figurec

# Enhanced Itinerant Ferromagnetism in Hole-doped Transition Metal Oxides:
Beyond the Canonical Double Exchange Mechanism

Zhao Liu<EMAIL_ADDRESS>Department of Materials Science and Engineering,
Monash University, Victoria 3800, Australia  ARC Centre of Excellent in Future
Low-Energy Electronics Technologies, Monash University, Victoria 3800,
Australia  Nikhil V. Medhekar<EMAIL_ADDRESS>Department of
Materials Science and Engineering, Monash University, Victoria 3800, Australia
ARC Centre of Excellent in Future Low-Energy Electronics Technologies, Monash
University, Victoria 3800, Australia

###### Abstract

Here we demonstrate the occurrence of robust itinerant ferromagnetism in Mott-
Hubbard systems at both low and high doping concentrations. Specifically, we
study the effect of hole doping on the experimentally synthesized LaCrAsO via
first-principles calculations and observe that the parent G-type
antiferromagnetism vanishes quickly at low doping concentration ($x$ $\sim$
0.20) and the system becomes ferromagnetic metal due to the canonical double
exchange (CDE) mechanism. As $x$ continues to increase, the onsite energy
difference between Cr 3$d$ and As 4$p$ orbitals decreases and the system
transitions to a ferromagnetic negative charge-transfer energy metal.
Therefore, the itinerant ferromagnetism doesn’t terminate at intermediate $x$
as CDE mechanism usually predicts. Furthermore, our calculations reveal that
both nearest and next-nearest ferromagnetic exchange coupling strengths keep
growing with $x$, showing that ferromagnetism caused by negative charge-
transfer energy state is "stronger" than that of CDE picture. Our work not
only unveils an alternative mechanism of itinerant ferromagnetism, but also
has the potential to attract immediate interest among experimentalists.

Introduction.-One of the oldest but recurrent topics in condensed matter
physics is itinerant ferromagnetism (FM) Moriya1985 ; Belitz2005 ;
Katsnelson2008 ; Brando2016 ; Li2014 ; Deng2018 ; Bonilla2018 ; Sharpe2019 ;
Park2021 ; Bao2022 . Discovered since ancient time in materials such as
elemental iron and nickel, itinerant FM now plays a crucial role in various
technological applications, including modern-day data processing and storage
Prinz1998 . Several mechanisms have been proposed for itinerant FM, including
Stoner criterion Stoner1938 , Nagaoka’s theorem Nagaoka1966 ; Tasaki1989 ;
Bobrow2018 , flat-band model Mielke1991_1 ; Mielke1991_2 ; Tasaki1992 ;
Mielke1993 , multi-orbital Hubbard models Shen1989 ; Sakai2007 ; LiY2014 ,
canonical double exchange (CDE) Zener1951 ; Anderson1955 , etc. Among these
mechanisms, CDE is a prominent one in both mixed valence transition metal
oxides (such as magnetites Hu2002  and spinel ferrites Ramirez1997 ) and doped
Mott-Hubbard insulators Khomskii2014 . In the latter system, itinerant charge
carriers arise from cation’s broad $d$ bands while the anion’s $p$ orbitals
remain inactive. To maximize the kinetic energy, an antiferromagnetism
(AFM)-FM transition occurs at low doping concentration. In manganite
perovskites, the occurrence of itinerant FM is marked by a strong suppression
of resistivity by magnetic field during the FM-paramagnetism phase transition
Cheong1999 . This is now known as the colossal negative magnetoresistance, a
crucial concept in spintronics. Nevertheless, a transition from FM to AFM
always occurs at high doping concentration, limiting the application of
itinerant FM in spintronics.

This work introduces a mechanism for persistent itinerant FM that can survive
at high doping concentration in Mott-Hubbard systems. To set the stage, we
first discuss the prototypical Mott-Hubbard system: LaCrAsO, which has been
experimentally synthesized Park2013 . While LaCrAsO is metallic, several
indirect observations suggest that it is in close proximity to a Mott-Hubbard
insulator Park2013 . Firstly, the measured room-temperature electrical
resistivity $\rho$ $\sim$ 3.8 m$\Omega\cdot cm$, which corresponds to a
normalized mean free path $k_{F}l\sim hc/e^{2}\rho\sim$ 0.6 (where c = 8.98 Å
is the lattice constant along the c direction). The fact that $k_{F}l$ < 1
strongly indicates LaCrAsO is a bad metal. Secondly, a local magnetic momentum
of 1.57 $\mu_{B}$/Cr is reported at room temperature—the existence of local
magnetic momenta implies a strong correlation effect. Finally, the ground
state of LaCrAsO is G-type AFM, such a long-range order is the result of
superexchange mechanism. Taking all of these experimental results together,
even if LaCrAsO is not fully Mott-Hubbard insulating, it should not be far
away from it. Consequently, various theoretical studies have explored the
possibility of high-temperature superconductivity in electron doped LaCrAsO
Pizarro2017 ; Wang2017  and BaCr2As2 Edelmann2017 , aiming at constructing a
similar 3$d$ orbital filling to Fe2+ in the well-known superconductor LaFeAsO
Kamihara2008 .

