of Public Health, The Ohio State
University
3\. Department of Statistics & Data Science, Yale University
4\. Department of Ecology and Evolutionary Biology, Yale University
5\. Yale School of Management

###### Abstract

Causal identification of treatment effects for infectious disease outcomes in
interconnected populations is challenging because infection outcomes may be
transmissible to others, and treatment given to one individual may affect
others’ outcomes. Contagion, or transmissibility of outcomes, complicates
standard conceptions of treatment interference in which an intervention
delivered to one individual can affect outcomes of others. For example, a
vaccine given to an individual may affect their risk of infection given
exposure to an infectious source as well as their infectiousness if they
become infected. Several statistical frameworks have been proposed to measure
causal treatment effects in this setting, including structural transmission
models, mediation-based partnership models, and randomized trial designs.
However, existing estimands for infectious disease intervention effects are of
limited conceptual usefulness: Some are parameters in a structural model whose
causal interpretation is unclear, others are causal effects defined only in a
restricted two-person setting, and still others are nonparametric estimands
that arise naturally in the context of a randomized trial but may not measure
any biologically meaningful effect. In this paper, we describe a unifying
formalism for defining nonparametric structural causal estimands and an
identification strategy for learning about infectious disease intervention
effects in clusters of interacting individuals when infection times are
observed. The estimands generalize existing quantities and provide a framework
for causal identification in randomized and observational studies, including
situations where only binary infection outcomes are observed. A semiparametric
class of pairwise Cox-type transmission hazard models is used to facilitate
statistical inference in finite samples. A comprehensive simulation study
compares existing and proposed estimands under a variety of randomized and
observational vaccine trial designs.
Keywords: causal inference, contagion, direct effect, indirect effect,
interference, vaccine

##  1 Introduction

Infectious diseases represent a leading cause of illness, disability, and
death worldwide [1, 2]. Major successes in the control and prevention of
infectious diseases have been achieved by vaccination programs, and evaluation
of vaccine efficacy is of major interest for epidemiologists and statisticians
[3]. However, when trial participants come from the same population, the
infection outcomes of study participants may not be independent, posing
problems for definition and identification of causal vaccine effects. This
dependence arises from two distinct sources. First, infection itself—with or
without vaccination—may be transmissible between subjects. Second, vaccination
of an individual may alter both their susceptibility to infection and their
infectiousness if infected. The complex interaction of vaccination patterns
with individual and group-level risks of infection makes causal inference for
vaccine effects a uniquely challenging problem.

Statisticians and infectious disease epidemiologists have devoted a great deal
of effort to studying this problem. Causal dependence in infectious disease
outcomes arises in several different but overlapping forms, including herd
immunity [4, 5, 6, 7], spillover [8, 9], dependent happenings [5, 10, 6, 11,
12, 13, 9, 7], interference [14, 10, 8, 15, 13, 16, 9, 17, 7], and indirect
effects [10, 6, 11, 15, 18, 9, 19, 20]. Specifically, treatment of one
individual may affect the distribution of their infection time when they are
exposed to an infectious source. If this individual becomes infected, their
infection may subsequently increase the risk of infection in others, and this
change in risk may depend on treatment in both the infectious and susceptible
individuals. This dependence between infection outcomes creates identification
problems in causal inference for intervention effects. Using principles of
causal reasoning [11, 8, 9, 7, 21, 22], evidence from simulation [5, 23, 24,
12, 25, 26], and the behavior of dynamic transmission models [24, 27, 28, 23,
12, 29, 18, 30, 26], researchers have shown that ignoring this dependence may
result in biased estimates of causal vaccine effects.

Some researchers have attempted to circumvent this problem by randomization,
proposing design-based inferential frameworks for estimating vaccine effects.
Halloran and colleagues proposed a two-stage block randomization design and
formalized the “direct” and “indirect” effects of treatment [5, 6, 10, 11, 31,
13]. Following ideas from mediation analysis, VanderWeele and Tchetgen
Tchetgen [9], VanderWeele et al. [19], Ogburn and VanderWeele [20] and
Halloran and Hudgens [32] restricted the identification problem to an
asymmetric partnership setting (i.e., pairs where infection can be transmitted
in only one direction), and they defined a causal direct effect, contagion
effect, and infectiousness effect. Hudgens and Halloran [31] and Halloran and
Hudgens [32] applied a principal stratification strategy to define a causal
infectiousness effect, by contrasting the secondary attack risks of
individuals exposed to treated versus untreated infected neighbors, providing
a non-parametric bound for this infectiousness effect. These methods do not
require complete knowledge of the causal structure between outcomes, but they
also ignore individual characteristics and the dynamics of of infectious
disease transmission. Cai et al. [22] study causal identification in the
symmetric partnership setting, where infection can be transmitted in both
directions. Their continuous-time strategy clarifies the causal structure of
contagion and vaccine effects in groups of two, when both individuals have
covariates, treatments, and can transmit infections to each other, but its
usefulness is limited to the partnership scenario. The identification problem
remains open for clusters of size greater than two.

Statistical measures of association have long been used to summarize vaccine
effects. Halloran et al. [6, 24], Halloran and Struchiner [11], Rhodes et al.
[30], Halloran et al. [27], O’Hagan et al. [7] described summary statistics
based on attack risks, secondary attack risks, and per-contact-per-time
conditional (secondary) attack rates of different treatment groups. However,
the causal interpretation of these quantities depends on assumptions of
homogeneous mixing [6, 27, 30], equal contact rates [27, 30], constant
prevalence of infection [27], comparable exposure to infection between treated
and untreated [6, 11, 30], or detailed information about “who-infected-whom”
[28, 30, 11], and it remains unclear what mechanistic or causal features of
the infection process are revealed by these measures.

Mechanistic models serve as another major tool to represent transmission
dynamics of an infectious disease, but it is unclear whether parameters in
these models are interpretable as causal effects. Longini et al. [28, 4] and
Longini and Koopman [33] proposed the chain binomial model for longitudinal
data and characterized disease transmission using two terms: exogenous risk
from outside the cluster and endogenous risk per infective-susceptible contact
within the cluster. However, the chain binomial model only applies to
longitudinal data of fixed time intervals, and it can be difficult for these
models to deal with individual heterogeneity. Halloran et al. [24, 27] and
Rhodes et al. [30] further consider models for infection hazards on continuous
timescale, also assuming homogeneous mixing [6, 30, 27], equal contact rates
[30, 27], constant prevalence [27], or detailed information about “who-
infected-whom”[28, 30]. Kenah [34, 35, 36] and Kenah et al. [37] proposed
pairwise hazard models for transmission, which account for individual
covariates and transmission dynamics over time. However, it remains unclear
whether parameters (i.e. a coefficient on the vaccination status of an
individual) in mechanistic transmission models may have a causal
interpretation.

In this paper, we describe a nonparametric causal framework for defining and
identifying intervention effects for transmissible infectious disease outcomes
in clusters. These methods incorporate individual treatments and covariates
that may affect both susceptibility and infectiousness. We propose new causal
estimands—contagion, susceptibility, and infectiousness effects—using
contrasts of potential infection outcomes under different treatment
conditions. Two key insights distinguish this approach from others: First, we
define potential infection outcomes holding constant infection times of other
individuals, thereby blocking feedback from the infection status of a given
individual to that of others who might infect them. Second, we define marginal
effects by integrating over the infection times of others using a variant of
Robins’s G-computation formula [38] to obtain average potential infection
outcomes under the infection risk an individual would experience _if they were
unable to infect others_. We describe a semiparametric hazard framework
developed by Kenah et al. [37] and Kenah [34, 36, 35] for statistical
estimation of parameters and causal effects. The performance of the proposed
causal estimands is illustrated and compared with several other methods. We
conclude with practical recommendations for targeting meaningful causal
intervention effects for infectious disease outcomes.

##  2 Setting and notation

Consider a population of independent clusters of individuals susceptible to
infection by an infectious disease. Following the conventional partial
interference assumption [8, 13], we assume there is no transmission between
clusters. We develop notation and definitions for a generic cluster of $n$
individuals. Let $T_{i}$ be the random time at which $i$ becomes infected, and
let $\mathbf{T}=(T_{1},\ldots,T_{n})$ be the vector of infection times of the
cluster. Let $Y_{i}(t)=\mathds{1}\left\\{T_{i}\leq t\right\\}$ be the binary
indicator of prior infection of $i$ at time $t$, and let
$\mathbf{Y}(t)=\big{(}Y_{1}(t),\ldots,Y_{n}(t)\big{)}$ be the vector of
infection indicators at time $t$ of the cluster. Let $X_{i}$ be the binary
treatment (e.g. vaccination) status of $i$, and let
$\mathbf{X}=(X_{1},\ldots,X_{n})$ be the treatment vector for the cluster.
Similarly, let $\mathbf{L}_{i}$ be the vector of measured covariates of $i$,
and let $\mathbf{L}=(\mathbf{L}_{1},\ldots,\mathbf{L}_{n})$ be the covariates
for the cluster. Denote the vector $\mathbf{T}$ excluding $T_{i}$ as
$\mathbf{T}_{(i)}=(T_{1},\ldots,T_{i},T_{i+1},\ldots,T_{n})$, and define the
vectors $\mathbf{Y}_{(i)}(t)$, $\mathbf{X}_{(i)}$, and $\mathbf{L}_{(i)}$
similarly.

It is useful to represent the history of infections in the cluster up to a
particular time $t$. For a time $t>0$, let $H_{i}(t)=\\{Y_{i}(s):0\leq s<t\\}$
be the infection history of individual $i$ up to time $t$, with a realization
denoted by $h_{i}(t)$, and let $\mathbfcal{H}(t)=\\{Y_{i}(s):0\leq
s<t,i=1,\ldots,n\\}$ be the binary infection history of the cluster up to time
$t$, with a realization denoted by $\mathbf{h}(t)$. Specifically,
$\mathbfcal{H}(t)$ represents the pattern of infections that have occurred in
the cluster, up to time $t$. Let $\mathbfcal{H}_{(i)}(t)=\\{Y_{j}(s):0\leq
s<t,j\neq i\\}$ be the infection history of all individuals except $i$ up to
time $t$, with its realization $\mathbf{h}_{(i)}(t)$. To specify a complete
infection history for all times $0\leq t<\infty$, we will write
$H_{i}=H_{i}(\infty)$, $\mathbfcal{H}=\mathbfcal{H}(\infty)$, and
$\mathbfcal{H}_{(i)}=\mathbfcal{H}_{(i)}(\infty)$. We can partition the joint
infection history in the cluster into
$\mathbfcal{H}(t)=\big{(}H_{i}(t),\mathbfcal{H}_{(i)}(t)\big{)}$.

Informally, every individual’s potential infection time depends on their own
treatment as well as the treatments and infection times of others in the same
cluster. To express these simultaneous relationships, we define the potential
infection outcome $Y_{i}(t)$ (or $T_{i}$) under the joint treatment
$\mathbf{X}=\mathbf{x}$ and others’ infection histories
$\mathbfcal{H}_{(i)}=\mathbf{h}_{(i)}$. While $Y_{i}(t)$ and $T_{i}$ may
depend on $\mathbfcal{H}_{(i)}$, the history $\mathbfcal{H}_{(i)}$ also
depends on $Y_{i}(t)$ (or equivalently, $T_{i}$ or $H_{i}$) because infection
of $i$ prior to $t$ might have caused infection of other individuals,
affecting $\mathbfcal{H}_{(i)}(t)$. For this reason, we will define the
potential infection outcomes $T_{i}\left(\mathbf{x},\mathbf{h}_{(i)}\right)$
and
$Y_{i}(t;\mathbf{x},\mathbf{h}_{(i)})=\mathds{1}\left\\{T_{i}(\mathbf{x},\mathbf{h}_{(i)})\leq
t\right\\}$ under joint treatments $\mathbf{x}$ and a deterministic infection
history $\mathbf{h}_{(i)}$, where $\mathbf{h}_{(i)}$ is chosen in advance and
cannot be affected by $T_{i}$ or $Y_{i}(t)$. By fixing the infection histories
$\mathbf{h}_{(i)}$ of individuals other than $i$, we block feedback from the
infection time of $i$ to the infection histories of others.

Following the strategy developed for two-person partnerships by Cai et al.
[22], we decompose the potential infection time
$T_{i}(\mathbf{x},\mathbf{h}_{(i)})$ of a given individual $i$ into a series
of latent infection waiting times following the infections of individuals
other than $i$. To illustrate, let
$(t_{(i)}^{1},t_{(i)}^{2},\ldots,t_{(i)}^{n-1})$ be the ordered history of
others’ infection times specified by $\mathbf{h}_{(i)}$, where $t_{(i)}^{k}$
is the $k^{\text{th}}$ infection time among individuals other than $i$,
ordered from earliest to latest. Denote the individual whose infection time is
$t_{(i)}^{k}$ by $\varphi^{k}_{i}$, indexing the $k^{\text{th}}$ infected
cluster member for $i$. Similarly, we rearrange the treatments for individuals
except $i$ by the order of their infection times as
$(x_{(i)}^{1},\ldots,x_{(i)}^{n-1})$, so that
$x_{(i)}^{k}=x_{\varphi^{k}_{i}}$. For ease of notation, we denote
$t_{(i)}^{0}=0$ and $t_{(i)}^{n}=\infty$, for all $i$. The necessary
reordering is contained in the history $\mathbf{h}_{(i)}$.

Because individuals are assumed to be uninfected at time $0$, let
$I_{i}^{0}(\mathbf{x},\mathbf{h}_{(i)})$ be the potential time to infection of
$i$ when no other cluster members are infected. If $i$ is infected before
$t_{(i)}^{1}$, then $i$ is the first cluster member to be infected, and $i$
realizes the infection time
$T_{i}(\mathbf{x},\mathbf{h}_{(i)})=I_{i}^{0}(\mathbf{x},\mathbf{h}_{(i)})$.
Otherwise, the initial infection time $I_{i}^{0}(\mathbf{x},\mathbf{h}_{(i)})$
is censored by the infection of individual $\varphi^{1}_{i}$ at time
$t_{(i)}^{1}$. If $I_{i}^{0}(\mathbf{x},\mathbf{h}_{(i)})$ is censored at
$t_{(i)}^{1}$, let $I_{i}^{1}(\mathbf{x},\mathbf{h}_{(i)})$ be the additional
time to infection of $i$ beyond $t_{(i)}^{1}$. If $i$ is subsequently infected
before the next infection among its fellow cluster members, which occurs in
individual $\varphi^{2}_{i}$ at time $t_{(i)}^{2}$, then $i$ realizes the
infection time
$T_{i}(\mathbf{x},\mathbf{h}_{(i)})=t_{(i)}^{1}+I_{i}^{1}(\mathbf{x},\mathbf{h}_{(i)})$.
Otherwise, the waiting time $I_{i}^{1}(\mathbf{x},\mathbf{h}_{(i)})$ is
censored by infection of $\varphi^{2}_{i}$ at $t_{(i)}^{2}$. Following the
same reasoning, let $I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)})$ be the additional
time to infection of $i$ after $t_{(i)}^{k}$. If $i$ is infected before the
$(k+1)^{\text{th}}$ infection in fellow cluster members at $t_{(i)}^{k+1}$,
then $i$ realizes the infection time
$T_{i}(\mathbf{x},\mathbf{h}_{(i)})=t_{(i)}^{k}+I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)})$,
and otherwise $I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)})$ is censored. Finally,
if $i$ is last to be infected, then $I_{i}^{n-1}(\mathbf{x},\mathbf{h}_{(i)})$
is the remaining time to infection beyond $t_{(i)}^{n-1}$. We can rewrite
$T_{i}(\mathbf{x},\mathbf{h}_{(i)})$ in terms of its relationship to others’
specified infection times in $\mathbf{h}_{(i)}$:

$T_{i}(\mathbf{x},\mathbf{h}_{(i)})=\begin{cases}I^{0}_{i}(\mathbf{x},\mathbf{h}_{(i)})&\textrm{if
}I^{0}_{i}(\mathbf{x},\mathbf{h}_{(i)})<t_{(i)}^{1}\\\
t_{(i)}^{1}+I^{1}_{i}(\mathbf{x},\mathbf{h}_{(i)})&\textrm{if
}I^{0}_{i}(\mathbf{x},\mathbf{h}_{(i)})\geq
t_{(i)}^{1},I^{1}_{i}(\mathbf{x},\mathbf{h}_{(i)})<t_{(i)}^{2}-t_{(i)}^{1}\\\
\vdots&\vdots\\\
t_{(i)}^{n-1}+I^{n-1}_{i}(\mathbf{x},\mathbf{h}_{(i)})&\textrm{if
}I^{0}_{i}(\mathbf{x},\mathbf{h}_{(i)})\geq t_{(i)}^{1},\ \ldots,\
I^{n-2}_{i}(\mathbf{x},\mathbf{h}_{(i)})\geq
t_{(i)}^{n-1}-t_{(i)}^{n-2}\end{cases}\\\\[6.99997pt]
=\sum_{j=0}^{n-1}\left[\left(t_{(i)}^{j}+I_{i}^{j}(\mathbf{x},\mathbf{h}_{(i)})\right)\mathds{1}\left\\{I_{i}^{j}(\mathbf{x},\mathbf{h}_{(i)})<t_{(i)}^{j+1}-t_{(i)}^{j}\right\\}\prod_{k=0}^{j-1}\mathds{1}\left\\{I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)})\geq
t_{(i)}^{k+1}-t_{(i)}^{k}\right\\}\right]$ (1)

We emphasize that the decomposition in (1) is purely notational and places no
restrictions on the joint distribution of infection times in the cluster.
Rather, the decomposition permits specification of causal assumptions for the
latent times $I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)})$ that permit
identification of intervention effects on potential infection outcomes.
Because $\mathbf{h}_{(i)}$ is a deterministic infection history, stochasticity
in $T_{i}(\mathbf{x},\mathbf{h}_{(i)})$ is entirely due to the random waiting
times $I_{i}^{k}(\mathbf{h}_{(i)},\mathbf{x})$. Figure 1 illustrates the
decomposition of $T_{i}(\mathbf{x},\mathbf{h}_{(i)})$ in terms of the waiting
times $I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)})$ in (1).