Here we investigate the other type of charge doping: hole doping (its
concentration is labelled by $x$), where the driving force is to upgrade the
valence state of Cr from 2+ to 3+. As described earlier, the CDE mechanism can
induce an AFM-FM transition, but only at small $x$, and at large $x$ the
system often returns to AFM phase as the superexchange mechanism dominates. On
the other hand, it is known that the higher valence state Cr3+ can lead to
negative charge-transfer energy states which greatly enhance FM Khomskii1997 ;
Liu2023 . Based on these arguments, there will be a competition between
superexchange mechanism and negative charge-transfer energy states in heavily
hole-doped LaCrAsO, which makes it intriguing to explore the evolution of
magnetic ground state as well as the exchange coupling strength ($J$) with
respect to $x$.

Figure 1: (a) Perspective view of 2 $\times$ 2 $\times$ 1 (La1-xSrx)CrAsO
supercell, here $x$ = 0.125 is shown as an example. (b) Lattice constants a,
b, c (black color) and angles $\alpha$, $\beta$, $\gamma$ (blue color) of the
ground state structures at different $x$ values. (c) Relative energies of
three long-range magnetic orders at different $x$ values.

In this work, via first-principles calculations, we demonstrate that the
parent G-type AFM quickly disappears at small $x$ values, as predicted by the
CDE mechanism. However, contrary to CDE mechanism, the FM doesn’t vanish at an
intermediate $x$ value but persists up to $x$ = 1.00, violating the
superexchange mechanism. Additionally, our calculations reveal that the
nearest and next-nearest ferromagnetic exchange coupling strengths (labelled
as $J_{1}$ and $J_{2}$) continue to grow with increasing $x$, indicating a
stronger itinerant FM than that of the CDE mechanism. We attribute this
enhanced itinerant FM at large $x$ values to the formation of negative charge-
transfer energy states .

First-principles calculations.-The first-principles calculations were
performed using the Vienna ab-initio simulation package (VASP) within the
framework of density functional theory (DFT) VASP . For geometric optimization
and electronic property calculations, a plane-wave cutoff 600 eV was used. All
our calculations were converged within 10-5 eV for energy and 0.01 eV/Å for
Hellman-Feynman forces. For the LaCrAsO unitcell, the Brillouin zone
integration was carried out with 14 $\times$ 14 $\times$ 10 k-point sampling
for self-consistency. The majority of calculations were based on the non-
empirical, strongly constrained and appropriately normed (SCAN) functional
Sun2015 ; Sun2016 , which is a parameter-free functional and can treat charge,
spin and lattice degrees of freedom on equal footing. For the interlayer van
der Waals (vdW) interactions, the revised Vydrov-van Voorhis nonlocal
correlation functional ($r$VV10) was employed Peng2016 . It is found that SCAN
+ $r$VV10 shows a good performance on LaCrAsO by reproducing several key
experimental results (see Sec. A of Supplementary Materials SI  for details).
In addition, we also employed the hybrid HSE06 functional HSE2003 ; HSE2004
to correct the magnetic band structure.

The hole doping in LaCrAsO is achieved by partially replacing La with Sr in a
supercell. The ionic size of Sr2+ is almost identical to that of La2+, thus, a
true solid solution should be formed over a large $x$ range. This substitution
method has been widely used in hole-doped materials, such as cuprates
Bednorz1986 , iron-based pnictides Wen2008 , infinite-layer nickelate Li2019 ,
and magnanites Cheong1999 . Since charge/spin/orbital anomalies have been
reported at doping concentration $x$ = n/8 (n is an integer) for both square
lattice based cuprates Moodenbaugh1988 ; Tranquada1995  and perovskite based
magnanites Cheong1999 , a 2 $\times$ 2 $\times$ 1 supercell was adopted in
this work (see  Fig. 1(a)). By replacing n (n = 1–7) La atoms with Sr, $x$
value from 0.00 to 1.00 can be simulated. To account for any possible
distortion introduced by hole doping, both the supercell and atomic
coordinates were allowed to optimize freely. For all $x$ values, our
calculations show that (La1-xSrx)CrAsO alloy is energetically preferred (see
Sec. B of Supplementary Materials SI ). The optimized lattice constants and
angles of the ground states are presented in  Fig. 1(b), from which it is
clear that all the supercells have negligible distortions as the three angles
$\alpha$, $\beta$ and $\gamma$ are all close to 90∘. Such small distortions
can be traced back to the similar ionic radius of La3+ and Sr2+. To provide a
better description of the electronic structures at $x$ = 0.50, it was also
simulated in a unitcell.

Results and discussion.-First we explore the effect of hole doping on the
fundamental electronic structures and in particular, the charge-transfer
energy. The charge-transfer energy composes of two parts: single-particle part
$\varepsilon_{dp}$ which is the onsite energy difference between Cr 3$d$ and
As 4$p$ orbitals, and interacting part Eint which is a function of the
interaction parameters and electron fillings. As shown in  Fig. S3, as $x$
increases, the position of As 4$p$ orbitals gradually shift to higher energy
compared with Cr 3$d$, leading to a reduced $\varepsilon_{dp}$. Such a
phenomenon has also been reported in hole doped infinite-layer nickelate
recently Liu2021 . Additionally, the charge-transfer energy of an electron
from As 4$p$ to Cr 3$d$ orbitals includes the Hubbard repulsion of the
transferred electron with the $d$-electrons already present on the Cr ions.
Therefore, Eint is reduced with a decrease in the number of $d$-electrons,
following approximately Eint(Cr3+) = Eint(Cr2+) - Ud, where Ud is the average
Hubbard interaction of Cr 3$d$ orbitals Khomskii2014 . Taken together, with
high cationic valence states, charge-transfer energy is significantly reduced.

Figure 2:  (a,b) Orbital-resolved magnetic band structure of SrCrAsO for spin
up and spin down channel. The high symmetry k-path is
R-A-Z-$\Gamma$-X-M-$\Gamma$: (0.5, 0.5, 0.0)-(0.5, 0.5, 0.5)-(0.0, 0.0,
0.5)-(0.0, 0.0, 0.0)-(0.5, 0.0, 0.0)-(0.5, 0.5, 0.0)-(0.0, 0.0, 0.0). (c)
Magnetic band structure of (La0.5Sr0.5)CrAsO (in a unitcell) in the spin up
channel (black). Red: the Wannier fitted band structure. (d) Topview of the
four maximally localized Wannier functions downfolded from the grey color
shaded bands in (c). The (La0.5Sr0.5)O sublayer is omitted for a better view
and the isovalue is $\pm$ 0.60 Å-3/2.

To investigate the magnetic ground state, three long-range magnetic orders
were considered: FM with magnetic ordering momentum q = (0, 0, 0), checkboard
AFM (C-type AFM) with q = ($\pi$, $\pi$, 0) and strip AFM (S-type AFM) with q
= ($\pi$, 0, 0). The relative energies of these magnetic orders at different
$x$ values are shown in  Fig. 1(c). At $x$ = 0.00, C-type AFM is the ground
state, which is ascribed to the superexchange mechanism (see Sec. D of
Supplementary Materials SI  for more details). This magnetic order persists up
to $x\sim$ 0.20, beyond which the FM order becomes the lowest one. As $x$
further increases, the energy difference between FM and other AFM magnetic
order grows larger, in accordance with the CDE picture. However, it is
unexpected that the energy difference continues to grow without saturating
even at $x$ = 1.00. This behavior significantly deviates from the CDE picture
in the sense that there is no optimal doping concentration ($x_{c}$). In the
CDE mechanism, the gain from kinetic energy reaches the most at $x_{c}$ with
$T_{c}$ the highest. After $x_{c}$, the gain from exchange energy gradually
prevails and the superexchange mechanism should make SrCrAsO AFM (see Sec. D
of Supplementary Materials SI  for details). In order to explain this
unconventional doping behaviour, we now turn to the electronic structures.

Figure 3: (a) $J_{1}$ and $J_{2}$ at different $x$ values. (b) Phase
transition temperature at different $x$ values. The orange colored region near
$x$ = 0.20 is the critical point region. (c) Total density of states at $x$ >
0.20, red/blue represents spin dn/up channel respectively. (d) O 2$p$ orbital-
resolved magnetic band structure of SrCrAsO in spin up channel. (e) Evolution
of magnetic susceptibility with respect to temperature for different
Heisenberg model in SrCrAsO.

Fig. 2(a)-(b) presents the orbital- and spin-resolved FM band structures of
SrCrAsO ($x$ = 1.00). We observe that the spin up channel is metallic while
spin down channel is insulating, indicating that SrCrAsO is a half metal
(HSE06 functional also confirms this, see  Fig. S7). Moreover, there is a
significant disparity in orbital compositions for the two spin channels.
Specifically, in the spin down channel, the valence and conduction bands
around Fermi level (Ef) are contributed by As 4$p$/O 2$p$ and Cr 3$d$ orbitals
respectively. Nevertheless, in the spin up channel, there are states of
predominantly As 4$p$ character above Ef along Z-$\rm\Gamma$-X direction at
around 2.60 eV. We stress that these As 4$p$ states are not originated from
$d-p$ hybridization. In the case here, the 4$p$ states would be locating at Ef
even with the hybridization switching off, indicating the negative charge-
transfer energy nature Liu2023 . These negative charge-transfer energy states
make the magnetic molecular orbitals as the underlying building blocks rather
than the localized atomic orbitals, resulting in a large spreading of magnetic
orbitals. This, in turn, strongly enhances the FM exchange interaction, as
both $J_{1}$ and $J_{2}$ will become FM Liu2023 .