Figure 1: Illustration of the decomposition of the infection time
$T_{i}(\mathbf{x},\mathbf{h}_{(i)})$ of $i$ into a sequence of waiting times
according to (1) given infection history $\mathbf{h}_{(i)}$. If
$I_{i}^{0}(\mathbf{x},\mathbf{h}_{(i)})$ occurs before $t_{(i)}^{1}$, then
$T_{i}(\mathbf{x},\mathbf{h}_{(i)})=I_{i}^{0}(\mathbf{x},\mathbf{h}_{(i)})$ is
observed. If $I_{i}^{0}(\mathbf{x},\mathbf{h}_{(i)})\geq t_{(i)}^{1}$, then
$I_{i}^{0}(\mathbf{x},\mathbf{h}_{(i)})$ is censored, and the waiting time
$I_{i}^{1}(\mathbf{x},\mathbf{h}_{(i)})$ elapses following $t_{(i)}^{1}$. If
$I_{i}^{1}(\mathbf{x},\mathbf{h}_{(i)})<t_{(i)}^{2}$, then
$T_{i}(\mathbf{x},\mathbf{h}_{(i)})=t_{(i)}^{1}+I_{i}^{1}(\mathbf{x},\mathbf{h}_{(i)})$
is observed. If $I_{i}^{1}(\mathbf{x},\mathbf{h}_{(i)})\geq t_{(i)}^{2}$, then
$I_{i}^{1}(\mathbf{x},\mathbf{h}_{(i)})$ is censored, and the waiting time
$I_{i}^{2}(\mathbf{x},\mathbf{h}_{(i)})$ elapses following $t_{(i)}^{2}$.
Likewise, if $I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)})$ is censored for every
$k<n-1$, then
$T_{i}(\mathbf{x},\mathbf{h}_{(i)})=t_{(i)}^{n-1}+I_{i}^{n-1}(\mathbf{x},\mathbf{h}_{(i)})$
is observed following $t_{(i)}^{n-1}$.

If our interest is in evaluating potential infection outcomes as a function of
treatments alone, we may view the history $\mathbf{h}_{(i)}$ as a nuisance
variable, to be averaged out. If we treat $\mathbf{h}_{(i)}$ as a realization
of the random infection times $\mathbfcal{H}_{(i)}$ of others, what
distribution should it have? In non-interventional settings, the distribution
of $\mathbfcal{H}_{(i)}$ depends on $H_{i}$ (equivalently, $T_{i}$), because
$i$ might infect others following their own infection. Therefore the marginal
distribution of $\mathbfcal{H}_{(i)}$ is not of interest as a marginalizing
distribution for the potential infection outcome of $i$, because it contains
information about the outcome of $i$. Instead, we block this feedback by
constructing an alternative marginalizing variable, denoted
$\mathbfcal{H}_{(i)}^{*}$, whose distribution is invariant to the infection
time of $i$.

Let $\mathbfcal{H}_{(i)}^{*}(t)$ be the random history of infection times in
individuals other than $i$ up to time $t$ under the condition that $i$ cannot
transmit infection to any other cluster member. Let
$\mathbfcal{H}_{(i)}^{*}(t;\mathbf{x}_{(i)})$ be the corresponding potential
infection history under this circumstance. The alternative history
$\mathbfcal{H}_{(i)}^{*}(t;\mathbf{x}_{(i)})$ has a special connection to the
observational infection history $\mathbfcal{H}_{(i)}(\mathbf{x})$: Before $i$
is infected, $i$ cannot transmit infection to others, so
$\mathbfcal{H}_{(i)}(\mathbf{x})$ is equivalent to
$\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})$ in distribution. Once $i$ is
infected and can transmit infection to others, this alters the distribution of
$\mathbfcal{H}_{(i)}(\mathbf{x})$ but not
$\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})$, so they are no longer equal in
distribution. Since we are evaluating the potential infection outcome of $i$,
the difference in the evolution of the cluster-level infection after the
infection of $i$ is inconsequential. Therefore,
$\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})$, the infection history in an
otherwise identical group where $i$ cannot transmit infection, is equal in
distribution to the observed $\mathbfcal{H}_{(i)}(\mathbf{x})$ before the
infection of $i$, and it imitates the observable exposure to infection for $i$
due to others’ infections. We formalize this intuition in identification
results presented in the next Section.

##  3 Identification of average potential infection outcomes

###  3.1 Assumptions

By the end of the observation at time $t$, we observe
$\big{(}\mathbf{X},\mathbf{L},\mathbfcal{H}(t)\big{)}$. We state the following
assumptions for a generic individual $i$.

######  Assumption 1 (Exclusion restriction for future infection times).

For $k=1,\ldots,n-1$, we have
$I_{i}^{0}(\mathbf{x},\mathbf{h}_{(i)})=I_{i}^{0}(x_{i})$ and
$I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)})=I_{i}^{k}\big{(}x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}(t_{(i)}^{k})\big{)}$.

Assumption 1 states that the latent time to infection of $i$ following the
$k$th infection time $t_{(i)}^{k}$ depends only on the treatments $\mathbf{x}$
assigned at baseline and others’ infection history up to $t_{(i)}^{k}$,
$\mathbf{h}_{(i)}(t_{(i)}^{k})$. It does not depend on the infection history
after $t_{(i)}^{k}$.

######  Assumption 2 (Independence of latent times to infection).

For a given history $\mathbf{h}_{(i)}(t_{(i)}^{k})$ up to the $k$th infection
and each $j=\varphi_{i}^{k+1},\ldots,\varphi_{i}^{n-1}$ infected after time
$t_{(i)}^{k}$,

$I_{i}^{k}(x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}(t_{(i)}^{k}))\mathbin{\rotatebox[origin={c}]{90.0}{$\models$}}I_{j}^{k}(x_{j},\mathbf{x}_{(j)},\mathbf{h}_{(i)}(t_{(i)}^{k}))\mid\mathbf{L}.$

Assumption 2 states that under a given infection history up to the
$k^{\text{th}}$ infection, the latent time to infection for individual $i$ is
conditionally independent of the latent time to infection for any other
uninfected individual $j$ given $\mathbf{L}$. In other words, the competing
(simultaneously elapsing) infection waiting times
$I_{i}^{k}(x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}(t_{(i)}^{k}))$ and
$I_{j}^{k}(x_{j},\mathbf{x}_{(j)},\mathbf{h}_{(i)}(t_{(i)}^{k}))$ are
conditionally independent.

######  Assumption 3 (Previous exposure exchangeability).

For all $t>0$ and each $k=0,\ldots,n-1$,

$I_{i}^{k}(x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}(t_{(i)}^{k}))\mathbin{\rotatebox[origin={c}]{90.0}{$\models$}}\mathbfcal{H}_{(i)}^{*}(t_{(i)}^{k};\mathbf{x}_{(i)})\mid\mathbf{L}.$

Assumption 3 states that the latent time to infection
$I_{i}^{k}\big{(}x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}(t_{(i)}^{k})\big{)}$
under the deterministic infection history $\mathbf{h}_{(i)}(t_{(i)}^{k})$ up
to $t_{(i)}^{k}$ is conditionally independent of the random infection history
$\mathbfcal{H}_{(i)}^{*}(t_{(i)}^{k};\mathbf{x}_{(i)})$ given $\mathbf{L}$. If
we interpret $\mathbfcal{H}_{(i)}^{*}(t;\mathbf{x}_{(i)})$ as the exposure of
$i$ to infection up to time $t$, then this assumption corresponds to no
unmeasured confounding for the effect of prior exposure to infection on the
latent time to infection. In this way, Assumption 3 is a conventional
treatment exchangeability assumption, where the “treatment” is prior exposure
to infection.

######  Assumption 4 (Treatment exchangeability).

For $k=0,1,\ldots,n-1$,

$I_{i}^{k}(x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}(t_{(i)}^{k})\big{)}\mathbin{\rotatebox[origin={c}]{90.0}{$\models$}}\mathbf{X}\mid\mathbf{L}.$

Assumption 4 states that the potential latent time to infection
$I_{i}^{k}\big{(}x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}(t_{(i)}^{k})\big{)}$
is conditionally independent of the joint treatment assignment $\mathbf{X}$
given $\mathbf{L}$. This corresponds to no unmeasured confounding of the
effect of treatment on the latent time to infection.

######  Assumption 5 (Consistency).

When $X_{i}=x_{i}$, $\mathbf{X}_{(i)}=\mathbf{x}_{(i)}$,
$\mathbfcal{H}_{(i)}(t_{(i)}^{k})=\mathbf{h}_{(i)}(t_{(i)}^{k})$ and
$T_{i}>t_{(i)}^{k}$,

$I_{i}^{k}=I_{i}^{k}(x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}(t_{(i)}^{k})).$

Assumption 5 states that
$I_{i}^{k}(x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}(t_{(i)}^{k}))$ is the
realized latent time to infection when the observed treatments and infection
history up to $t_{(i)}^{k}$ match those specified and $i$ remains susceptible
at $t_{(i)}^{k}$.

######  Assumption 6 (Positivity).

$\Pr\big{(}I_{i}^{k}\leq
t\,\big{|}\,\mathbf{X}=\mathbf{x},\mathbfcal{H}_{(i)}(t_{(i)}^{k})=\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{L}=\mathbf{l}\big{)}$
is positive and strictly increasing in $t$ and

$\Pr\big{(}\mathbf{X}=\mathbf{x},\mathbfcal{H}_{(i)}(t_{(i)}^{k})=\mathbf{h}_{(i)}(t_{(i)}^{k})\,\big{|}\,\mathbf{L}=\mathbf{l}\big{)}\in(0,1)$

for $k=0,\ldots,n-1$ and all $\mathbf{x}$, $\mathbf{h}_{(i)}$, and
$\mathbf{l}$.

Assumption 6 states that it is possible for the latent potential infection
times to take any positive value and that any treatment allocation is
possible.

A final assumption, which is only used to identify certain exposure-
marginalized estimands, is given below.

######  Assumption 7 (Cross-world independence).

For $\mathbf{x}_{(i)}\neq\mathbf{x}_{(i)}^{\prime}$ for $k=0,\ldots,n-1$,

$I_{i}^{k}(x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}(t_{(i)}^{k}))\mathbin{\rotatebox[origin={c}]{90.0}{$\models$}}\mathbfcal{H}_{(i)}^{*}(t_{(i)}^{k};\mathbf{x}_{(i)}^{\prime})\mid\mathbf{L}$

Assumption 7 states that, given $\mathbf{L}=\mathbf{l}$, the potential latent
time to infection
$I_{i}^{k}(x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}(t_{(i)}^{k}))$ is
conditionally independent of the the infection history among individuals other
than $i$ under treatments different from $\mathbf{x}_{(i)}$.

###  3.2 Identification of average potential infection outcomes

We first seek to identify the distribution of the infection time
$I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)})$ of $i$ following the $k$th infection
among other cluster members. Let

$\begin{split}F_{I_{i}^{k}}(s\,|\,\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})&=\Pr\big{(}I_{i}^{k}(x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)})<s\,\big{|}\,\mathbf{L}=\mathbf{l}\big{)}\\\
&=\Pr\big{(}I_{i}^{k}(x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}(t_{(i)}^{k}))<s\,\big{|}\,\mathbf{L}=\mathbf{l}\big{)}\end{split}$

be the cumulative distribution function of
$I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)})$ given $\mathbf{L}=\mathbf{l}$ for
$k=1,\ldots,n-1$, and let

$\begin{split}F_{I_{i}^{0}}(s|\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})&=\Pr\big{(}I_{i}^{0}(\mathbf{x},\mathbf{h}_{(i)})<s\,\big{|}\,\mathbf{L}_{i}=\mathbf{l}_{i})\\\
&=\Pr\big{(}I_{i}^{0}(x_{i})<s\,\big{|}\,\mathbf{L}_{i}=\mathbf{l}_{i})\end{split}$

be the cumulative distribution of $I_{i}^{0}(\mathbf{x},\mathbf{h}_{(i)})$
under $\mathbf{L}=\mathbf{l}$. The following result shows that these
distributions are identified via observation of
$\big{(}\mathbf{X},\mathbf{L},\mathbfcal{H}(t)\big{)}$.

######  Lemma 1 (Identification of latent time to infection under competing
risks).

Suppose Assumptions 1-6 hold and $\mathbf{h}_{(i)}$ is a deterministic
infection history with ordered infection times
$t_{(i)}^{1},\ldots,t_{(i)}^{n-1}$. Then the distribution of
$I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)})$ given $\mathbf{L}=\mathbf{l}$ is
identified by

$F_{I_{i}^{k}}(s|\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})=1-\exp\left[-\int_{t_{(i)}^{k}}^{t_{(i)}^{k}+s}\frac{f_{i}^{k}(u|\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})}{S_{i}^{k}(u|\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})}du\right]\text{
for $k=0,\ldots,n-1$}\\\ $ (2)

where

$\displaystyle S_{i}^{0}(u\,|\,\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})$
$\displaystyle=\Pr\big{(}T_{i}>u,T_{\varphi_{i}^{1}}>u,\ldots,T_{\varphi_{i}^{n-1}}>u\,\big{|}\,\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\big{)}$
$\displaystyle S_{i}^{k}(u\,|\,\mathbf{x},\mathbf{h}_{(i)},\mathbf{l}\big{)}$
$\displaystyle=\Pr\big{(}T_{i}>u,T_{\varphi_{i}^{k+1}}>u,\ldots,T_{\varphi_{i}^{n-1}}>u\,\big{|}\,T_{\varphi_{i}^{1}}=t_{(i)}^{1},\ldots,T_{\varphi_{i}^{k}}=t_{(i)}^{k},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\big{)}$

and

$\displaystyle f_{i}^{0}(u\,|\,\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})$
$\displaystyle=p\big{(}T_{i}=u,T_{\varphi_{i}^{1}}>u,\ldots,T_{\varphi_{i}^{n-1}}>u\,\big{|}\,\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\big{)}$
$\displaystyle f_{i}^{k}(u\,|\,\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})$
$\displaystyle=p\big{(}T_{i}=u,T_{\varphi_{i}^{k+1}}>u,\ldots,T_{\varphi_{i}^{n-1}}>u\,\big{|}\,T_{\varphi_{i}^{1}}=t_{(i)}^{1},\ldots,T_{\varphi_{i}^{k}}=t_{(i)}^{k},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l})$

for any fixed values of $\mathbf{x}$, $\mathbf{h}_{(i)}$ and $\mathbf{l}$.

For $k=n-1$, (2) simplifies to

$F_{I_{i}^{n-1}}(s\,\big{|}\,\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})=\Pr\big{(}T_{i}\leq
t_{(i)}^{n-1}+s\,\big{|}\,T_{\varphi_{i}^{1}}=t_{(i)}^{1},\ldots,T_{\varphi_{i}^{n-1}}=t_{(i)}^{n-1},T_{i}>t_{(i)}^{n-1},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\big{)}.$

Lemma 1 can be understood as a standard competing risks identification result:
after the $i^{\text{th}}$ infection at $t_{(i)}^{k}$, the infection of $i$ and
the infections of the other still-uninfected individuals serve as competing
events; in order to observe $I_{i}^{k}$ and estimate $F_{I_{i}^{k}}$, $i$ must
be infected first among the individuals in the cluster who remain uninfected.

In the following main result, we show that the average potential outcome
$\mathbb{E}[Y_{i}(t;\mathbf{h}_{(i)},\mathbf{x})]$ under a particular
deterministic exposure history $\mathbf{h}_{(i)}$ and treatment allocation
$\mathbf{x}$ can be identified nonparametrically.

######  Theorem 1 (Identification of exposure-and-treatment-controlled average
potential infection outcomes).

Suppose Assumptions 1–6 hold. Then

$\mathbb{E}\big{[}Y_{i}(t;\mathbf{h}_{(i)},\mathbf{x})\,\big{|}\,\mathbf{L}=\mathbf{l}\big{]}=\\\
\sum_{j=0}^{n-1}\bigg{[}F_{I_{i}^{j}}(\min\\{t,t_{(i)}^{j+1}\\}-t_{(i)}^{j}\,|\,\mathbf{x},\mathbf{h}_{(i)},\mathbf{l}\big{)}\prod_{k=0}^{j-1}\big{(}1-F_{I_{i}^{k}}(t_{(i)}^{k+1}-t_{(i)}^{k}\,|\,\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})\big{)}\bigg{]}\\\
$

where $F_{I^{j}_{i}}(s\,|\,\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})$ is given
by Lemma 1, $F_{I^{j}_{i}}(s\,|\,\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})=0$
when $s\leq 0$, and the product term is one when $j=0$.

A simplified version of Theorem 1, derived under a two-person partnership
model, is given in Theorem 1 of Cai et al. [22], which we restate here.

######  Corollary 1 (Identification of exposure-and-treatment-controlled
average potential infection outcomes in a two-person partnership).

Let $n=2$ and label the individuals $i$ and $j$. Let
$\mathbf{h}_{(i)}=\\{t_{(i)}^{1}=t_{j}\\}$ where $t_{j}$ is the infection time
of $j$ in Theorem 1, and suppose Assumptions 1–6 hold. Then for individual
$i$, if $t>t_{j}$, we have

$\mathbb{E}[Y_{i}(t;\mathbf{h}_{(i)},\mathbf{x})|\mathbf{L}=\mathbf{l}]=F_{I_{i}^{0}}(t_{j}|x_{i},\mathbf{l}_{i})+\big{(}1-F_{I_{i}^{0}}(t_{j}|x_{i},\mathbf{l}_{i})\big{)}\Pr[T_{i}<t|T_{i}\geq
t_{j},T_{j}=t_{j},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}]$

and if $t\leq t_{j}$, we have
$\mathbb{E}[Y_{i}(t;\mathbf{h}_{(i)},\mathbf{x})|\mathbf{L}=\mathbf{l}]=F_{I_{i}^{0}}(t|x_{i},\mathbf{l}_{i})$.

###  3.3 Identification of average exposure-marginalized potential infection
outcomes

Theorem 1 identifies average potential infection outcomes under a particular
deterministic infection history $\mathbf{h}_{(i)}$. But this history
$\mathbf{h}_{(i)}$ may not be of specific interest; rather, we often wish to
learn about potential infection outcomes as a function of treatments alone, by
marginalizing over an appropriate distribution on infection histories. To this
end, we show that the distribution of the potential infection history
$\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})$ of individuals other than $i$ is
identified.

Define the _potential exposure distribution_
$G_{(i)}^{*}(\mathbf{h}_{(i)}|\mathbf{x}_{(i)},\mathbf{l}_{(i)})$ as the
distribution of the potential infection history
$\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})$ given
$\mathbf{L}_{(i)}=\mathbf{l}_{(i)}$. For a particular deterministic infection
history $\mathbf{h}_{(i)}$, we denote $\mathbf{h}_{(i,j)}$ as the elements of
this history without the history of $j$. To identify the potential exposure
distribution, we will need to represent the history of the group, absent $j$,
where $i$ remains uninfected. Therefore, for a deterministic history
$\mathbf{h}_{(i)}$, let $\mathbf{h}_{(j)}^{i}=(\mathbf{h}_{(i,j)},h_{i})$
where $h_{i}$ corresponds to $y_{i}(t)=0$ for all $t>0$, so that $i$ is
uninfected. As before, we order individuals by their infection times so that
$t_{\varphi_{i}^{k}}=t_{(i)}^{k}$ for $k=1,\ldots,n-1$. Then individual
$\varphi_{i}^{1}$ is infected first (via some exogenous source of infection)
at time $t_{(i)}^{1}$, which means $I_{\varphi_{i}^{1}}^{0}=t_{(i)}^{1}$ and
$I_{j}^{0}>t_{(i)}^{1}$ for $j=\varphi_{i}^{2},\ldots,\varphi_{i}^{n-1}$.
After the first infection, $\varphi_{i}^{2}$ wins the competition to be
infected at time $t_{(i)}^{2}$, which means
$I_{\varphi_{i}^{2}}^{1}=t_{(i)}^{2}-t_{(i)}^{1}$ and
$I_{\varphi_{j}}^{1}>t_{(i)}^{2}-t_{(i)}^{1}$ for
$j=\varphi_{i}^{3},\ldots,\varphi_{i}^{n-1}$. This continues until the last
individual $\varphi_{i}^{n-1}$ gets infected at $t_{(i)}^{n-1}$ with the
additional time to infection after the previous infection as
$I_{\varphi_{i}^{n-1}}^{n-2}=t_{(i)}^{n-1}-t_{(i)}^{n-2}$. Let
$f_{I_{\varphi_{i}^{j}}^{j-1}}(\cdot|\mathbf{x},\mathbf{h}^{i}_{(\varphi_{i}^{j})},\mathbf{l})$
be the density of $I_{\varphi_{i}^{j}}^{j-1}$, which specifies the density of
additional time to infection for the $j^{\text{th}}$ infected subject
$\varphi_{i}^{j}$ after the previous $j-1$ infections, and let
$S_{I_{\varphi_{i}^{k}}^{j-1}}(\cdot|\mathbf{x},\mathbf{h}^{j}_{(\varphi_{i}^{k})},\mathbf{l})$
be the corresponding survival function of $I_{\varphi_{i}^{j}}^{j-1}$,
identified by Lemma 1. By Lemma 1, we can identify the distribution of
$\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})$.