To visualize the magnetic molecular orbitals, we focus on the system at $x$ =
0.50 in an unitcell instead of $x$ = 1.00 for there is no local gap structure
around Ef in Fig. 2(a). In  Fig. 2(c), we observe four bands (grey color
shaded) are isolated from the others, which can be downfolded to obtain
maximally localized Wannier orbitals (MLWFs) Wanpac . The overall downfolding
is satisfactory as evidenced by the good agreement between the DFT and the
Wannier fitted bands (see  Fig. 2(c)).  Fig. 2(d) displays the four MLWFs, and
it is clear that they are composed of small CrAs clusters rather than
localized atomic orbitals. This feature can be reflected in the magnetic form
factor through inelastic neutron scattering, as observed in itinerant chiral
magnet MnSi recently Jin2023 . Similar to the FM CrAs monolayer, these MLWFs
are of anti-bonding type, so in principle, with further hole doping, the FM
phase should become stronger as the occupation of anti-bonding orbitals
decreases Liu2023 . This actually explains the anomaly we observe in  Fig.
1(c), where the FM order becomes increasingly stable as $x$ approaches 1.00.
Hence, at large $x$, (La1-xSrx)CrAsO becomes a FM negative charge-transfer
energy metal and it is noted that the itinerant FM here can’t be explained by
other exchange mechanisms (see Sec. D of Supplementary Materials SI  for
detailed elaboration).

After understanding the origin of FM in both small and large $x$ values, we
next evaluate the phase transition temperatures ($T_{N}/T_{c}$ for Neel/Curie
temperature respectively). By mapping the relative energies of FM, C-type AFM
and S-type AFM to the Heisenberg model with $S$ = 3/2 (see SI  for detailed
information), both $J_{1}$ and $J_{2}$ can be obtained as shown in  Fig. 3(a).
At $x$ = 0.00 and 0.125, $J_{1}$ is AFM (positive value) while $J_{2}$ is FM
(negative value), therefore C-type AFM is the ground state as confirmed by the
DFT calculations. At $x$ $\sim$ 0.20, $J_{1}$ changes its sign and becomes FM,
and then both $J_{1}$ and $J_{2}$ are FM. As $x$ increases, the magnitude of
both $J_{1}$ and $J_{2}$ increase without saturation. As mentioned before,
such a feature stems from the fact that the system is closer to "ideal
filling" as $x$ approaches 1.00. To determine $T_{N}/T_{c}$, classical Monte
Carlo (MC) simulations were preformed on a 40 $\times$ 40 $\times$ 1 supercell
based on Heisenberg Hamiltonian with $J_{1}$ and $J_{2}$ PASP . The phase
transition temperatures obtained are presented in  Fig. 3(b). Initially,
$T_{N}$ gradually decreases until a critical point region (labelled by orange
color). After that $T_{c}$ continuously increases to $\sim$ 920 K at $x$ =
1.00. The total density of states for $x$ > 0.20 are displayed in  Fig. 3(c).
It is evident that most of these FM phases are half metal, except for $x$ =
0.25. The half-metallic gap reaches its maximum value of $\sim$ 1.88 eV at $x$
= 0.375 and reduces to $\sim$ 0.80 eV at $x$ = 1.00. If HSE06 functional is
further considered, there is a $\sim$ +0.80 eV correction to the half-metallic
gap (see  Fig. S7). The high $T_{c}$ and the large half-metallic gap makes
(La1-xSrx)CrAsO ($x$ > 0.25) a promising candidate for half-metallic
ferromagnets.

In the above discussions, we concentrated solely on the intralayer coupling.
However, we found that at high $x$ values, not only As 4$p$, but also O 2$p$
orbitals are polarized. In  Fig. 3(d), we present the O 2$p$ orbital-resolved
magnetic band structure in the spin up channel, and it is obvious that O 2$p$
bands cross Ef. This means that these O 2$p$ states can be regarded as
electron or hole reservoirs between CrAs sublayers and promote a three-
dimensional (3D) magnet behavior. To investigate the 2D-3D crossover at $x$ =
1.00, we calculated the nearest out-of-plane exchange coupling ($J_{\perp}$)
to be -6.40 meV and simulated the magnetic susceptibility with/without
$J_{\perp}$, as plotted in  Fig. 3(e). The introduction of $J_{\perp}$ not
only pushes $T_{c}$ up to a higher temperature ($\sim$ 1080 K), but also gives
a sharper peak with smaller full width at half maximum, suggesting the 3D
nature of SrCrAsO.