######  Theorem 2 (Identification of the potential exposure distribution).

The density function
$\mathrm{d}G_{(i)}^{*}(\mathbf{h}_{(i)}\,|\,\mathbf{x}_{(i)},\mathbf{l}_{(i)})$
is identified by

$\begin{split}\text{d}G_{(i)}^{*}(\mathbf{h}_{(i)}\,|\,\mathbf{x}_{(i)},\mathbf{l}_{(i)})&=\prod_{j=1}^{n-1}\bigg{[}f_{I_{\varphi_{i}^{j}}^{j-1}}\big{(}t_{(i)}^{j}-t_{(i)}^{j-1}\,\big{|}\,\mathbf{x},\mathbf{h}^{i}_{(\varphi_{i}^{j})},\mathbf{l}\big{)}\prod_{k=j+1}^{n-1}S_{I_{\varphi_{i}^{k}}^{j-1}}\big{(}t_{(i)}^{j}-t_{(i)}^{j-1}\,\big{|}\,\mathbf{x},\mathbf{h}^{j}_{(\varphi_{i}^{k})},\mathbf{l}\big{)}\bigg{]}\\\
\end{split}$

where the ordered infection times $t_{(i)}^{1},\ldots,t_{(i)}^{n-1}$ and the
histories $\mathbf{h}^{i}_{(\varphi_{i}^{j})}$ are specified by the infection
history $\mathbf{h}_{(i)}$.

Using the expected potential infection outcome identified by Theorem 2, we can
describe how to integrate, marginalize, or standardize these conditional
estimands so that they do not depend on any particular value of exposure
history $\mathbf{h}_{(i)}$.

######  Proposition 1 (Exposure-marginalized average potential infection
outcomes).

Under Assumptions 5–6, the exposure-marginalized average potential infection
outcome under $\mathbf{X}=\mathbf{x}$ is

$\mathbb{E}\big{[}Y_{i}\big{(}t;x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})\big{)}\,\big{|}\,\mathbf{L}=\mathbf{l}\big{]}=\int\mathbb{E}[Y_{i}(t;x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)})\,|\,\mathbf{L}=\mathbf{l}]\,\mathrm{d}G_{(i)}^{*}(\mathbf{h}_{(i)}\,|\,\mathbf{x}_{(i)},\mathbf{l}_{(i)})$
(3)

where
$\mathbb{E}[Y_{i}(t;x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)})\,|\,\mathbf{L}=\mathbf{l}]$
is given by Theorem 1 and
$\mathrm{d}G_{(i)}^{*}(\mathbf{h}_{(i)}|\mathbf{x}_{(i)},\mathbf{l}_{(i)})$ is
given by Theorem 2. If Assumption 7 also holds and
$\mathbf{x}_{(i)}\neq\mathbf{x}_{(i)}^{\prime}$ then the exposure-marginalized
average potential infection outcome under $\mathbfcal{H}_{(i)}^{*}$, hold to
be realized under $\mathbf{x}_{(i)}^{\prime}\neq\mathbf{x}_{(i)}$, and
$\mathbf{X}=\mathbf{x}$ is

$\mathbb{E}\big{[}Y_{i}\big{(}t;x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)}^{\prime})\big{)}\,\big{|}\,\mathbf{L}=\mathbf{l}]=\int\mathbb{E}[Y_{i}(t;x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)})\,|\,\mathbf{L}=\mathbf{l}]\
\mathrm{d}G_{(i)}^{*}(\mathbf{h}_{(i)}\,|\,\mathbf{x}_{(i)}^{\prime},\mathbf{l}_{(i)})$
(4)

This result follows directly from Assumption 6 and Theorems 1 and 2. Note that
when $\mathbf{x}_{(i)}\neq\mathbf{x}_{(i)}^{\prime}$, (4) is a “cross-world”
potential outcome because the integrand is evaluated under the treatment
allocation $\mathbf{X}=(x_{i},\mathbf{x}_{(i)})$, while the marginalization is
with respect to infection histories under a different treatment allocation
$\mathbf{X}_{(i)}=\mathbf{x}_{(i)}^{\prime}$.

Theorems 1 and 2 come together in Proposition 4, which can be interpreted as a
special case of Robins’ G-formula [38]: it decomposes the average potential
outcome into an integral over a fully conditional expectation, with respect to
a modified distribution over prior outcomes, treatments, and covariates that
avoids post-treatment confounding. By integrating with respect to the modified
history $\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})$ in the marginalization, we
circumvent the dependence problem between the infection outcome of $i$ and the
infection history of others. In fact, a stronger result shows that the
exposure-marginalized average potential infection outcome, given by the
integral in (3), is equal to the observed conditional expectation.

######  Corollary 2 (Equivalence of observed and potential infection
outcomes).

By Assumptions 1, 3, 4 and 5,

$\mathbb{E}[Y_{i}(t)\,|\,\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}]=\mathbb{E}[Y_{i}\big{(}t;x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})\big{)}\,|\,\mathbf{L}=\mathbf{l}]$

Proposition 4 and Corollary 2 provide two ways of identifying potential
infection outcomes. The key insight is that we can decompose the observational
quantity
$\mathbb{E}[Y_{i}(t)\,|\,\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}]$ into a
integral of potential infection outcomes with respect to an infection history
$\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})$ that omits the role of $i$ in
infecting others. The explicit decomposition in Proposition 4 is especially
useful in statistical settings when data are sparse: by estimating the
integrand and marginalizing distribution, we can evaluate average potential
infection outcomes under joint treatments or covariates that were not observed
in the data.

###  3.4 Treatment- and covariate-marginalized estimands

Proposition 4 identifies potential infection outcomes that are a function of
the joint treatment vector $\mathbf{X}=\mathbf{x}$, conditional on covariates
$\mathbf{L}=\mathbf{l}$. To represent potential outcomes as a function only of
treatment assigned to the cluster, or of treatment assigned to a single
cluster member, it is necessary to marginalize (integrate) with respect to
$\mathbf{X}_{(i)}$ and/or $\mathbf{L}$.

To define exposure-and-treatment-marginalized average potential outcomes, we
integrate the expected exposure-marginalized potential infection outcomes over
a distribution for others’ treatments conditional on covariates
$\mathbf{L}=\mathbf{l}$. An additional assumption ensures that that the
cluster-level covariates $\mathbf{L}$ suffice to render $X_{i}$ and
$\mathbf{X}_{(i)}$ conditionally independent.

######  Assumption 8 (Conditional independence of treatments).

For $i=1,\ldots,n$,
$X_{i}\mathbin{\rotatebox[origin={c}]{90.0}{$\models$}}\mathbf{X}_{(i)}\mid\mathbf{L}$.

In other words, Assumption 8 makes
$\Pr(\mathbf{X}_{(i)}=\mathbf{x}_{(i)}|\mathbf{L}=\mathbf{l})$ invariant to
the value of $X_{i}=x_{i}$. Let
$p(\mathbf{x}_{(i)}\,|\,\mathbf{L}=\mathbf{l})$ be a distribution over others’
treatments, conditional on cluster-level covariates. When this is the
distribution of others’ treatments imposed by the data-generating process,
then
$p(\mathbf{x}_{(i)}\,|\,\mathbf{L}=\mathbf{l})=\Pr(\mathbf{X}_{(i)}=\mathbf{x}_{(i)}|\mathbf{L}=\mathbf{l})$.
Importantly, $p(\mathbf{x}_{(i)}\,|\,\mathbf{L}=\mathbf{l})$ must be invariant
to the treatment $X_{i}=x_{i}$ of $i$ [39, 40, 41]. If in addition Assumption
8 holds, then we can identify the potential infection outcome of $i$ under
treatment $x_{i}$, integrated with respect to the marginal distribution of
treatment to others, given covariates. Let $\mathcal{X}^{n-1}=\\{0,1\\}^{n-1}$
be the set of all binary vectors of length $n-1$.

######  Proposition 2 (Exposure-and-treatment-marginalized average potential
infection outcomes).

Under Assumption 8,

$\displaystyle\overline{Y}_{i}^{p}(t;x_{i}\,|\,\mathbf{L}=\mathbf{l})$
$\displaystyle=\sum_{x_{(i)}\in\mathcal{X}^{n-1}}\mathbb{E}\big{[}Y_{i}\big{(}t;x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})\big{)}\mid\mathbf{L}=\mathbf{l}\big{]}p(\mathbf{x}_{(i)}\,|\,\mathbf{L}=\mathbf{l})$
$\displaystyle\overline{Y}_{i}^{p}(t;x_{i},\mathbf{x}_{(i)}^{\prime}\,|\,\mathbf{L}=\mathbf{l})$
$\displaystyle=\sum_{x_{(i)}\in\mathcal{X}^{n-1}}\mathbb{E}\big{[}Y_{i}\big{(}t;x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)}^{\prime})\big{)}\mid\mathbf{L}=\mathbf{l}\big{]}p(\mathbf{x}_{(i)}\,|\,\mathbf{L}=\mathbf{l})$

In Proposition 2 we have specified an arbitrary distribution
$p(\mathbf{x}_{(i)}\,|\,\mathbf{L}=\mathbf{l})$ which may differ from the
distribution by which treatment was assigned. We define covariate-marginalized
potential outcomes over the marginal distribution of covariates $\mathbf{L}$
in the cluster, denoted by $Q(\mathbf{l})$.

######  Definition 1 (Covariate adjustment).

Under Assumptions 1–6,

$\displaystyle\mathbb{E}[Y_{i}(t;x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)})]$
$\displaystyle=\int\mathbb{E}[Y_{i}(t;x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)})\,|\,\mathbf{L}=\mathbf{l}]\,\mathrm{d}Q(\mathbf{l})$
$\displaystyle\mathbb{E}\big{[}Y_{i}(t;x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})\big{)}\big{]}$
$\displaystyle=\int\mathbb{E}\big{[}Y_{i}(t;x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})\big{)}\mid\mathbf{L}=\mathbf{l}\big{]}\,\mathrm{d}Q(\mathbf{l})$
$\displaystyle\overline{Y}_{i}^{p}(t;x_{i})$
$\displaystyle=\int\overline{Y}_{i}^{p}(t;x_{i}\,|\,\mathbf{L}=\mathbf{l})\,\mathrm{d}Q(\mathbf{l})$

Under Assumptions 1–7 when $\mathbf{x}_{(i)}\neq\mathbf{x}_{(i)}^{\prime}$,

$\displaystyle\mathbb{E}\big{[}Y_{i}(t;x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)}^{\prime})\big{)}\big{]}$
$\displaystyle=\int\mathbb{E}\big{[}Y_{i}(t;x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)}^{\prime})\big{)}\,\big{|}\,\mathbf{L}=\mathbf{l}\big{]}\,\mathrm{d}Q(\mathbf{l})$
$\displaystyle\overline{Y}_{i}^{p}(t;x_{i},\mathbf{x}_{(i)}^{\prime})$
$\displaystyle=\int\overline{Y}_{i}^{p}(t;x_{i},\mathbf{x}_{(i)}^{\prime}\,|\,\mathbf{L}=\mathbf{l})\,\mathrm{d}Q(\mathbf{l})$

where
$\mathbb{E}[Y_{i}(t;x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}))\,|\,\mathbf{L}=\mathbf{l}]$
is given by Theorem 1,
$\mathbb{E}\big{[}Y_{i}\big{(}t,x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})\big{)}\,\big{|}\,\mathbf{L}=\mathbf{l}\big{]}$
and
$\mathbb{E}\big{[}Y_{i}\big{(}t,x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}(\mathbf{x}_{(i)}^{\prime})\big{)}\,\big{|}\,\mathbf{L}=\mathbf{l}\big{]}$
are given by Proposition 4, and the definitions of
$\overline{Y}_{i}^{p}(t,x_{i}\,|\,\mathbf{L}=\mathbf{l})$ and
$\overline{Y}_{i}^{p}(t,x_{i},\mathbf{x}_{(i)}^{\prime}\,|\,\mathbf{L}=\mathbf{l})$
are given by Proposition 2.

##  4 Causal effects based on risk differences

With these identification results in hand, we present biologically meaningful
causal estimands summarizing the transmissibility of disease, as well as the
effects of treatment on infection and transmision. We define causal effects in
terms of infection risk differences, but similar effects may be defined in
terms of risk ratios or odds ratios. We focus on the potential infection
outcomes of a particular individual $i$.

First, we define what it means for the infection outcome to be contagious, or
transmissible.

######  Definition 2 (Contagion effect).

The exposure-controlled contagion effect is

$CE_{i}(t,\mathbf{x},\mathbf{h}_{(i)},\mathbf{h}_{(i)}^{\prime})=\mathbb{E}[Y_{i}(t;\mathbf{x},\mathbf{h}_{(i)})-Y_{i}(t;\mathbf{x},\mathbf{h}_{(i)}^{\prime})].$

The exposure-marginalized contagion effect is

$CE_{i}(t,x_{i},\mathbf{x}_{(i)},\mathbf{x}^{\prime}_{(i)})=\mathbb{E}\big{[}Y_{i}\big{(}t;x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})\big{)}-Y_{i}\big{(}t;x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)}^{\prime})\big{)}\big{]}.$

Informally, the infection outcome is “contagious” if it depends on the
infection history of others, holding treatments constant. To make this idea
more concrete, for $\mathbf{h}_{(i)}=(t_{i}^{(1)},\ldots,t_{i}^{(n-1)})$ and
$\mathbf{h}_{(i)}^{\prime}=(t_{i}^{{}^{\prime}(1)},\ldots,t_{i}^{{}^{\prime}(n-1)})$,
we denote $\mathbf{h}_{(i)}\prec\mathbf{h}_{(i)}^{\prime}$ if $t_{i}^{(k)}\leq
t_{i}^{{}^{\prime}(k)}$ element-wise for $k=1,\ldots,n-1$. We say the
infection outcome is _positively contagious_ absent treatment if earlier
infection of cluster members increases the average potential infection risk,
so $CE(t,\mathbf{0},\mathbf{h}_{(i)}(t),\mathbf{h}_{(i)}^{\prime}(t))>0$
whenever $\mathbf{h}_{(i)}\prec\mathbf{h}_{(i)}^{\prime}$, when no cluster
members receive treatment [22].

Susceptibility effects compare potential infection outcomes of $i$ under
treatment versus no treatment of $i$, while holding infection history and
treatment to others constant.

######  Definition 3 (Susceptibility effect).

The exposure-controlled susceptibility effect is

$SE_{i}(t,\mathbf{x}_{(i)},\mathbf{h}_{(i)})=\mathbb{E}[Y_{i}(t;1,\mathbf{x}_{(i)},\mathbf{h}_{(i)})-Y_{i}(t;0,\mathbf{x}_{(i)},\mathbf{h}_{(i)})],$

the exposure-marginalized susceptibility effect is

$SE_{i}(t,\mathbf{x}_{(i)})=\mathbb{E}\big{[}Y_{i}\big{(}t;1,\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})\big{)}-Y_{i}\big{(}t;0,\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})\big{)}\big{]},$

and the exposure-and-treatment-marginalized susceptibility effect is

$SE_{i}^{p}(t)=\overline{Y}_{i}^{p}(t;1)-\overline{Y}_{i}^{p}(t;0).$

This susceptibility effect summarizes the effect of changing only the
treatment of $i$, holding all else constant; if it is negative, then the
treatment reduces the chance of infection.

Infectiousness effects compare the potential infection outcome of $i$ under
different treatments to others while holding constant treatment and exposure
to infection of $i$.

######  Definition 4 (Infectiousness effect).

The exposure-controlled infectiousness effect is

$IE_{i}(t,x_{i},\mathbf{x}_{(i)},\mathbf{x}_{(i)}^{\prime},\mathbf{h}_{(i)})=\mathbb{E}[Y_{i}(t;x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)})-Y_{i}(t;x_{i},\mathbf{x}_{(i)}^{\prime},\mathbf{h}_{(i)})],$

and the exposure-marginalized infectiousness effect is

$IE_{i}(t,x_{i},\mathbf{x}_{(i)},\mathbf{x}_{(i)}^{\prime})=\mathbb{E}\big{[}Y_{i}\big{(}t;x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})\big{)}-Y_{i}\big{(}t;x_{i},\mathbf{x}_{(i)}^{\prime},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})\big{)}\big{]}$

where we have held the marginalizing distribution
$\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})$ constant under treatment
$\mathbf{x}_{(i)}$. The treatment-and-exposure-marginalized infectiousness
effect is:

$IE^{p}(t)=\sum_{\mathbf{x}\in\mathcal{X}^{n}}IE_{i}(t,x_{i},\mathbf{x}_{(i)})p(\mathbf{x}).$

Informally, the infectiousness effect summarizes the effect of treatment on an
infected individual’s ability to infect others. Note that the distribution of
$\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})$ is the same in both parts of the
contrast, and we are only changing the integrand, not the marginalizing
distribution.

##  5 Statistical identification using pairwise hazard models

Non-parametric identification of exposure-controlled potential infection
outcomes via Lemma 1 and Theorem 1 can be difficult in finite samples because
realized infection histories may be different from the history of interest, or
we may not have enough realizations to adequately capture the natural
distribution over infection histories. Fortunately, flexible statistical
modeling assumptions can help parameterize distribution functions
$F_{I_{i}^{k}}(s|\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})$ so that these can be
efficiently estimated, facilitating computation of expected potential
infection outcomes. Kenah [36, 35] developed a semiparametric class of
infection transmission models that provides the tools needed to dramatically
simplify inference of identified estimands while accommodating individual-
level treatment and covariates. The approach is based on extensive prior work
modeling time-to-infection data [24, 30, 37, 34, 36, 21].

###  5.1 Hazards for time-to-transmission

We seek to parameterize the distribution of the potential waiting time
$I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)})$ from the $k$th infection to the
infection of $i$. The key idea is that previously infected individuals, along
with a possibly exogenous source of infection, impose competing risks of
disease transmission to remaining uninfected individuals. We view
$I_{i}^{k}(x_{i},\mathbf{x}_{(i)}^{\mathbf{k}},\mathbf{h}_{(i)}^{\mathbf{k}})$
as the minimum of these $k+1$ competing transmission times, starting at the
time of the $k$th infection $t_{(i)}^{k}$.

To formalize this intuition, let $\tau_{ji}$ be the time for $j$ to infect
$i$, following infection of $j$, and suppose all $\tau_{ji}$ are conditionally
independent, given relevant treatments and covariates. Let
$\tau_{ji}(x_{j},x_{i})$ be the potential transmission time as a function of
treatments to $j$ and $i$, and let
$\lambda_{ji}(\cdot|x_{j},x_{i},\mathbf{l}_{j},\mathbf{l}_{i})$ be the hazard
function of $\tau_{ji}(x_{j},x_{i})$, where we interpret $\tau_{0i}(x_{i})$ as
the potential transmission time to $i$ from an outside (exogenous) source of
transmission, and let $\lambda_{0i}(t|x_{i},\mathbf{l}_{i})$ be the
corresponding transmission hazard.

######  Assumption 9 (Independence of competing transmission times).

For all $i$ and all $j\neq i$, $k\neq l$

$\begin{split}\tau_{ji}(x_{j},x_{i})&\mathbin{\rotatebox[origin={c}]{90.0}{$\models$}}\tau_{kl}(x_{k},x_{l})\mid\mathbf{L},\\\
\lambda_{0i}(t\,|\,\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})&=\lambda_{0i}(t\,|\,x_{i},\mathbf{l}_{i}),\\\
\lambda_{ji}(t\,|\,\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})&=\lambda_{ji}(t-t_{j}\,|\,x_{i},x_{j},\mathbf{l}_{i},\mathbf{l}_{j}).\end{split}$
(5)

Assumption 5 states that the latent waiting times to infection from different
sources to $i$ are conditionally independent given treatments and covariates
and that their hazard functions are determined by treatments and covariates in
the target individual and, if needed, the infected source individual.

This assumption yields a convenient additive hazard model for the infection
time of $i$. At time $t>t_{(i)}^{k}$, the infection waiting time $I_{i}^{k}$
has hazard

$\lambda_{0i}(t\,|\,x_{i},\mathbf{l}_{i})+\sum_{j=1}^{k}\lambda_{\varphi_{i}^{j},i}(t-t_{(i)}^{k}\,|\,x_{\varphi_{i}^{j}},x_{i},\mathbf{l}_{j},\mathbf{l}_{i})$
(6)

which is the sum of the transmission hazards from existing sources of
infection. Therefore the infection hazard of $i$ at an arbitrary time $t$,
derived from (6), can be written as

$\lambda_{i}(t\,|\,\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})=\lambda_{0i}(t\,|\,x_{i},\mathbf{l}_{i})+\sum_{j\neq
i}y_{j}(t)\lambda_{ji}(t-t_{j}\,|\,x_{i},x_{j},\mathbf{l}_{i},\mathbf{l}_{j}).$
(7)

Assumption 5 dramatically simplifies representation and estimation of causal
estimands in terms of time integrals of these hazard functions. Figure 2 shows
how $\lambda_{i}(t|\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})$ changes under a
deterministic history $\mathbf{h}_{(i)}=(t_{i}^{(1)},\ldots,t_{i}^{(n-1)})$.

Figure 2: Illustration of the infection hazard experienced by $i$ under the
hazard model (7) and a deterministic infection history
$\mathbf{h}_{(i)}=(t_{i}^{(1)},\ldots,t_{i}^{(n-1)})$. For simplicity, we
assume $\lambda_{ji}(t|\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})$ is constant
for $t>0$. Before the first infection at $t_{(i)}^{1}$, $i$ experiences only
the exogenous force of infection $\lambda_{0i}(t\,|\,x_{i},\mathbf{l}_{i})$.
Following the first infection at $t_{(i)}^{1}$, $i$ experiences
$\lambda_{0i}(t\,|\,x_{i},\mathbf{l}_{i})+\lambda_{\varphi_{i}^{1},i}(t-t_{(i)}^{1}|x_{i},x_{\varphi_{i}^{1}},\mathbf{l}_{i},\mathbf{l}_{\varphi_{i}^{1}})$.
This process continues until all $n-1$ other cluster members are infected.

The hazard representation (7) has an important consequence fo identification
of expected potential infection outcomes.

######  Theorem 3 (Identification of exposure-controlled potential outcomes
under the pairwise hazard model).

Under Assumption 5, Theorem 1 reduces to

$\mathbb{E}[Y_{i}(t,\mathbf{x},\mathbf{h}_{(i)})\,|\,\mathbf{L}=\mathbf{l}]=1-\exp\left[-\int_{0}^{t}\left(\lambda_{0i}(s|x_{i},\mathbf{l}_{i})+\sum_{j\neq
i}y_{j}(s)\lambda_{ji}(s-t_{j}|x_{i},x_{j},\mathbf{l}_{i},\mathbf{l}_{j})\right)\mathrm{d}s\right]$

where $t_{j}$ is the infection time of $j$ in the history $\mathbf{h}_{(i)}$
and $y_{j}(s)$ is the indicator that $s>t_{(i)}^{j}$.

This result is a consequence of the hazard representation (7), and makes clear
that the expected exposure-controlled potential infection outcome is simply
the probability of infection (failure) prior to time $t$ under the specified
treatments and infection history of others.

Finally, exposure-marginalized potential infection outcomes can be computed
using the identified distribution of the potential exposure distribution.

######  Theorem 4 (Identification of exposure-marginalized potential outcomes
under the pairwise hazard model).

Under Assumption 5, Theorem 2 reduces to

$\text{d}G_{(i)}^{*}(\mathbf{h}_{(i)}\,|\,\mathbf{x}_{(i)},\mathbf{l}_{(i)})=\prod_{j=1}^{n-1}\bigg{[}\lambda_{\varphi_{i}^{j}}(t_{(i)}^{j}\,|\,\mathbf{x},\mathbf{h}^{i}_{(\varphi_{i}^{j})},\mathbf{l})\prod_{k=j}^{n-1}\exp\bigg{(}-\int_{t_{(i)}^{j-1}}^{t_{(i)}^{j}}\lambda_{\varphi_{i}^{k}}(s\,|\,\mathbf{x},\mathbf{h}^{i}_{(\varphi_{i}^{k})},\mathbf{l})\,\mathrm{d}s\bigg{)}\bigg{]}\\\
$ (8)

For the terms inside the square bracket, we have used the fact that the
probability density for the infection time of $j$ equals the product of its
hazard and survival functions. For the product term inside the square
brackets, each individual $k$ is included for $j=1,\ldots,k-1$. Thus, (8)
simplifies to

$\text{d}G_{(i)}^{*}(\mathbf{h}_{(i)}\,|\,\mathbf{x}_{(i)},\mathbf{l}_{(i)})=\prod_{j=1}^{n-1}\lambda_{\varphi_{i}^{j}}(t_{(i)}^{j}\,|\,\mathbf{x},\mathbf{h}^{i}_{(\varphi_{i}^{j})},\mathbf{l})\times\prod_{k=1}^{n-1}\exp\bigg{(}-\int_{0}^{t_{(i)}^{k}}\lambda_{\varphi_{i}^{k}}(s\,|\,\mathbf{x},\mathbf{h}^{i}_{(\varphi_{i}^{k})},\mathbf{l})\,\mathrm{d}s\bigg{)}$
(9)

###  5.2 Statistical infection hazard model

Following Kenah [35], we consider a Cox-type hazard model for the pairwise
infection times:

$\begin{split}\lambda_{0i}(t\,|\,x_{i},\mathbf{l}_{i})&=\alpha(t)\exp[\beta_{1}x_{i}+\theta_{1}\mathbf{l}_{i}]\\\
\lambda_{ji}(t\,|\,x_{i},x_{j},\mathbf{l}_{i})&=\gamma(t-t_{j})\exp[\beta_{1}x_{i}+\beta_{2}x_{j}+\theta_{1}\mathbf{l}_{i}+\theta_{2}\mathbf{l}_{j}]\end{split}$
(10)

which has non-parametric baseline hazards $\alpha(t)$ for the (time-varying)
risk of infection from outside the cluster, and $\gamma(t-t_{j})$ for the
(time-varying) infection risks from already-infected cluster members $j$, with
the effects of treatments $\mathbf{X}=x$ and covariates $\mathbf{L}$
accommodated in a log-linear model with coefficients $\beta_{1}$, $\beta_{2}$,
$\theta_{1}$ and $\theta_{2}$. This infection hazard model has been used by
many authors to represent transmission dynamics in groups [30, 42, 36, 35, 26,
21, 43].

The coefficients of the infection hazard model (10) have inutitive
interpretations. If $\beta_{1}<0$, vaccination decreases the hazard of
infection for treated individuals against all sources of infection by a factor
of $e^{\beta_{1}}$. If $\beta_{2}<0$, vaccination of other individuals
decreases their hazard of transmission to $i$ by a factor of $e^{\beta 2}$
once they become infectious. Plugging in (10), the hazard model (7) takes a
simple form,

$\lambda_{i}(t\,|\,\mathbf{x},\mathbf{h}_{(i)},\mathbf{l})=\exp[\beta_{1}x_{i}+\theta_{1}\mathbf{l}_{i}]\Big{(}\alpha(t)+\sum_{j\neq
i}y_{j}(t)\gamma(t-t_{j})\exp[\beta_{2}x_{j}+\theta_{2}\mathbf{l}_{j}]\Big{)}.$
(11)

###  5.3 Hazard ratio estimands

The parameters of the hazard model (7) are closely related to causal effects
defined above. For example, the contagion effect is positive whenever
$\gamma(t)>0$, and the sign of the susceptibility effects and infectiousness
effects depend only on $\beta_{1}$ and $\beta_{2}$, respectively (details are
given in the Appendix). These facts suggest that the parameters of this model
may have a causal interpretation in terms of their relationship to the causal
contagion, susceptibility, and infectiousness effects defined above. We
construct hazard ratio estimands by contrasting potential infection hazard
under different exposure history and treatment.

######  Definition 5 (Controlled susceptibility hazard ratio).

The controlled susceptibility hazard ratio is defined as:

$HSE^{C}(t,\mathbf{x}_{(i)},\mathbf{h}_{(i)},\mathbf{l})=\frac{\lambda_{i}(t\,|\,x_{i}=1,\mathbf{x}_{(i)},\mathbf{h}_{(i)},\mathbf{l})}{\lambda_{i}(t\,|\,x_{i}=0,\mathbf{x}_{(i)},\mathbf{h}_{(i)},\mathbf{l})}$

The controlled hazard susceptibility ratio is the ratio of counterfactual
hazards of $i$ when $i$ is treated versus untreated, while holding the
treatments, infection histories, and covariates of others constant. Plugging
(10) into the controlled hazard susceptibility effect, we find that

$HSE^{C}(t,\mathbf{x}_{(i)},\mathbf{h}_{(i)},\mathbf{l})=e^{\beta_{1}}$ (12)

regardless of the values of $t$, $\mathbf{x}_{(j)}$, $\mathbf{h}_{(i)}$, and
cluster size. If $\beta_{1}<0$, vaccination is beneficial because it decreases
the hazard of infection from all sources of infection by a factor of
$e^{\beta_{1}}<1$.

######  Definition 6 (Controlled infectiousness hazard ratio).

The controlled infectiousness hazard ratio is defined as:

$HIE^{C}(t,h_{j},h^{\prime}_{j},\mathbf{x}_{(j)},\mathbf{h}_{(i,j)},\mathbf{l})=\frac{\lambda_{i}(t|x_{j}=1,\mathbf{x}_{(j)},h^{\prime}_{j},\mathbf{h}_{(i,j)},\mathbf{l})-\lambda_{i}(t|x_{j}=1,\mathbf{x}_{(j)},h_{j},\mathbf{h}_{(i,j)},\mathbf{l})}{\lambda_{i}(t|x_{j}=0,\mathbf{x}_{(j)},h^{\prime}_{j},\mathbf{h}_{(i,j)},\mathbf{l})-\lambda_{i}(t|x_{j}=0,\mathbf{x}_{(j)},h_{j},\mathbf{h}_{(i,j)},\mathbf{l})}$

where $y_{j}(t)=1$ in $h^{\prime}_{j}$ and $y_{j}(t)=0$ in $h_{j}$.

The controlled infectiousness hazard ratio is the ratio of the hazard of $i$
being infected by a given infectious neighbor $j$, when $j$ is vaccinated
versus unvaccinated, holding $\mathbf{x}_{(j)}$, exposure to infection from
sources other than $j$, and all covariates constant. Applying the hazard model
(10) to the controlled infectiousness effect, we have

$HIE^{C}(t,h_{j},h^{\prime}_{j},\mathbf{x}_{(j)},\mathbf{h}_{(i,j)},\mathbf{l})=e^{\beta_{2}}$
(13)

regardless of the values of $t$, $\mathbf{x}_{(j)}$, $\mathbf{h}_{(i,j)}$, and
cluster size. If $\beta_{2}<0$, the infectiousness effect is beneficial
because it decreases the instantaneous infectiousness of $j$ by a factor of
$e^{\beta_{2}}<1$.

To determine whether the disease is contagious, we focus on one transmission
contact between $i$ and an infectious individual $j$, and evaluate if coming
into contact with an infectious individual increases the infection risk. Since
treatments are irrelevant to the intrinsic transmissibility of a disease, we
define the contagion effect under null treatments
$\mathbf{X}=(0,\ldots,0)=\mathbf{0}$.

######  Definition 7 (Controlled contagion cumulative hazard ratio).

The controlled contagion cumulative hazard ratio is

$HCE^{C}(t,h^{{}^{\prime\prime}}_{j},h^{\prime}_{j},\mathbf{h}_{(i,j)},\mathbf{l})=\frac{\int_{0}^{t}\big{[}\lambda_{i}(u;\mathbf{x}=\mathbf{0},h^{\prime}_{j}(t),\mathbf{h}_{(i,j)},\mathbf{l})-\lambda_{i}(u;\mathbf{x}=\mathbf{0},h_{j}(t),\mathbf{h}_{(i,j)},\mathbf{l})\big{]}du}{\int_{0}^{t}\big{[}\lambda_{i}(u;\mathbf{x}=\mathbf{0},h^{{}^{\prime\prime}}_{j}(t),\mathbf{h}_{(i,j)},\mathbf{l})-\lambda_{i}(u;\mathbf{x}=\mathbf{0},h_{j}(t),\mathbf{h}_{(i,j)},\mathbf{l})\big{]}du}$

where the corresponding infection times $t^{\prime}_{j}$ and
$t^{\prime\prime}_{j}$ contained in $h^{\prime}_{j}$ and
$h^{\prime\prime}_{j}$ has the relationship of
$t^{\prime}_{j}<t^{{}^{\prime\prime}}_{j}<t$, and $y_{j}(t)=0$ in $h_{j}(t)$.

Applying the hazard model (10) to the controlled contagion effect, we have:

$HCE^{C}(t,h^{{}^{\prime\prime}}_{j},h^{\prime}_{j},\mathbf{h}_{(i,j)})=\frac{\int_{t_{j}^{\prime}}^{t}\gamma(u)du}{\int_{t_{j}^{\prime\prime}}^{t}\gamma(u)du}$
(14)

Therefore, $\gamma(t)>0$ for any $t>0$ implies positive contagion, since
earlier exposure to infection increases the infection risk.

###  5.4 Statistical estimation

The hazard model (10) can be estimated parametrically [34] or
semiparametrically [36, 35]. We illustrate identification and estimation in
the parametric case when the functional forms for $\alpha(t)$ and $\gamma(t)$
are known to obtain maximum likelihood estimates (MLEs) for model parameters.
We then compute causal estimands. Specifying particular functional forms for
$\alpha(t)$ and $\gamma(t)$ is a relatively strong assumption, and if we do
not want to specify the functional form of the baseline hazards, we can also
estimate them non-parametrically by martingale techniques in counting process.
Nevertheless, regardless of known or unknown baseline hazards, the meanings of
$e^{\beta_{1}}$ and $e^{\beta_{2}}$ as causal estimands for the susceptibility
and infectiousness effects are preserved, as long as the hazard model (10)
holds.

With independent (or administrative) censoring at $C_{i}$ for individual $i$,
we denote $\Delta_{i}=\mathds{1}\left\\{T_{i}<C_{i}\right\\}$, with
$\Delta_{i}=1$ indicating $i$’s infection time being observed, and
$\Delta_{i}=0$ indicating $i$’s infection time being censored. Denote
$T^{*}_{i}=\min(T_{i},C_{i})$. For each cluster, we observe
$(T^{*}_{i},\Delta_{i},X_{i},\mathbf{L}_{i})$ for all $i$. Given the hazard
model for $\lambda_{i}$ specified in (10), we can derive the full likelihood
of infection times of all clusters as follows:

$\begin{split}L(\beta_{1},\beta_{2},\theta_{1},\theta_{2})&=\prod_{i=1}^{n}\left[\lambda_{i}(t^{*}_{i})\right]^{\Delta_{i}}\exp\left[-\int_{0}^{t^{*}_{i}}\lambda_{i}(u)\
\text{d}u\right]\\\
&=\prod_{i=1}^{n}\left[e^{\beta_{1}x_{i}+\theta_{1}\mathbf{l}_{i}}\Big{(}\alpha(t^{*}_{i})+\sum_{j\neq
i}y_{j}(t^{*}_{i})\gamma(t^{*}_{i}-t_{j})e^{\beta_{2}x_{j}+\theta_{2}\mathbf{l}_{j}}\Big{)}\right]^{\Delta_{i}}\\\
&\quad\times\exp\left(-\int_{0}^{t^{*}_{i}}e^{\beta_{1}x_{i}+\theta_{1}\mathbf{l}_{i}}\Big{(}\alpha(u)+\sum_{j\neq
i}y_{j}(u)\gamma(u-t_{j})e^{\beta_{2}x_{j}+\theta_{2}\mathbf{l}_{j}}\Big{)}\
\text{d}u\right)\end{split}$ (15)

##  6 Simulations and comparisons to other estimands

Infectious disease treatment effects have been evaluated using both randomized
trials and observational studies. For infectious diseases that are
transmissible within pairs or partnerships (e.g., HIV), researchers have
designed partnership randomization studies based on sexual partners [44, 45,
46, 47] or mother-child pairs [48, 49]. In outbreak or endemic settings,
researchers have proposed trial designs for clusters when individuals are
randomized to receive treatment and placebo [27, 50, 13, 3, 51, 52], and
defined a series of direct, indirect, susceptibility and infectiousness
vaccine effects by different summary measurements [6, 27, 50]. Generalized
estimation equation (GEE) regression approaches are popular for confounder
adjustment when outcomes are correlated. These methods have been applied to
assess treatment effects in cluster randomized trials [53] and network-
randomized trials [51, 54], and model the spread of contagious outcomes across
network ties [55, 56, 57].