Because As 4$p$ orbitals play a vital role in stabilizing the FM order, lastly
we discuss how to track them experimentally. The contribution of As 4$p$
orbitals can be identified in both energy space and real space. In energy
space, a "shoulder" or even a "peak" structure will occur at As $L_{2/3}$ edge
in the electron energy loss spectroscopy as $x$ increases, just like the case
in cuprates Romberg1990 . In real space, the valence charge density around the
As site should gradually reduce with increasing $x$, which can be observed by
synchrotron X-ray diffraction. We also note that similar observations have
been made for the ligand hole in cubic perovskite SrFeO3 in a recent study
Kitou2023 .

Conclusions.\- In summary, we propose an alternative phase diagram for doped
Mott-Hubbard system. As the concentration of hole doping increases, the
valence state of the metallic ions becomes higher and the charge-transfer gap
gradually turns to negative. The negative charge-transfer energy states result
in strong ferromagnetism, causing the overdoped system to be ferromagnetic
rather than antiferromagnetic described by superexchange mechanism.
Furthermore, the ferromagnetism at large doping concentration is so robust
that it is even stronger than that of canonical double exchange mechanism at
low doping. Based on this proposed phase diagram, we expect that a colossal
negative magnetoresistance will occur at large doping concentration, similar
to that observed in double exchange systems. Therefore, our work not only
facilitates the discovery of half-metallic ferromagnets but also expands the
search for materials with colossal negative magnetoresistance, which has
important practical applications.

We thank J.-W. Li, G. Su and X. Cui for useful discussions. Z. L. and N. V. M.
gratefully acknowledge the support from National Computing Infrastructure,
Pawsey Supercomputing Facility and the Australian Research Council’s Centre of
Excellence in Future Low-Energy Electronic Technologies (CE170100039).

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When $n=2$, Theorem 1 for individual $i=1$ becomes,

$\begin{split}&\mathbb{E}[Y_{1}(t;\mathbf{h}_{(1)},\mathbf{x})|\mathbf{L}=\mathbf{l}]=\mathbb{E}[Y_{1}(t;t_{2},\mathbf{x})|\mathbf{L}=\mathbf{l}]\\\
&=F_{I_{i}^{0}}(\min\\{t,t_{2}\\}\mid\mathbf{x},t_{2},\mathbf{l})+F_{I_{i}^{1}}(t-t_{2}\mid\mathbf{x},t_{2},\mathbf{l})\left(1-F_{I_{i}^{0}}(t_{2}\mid\mathbf{x},t_{2},\mathbf{l})\right)\text{by
Lemma \ref{Iidentification}}\\\ &=F_{I_{i}^{0}}(\min\\{t,t_{2}\\}\mid
x_{1},\mathbf{l}_{1})+F_{I_{i}^{1}}(t-t_{2}\mid\mathbf{x},t_{2},\mathbf{l})\left(1-F_{I_{i}^{0}}(t_{2}\mid
x_{1},\mathbf{l}_{1})\right)\text{by Assumption \ref{as:exclusion}}\\\
&=F_{I_{i}^{0}}(\min\\{t,t_{2}\\}\mid
x_{1},\mathbf{l}_{1})+\Pr[T_{1}<t|T_{2}=t_{2},T_{1}>t_{2},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}]\left(1-F_{I_{i}^{0}}(t_{2}\mid
x_{1},\mathbf{l}_{1})\right)\\\ \intertext{by Lemma
\ref{Iidentification}}&=F_{1}(\min\\{t,t_{2}\\}|x_{1},\mathbf{l}_{1})+\Pr[T_{1}<t|T_{2}=t_{2},T_{1}>t_{2},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}]\left(1-F_{1}(t_{2}|x_{1},\mathbf{l}_{1})\right)\\\
\end{split}$

Therefore, when $t>t_{2}$,

$\mathbb{E}[Y_{1}(t;\mathbf{h}_{(1)},\mathbf{x})|\mathbf{L}=\mathbf{l}]=F_{1}(t_{2}|x_{1},\mathbf{l}_{1})+\Pr[T_{1}<t|T_{2}=t_{2},T_{1}>t_{2},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}]\big{(}1-F_{1}(t_{2}|x_{1},\mathbf{l}_{1})\big{)}$

and when $t\leq t_{2}$,

$\mathbb{E}[Y_{i}(t;\mathbf{h}_{(i)},\mathbf{x})|\mathbf{L}=\mathbf{l}]=F_{i}(t|x_{i},\mathbf{l}_{i})$