###  6.1 Simulation of seasonal influenza vaccine trials

We illustrate the performance of the controlled and exposure-marginalized
causal estimands under an extensive collection of simulations that mimic a
hypothetical influenza trial. We assume all individuals are uninfected at
baseline, and each individual is susceptible to infection from a source
outside the cluster and from any of their infected fellow cluster members.
Infections can be transmitted between individuals in either direction and
treatments are randomized under multiple strategies or correlated as in
observational studies. We assume individuals remain infectious after
infection, with their transmissibility governed by the hazard $\gamma(t)>0$.

We consider an influenza vaccine that is protective in its susceptibility
effect ($\beta_{1}<0$) and infectiousness effect ($\beta_{2}<0$), in clusters
of 2, 4 and 8 individuals. We consider simulations under varied transmission
dynamics, including: (1) constant external and internal baseline hazards of
$\alpha(t)=\alpha$ and $\gamma(t)=\gamma$, which reduces to a Markov
susceptible-infective process with an external force of infection [e.g 26,
21], (2) a constant external force of infection $\alpha(t)=\alpha$ and
decreasing internal baseline hazards of $\gamma(t)=b\exp[-\omega(t-t_{j})]$,
and (3) a seasonally varying external baseline hazard of
$\alpha(t)=a(1+\sin(2\pi t+\phi))$ and decreasing internal baseline hazards of
$\gamma(t)=b\exp[-\omega(t-t_{j})]$.

In randomized trial simulations, vaccine allocation is randomized in
accordance with a specified distribution – Bernoulli, block, or cluster
randomization. For Bernoulli randomization, each individual within the cluster
is independently randomized with $\Pr(\mathbf{X}_{i}=1)=\frac{1}{2}$; for
block randomization, exactly half of the individuals of the cluster are
randomized to the treatment and the remaining half are randomized to the
placebo with $\sum_{i=1}^{n}X_{i}=\frac{n}{2}$; for cluster randomization, the
whole cluster is either randomized to all receiving treatments or all
receiving placebo with
$\Pr(\mathbf{X}=\mathbf{0})=\Pr(\mathbf{X}=\mathbf{1})=\frac{1}{2}$. In the
observational study simulation, the traits $\mathbf{L}=\mathbf{l}$ together
determine the joint distribution of vaccine in the cluster as

$\Pr(X_{i}=1|L_{i}=l_{i})=\frac{1}{1+e^{l_{i}}}$

where

$\begin{pmatrix}L_{1}\\\ L_{2}\\\ ...\\\
L_{n}\end{pmatrix}\sim\text{Normal}\left(\begin{pmatrix}0\\\ 0\\\ ...\\\
0\end{pmatrix},v\begin{pmatrix}1&\rho&...&\rho\\\ \rho&1&...&\rho\\\ ...\\\
\rho&\rho&...&1\end{pmatrix}\right)$ (16)

with $v>0$.

Tables 1–4 show estimates of the hazard model coefficients $\beta_{1}$ and
$\beta_{2}$, along with exposure-marginalized effects $CE$, $SE$, and $IE$,
under different cluster sizes, transmission dynamics, and vaccine allocation
distributions. We set $e^{\beta_{1}}=0.5$ so that influenza vaccination
reduces the hazard of infection for vaccinated individuals by $50\%$ [58] in
Tables 1–3. There is little information about how much vaccines reduces the
infectiousness of influenza, so we vary this effect in simulations: we
simulate $e^{\beta_{2}}=0.5$ in Tables 1–2 and $e^{\beta_{2}}=0.9$ in Table 3,
so that vaccine reduces the transmission by $50\%$ and $10\%$. The covariates
are set to be negatively correlated with $\rho=-0.1$ in Tables 1 and 3 and
positively correlated with $\rho=0.1$ in Tables 2 and 4. In Table 4 we show a
special scenario when the infectiousness effect of a vaccine is much stronger
than the susceptibility effect with $e^{\beta_{1}}=0.9$ and
$e^{\beta_{2}}=0.1$ so that vaccination reduces the rate of infection in a
treated individual by $10\%$ but reduces transmission to others by $90\%$. The
transmission dynamics for the baseline hazards are: 1) $\alpha(t)=0.3$ and
$\gamma(t)=3$ for the constant hazard scenario, 2) $\alpha(t)=0.3$ and
$\gamma(t)=6e^{-0.5(t-t_{j})}$ for the constant external hazard and decreasing
internal hazard scenario, and 3) $\alpha(t)=0.2\big{(}1+\sin(2\pi
t+\frac{\pi}{2})\big{)}$ and $\gamma(t)=6e^{-0.5(t-t_{j})}$ for the seasonal
external hazard and decreasing internal hazard scenario, so that the average
infection rate in the population is $5\%$-$20\%$ [58]. The observational
duration for clusters of $2$, $4$ and $8$ is approximately $4$ months, $2$
months, and $1$ month respectively from the beginning of a flu season,
assuming that a flu season typically lasts for 6 months.

From Tables 1–4, we find that the signs of the estimated exposure-marginalized
susceptibility effect $\widehat{SE}$ and infectiousness effect $\widehat{IE}$
match the signs of $\beta_{1}$ and $\beta_{2}$, and $\widehat{CE}>0$ confirms
that infection is contagious. The causal estimands $CE$, $SE$, and $IE$ are
difficult to identify non-parametrically by Theorem 1–2 and Proposition 4.
Additionally, these estimands are unidentifiable under block and cluster
randomization due to lack of outcomes under certain treatments hence a
violation of the positivity assumption. Although the positivity assumption is
violated (not all treatment assignments are possible), we show how to estimate
these estimands using the hazard model, and recover reasonable estimates for
block and cluster randomizations. We also see that $\widehat{CE}$,
$\widehat{SE}$, and $\widehat{IE}$ are summary measurements for certain
transmission process at a particular time and change with the cluster size
$n$, baseline hazard and randomization strategy, as shown in Figure 3. In
contrast, $\beta_{1}$ and $\beta_{2}$ are mechanistic features of the vaccine
effect that, under the specified hazard model, do not depend on the
randomization design, or other features of transmission or the observation
time. Thus, their estimates are relatively stable across all scenarios.

Cluster | Treatment | Hazard estimands | Probability estimands
---|---|---|---
$\hat{\beta}_{1}$ | $\hat{\beta}_{2}$ | $\hat{CE}(t,0,\mathbf{0},\mathbf{1})$ | $\hat{SE}(t,\mathbf{0})$ | $\hat{IE}(t,0,\mathbf{0})$ | $DE(t)$ | $IDE(t)$
Constant external and internal hazards |  |  |  |
2 | Obs. | -0.726 | -0.565 | 0.024 | -0.077 | -0.015 | -0.067 | -0.006
| Bernoulli | -0.722 | -0.713 | 0.023 | -0.075 | -0.018 | -0.068 | -0.032
| Block | -0.742 | -0.832 | 0.024 | -0.076 | -0.021 | -0.045 | -
| Cluster | -0.755 | -0.732 | 0.024 | -0.077 | -0.020 | -0.097 | -
4 | Obs. | -0.656 | -0.708 | 0.050 | -0.069 | -0.035 | -0.061 | -0.032
| Bernoulli | -0.682 | -0.724 | 0.052 | -0.073 | -0.034 | -0.062 | -0.051
| Block | -0.703 | -0.779 | 0.054 | -0.074 | -0.036 | -0.044 | -
| Cluster | -0.677 | -0.808 | 0.053 | -0.073 | -0.039 | -0.127 | -
8 | Obs. | -0.683 | -0.764 | 0.100 | -0.078 | -0.060 | -0.061 | 0.016
| Bernoulli | -0.678 | -0.710 | 0.098 | -0.074 | -0.053 | -0.061 | -0.074
| Block | -0.691 | -0.680 | 0.099 | -0.078 | -0.052 | -0.049 | -
| Cluster | -0.648 | -0.833 | 0.095 | -0.071 | -0.061 | -0.167 | -
Constant external hazard and time-varying internal hazard |  |  |  |
2 | Obs. | -0.650 | -0.688 | 0.029 | -0.072 | -0.022 | -0.068 | -0.047
| Bernoulli | -0.714 | -0.665 | 0.032 | -0.077 | -0.022 | -0.066 | -0.047
| Block | -0.575 | -0.685 | 0.025 | -0.063 | -0.021 | -0.026 | -
| Cluster | -0.721 | -0.630 | 0.032 | -0.078 | -0.021 | -0.104 | -
4 | Obs. | -0.704 | -0.662 | 0.082 | -0.080 | -0.039 | -0.070 | -0.058
| Bernoulli | -0.694 | -0.685 | 0.084 | -0.082 | -0.044 | -0.071 | -0.065
| Block | -0.721 | -0.622 | 0.088 | -0.086 | -0.042 | -0.044 | -
| Cluster | -0.700 | -0.706 | 0.086 | -0.084 | -0.046 | -0.161 | -
8 | Obs. | -0.669 | -0.722 | 0.136 | -0.065 | -0.046 | -0.070 | -0.233
| Bernoulli | -0.681 | -0.633 | 0.133 | -0.065 | -0.039 | -0.068 | -0.098
| Block | -0.703 | -0.706 | 0.138 | -0.067 | -0.043 | -0.044 | -
| Cluster | -0.731 | -0.611 | 0.141 | -0.071 | -0.038 | -0.223 | -
Time-varying external and internal hazards |  |  |  |
2 | Obs. | -0.650 | -0.688 | 0.029 | -0.072 | -0.022 | -0.068 | -0.047
| Bernoulli | -0.714 | -0.665 | 0.032 | -0.077 | -0.022 | -0.066 | -0.047
| Block | -0.575 | -0.685 | 0.025 | -0.063 | -0.021 | -0.026 | -
| Cluster | -0.721 | -0.630 | 0.032 | -0.078 | -0.021 | -0.104 | -
4 | Obs. | -0.704 | -0.662 | 0.082 | -0.080 | -0.039 | -0.070 | -0.058
| Bernoulli | -0.694 | -0.685 | 0.084 | -0.082 | -0.044 | -0.071 | -0.065
| Block | -0.721 | -0.622 | 0.088 | -0.086 | -0.042 | -0.044 | -
| Cluster | -0.700 | -0.706 | 0.086 | -0.084 | -0.046 | -0.161 | -
8 | Obs. | -0.669 | -0.722 | 0.136 | -0.065 | -0.046 | -0.070 | -0.233
| Bernoulli | -0.681 | -0.633 | 0.133 | -0.065 | -0.039 | -0.068 | -0.098
| Block | -0.703 | -0.706 | 0.138 | -0.067 | -0.043 | -0.044 | -
| Cluster | -0.731 | -0.611 | 0.141 | -0.071 | -0.038 | -0.223 | -
Table 1: Simulation results showing estimates of hazard coefficients, exposure-marginalized causal estimands, and direct and indirect effect under $e^{\beta_{1}}=0.5$ and $e^{\beta_{2}}=0.5$. Estimands are evaluated under three transmission scenarios – constant external and internal hazards of $\alpha(t)=0.3$ and $\gamma(t)=3$, constant external hazard and time-varying internal hazard of $\alpha(t)=0.3$ and $\gamma(t)=6e^{-0.5(t-t_{j})}$, and time-varying internal and external hazard of $\alpha(t)=0.2(1+\sin(2\pi t+\frac{\pi}{2}))$ and $\gamma(t)=6e^{-0.5(t-t_{j})}$. The individual covariates are correlated with $\rho=-0.1$ and have coefficients $e^{\theta_{1}}=e^{\theta_{2}}=0.9$. Clusters of $2$, $4$, and $8$ are observed at $0.4$ year, $0.3$ year, and $0.2$ year, respectively.  Cluster | Treatment | Hazard estimands | Probability estimands
---|---|---|---
$\hat{\beta}_{1}$ | $\hat{\beta}_{2}$ | $\hat{CE}(t,0,\mathbf{0},\mathbf{1})$ | $\hat{SE}(t,\mathbf{0})$ | $\hat{IE}(t,0,\mathbf{0})$ | $DE(t)$ | $IDE(t)$
Constant external and internal hazards |  |  |  |
2 | Obs. | -0.719 | -0.772 | 0.024 | -0.074 | -0.021 | -0.069 | -0.050
| Bernoulli | -0.722 | -0.713 | 0.023 | -0.075 | -0.018 | -0.068 | -0.032
| Block | -0.742 | -0.832 | 0.024 | -0.076 | -0.021 | -0.045 | -
| Cluster | -0.755 | -0.732 | 0.024 | -0.077 | -0.020 | -0.097 | -
4 | Obs. | -0.715 | -0.730 | 0.055 | -0.074 | -0.034 | -0.067 | -0.054
| Bernoulli | -0.682 | -0.724 | 0.052 | -0.073 | -0.034 | -0.062 | -0.051
| Block | -0.703 | -0.779 | 0.054 | -0.074 | -0.036 | -0.044 | -
| Cluster | -0.677 | -0.808 | 0.053 | -0.073 | -0.039 | -0.127 | -
8 | Obs. | -0.703 | -0.710 | 0.101 | -0.082 | -0.056 | -0.062 | -0.165
| Bernoulli | -0.678 | -0.710 | 0.098 | -0.074 | -0.053 | -0.061 | -0.074
| Block | -0.691 | -0.680 | 0.099 | -0.078 | -0.052 | -0.049 | -
| Cluster | -0.648 | -0.833 | 0.095 | -0.071 | -0.061 | -0.167 | -
Constant external hazard and time-varying internal hazard |  |  |  |
2 | Obs. | -0.596 | -0.552 | 0.025 | -0.064 | -0.017 | -0.055 | -0.014
| Bernoulli | -0.714 | -0.665 | 0.032 | -0.077 | -0.022 | -0.066 | -0.047
| Block | -0.575 | -0.685 | 0.025 | -0.063 | -0.021 | -0.026 | -
| Cluster | -0.721 | -0.630 | 0.032 | -0.078 | -0.021 | -0.104 | -
4 | Obs. | -0.706 | -0.739 | 0.087 | -0.084 | -0.046 | -0.072 | -0.061
| Bernoulli | -0.694 | -0.685 | 0.084 | -0.082 | -0.044 | -0.071 | -0.065
| Block | -0.721 | -0.622 | 0.088 | -0.086 | -0.042 | -0.044 | -
| Cluster | -0.700 | -0.706 | 0.086 | -0.084 | -0.046 | -0.161 | -
8 | Obs. | -0.686 | -0.715 | 0.138 | -0.063 | -0.043 | -0.065 | -0.153
| Bernoulli | -0.681 | -0.633 | 0.133 | -0.065 | -0.039 | -0.068 | -0.098
| Block | -0.703 | -0.706 | 0.138 | -0.067 | -0.043 | -0.044 | -
| Cluster | -0.731 | -0.611 | 0.141 | -0.071 | -0.038 | -0.223 | -
Time-varying external and internal hazards |  |  |  |
2 | Obs. | -0.596 | -0.552 | 0.025 | -0.064 | -0.017 | -0.055 | -0.014
| Bernoulli | -0.714 | -0.665 | 0.032 | -0.077 | -0.022 | -0.066 | -0.047
| Block | -0.575 | -0.685 | 0.025 | -0.063 | -0.021 | -0.026 | -
| Cluster | -0.721 | -0.630 | 0.032 | -0.078 | -0.021 | -0.104 | -
4 | Obs. | -0.706 | -0.739 | 0.087 | -0.084 | -0.046 | -0.072 | -0.061
| Bernoulli | -0.694 | -0.685 | 0.084 | -0.082 | -0.044 | -0.071 | -0.065
| Block | -0.721 | -0.622 | 0.088 | -0.086 | -0.042 | -0.044 | -
| Cluster | -0.700 | -0.706 | 0.086 | -0.084 | -0.046 | -0.161 | -
8 | Obs. | -0.686 | -0.715 | 0.138 | -0.063 | -0.043 | -0.065 | -0.153
| Bernoulli | -0.681 | -0.633 | 0.133 | -0.065 | -0.039 | -0.068 | -0.098
| Block | -0.703 | -0.706 | 0.138 | -0.067 | -0.043 | -0.044 | -
| Cluster | -0.731 | -0.611 | 0.141 | -0.071 | -0.038 | -0.223 | -
Table 2: Simulation results showing estimates of hazard coefficients, exposure-marginalized causal estimands, and direct and indirect effect under $e^{\beta_{1}}=0.5$ and $e^{\beta_{2}}=0.5$. Estimands are evaluated under three transmission scenarios – constant external and internal hazards of $\alpha(t)=0.3$ and $\gamma(t)=3$, constant external hazard and time-varying internal hazard of $\alpha(t)=0.3$ and $\gamma(t)=6e^{-0.5(t-t_{j})}$, and time-varying internal and external hazard of $\alpha(t)=0.2(1+\sin(2\pi t+\frac{\pi}{2}))$ and $\gamma(t)=6e^{-0.5(t-t_{j})}$. The individual covariates are correlated with $\rho=0.1$ and have coefficients $e^{\theta_{1}}=e^{\theta_{2}}=0.9$. Clusters of $2$, $4$, and $8$ are observed at $0.4$ year, $0.3$ year, and $0.2$ year, respectively.  Cluster | Treatment | Hazard estimands | Probability estimands
---|---|---|---
$\hat{\beta}_{1}$ | $\hat{\beta}_{2}$ | $\hat{CE}(t,0,\mathbf{0},\mathbf{1})$ | $\hat{SE}(t,\mathbf{0})$ | $\hat{IE}(t,0,\mathbf{0})$ | $DE(t)$ | $IDE(t)$
Constant external and internal hazards |  |  |  |
2 | Obs. | -0.721 | 0.029 | 0.024 | -0.077 | 0.001 | -0.068 | -0.002
| Bernoulli | -0.715 | -0.105 | 0.022 | -0.074 | -0.005 | -0.068 | -0.025
| Block | -0.742 | -0.203 | 0.024 | -0.076 | -0.008 | -0.052 | -
| Cluster | -0.755 | -0.028 | 0.024 | -0.077 | 0.000 | -0.091 | -
4 | Obs. | -0.671 | -0.039 | 0.048 | -0.070 | -0.002 | -0.066 | -0.016
| Bernoulli | -0.686 | -0.102 | 0.050 | -0.073 | -0.006 | -0.065 | -0.039
| Block | -0.700 | -0.159 | 0.052 | -0.074 | -0.008 | -0.051 | -
| Cluster | -0.677 | -0.195 | 0.051 | -0.073 | -0.011 | -0.115 | -
8 | Obs. | -0.691 | -0.161 | 0.093 | -0.080 | -0.014 | -0.066 | 0.038
| Bernoulli | -0.688 | -0.150 | 0.092 | -0.076 | -0.011 | -0.065 | -0.055
| Block | -0.690 | -0.098 | 0.091 | -0.078 | -0.007 | -0.056 | -
| Cluster | -0.648 | -0.216 | 0.088 | -0.071 | -0.015 | -0.149 | -
Constant external hazard and time-varying internal hazard |  |  |  |
2 | Obs. | -0.648 | -0.090 | 0.028 | -0.071 | -0.002 | -0.069 | -0.037
| Bernoulli | -0.722 | -0.083 | 0.032 | -0.078 | -0.003 | -0.067 | -0.037
| Block | -0.576 | -0.089 | 0.025 | -0.063 | -0.003 | -0.036 | -
| Cluster | -0.721 | -0.028 | 0.032 | -0.078 | -0.001 | -0.097 | -
4 | Obs. | -0.703 | -0.066 | 0.078 | -0.080 | -0.004 | -0.071 | -0.043
| Bernoulli | -0.686 | -0.123 | 0.081 | -0.081 | -0.008 | -0.071 | -0.051
| Block | -0.720 | -0.040 | 0.085 | -0.087 | -0.003 | -0.050 | -
| Cluster | -0.700 | -0.086 | 0.082 | -0.083 | -0.006 | -0.141 | -
8 | Obs. | -0.681 | -0.144 | 0.127 | -0.066 | -0.008 | -0.071 | -0.210
| Bernoulli | -0.681 | -0.059 | 0.123 | -0.065 | -0.003 | -0.067 | -0.078
| Block | -0.695 | -0.121 | 0.127 | -0.067 | -0.006 | -0.047 | -
| Cluster | -0.731 | -0.035 | 0.131 | -0.072 | -0.002 | -0.190 | -
Time-varying external and internal hazards |  |  |  |
2 | Obs. | -0.648 | -0.090 | 0.028 | -0.071 | -0.002 | -0.069 | -0.037
| Bernoulli | -0.722 | -0.083 | 0.032 | -0.078 | -0.003 | -0.067 | -0.037
| Block | -0.576 | -0.089 | 0.025 | -0.063 | -0.003 | -0.036 | -
| Cluster | -0.721 | -0.028 | 0.032 | -0.078 | -0.001 | -0.097 | -
4 | Obs. | -0.703 | -0.066 | 0.078 | -0.080 | -0.004 | -0.071 | -0.043
| Bernoulli | -0.686 | -0.123 | 0.081 | -0.081 | -0.008 | -0.071 | -0.051
| Block | -0.720 | -0.040 | 0.085 | -0.087 | -0.003 | -0.050 | -
| Cluster | -0.700 | -0.086 | 0.082 | -0.083 | -0.006 | -0.141 | -
8 | Obs. | -0.681 | -0.144 | 0.127 | -0.066 | -0.008 | -0.071 | -0.210
| Bernoulli | -0.681 | -0.059 | 0.123 | -0.065 | -0.003 | -0.067 | -0.078
| Block | -0.695 | -0.121 | 0.127 | -0.067 | -0.006 | -0.047 | -
| Cluster | -0.731 | -0.035 | 0.131 | -0.072 | -0.002 | -0.190 | -
Table 3: Simulation results showing estimates of hazard coefficients, exposure-marginalized causal estimands, and direct and indirect effect under $e^{\beta_{1}}=0.5$ and $e^{\beta_{2}}=0.9$. Estimands are evaluated under three transmission scenarios – constant external and internal hazards of $\alpha(t)=0.3$ and $\gamma(t)=3$, constant external hazard and time-varying internal hazard of $\alpha(t)=0.3$ and $\gamma(t)=6e^{-0.5(t-t_{j})}$, and time-varying internal and external hazard of $\alpha(t)=0.2(1+\sin(2\pi t+\frac{\pi}{2}))$ and $\gamma(t)=6e^{-0.5(t-t_{j})}$. The individual covariates are correlated with $\rho=-0.1$ and have coefficients $e^{\theta_{1}}=e^{\theta_{2}}=0.9$. Clusters of $2$, $4$, and $8$ are observed at $0.4$ year, $0.3$ year, and $0.2$ year, respectively.  Cluster | Treatment | Hazard estimands | Probability estimands
---|---|---|---
$\hat{\beta}_{1}$ | $\hat{\beta}_{2}$ | $\hat{CE}(t,0,\mathbf{0},\mathbf{1})$ | $\hat{SE}(t,\mathbf{0})$ | $\hat{IE}(t,0,\mathbf{0})$ | $DE(t)$ | $IDE(t)$
Constant external and internal hazards |  |  |  |
2 | Obs. | -0.122 | -2.481 | 0.006 | -0.016 | -0.038 | -0.016 | -0.056
| Bernoulli | -0.115 | -2.276 | 0.005 | -0.015 | -0.035 | -0.012 | -0.039
| Block | -0.113 | -2.265 | 0.005 | -0.015 | -0.036 | 0.021 | -
| Cluster | -0.139 | -2.477 | 0.006 | -0.016 | -0.038 | -0.054 | -
4 | Obs. | -0.111 | -2.412 | 0.023 | -0.013 | -0.083 | -0.011 | -0.067
| Bernoulli | -0.077 | -2.287 | 0.022 | -0.010 | -0.078 | -0.007 | -0.063
| Block | -0.084 | -2.326 | 0.023 | -0.011 | -0.080 | 0.019 | -
| Cluster | -0.083 | -2.377 | 0.024 | -0.011 | -0.082 | -0.097 | -
8 | Obs. | -0.094 | -2.242 | 0.065 | -0.013 | -0.128 | -0.008 | -0.183
| Bernoulli | -0.065 | -2.360 | 0.060 | -0.008 | -0.127 | -0.006 | -0.092
| Block | -0.120 | -2.319 | 0.066 | -0.014 | -0.129 | 0.009 | -
| Cluster | -0.050 | -2.554 | 0.060 | -0.005 | -0.133 | -0.151 | -
Constant external hazard and time-varying internal hazard |  |  |  |
2 | Obs. | 0.006 | -2.241 | 0.000 | 0.000 | -0.046 | 0.004 | -0.030
| Bernoulli | -0.097 | -2.268 | 0.006 | -0.011 | -0.050 | -0.008 | -0.058
| Block | -0.009 | -2.267 | 0.001 | -0.001 | -0.046 | 0.044 | -
| Cluster | -0.136 | -2.310 | 0.008 | -0.017 | -0.051 | -0.066 | -
4 | Obs. | -0.114 | -2.357 | 0.040 | -0.014 | -0.119 | -0.015 | -0.079
| Bernoulli | -0.109 | -2.319 | 0.042 | -0.016 | -0.118 | -0.015 | -0.088
| Block | -0.105 | -2.294 | 0.041 | -0.015 | -0.115 | 0.030 | -
| Cluster | -0.115 | -2.284 | 0.042 | -0.017 | -0.117 | -0.136 | -
8 | Obs. | -0.108 | -2.286 | 0.089 | -0.010 | -0.163 | -0.011 | -0.180
| Bernoulli | -0.074 | -2.260 | 0.082 | -0.007 | -0.160 | -0.009 | -0.126
| Block | -0.114 | -2.412 | 0.090 | -0.013 | -0.171 | 0.021 | -
| Cluster | -0.127 | -2.225 | 0.090 | -0.013 | -0.160 | -0.217 | -
Time-varying external and internal hazards |  |  |  |
2 | Obs. | 0.006 | -2.241 | 0.000 | 0.000 | -0.046 | 0.004 | -0.030
| Bernoulli | -0.097 | -2.268 | 0.006 | -0.011 | -0.050 | -0.008 | -0.058
| Block | -0.009 | -2.267 | 0.001 | -0.001 | -0.046 | 0.044 | -
| Cluster | -0.136 | -2.310 | 0.008 | -0.017 | -0.051 | -0.066 | -
4 | Obs. | -0.114 | -2.357 | 0.040 | -0.014 | -0.119 | -0.015 | -0.079
| Bernoulli | -0.109 | -2.319 | 0.042 | -0.016 | -0.118 | -0.015 | -0.088
| Block | -0.105 | -2.294 | 0.041 | -0.015 | -0.115 | 0.030 | -
| Cluster | -0.115 | -2.284 | 0.042 | -0.017 | -0.117 | -0.136 | -
8 | Obs. | -0.108 | -2.286 | 0.089 | -0.010 | -0.163 | -0.011 | -0.180
| Bernoulli | -0.074 | -2.260 | 0.082 | -0.007 | -0.160 | -0.009 | -0.126
| Block | -0.114 | -2.412 | 0.090 | -0.013 | -0.171 | 0.021 | -
| Cluster | -0.127 | -2.225 | 0.090 | -0.013 | -0.160 | -0.217 | -
Table 4: Simulation results showing estimates of hazard coefficients,
exposure-marginalized causal estimands, and direct and indirect effect under
$e^{\beta_{1}}=0.9$ and $e^{\beta_{2}}=0.1$. Estimands are evaluated under
three transmission scenarios – constant external and internal hazards of
$\alpha(t)=0.3$ and $\gamma(t)=3$, constant external hazard and time-varying
internal hazard of $\alpha(t)=0.3$ and $\gamma(t)=6e^{-0.5(t-t_{j})}$, and
time-varying internal and external hazard of $\alpha(t)=0.2(1+\sin(2\pi
t+\frac{\pi}{2}))$. The individual covariates are correlated with $\rho=0.1$
and have coefficients $e^{\theta_{1}}=e^{\theta_{2}}=0.9$. Clusters of $2$,
$4$, and $8$ are observed at $0.4$ year, $0.3$ year, and $0.2$ year,
respectively.

Figure 3: Illustration of $DE(t)$ under three randomization schemes and
$IDE(t)$, when the infectiousness effect is relatively strong, moderate and
weak compared to the susceptibility effect. All graphs are generated under
constant hazard scenario of $\alpha=0.3$, $\gamma=3$ in clusters of 2
individuals. Figures a), b) and c) show the $DE(t)$ over time when individuals
in clusters are under Bernoulli, Block, and Cluster randomization,
respectively. Figure d) shows the $IDE(t)$ over time in the two-stage
randomization scheme. $DE(t)$ performs poorly under block randomization,
giving an effect with sign opposite that of the true susceptibility effect
under one parameter setting. Related results are shown in Figures 4 and 5.

Figure 4: Illustration of $DE(t)$ under block randomization and $IDE(t)$ under
two-stage randomization in clusters of $2$, $4$, and $8$. Simulations are
generated under $\alpha(t)=0.3$, $\gamma(t)=3$, $e^{\beta_{2}}=0.1$ and
$e^{\beta_{1}}=0.9$, so that the infectiousness effect is stronger than
susceptibility effect of the vaccine. Figure a) shows $DE(t)$ may have an
opposite sign to $\beta_{1}<0$ under block randomization, and how the peak of
this directional bias shifts with cluster size. As cluster size increases,
each individual is generally exposed to greater number of infected neighbors
and thus the infection process is accelerated. Figure b) shows the $IDE(t)$
has the same sign as $\beta_{2}<0$, and how its peak shifts with cluster size.

###  6.2 Comparison to other estimands

Statisticians and epidemiologists have proposed a wide variety of estimands
summarizing the effect of interventions for infectious disease outcomes. In
this section, we evaluate the performance of two commonly used estimands –
direct effect ($DE$) and indirect effect ($IDE$) defined in the two-stage
cluster randomization design [27, 50, 13, 3, 51, 52] – and compare them to our
estimands under the same simulation scenarios. We also employ GEE regression
models to adjust for covariates in the estimation of $DE$ and $IDE$ in the
observational study setting.

The “attack risk” of an infectious disease for individuals with treatment $x$
is defined as $AR_{x}(t)=\mathbb{E}[Y_{i}(t)|X_{i}=x]$. A ratio of attack
risks under different treatments (sometimes called “relative cumulative
incidence”) has been used to measure the vaccine effect on susceptibility,
defined as $VE_{AR}(t)=1-AR_{1}(t)/AR_{0}(t)$ [59, 60]. A related estimand
defined on the difference scale is referred to as the “direct effect” [13],
$DE(t)=AR_{1}(t)-AR_{0}(t)$. Put simply, $DE(t)$ is the contrast of
expectations of infection outcomes between treated individuals versus
untreated individuals. In the Bernoulli setting, attack risks $AR_{x}(t)$
conditioning only on treatment $X_{i}$ implicitly marginalize over the
treatments assigned to other individuals within the same cluster. In the block
randomization setting, we compare outcomes of treated individuals to those of
untreated individuals in clusters where exactly half of individuals are
treated and untreated. In the cluster randomization setting, we compare the
outcome of individuals in treated clusters to those in control clusters. In
the observational setting, $DE(t)$ is estimated with GEE regression model
under the logit link function and exchangeable correlation structure to adjust
for covariates. In Tables 1–4, we show the estimated $DE(t)$ under three
randomization strategies and under an observational setting. Figure 3 (a)–(c)
show $DE(t)$ over time when the infectiousness effect is relatively strong,
moderate and weak compared to the susceptibility effect. Figure 4 (a) shows
how $DE(t)$ changes with cluster size under block randomization when the
infectiousness effect is much stronger than the susceptibility effect.

Halloran et al. [6, 24, 27] and Rhodes et al. [30] warned that $VE_{AR}(t)$
(or equivalently, $DE(t)$) may be a biased approximation to the susceptibility
effect due to differential exposure to infection between treated and untreated
individuals in clusters. This differential exposure to infection is due to
different distributions of neighbors’ infection times among treated and
untreated, which cannot always be controlled by randomization at baseline.
Morozova et al. [26], Eck et al. [21] and Cai et al. [22] proved similar
results.

Our simulation results also demonstrate the potential bias in $DE(t)$. In
particular, we show that under block randomization, $DE(t)$ can have a
different sign from that of $\hat{\beta_{1}}$ and $\hat{SE}(t,\mathbf{0})$ in
scenarios in Table 4 where the infectiousness effect is much stronger than the
susceptibility effect. Figures 3 (b) and 4 (a) show how this bias changes with
time and cluster size. We see that this bias can still be substantial in
clusters of larger size. To better illustrate the reasons behind for
misleading performance of $DE(t)$, we simulate outcomes under an intervention
whose true susceptibility effect is null with $\beta_{1}=0$ and whose true
infectiousness effect is negative (protective) with $\beta_{2}<0$, and
simulate an infectious disease with $\alpha(t)=0.3$ and $\gamma(t)=3$. In
Figure 5, we show the calculated $DE(t)$ over time for different cluster sizes
when treatment assignment for $X_{i}$ and $X_{j}$ ($i\neq j$) is positively
correlated and when treatment assignment for $X_{i}$ and $X_{j}$ ($i\neq j$)
is negatively correlated, achieved by making $\rho$ positive and negative
respectively in (16). We see that for positively correlated treatment
assignment, $DE(t)>0$ for all $t>0$ and all cluster sizes, and similarly for
negatively correlated treatment assignment, $DE(t)<0$ for all $t>0$ and all
cluster sizes. Theorem 2 of Cai et al. [22] provides a nonparametric proof of
this apparent paradox in the two-person partnership scenario.

Figure 5: Illustration of $DE(t)$ when treatment assignment is correlated
randomized in clusters of $2$, $4$, and $8$. Simulations are generated under
$\alpha(t)=0.3$, $\gamma(t)=3$, $e^{\beta_{1}}=1$ and $e^{\beta_{2}}=0.1$, so
that the susceptibility effect of the vaccine is null and the infectiousness
effect decreases the transmission ability by $90\%$. Figure a) shows $DE(t)$
is strictly positive, contrary to $\beta_{1}=0$ when treatment assignment is
negatively correlated with $\rho=-0.9$ for cluster size of 2, $\rho=-0.3$ for
cluster size of 4, and $\rho=-0.1$ for cluster size of 8. Figure b) shows the
$DE(t)$ is strictly negative, contrary to $\beta_{1}=0$ when treatment
assignment is positively correlated with $\rho=0.9$ for cluster size of 2,
$\rho=0.3$ for cluster size of 4, and $\rho=0.1$ for cluster size of 8.

To capture the indirect effect of others’ treatments on individual $i$,
researchers have defined indirect effects under a two-stage randomization
design [5, 6, 10, 11, 31, 13]. In the two-stage randomization design, clusters
are first randomized to receive either high or low coverage of vaccines, then
individuals in each cluster are randomized to receive vaccines according to
the coverage rate chosen for this cluster. In Tables 1–4, we estimate

$IDE(t)=\sum_{|\mathbf{x}_{(i)}|=\frac{n}{2}}\mathbb{E}[Y_{i}(t)|X_{i}=0,\mathbf{X}_{(i)}=\mathbf{x}_{(i)}]p(\mathbf{x}_{(i)})-\sum_{|\mathbf{x}_{(i)}|=0}\mathbb{E}[Y_{i}(t)|X_{i}=0,\mathbf{X}_{(i)}=\mathbf{x}_{(i)}]p(\mathbf{x}_{(i)})$

to compare the infection rates between untreated individuals in clusters of
$50\%$ coverage rate and untreated individuals in clusters of $0\%$ coverage
rate. In the observational setting, we apply the GEE model under the logit
link function and exchangeable correlation structure to estimate $IDE(t)$. The
$IDE(t)$ estimand preserves the sign of $\beta_{2}$, as shown in Tables 1-4.
In Figure 3 (d), we show $IDE(t)$ over time under the circumstances when the
infectiousness effect is relatively strong, moderate and weak compared to the
susceptibility effect. In Figure 4 (b), we show how $IDE(t)$ changes over
study time across different cluster sizes.

##  7 Discussion

Causal dependence between infection outcomes is a major obstacle to
identification of meaningful treatment effects in infectious disease
epidemiology [5, 10, 6, 11, 12, 13, 9, 7]. Inferential approaches that neglect
this dependence can result in misleading estimates of vaccine effects, even in
randomized trials [26, 21, 22]. In this paper, we have presented a framework
for estimating meaningful treatment effects under contagion by clarifying the
temporal relationship between infection outcomes in clusters. We describe a
new strategy for identifying average potential infection outcomes. The
approach proceeds by first identifying the average infection outcome under a
deterministic infection exposure history, then integrating this history with
respect to a distribution over others’ infections, in the situation where the
focal individual is not present. This work is a direct extension of prior work
by Cai et al. [22] and Ogburn and VanderWeele [20] to the general cluster
setting, with no restrictions on the direction of transmission or treatment
assignment.

We define effects that contrast potential infection outcomes when the
infection history is either held fixed at a deterministic value (controlled)
or integrated (marginalized). One advantage of this approach is that it
permits definition of causal estimands in which the infection history of
others (or its distribution) is held constant in a contrast of individual
treatments. In this way, the susceptibility effect of Definition 3 avoids the
problem of differential exposure to infection between treated and untreated
individuals which arises in the definition of the “direct effect” proposed by
Hudgens and Halloran [13]. Simulation results show that while crude contrasts
like $DE(t)$ may sometimes have the same sign as the true effect when the
susceptibility effect is different from zero, they may be biased under certain
randomization designs or when the infectiousness effect is strong.

In this work, we have drawn a connection between nonparametric causal
estimands [e.g. 20] for infectious disease intervention effects and
statistical models for learning about the dynamics of tranmission [34, 35, 36,
37]. Hazard models provide a natural and well studied class of structural
models for time-to-event data. The decomposition (1) and corresponding
independence assumptions provide a natural way of turning a complex sequence
of infection times into conditionally independent waiting times. These
assumptions can endow certain hazard model parameters, corresponding to
covariate effects, with causal interpretations. The hazard model respects the
logic of infection disease transmission: no transmission can occur before
infection, and multiple waiting times to transmission can elapse concurrently.
Beyond the causal interpretation of hazard ratios described in Section 5.3,
the fitted hazard models provide a convenient statistical framework in which
to learn about both exposure-controlled and exposure-marginalized estimands
via Theorems 1 and 2.