∎

###### Proof of Theorem 2.

$\begin{split}&dG^{*}_{(i)}(\mathbf{h}_{(i)}|\mathbf{x}_{(i)},\mathbf{l}_{(i)})=p(\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})=\mathbf{h}_{(i)}|\mathbf{L}_{(i)}=\mathbf{l}_{(i)})\\\
&\quad=p(T_{\varphi_{i}^{1}}(\mathbf{h}^{i}_{(\varphi_{i}^{1})},\mathbf{x})=t^{1}_{(i)},\ldots,T_{\varphi_{i}^{n-1}}(\mathbf{h}^{i}_{(\varphi_{i}^{n-1})},\mathbf{x})=t^{n-1}_{(i)}|\mathbf{L}=\mathbf{l})\\\
\intertext{by the definitions of $\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})$
and $\mathbf{h}^{i}_{(j)}$, and Assumption
\ref{as:exclusion}}&\quad=p(I^{0}_{\varphi_{i}^{1}}(\mathbf{h}^{i}_{(\varphi_{i}^{1})},\mathbf{x})=t^{1}_{(i)},I^{0}_{\varphi_{i}^{2}}(\mathbf{h}^{i}_{(\varphi_{i}^{1})},\mathbf{x})>t^{1}_{(i)},\ldots,I^{0}_{\varphi_{i}^{n-1}}(\mathbf{h}^{i}_{(\varphi_{i}^{1})},\mathbf{x})>t^{1}_{(i)},\\\
&\quad\quad\quad
I^{1}_{\varphi_{i}^{2}}(\mathbf{h}^{i}_{(\varphi_{i}^{2})},\mathbf{x})=t^{2}_{(i)}-t^{1}_{(i)},I^{1}_{\varphi_{i}^{3}}(\mathbf{h}^{i}_{(\varphi_{i}^{3})},\mathbf{x})>t^{2}_{(i)}-t^{1}_{(i)},\ldots,I^{1}_{\varphi_{i}^{n-1}}(\mathbf{h}^{i}_{(\varphi_{i}^{n-1})},\mathbf{x})>t^{2}_{(i)}-t^{1}_{(i)},\\\
&\quad\quad\quad,\ldots,\\\ &\quad\quad\quad
I^{n-2}_{\varphi_{i}^{n-1}}(\mathbf{h}^{i}_{(\varphi_{i}^{n-1})},\mathbf{x})=t^{n-1}_{(i)}-t^{n-2}_{(i)}\mid\mathbf{L}_{(i)}=\mathbf{l}_{(i)})\\\
\intertext{by Definition
\eqref{eq:Tdecomposition}}&\quad=\prod_{j=1}^{n-1}\left[f_{I_{\varphi_{i}^{j}}^{j-1}}\big{(}t_{(i)}^{j}-t_{(i)}^{j-1}\mid\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}\mid\mathbf{x},\mathbf{h}^{j}_{(\varphi_{i}^{k})},\mathbf{l}\big{)}\right]\\\
\intertext{by Assumption \ref{as:independence}}\end{split}$

∎

###### Proof of Corollary 2.

$\begin{split}\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;\mathbf{x},\mathbf{h}_{(i)})|\mathbf{L}=\mathbf{l}]\text{
d}G^{*}_{(i)}(\mathbf{h}_{(i)}|\mathbf{x},\mathbf{l})\\\ \intertext{by
Definition
\ref{eq:yitx}}&=\int\mathbb{E}[Y_{i}(t;\mathbf{x},\mathbf{h}_{(i)})|\mathbfcal{H}_{(i)}^{*}=\mathbf{h}_{(i)},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}]\text{
d}G^{*}_{(i)}(\mathbf{h}_{(i)}|\mathbf{x},\mathbf{l})\\\ \intertext{by
Equation \eqref{eq:exchangeabilityforT} under Assumptions
\ref{as:prevexchangeability} --
\ref{as:txexchangeability}}&=\int\mathbb{E}[Y_{i}(t)\,|\,\mathbfcal{H}_{(i)}^{*}=\mathbf{h}_{(i)},\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}]\text{
d}G^{*}_{(i)}(\mathbf{h}_{(i)}|\mathbf{x},\mathbf{l})\\\ \intertext{by
Assumption
\ref{as:consistency}}&=\mathbb{E}[Y_{i}(t)|\mathbf{X}=\mathbf{x},\mathbf{L}=\mathbf{l}]\intertext{as
$G^{*}_{(i)}(\mathbf{h}_{(i)}|\mathbf{x},\mathbf{l})$ does not depend on
$Y_{i}(t)$}\end{split}$