Despite the generality of the proposed framework, the approach has several
limitations. First, it relies on exact observation of infection times. In
reality, binary infection status outcomes may be measured either at regular
intervals or at follow-up. Statistical tools for interval censoring and/or
tied responses could be employed to fit hazard models of the kind proposed
here, but identifiability of model parameters could become more challenging.
However, some statistical estimands do not require observation of exact
infection times. In particular, Corollary 2 shows that under some
circumstances, meaningful causal contrasts can be computed by conditioning on
the full vector of treatments within the cluster, even when infection times
are unobserved. This suggest that randomized trials that identify average
outcomes under fully specified treatment allocations within clusters may
reveal causal effects of the kind studied here. Another limitation of this
work is that practical statistical estimation requires specification of a
hazard model with parametric or semiparametric structure. Lack of knowledge
about the true structure of the distribution of pairwise transmission waiting
times could result in bias. A potential mitigating factor is that
statisticians have access to a wide variety of models for time-to-event data.
Almost any of these could be usefully employed in (10) if the requisite
conditional independence assumptions are met.

Acknowledgements: This work was supported by National Institutes of Health
(NIH) grants 1DP2HD09179 and R01 AI116770. We are grateful to Peter M. Aronow,
Olga Morozova, Elizabeth Ogburn, and Virginia E. Pitzer.

## References

  * Organization [2009] World Health Organization.  _Global health risks: mortality and burden of disease attributable to selected major risks_.  World Health Organization, 2009.
  * for Disease Control and Prevention [2011] Centers for Disease Control and Prevention.  A CDC framework for preventing infectious disease: sustaining the essentials and innovating for the future.  _Centers for Disease Control and Prevention, Atlanta_ , 2011.
  * Halloran et al. [2010] M. Elizabeth Halloran, Ira M. Longini, Jr, and Claudio J. Struchiner.  _Design and Analysis of Vaccine Studies_.  Springer, 2010.
  * Longini et al. [1988] Ira M. Longini, Jr, James S Koopman, Michael Haber, and George A Cotsonis.  Statistical inference for infectious diseases: risk-specific household and community transmission parameters.  _American Journal of Epidemiology_ , 128(4):845–859, 1988.
  * Struchiner et al. [1990] Claudio J Struchiner, Mary Elizabeth Halloran, James M Robins, and Andrew Spielman.  The behaviour of common measures of association used to assess a vaccination programme under complex disease transmission patterns-a computer simulation study of malaria vaccines.  _International Journal of Epidemiology_ , 19(1):187–196, 1990.
  * Halloran et al. [1991] M. Elizabeth Halloran, Michael Haber, Ira M. Longini, Jr, and Claudio J. Struchiner.  Direct and indirect effects in vaccine efficacy and effectiveness.  _American Journal of Epidemiology_ , 133(4):323–331, 1991.
  * O’Hagan et al. [2014] Justin J. O’Hagan, Marc Lipsitch, and Miguel A. Hernán.  Estimating the per-exposure effect of infectious disease interventions.  _Epidemiology_ , 25(1):134–138, 2014.
  * Sobel [2006] Michael E Sobel.  What do randomized studies of housing mobility demonstrate?: Causal inference in the face of interference.  _Journal of the American Statistical Association_ , 101(476):1398–1407, 2006.
  * VanderWeele and Tchetgen Tchetgen [2011a] Tyler J. VanderWeele and Eric J. Tchetgen Tchetgen.  Bounding the infectiousness effect in vaccine trials.  _Epidemiology_ , 22(5):686–693, 2011a.
  * Halloran and Struchiner [1991] M. Elizabeth Halloran and Claudio J. Struchiner.  Study designs for dependent happenings.  _Epidemiology_ , 2(5):331–338, 1991.
  * Halloran and Struchiner [1995] M. Elizabeth Halloran and Claudio J. Struchiner.  Causal inference in infectious diseases.  _Epidemiology_ , 6(2):142–151, 1995.
  * Eisenberg et al. [2003] Joseph N.S. Eisenberg, Bryan L. Lewis, Travis C. Porco, Alan H. Hubbard, and John M. Colford, Jr.  Bias due to secondary transmission in estimation of attributable risk from intervention trials.  _Epidemiology_ , 14(4):442–450, 2003.
  * Hudgens and Halloran [2008] Michael G. Hudgens and M. Elizabeth Halloran.  Toward causal inference with interference.  _Journal of the American Statistical Association_ , 103(482):832–842, 2008.
  * Cox [1958] D.R. Cox.  _Planning of Experiments_.  John Wiley & Sons, New York, 1958.
  * Rosenbaum [2007] Paul R Rosenbaum.  Interference between units in randomized experiments.  _Journal of the American Statistical Association_ , 102(477):191–200, 2007.
  * VanderWeele [2010] Tyler J. VanderWeele.  Direct and indirect effects for neighborhood-based clustered and longitudinal data.  _Sociological Methods & Research_, 38(4):515–544, 2010.
  * Tchetgen Tchetgen and VanderWeele [2012] Eric J Tchetgen Tchetgen and Tyler J VanderWeele.  On causal inference in the presence of interference.  _Statistical Methods in Medical Research_ , 21(1):55–75, 2012.
  * Pitzer and Basta [2012] Virginia E. Pitzer and Nicole E. Basta.  Linking data and models: The importance of statistical analyses to inform models for the transmission dynamics of infections.  _Epidemiology_ , 23(4):520–522, 2012.
  * VanderWeele et al. [2012] Tyler J. VanderWeele, Eric J. Tchetgen Tchetgen, and M. Elizabeth Halloran.  Components of the indirect effect in vaccine trials: identification of contagion and infectiousness effects.  _Epidemiology_ , 23(5):751–761, 2012.
  * Ogburn and VanderWeele [2017] Elizabeth L. Ogburn and Tyler J. VanderWeele.  Vaccines, contagion, and social networks.  _The Annals of Applied Statistics_ , 11(2):919–948, 2017.
  * Eck et al. [2019] Daniel J. Eck, Olga Morozova, and Forrest W. Crawford.  Randomization for the susceptibility effect of an infectious disease intervention in a clustered study population.  _arXiv:1808.05593_ , 2019.
  * Cai et al. [2021] X Cai, W W Loh, and F W Crawford.  Identification of causal intervention effects under contagion.  _Journal of Causal Inference_ , 9:9–38, 2021.
  * Koopman et al. [1991] James S. Koopman, Ira M. Longini, Jr, John A. Jacquez, Carl P. Simon, David G. Ostrow, William R. Martin, and David M. Woodcock.  Assessing risk factors for transmission of infection.  _American Journal of Epidemiology_ , 133(12):1199–1209, 1991.
  * Halloran et al. [1994] M. Elizabeth Halloran, Ira M. Longini, Jr, Michael J. Haber, Claudio J. Struchiner, and Robert C. Brunet.  Exposure efficacy and change in contact rates in evaluating prophylactic hiv vaccines in the field.  _Statistics in Medicine_ , 13(4):357–377, 1994\.
  * Staples et al. [2015] Patrick C. Staples, Elizabeth L. Ogburn, and Jukka-Pekka Onnela.  Incorporating contact network structure in cluster randomized trials.  _Scientific Reports_ , 5:17581, 2015.
  * Morozova et al. [2018] Olga Morozova, Ted Cohen, and Forrest W. Crawford.  Risk ratios for contagious outcomes.  _Journal of The Royal Society Interface_ , 15:1–12, 2018\.
  * Halloran et al. [1997] M. Elizabeth Halloran, Claudio J. Struchiner, and Ira M. Longini, Jr.  Study designs for evaluating different efficacy and effectiveness aspects of vaccines.  _American Journal of Epidemiology_ , 146(10):789–803, 1997.
  * Longini et al. [1982] Ira M Longini, Jr, James S Koopman, Arnold S Monto, and John P Fox.  Estimating household and community transmission parameters for influenza.  _American Journal of Epidemiology_ , 115(5):736–751, 1982.
  * Koopman [2004] Jim Koopman.  Modeling infection transmission.  _Annual Reviews of Public Health_ , 25(1):303–326, 2004.
  * Rhodes et al. [1996] Philip H. Rhodes, M. Elizabeth Halloran, and Ira M. Longini Jr.  Counting process models for infectious disease data: Distinguishing exposure to infection from susceptibility.  _Journal of the Royal Statistical Society: Series B (Methodological)_ , 58(4):751–762, 1996.
  * Hudgens and Halloran [2006] Michael G Hudgens and M. Elizabeth Halloran.  Causal vaccine effects on binary postinfection outcomes.  _Journal of the American Statistical Association_ , 101(473):51–64, 2006.
  * Halloran and Hudgens [2012] M. Elizabeth Halloran and Michael G. Hudgens.  Causal inference for vaccine effects on infectiousness.  _The International Journal of Biostatistics_ , 8(2), 2012\.
  * Longini and Koopman [1982] Ira M. Longini, Jr and James S. Koopman.  Household and community transmission parameters from final distributions of infections in households.  _Biometrics_ , 38(1):115–126, 1982.
  * Kenah [2011] Eben Kenah.  Contact intervals, survival analysis of epidemic data, and estimation of $r_{0}$.  _Biostatistics_ , 12(3):548–566, 2011.
  * Kenah [2015] Eben Kenah.  Semiparametric relative-risk regression for infectious disease transmission data.  _Journal of the American Statistical Association_ , 110(509):313–325, 2015.
  * Kenah [2013] Eben Kenah.  Non-parametric survival analysis of infectious disease data.  _Journal of the Royal Statistical Society: Series B (Statistical Methodology)_ , 75(2):277–303, 2013.
  * Kenah et al. [2008] Eben Kenah, Marc Lipsitch, and James M. Robins.  Generation interval contraction and epidemic data analysis.  _Mathematical Biosciences_ , 213(1):71–79, 2008\.
  * Robins [1986] James Robins.  A new approach to causal inference in mortality studies with a sustained exposure period -application to control of the healthy worker survivor effect.  _Mathematical Modelling_ , 7(9):1393–1512, 1986\.
  * VanderWeele and Tchetgen Tchetgen [2011b] Tyler J. VanderWeele and Eric J. Tchetgen Tchetgen.  Effect partitioning under interference in two-stage randomized vaccine trials.  _Statistics & probability letters_, 81(7):861–869, 2011b.
  * Crawford et al. [2019] Forrest W. Crawford, O Morozova, A L Buchanan, and D Spiegelman.  Interpretation of the individual effect under treatment spillover.  _American Journal of Epidemiology_ , 188:1407–1409, 2019\.
  * Sävje et al. [2021] Fredrik Sävje, Peter M Aronow, and Michael G Hudgens.  Average treatment effects in the presence of unknown interference.  _The Annals of Statistics_ , 49(2):673–701, 2021\.
  * Cauchemez et al. [2004] Simon Cauchemez, F Carrat, C Viboud, AJ Valleron, and PY Boëlle.  A Bayesian MCMC approach to study transmission of influenza: application to household longitudinal data.  _Statistics in Medicine_ , 23(22):3469–3487, 2004\.
  * Crawford et al. [2020] Forrest W Crawford, Florian M Marx, Jon Zelner, and Ted Cohen.  Transmission modeling with regression adjustment for analyzing household-based studies of infectious disease: application to tuberculosis.  _Epidemiology_ , 31(2):238, 2020.
  * Kamenga et al. [1991] Munkolenkole Kamenga, Robert W Ryder, Muana Jingu, Nkashama Mbuyi, Lubamba Mbu, Frieda Behets, Christopher Brown, and William L Heyward.  Evidence of marked sexual behavior change associated with low HIV-1 seroconversion in 149 married couples with discordant HIV-1 serostatus: experience at an HIV counselling center in Zaire.  _AIDS_ , 5(1):61–67, 1991.
  * De Vincenzi [1994] Isabelle De Vincenzi.  A longitudinal study of human immunodeficiency virus transmission by heterosexual partners.  _New England Journal of Medicine_ , 331(6):341–346, 1994.
  * El-Bassel et al. [2005] Nabila El-Bassel, Susan S Witte, Louisa Gilbert, Elwin Wu, Mingway Chang, Jennifer Hill, and Peter Steinglass.  Long-term effects of an HIV/STI sexual risk reduction intervention for heterosexual couples.  _AIDS and Behavior_ , 9(1):1–13, 2005.
  * Gilbert et al. [2010] Louisa Gilbert, Nabila El-Bassel, Assel Terlikbayeva, Yelena Rozental, Mingway Chang, Anne Brisson, Elwin Wu, and Marat Bakpayev.  Couple-based HIV prevention for injecting drug users in Kazakhstan: a pilot intervention study.  _Journal of Prevention & Intervention in the Community_, 38(2):162–176, 2010.
  * McSherry et al. [1999] George D. McSherry, David E. Shapiro, Robert W. Coombs, Nuala McGrath, Lisa M. Frenkel, Paula Britto, Mary Culnane, and Rhoda S. Sperling.  The effects of zidovudine in the subset of infants infected with human immunodeficiency virus type-1 (Pediatric AIDS Clinical Trials Group Protocol 076).  _The Journal of Pediatrics_ , 134(6):717–724, 1999.
  * Brogly et al. [2010] Susan B. Brogly, Mark J. Abzug, D. Heather Watts, Coleen K. Cunningham, Paige L. Williams, James Oleske, Daniel Conway, Rhoda S. Sperling, Hans Spiegel, and Russell B. Van Dyke.  Birth defects among children born to HIV-infected women: Pediatric AIDS Clinical Trials Protocols 219 and 219C.  _The Pediatric Infectious Disease Journal_ , 29(8):721–727, 2010.
  * Halloran et al. [1999] M. Elizabeth Halloran, Ira M Longini, Jr, and Claudio J Struchiner.  Design and interpretation of vaccine field studies.  _Epidemiological Reviews_ , 21(1):73–88, 1999\.
  * Perez-Heydrich et al. [2014] Carolina Perez-Heydrich, Michael G. Hudgens, M. Elizabeth Halloran, John D. Clemens, Mohammad Ali, and Michael E. Emch.  Assessing effects of cholera vaccination in the presence of interference.  _Biometrics_ , 70(3):731–741, 2014.
  * Halloran and Hudgens [2016] M. Elizabeth Halloran and Michael G. Hudgens.  Dependent happenings: a recent methodological review.  _Current Epidemiology Reports_ , 3:297–305, 2016.
  * Ali et al. [2005] Mohammad Ali, Michael Emch, Lorenz Von Seidlein, Mohammad Yunus, David A Sack, Malla Rao, Jan Holmgren, and John D Clemens.  Herd immunity conferred by killed oral cholera vaccines in Bangladesh: a reanalysis.  _The Lancet_ , 366(9479):44–49, 2005.
  * Buchanan et al. [2018] Ashley L Buchanan, Sten H Vermund, Samuel R Friedman, and Donna Spiegelman.  Assessing individual and disseminated effects in network-randomized studies.  _American Journal of Epidemiology_ , 187(11):2449–2459, 2018.
  * Christakis and Fowler [2007] Nicholas A. Christakis and James H. Fowler.  The spread of obesity in a large social network over 32 years.  _New England Journal of Medicine_ , 357(4):370–379, 2007.
  * Christakis and Fowler [2008] Nicholas A. Christakis and James H. Fowler.  The collective dynamics of smoking in a large social network.  _New England Journal of Medicine_ , 358(21):2249–2258, 2008.
  * Cacioppo et al. [2009] John T Cacioppo, James H Fowler, and Nicholas A Christakis.  Alone in the crowd: the structure and spread of loneliness in a large social network.  _Journal of Personality and Social Psychology_ , 97(6):977, 2009.
  * for Disease Control and Prevention [2020] Centers for Disease Control and Prevention.  Vaccine effectiveness: How well do the flu vaccines work?, 2020.  URL https://www.cdc.gov/flu/vaccines-work/vaccineeffect.htm.
  * Greenwood and Yule [1915] Major Greenwood and G. Udny Yule.  The statistics of anti-typhoid and anti-cholera inoculations, and the interpretation of such statistics in general.  _Proceedings of the Royal Society of Medicine_ , 8(Sect Epidemiol State Med):113, 1915.
  * Francis Jr. [1955] Thomas Francis Jr.  Evaluation of the 1954 poliomyelitis vaccine field trial: Further studies of results determining the effectiveness of poliomyelitis vaccine (Salk) in preventing paralytic poliomyelitis.  _Journal of the American Medical Association_ , 158(14):1266–1270, 1955.