∎

###### Proof of Proposition 2.

$\displaystyle\overline{Y}_{i}^{p}(t;x_{i}\,|\,\mathbf{L}=\mathbf{l})$
$\displaystyle=\sum_{x_{(i)}\in\mathcal{X}^{n-1}}\mathbb{E}\big{[}Y_{i}\big{(}t;x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})\big{)}\mid\mathbf{L}=\mathbf{l}\big{]}p(\mathbf{x}_{(i)}\,|\,X_{i}=x_{i},\mathbf{L}=\mathbf{l})$
by Assumption 8
$\displaystyle=\sum_{x_{(i)}\in\mathcal{X}^{n-1}}\mathbb{E}\big{[}Y_{i}\big{(}t;x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)})\big{)}\mid\mathbf{L}=\mathbf{l}\big{]}p(\mathbf{x}_{(i)}\,|\,\mathbf{L}=\mathbf{l})$
$\displaystyle\overline{Y}_{i}^{p}(t;x_{i},\mathbf{x}_{(i)}^{\prime}\,|\,\mathbf{L}=\mathbf{l})$
$\displaystyle=\sum_{x_{(i)}\in\mathcal{X}^{n-1}}\mathbb{E}\big{[}Y_{i}\big{(}t;x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)}^{\prime})\big{)}\mid\mathbf{L}=\mathbf{l}\big{]}p(\mathbf{x}_{(i)}\,|\,X_{i}=x_{i},\mathbf{L}=\mathbf{l})$
by Assumption 8
$\displaystyle=\sum_{x_{(i)}\in\mathcal{X}^{n-1}}\mathbb{E}\big{[}Y_{i}\big{(}t;x_{i},\mathbf{x}_{(i)},\mathbfcal{H}_{(i)}^{*}(\mathbf{x}_{(i)}^{\prime})\big{)}\mid\mathbf{L}=\mathbf{l}\big{]}p(\mathbf{x}_{(i)}\,|\,\mathbf{L}=\mathbf{l})$

∎

###### Proof of Theorem 3.

$\begin{split}&\mathbb{E}[Y_{i}(t;\mathbf{h}_{(i)},\mathbf{x})|\mathbf{L}=\mathbf{l}]\\\
&={{\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)},\mathbf{l}\right)\prod_{k=0}^{j-1}\left(1-F_{I_{i}^{k}}(t_{(i)}^{k+1}-t_{(i)}^{k})|\mathbf{x},\mathbf{h}_{(i)},\mathbf{l}\right)\right]\\\
\intertext{by Theorem
\ref{thm:exposure}}&=\Pr\big{(}I_{i}^{0}(x_{i})\leq\min\\{t,t_{(i)}^{1}\\}|\mathbf{L}=\mathbf{l}\big{)}\\\
&\quad+\Pr\big{(}I_{i}^{1}(x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}(t_{(i)}^{1}))\leq\min\\{t,t_{(i)}^{2}\\}-t_{(i)}^{1}|\mathbf{L}=\mathbf{l}\big{)}\cdot\Pr\big{(}I_{i}^{0}(x_{i})>t_{(i)}^{1}|\mathbf{L}=\mathbf{l}\big{)}\\\
&\quad+\Pr\big{(}I_{i}^{2}(x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}(t_{(i)}^{2}))\leq\min\\{t,t_{(i)}^{3}\\}-t_{(i)}^{2}|\mathbf{L}=\mathbf{l}\big{)}\cdot\Pr\big{(}I_{i}^{0}(x_{i})>t_{i}^{(1)}|\mathbf{L}=\mathbf{l}\big{)}\\\
&\quad\quad\quad\cdot\Pr\big{(}I_{i}^{1}(x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}(t_{(i)}^{1}))>t_{(i)}^{2}-t_{(i)}^{1}|\mathbf{L}=\mathbf{l}\big{)}\\\
&\quad+\ldots\\\
&\quad+\Pr\big{(}I_{i}^{n-1}(x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}(t_{(i)}^{n-1}))\leq
t-t_{(i)}^{n-1}|\mathbf{L}=\mathbf{l}\big{)}\\\
&\quad\quad\quad\cdot\prod_{k=0}^{n-2}\Pr\big{(}I_{i}^{k}(x_{i},\mathbf{x}_{(i)},\mathbf{h}_{(i)}(t^{k}_{(i)}))>t_{i}^{k+1}-t_{i}^{k}|\mathbf{L}=\mathbf{l}\big{)}\\\
&=1-e^{-\int_{0}^{\min\\{t,t_{(i)}^{1}\\}}\lambda_{0i}(s|x_{i},\mathbf{l}_{i})ds}\\\
&\quad+\Big{(}1-e^{-\int_{t_{(i)}^{1}}^{\min\\{t,t_{(i)}^{2}\\}}\lambda_{0i}(s|x_{i},\mathbf{l}_{i})+\lambda_{\varphi_{i}^{1}i}(s-t_{(i)}^{1}|x_{i},x_{(i)}^{1},\mathbf{l}_{i},\mathbf{l}_{\varphi_{i}^{1}})ds}\Big{)}\cdot
e^{-\int_{0}^{t_{(i)}^{1}}\lambda_{i0}(s|x_{i},\mathbf{l}_{i})ds}\\\
&\quad+\Big{(}1-e^{-\int_{t_{(i)}^{2}}^{\min\\{t,t_{(i)}^{3}\\}}\lambda_{i0}(s|x_{i},\mathbf{l}_{i})+\lambda_{\varphi_{i}^{1}i}(s-t_{(i)}^{1}|x_{i},x_{(i)}^{1},\mathbf{l}_{i},\mathbf{l}_{\varphi_{i}^{1}})+\lambda_{\varphi_{i}^{2}i}(s-t^{2}_{(i)}|x_{i},x_{(i)}^{2},\mathbf{l}_{i},\mathbf{l}_{\varphi_{i}^{2}})ds}\Big{)}\\\
&\quad\quad\quad\quad\cdot
e^{-\int_{0}^{t_{(i)}^{1}}\lambda_{i0}(s|x_{i},\mathbf{l}_{i})ds}\cdot
e^{-\int_{t_{(i)}^{1}}^{t_{(i)}^{2}}\lambda_{0i}(s|x_{i},\mathbf{l}_{i})+\lambda_{\varphi_{i}^{1}i}(s-t_{(i)}^{1}|x_{i},x_{(i)}^{1},\mathbf{l}_{i},\mathbf{l}_{\varphi_{i}^{1}})ds}\\\
&\quad+\ldots\\\
&\quad+\Big{(}1-e^{-\int_{t_{(i)}^{n-1}}^{t}\lambda_{i0}(s|x_{i},\mathbf{l}_{i})+\lambda_{\varphi_{i}^{1}i}(s-t_{(i)}^{1}|x_{i},x_{(i)}^{1},\mathbf{l}_{i},\mathbf{l}_{\varphi_{i}^{1}})+\ldots+\lambda_{\varphi_{i}^{n-1}i}(s-t{(i)}^{n-1}|x_{i},x_{(i)}^{n-1},\mathbf{l}_{i},\mathbf{l}_{\varphi_{i}^{n-1}})ds}\Big{)}\\\
&\quad\quad\quad\cdot
e^{-\int_{0}^{t_{(i)}^{1}}\lambda_{i0}(s|x_{i},\mathbf{l}_{i})ds}\cdot
e^{-\int_{t_{(i)}^{1}}^{t_{(i)}^{2}}\lambda_{0i}(s|x_{i},\mathbf{l}_{i})+\lambda_{\varphi_{i}^{1}i}(s-t_{(i)}^{1}|x_{i},x_{(i)}^{1},\mathbf{l}_{i},\mathbf{l}_{\varphi_{i}^{1}})ds}\\\
&\quad\quad\quad\cdot\ldots\cdot
e^{-\int_{t_{(i)}^{n-2}}^{t_{(i)}^{n-1}}\lambda_{i0}(s|x_{i},\mathbf{l}_{i})+\lambda_{\varphi_{i}^{1}i}(s-t_{(i)}^{1}|x_{i},x_{(i)}^{1},\mathbf{l}_{i},\mathbf{l}_{\varphi_{i}^{1}})+\ldots+\lambda_{\varphi_{i}^{n-2}i}(s-t_{(i)}^{n-2}|x_{i},x_{(i)}^{(n-2)},\mathbf{l}_{i},\mathbf{l}_{\varphi_{i}^{n-1}})ds}\\\
\intertext{by plugging in Equation
\eqref{eq:sumhazard_forI}}&=1-\exp^{-\int_{0}^{t}\lambda_{0i}(s|x_{i},\mathbf{l}_{i})\text{d}s-\sum_{j=1}^{n-1}\int_{t_{(i)}^{j}}^{t}\lambda_{\varphi_{i}^{j}i}(s-t_{(i)}^{j}|x_{i},x_{\varphi_{i}^{j}},\mathbf{l}_{i},\mathbf{l}_{\varphi_{i}^{j}})\text{d}s}\\\
&=1-\exp^{-\int_{0}^{t}\Big{(}\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})\Big{)}\text{d}s}\\\
\end{split}$