##  Appendix A Proofs

###### Proof of Lemma 1.

By Assumptions 1 and 6, we can define $f_{I_{i}^{0}}(s|x_{i},\mathbf{l}_{i})$
be the density of $I_{i}^{0}(x_{i})$ when $\mathbf{L}_{i}=\mathbf{l}_{i}$. Let
$F_{I_{i}^{0}}(s|x_{i},\mathbf{l}_{i})$ be the corresponding cumulative
distribution function, and by Assumption 6, we have
$0<F_{I_{i}^{0}}(s|x_{i},\mathbf{l}_{i})<1$ for all $s>0$, $x_{i}$, and
$\mathbf{l}_{i}$. Thus, we can write,

$\frac{f_{I_{i}^{0}}(s|x_{i},\mathbf{l}_{i})}{1-F_{I_{i}^{0}}(s|x_{i},\mathbf{l}_{i})}=-\frac{d}{ds}\log(1-F_{I_{i}^{0}}(s|x_{i},\mathbf{l}_{i})).$

Then rearranging, we have:

$\begin{split}F_{I_{i}^{0}}(s|x_{i},\mathbf{l}_{i})&=1-\exp\left[-\int_{0}^{s}\frac{f_{I_{i}^{0}}(u|x_{i},\mathbf{l}_{i})}{1-F_{I_{i}^{0}}(u|x_{i},\mathbf{l}_{i})}du\right]\\\
&=1-\exp\left[-\int_{0}^{s}\frac{f_{I_{i}^{0}}(u|x_{i},\mathbf{l}_{i})\prod_{j\neq
i}\left(1-F_{I_{j}^{0}}(u|x_{j},\mathbf{l}_{j})\right)}{\big{(}1-F_{I_{i}^{0}}(u|x_{i},\mathbf{l}_{i})\big{)}\prod_{j\neq
i}\left(1-F_{j}(u|x_{j},\mathbf{l}_{j})\right)}du\right]\\\
&=1-\exp\left[-\int_{0}^{s}\frac{p(I_{i}^{0}(x_{i})=u,I_{j}^{0}(x_{j})>u\text{
for all }j\neq
i|\mathbf{L}=\mathbf{l})}{\Pr(I_{i}^{0}(x_{i})>u,I_{j}^{0}(x_{j})>u\text{ for
all }j\neq i|\mathbf{L}=\mathbf{l})}du\right]\\\ \intertext{by Assumption
\ref{as:independence}}&=1-\exp\left[-\int_{0}^{s}\frac{p(I_{i}^{0}(x_{i})=u,I_{j}^{0}(x_{j})>u\text{
for all }j\neq
i|\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l})}{\Pr(I_{i}^{0}(x_{i})>u,I_{j}^{0}(x_{j})>u\text{
for all }j\neq i|\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l})}du\right]\\\
\intertext{Assumption
\ref{as:txexchangeability}}&=1-\exp\left[-\int_{0}^{s}\frac{p(I_{i}^{0}=u,I_{j}^{0}>u\text{
for all }j\neq
i|\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l})}{\Pr(I_{i}^{0}>u,I_{j}^{0}>u\text{
for all }j\neq i|\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l})}du\right]\\\
\intertext{Assumption
\ref{as:consistency}}&=1-\exp\Big{[}-\int_{0}^{s}\frac{p(T_{i}=u,T_{\varphi_{i}^{1}}>u,\ldots,T_{\varphi_{i}^{n-1}}>u|\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l})}{\Pr(T_{i}>u,T_{\varphi_{i}^{1}}>u,\ldots,T_{\varphi_{i}^{n-1}}>u|\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l})}du\Big{]}\\\
\intertext{by Definition \eqref{eq:Tdecomposition}}\end{split}$

For $k=2,\ldots,n-2$, by Assumptions 1 and 6, we can define
$f_{I_{i}^{k}}(s|\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{l})$ as the
density of $I_{i}^{k}\big{(}\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k})\big{)}$
when $\mathbf{L}=\mathbf{l}$. Let
$F_{I_{i}^{k}}(s|\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{l})$ be the
corresponding cumulative distribution function, and by Assumption 6, we have
$0<F_{I_{i}^{k}}(s|\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{l})<1$ for
all $s>0$, $\mathbf{x}$, $\mathbf{h}_{(i)}(t_{(i)}^{k})$, and $\mathbf{l}$.
Thus, we can write:

$\frac{f_{I_{i}^{k}}(s|\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{l})}{1-F_{I_{i}^{k}}(s|\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{l})}=-\frac{d}{ds}\log(1-F_{I_{i}^{k}}(s|\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k},\mathbf{l}))$

Then rearranging, we have:

$\begin{split}&F_{I_{i}^{k}}(s|\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{l})=1-\exp\left[-\int_{0}^{s}\frac{\scriptstyle
f_{I_{i}^{k}}(u|\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{l})}{\scriptstyle
1-F_{I_{i}^{k}}(u|\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{l})}du\right]\\\
&=1-\exp\left[-\int_{0}^{s}\frac{\scriptstyle
f_{I_{i}^{k}}(t|\mathbf{x},\mathbf{h}_{(i)}(t^{k}_{(i)}),\mathbf{l})\prod_{j=\varphi_{i}^{k+1}}^{\varphi_{i}^{n-1}}\left(1-F_{I_{j}^{k}}(t|\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{l})\right)}{\scriptstyle\left(1-F_{I_{i}^{k}}(t|\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{l}\right)\prod_{j=\varphi_{i}^{k+1}}^{\varphi_{i}^{n-1}}\left(1-F_{I_{j}^{k}}(t|\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{l})\right)}du\right]\\\
&=1-\exp\left[-\int_{0}^{s}\frac{\scriptstyle
p\left(I_{i}^{k}\big{(}\mathbf{x},\mathbf{h}_{(i)}(t^{k}_{(i)})\big{)}=u,I_{j}^{k}\big{(}\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k})\big{)}>u\text{
for
}j=\varphi_{i}^{k+1},\ldots,\varphi_{i}^{n-1}|\mathbf{L}=\mathbf{l}\right)}{\scriptstyle\Pr\left(I_{i}^{k}\big{(}\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k})\big{)}>u,I_{j}^{k}\big{(}\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k})\big{)}>u\text{
for
}j=\varphi_{i}^{k+1},\ldots,\varphi_{i}^{n-1}|\mathbf{L}=\mathbf{l}\right)}du\right]\\\
\intertext{by Assumption \ref{as:independence}}&=1-\exp\Bigg{[}\\\
&-\int_{0}^{s}\frac{\scriptstyle
p\left(I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)}(t^{k}_{(i)}))=u,I_{j}^{k}(\mathbf{x},\mathbf{h}_{(i)}(t^{k}_{(i)}))>u\text{
for
}j=\varphi^{k+1},\ldots,\varphi^{n-1}|\mathbfcal{H}_{(i)}(t^{k}_{(i)})=\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{L}=\mathbf{l}\right)}{\scriptstyle\Pr\left(I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)}(t^{k}_{(i)}))>u,I_{j}^{k}(\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k}))>u\text{
for
}j=\varphi^{k+1},\ldots,\varphi^{n-1}|\mathbfcal{H}_{(i)}(t_{(i)}^{k})=\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{L}=\mathbf{l}\right)}du\Bigg{]}\\\
\intertext{by Assumption
\ref{as:prevexchangeability}}&=1-\exp\Bigg{[}-\int_{0}^{s}\frac{\scriptstyle
p\left(I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)}(t^{k}_{(i)}))=u,I_{j}^{k}(\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k}))>u\text{
for
}j=\varphi^{k+1},\ldots,\varphi^{n-1}|\mathbfcal{H}_{(i)}(t_{(i)}^{k})=\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)}{\scriptstyle\Pr\left(I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)}(t^{k}_{(i)}))>u,I_{j}^{k}(\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k}))>u\text{
for
}j=\varphi^{k+1},\ldots,\varphi^{n-1}|\mathbfcal{H}_{(i)}(t_{(i)}^{k})=\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)}du\Bigg{]}\\\
\intertext{by Assumption
\ref{as:txexchangeability}}&=1-\exp\left[-\int_{0}^{s}\frac{\scriptstyle
p\left(I_{i}^{k}=u,I_{j}^{k}>u\text{ for
}j=\varphi^{k+1},\ldots,\varphi^{n-1}|\mathbfcal{H}_{(i)}(t_{(i)}^{k})=\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)}{\scriptstyle\Pr\left(I_{i}^{k}>u,I_{j}^{k}>u\text{
for
}j=\varphi^{k+1},\ldots,\varphi^{n-1}|\mathbfcal{H}_{(i)}(t_{(i)}^{k})=\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)}du\right]\\\
\intertext{by Assumption
\ref{as:consistency}}&=1-\exp\Big{[}-\int_{t_{(i)}^{k}}^{t_{(i)}^{k}+s}\frac{\scriptstyle
p(T_{i}=u,T_{\varphi_{i}^{k+1}}>u,\ldots,T_{\varphi_{i}^{n-1}}>u|T_{\varphi_{i}^{1}}=t_{(i)}^{1},\ldots,T_{\varphi_{i}^{k}}=t_{(i)}^{k},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l})}{\scriptstyle\Pr(T_{i}>u,T_{\varphi_{i}^{k+1}}>u,\ldots,T_{\varphi_{i}^{n-1}}>u|T_{\varphi_{i}^{1}}=t_{(i)}^{1},\ldots,T_{\varphi_{i}^{k}}=t_{(i)}^{k},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l})}du\Big{]}\\\
\intertext{by Definition \eqref{eq:Tdecomposition} and the definition of
$\mathbfcal{H}_{(i)}(t)$.}\end{split}$ (17)

There is no competing risk problem for
$I_{i}^{n-1}(s|\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{n-1}),\mathbf{l})$. Let
$F_{I_{i}^{n-1}}(s|\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{n-1}),\mathbf{l})$ be
the cumulative distribution function for
$I_{i}^{n-1}(\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{n-1}))$ when
$\mathbf{L}=\mathbf{L}$, then we have:

$\begin{split}&F_{I_{i}^{n-1}}(s|\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{n-1}),\mathbf{l})=\Pr(I_{i}^{n-1}(\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{n-1}))<s|\mathbf{L}=\mathbf{l})\\\
&=\Pr(I_{i}^{n-1}(\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{n-1}))<s|\mathbfcal{H}_{(i)}(t_{(i)}^{n-1})=\mathbf{h}_{(i)}(t_{(i)}^{n-1}),\mathbf{L}=\mathbf{l})\text{
by Assumption \ref{as:prevexchangeability}}\\\
&=\Pr(I_{i}^{n-1}(\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{n-1}))<s|\mathbfcal{H}_{(i)}(t_{(i)}^{n-1})=\mathbf{h}_{(i)}(t_{(i)}^{n-1}),\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l})\text{
by Assumption \ref{as:txexchangeability}}\\\
&=\Pr(I_{i}^{n-1}<s|\mathbfcal{H}_{(i)}(t_{(i)}^{n-1})=\mathbf{h}_{(i)}(t_{(i)}^{n-1}),\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l})\text{
by Assumption \ref{as:consistency}}\\\
&=\Pr(T_{i}<s+t_{i}^{(n-1)}|\mathbf{X}=\mathbf{x},T_{\varphi_{i}^{1}}=t_{(i)}^{1},\ldots,T_{\varphi_{i}^{n-1}}=t_{(i)}^{n-1},T_{i}>t_{(i)}^{n-1},\mathbf{L}=\mathbf{l})\\\
\intertext{by Definition \eqref{eq:Tdecomposition} and the definition of
$\mathbfcal{H}_{(i)}(t)$.}\end{split}$

∎

###### Proof of Theorem 1.

$T_{i}(\mathbf{x},\mathbf{h}_{(i)})$ is composed of
$I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k})$ by Definition (1). Since
$I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k}))\mathbin{\rotatebox[origin={c}]{90.0}{$\models$}}\mathbfcal{H}_{(i)}^{*}(t;\mathbf{x}_{(i)})|\mathbf{L}$
and
$I_{i}^{k}(\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{k}))\mathbin{\rotatebox[origin={c}]{90.0}{$\models$}}\mathbf{X}|\mathbf{L}$
by Assumptions 3 and 4, we have
$T_{i}(\mathbf{x},\mathbf{h}_{(i)})\mathbin{\rotatebox[origin={c}]{90.0}{$\models$}}\mathbfcal{H}_{(i)}^{*}(t;\mathbf{x}_{(i)})|\mathbf{L}$
and
$T_{i}(\mathbf{x},\mathbf{h}_{(i)})\mathbin{\rotatebox[origin={c}]{90.0}{$\models$}}\mathbf{X}|\mathbf{L}$.
Thus,

$\mathbb{E}[Y_{i}(t;\mathbf{x},\mathbf{h}_{(i)})|\mathbf{L}=\mathbf{l}]=\mathbb{E}[Y_{i}(t;\mathbf{x},\mathbf{h}_{(i)})|\mathbfcal{H}_{(i)}^{*}=\mathbf{h}_{(i)},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}]$
(18)

Therefore,

$\begin{split}&\mathbb{E}[Y_{i}(t;\mathbf{x},\mathbf{h}_{(i)})|\mathbf{L}=\mathbf{l}]=\mathbb{E}[Y_{i}(t;\mathbf{x},\mathbf{h}_{(i)})|\mathbfcal{H}_{(i)}^{*}=\mathbf{h}_{(i)},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}]\\\
\intertext{ by Equation
\eqref{eq:exchangeabilityforT}}&=\Pr\left(T_{i}(\mathbf{x},\mathbf{h}_{(i)})<t|\mathbfcal{H}_{(i)}^{*}=\mathbf{h}_{(i)},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)\\\
\intertext{ by the definition of
$T_{i}(\mathbf{x},\mathbf{h}_{(i)})$}&=\sum_{j=0}^{n-1}\Pr\left(\scriptstyle
I_{i}^{k}\geq t_{(i)}^{k+1}-t_{(i)}^{k}\text{ for k=1,\ldots,j-1
},\mathbfcal{H}_{(i)}^{*}=\mathbf{h}_{(i)},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)\\\
&\quad\quad\quad\cdot\Pr\left(\scriptstyle
I_{i}^{j}<t_{(i)}^{j+1}-t_{(i)}^{j},I_{i}^{k}\geq
t_{(i)}^{k+1}-t_{(i)}^{k}\text{ for k=1,\ldots,j-1
}|\mathbfcal{H}_{(i)}^{*}=\mathbf{h}_{(i)},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)\\\
\intertext{by the Law of Total Probability with extra
conditioning}&=\sum_{j=0}^{n-1}\Pr\left(\scriptstyle
I_{i}^{j}(\mathbf{x},\mathbf{h}_{(i)})<t-t_{(i)}^{j}|I_{i}^{j}<t_{(i)}^{j+1}-t_{(i)}^{j},I_{i}^{k}\geq
t_{(i)}^{k+1}-t_{(i)}^{k}\text{ for k=1,\ldots,j-1
},\mathbfcal{H}_{(i)}^{*}=\mathbf{h}_{(i)},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)\\\
&\quad\quad\quad\cdot\Pr\left(\scriptstyle
I_{i}^{j}<t_{(i)}^{j+1}-t_{(i)}^{j},I_{i}^{k}\geq
t_{(i)}^{k+1}-t_{(i)}^{k}\text{ for k=1,\ldots,j-1
}|\mathbfcal{H}_{(i)}^{*}=\mathbf{h}_{(i)},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)\\\
\intertext{ by Definition
\eqref{eq:Tdecomposition}}&=\sum_{j=0}^{n-1}\Pr\Big{(}\scriptstyle
I_{i}^{j}(\mathbf{x},\mathbf{h}_{(i)})<t-t_{(i)}^{j}|I_{i}^{j}<t_{(i)}^{j+1}-t_{(i)}^{j},I_{i}^{k}\geq
t_{(i)}^{k+1}-t_{(i)}^{k}\text{ for k=1,\ldots,j-1
},\mathbfcal{H}_{(i)}^{*}=\mathbf{h}_{(i)},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\Big{)}\\\
\end{split}$ $\begin{split}&\quad\quad\quad\cdot\Pr\left(\scriptstyle
I_{i}^{j}<t_{(i)}^{j+1}-t_{(i)}^{j}|I_{i}^{k}\geq
t_{(i)}^{k+1}-t_{(i)}^{k}\text{ for k=1,\ldots,j-1
}|\mathbfcal{H}_{(i)}^{*}=\mathbf{h}_{(i)},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)\\\
&\quad\quad\quad\cdot\prod_{k=1}^{j-1}\Pr\left(\scriptstyle I_{i}^{k}\geq
t_{(i)}^{k+1}-t_{(i)}^{k}|I_{i}^{m}\geq t_{(i)}^{m+1}-t_{(i)}^{m}\text{ for
all
$m=1,\ldots,k-1$}|\mathbfcal{H}_{(i)}^{*}=\mathbf{h}_{(i)},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)\\\
\intertext{by the Chain Rule}&=\sum_{j=0}^{n-1}\Pr\Big{(}\scriptstyle
I_{i}^{j}(\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{j}))<t-t_{(i)}^{j}|I_{i}^{j}<t_{(i)}^{j+1}-t_{(i)}^{j},I_{i}^{k}\geq
t_{i}^{(k+1)}-t_{i}^{(k)}\text{ for k=1,\ldots,j-1
},\mathbfcal{H}_{(i)}(t_{(i)}^{j})=\mathbf{h}_{(i)}(t_{(i)}^{j}),\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\Big{)}\\\
&\quad\quad\quad\cdot\Pr\left(\scriptstyle
I_{i}^{j}<t_{(i)}^{j+1}-t_{(i)}^{j}|I_{i}^{k}\geq
t_{(i)}^{k+1}-t_{(i)}^{k}\text{ for k=1,\ldots,j-1
}|\mathbfcal{H}_{(i)}(t_{(i)}^{j})=\mathbf{h}_{(i)}(t_{(i)}^{j}),\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)\\\
&\quad\quad\quad\cdot\prod_{k=1}^{j-1}\Pr\left(\scriptstyle I_{i}^{k}\geq
t_{(i)}^{k+1}-t_{(i)}^{k}|I_{i}^{m}\geq t_{(i)}^{m+1}-t_{(i)}^{m}\text{ for
$m=1,\ldots,k-1$}|\mathbfcal{H}_{(i)}(t_{(i)}^{k})=\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)\\\
\intertext{by Assumption \ref{as:exclusion} and the equivalence of
$\mathbfcal{H}_{(i)}(t_{(i)}^{j})$ to $\mathbfcal{H}_{(i)}^{*}(t_{(i)}^{j})$
in distribution given
$T_{i}>t_{(i)}^{j}$}&=\sum_{j=0}^{n-1}\Pr\left(I_{i}^{j}(\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{j}))<t-t_{(i)}^{j}|\mathbfcal{H}_{(i)}(t_{(i)}^{j})=\mathbf{h}_{(i)}(t_{(i)}^{j}),\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)\\\
&\quad\quad\quad\cdot\Pr\left(I_{i}^{j}<t_{(i)}^{j+1}-t_{(i)}^{j}|\mathbfcal{H}_{(i)}(t_{(i)}^{j})=\mathbf{h}_{(i)}(t_{(i)}^{j}),\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)\\\
&\quad\quad\quad\cdot\prod_{k=1}^{j-1}\Pr\left(I_{i}^{k}\geq
t_{(i)}^{k+1}-t_{(i)}^{k}|\mathbfcal{H}_{(i)}(t_{(i)}^{k})=\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)\\\
\intertext{ by Assumption
\ref{as:independence}}&={\sum}_{j=0}^{n-1}\Pr\left(I_{i}^{j}<t-t_{(i)}^{j}|\mathbfcal{H}_{(i)}(t_{(i)}^{j})=\mathbf{h}_{(i)}(t_{(i)}^{j}),\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)\\\
&\quad\quad\quad\cdot\Pr\left(I_{i}^{j}<t_{(i)}^{j+1}-t_{(i)}^{j}|\mathbfcal{H}_{(i)}(t_{(i)}^{j})=\mathbf{h}_{(i)}(t_{(i)}^{j}),\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)\\\
&\quad\quad\quad\cdot\prod_{k=1}^{j-1}\Pr\left(I_{i}^{k}\geq
t_{(i)}^{k+1}-t_{(i)}^{k}|\mathbfcal{H}_{(i)}(t_{(i)}^{k})=\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)\\\
\intertext{ by Assumption
\ref{as:consistency}}&={\sum}_{j=0}^{n-1}\Pr\left(I_{i}^{j}<\min\\{t-t_{(i)}^{j},t_{(i)}^{j+1}-t_{(i)}^{j}\\}|\mathbfcal{H}_{(i)}(t_{(i)}^{j})=\mathbf{h}_{(i)}(t_{(i)}^{j}),\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)\\\
&\quad\quad\quad\cdot\prod_{k=1}^{j-1}\Pr\left(I_{i}^{k}\geq
t_{(i)}^{k+1}-t_{(i)}^{k}|\mathbfcal{H}_{(i)}(t_{(i)}^{k})=\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}\right)\\\
&={{\sum}}_{j=0}^{n-1}\left[F_{I_{i}^{j}}\left(\min\left\\{t,t_{(i)}^{j+1}\right\\}-t_{(i)}^{j}|\mathbf{x},\mathbf{h}_{(i)}(t_{(i)}^{j}),\mathbf{l}\right)\prod_{k=0}^{j-1}\left(1-F_{I_{i}^{k}}(t_{(i)}^{k+1}-t_{(i)}^{k}|x,\mathbf{h}_{(i)}(t_{(i)}^{k}),\mathbf{l})\right)\right]\end{split}$

∎

###### Proof of Corollary 1.