∎

###### Proof of Theorem 9.

$\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{]}\\\
\intertext{by Theorem
\ref{thm:exposure}}&\quad=f_{I_{\varphi_{i}^{1}}^{0}}\big{(}t_{(i)}^{1}\,\big{|}\,\mathbf{x},\mathbf{h}^{i}_{(\varphi_{i}^{1})},\mathbf{l}\big{)}\prod_{k=2}^{n-1}S_{I_{\varphi_{i}^{k}}^{0}}\big{(}t_{(i)}^{1}\,\big{|}\,\mathbf{x},\mathbf{h}^{j}_{(\varphi_{i}^{k})},\mathbf{l}\big{)}\\\
&\quad\quad\cdot
f_{I_{\varphi_{i}^{2}}^{1}}\big{(}t_{(i)}^{2}-t_{(i)}^{1}\,\big{|}\,\mathbf{x},\mathbf{h}^{i}_{(\varphi_{i}^{2})},\mathbf{l}\big{)}\prod_{k=3}^{n-1}S_{I_{\varphi_{i}^{k}}^{0}}\big{(}t_{(i)}^{2}-t_{(i)}^{1}\,\big{|}\,\mathbf{x},\mathbf{h}^{j}_{(\varphi_{i}^{k})},\mathbf{l}\big{)}\\\
&\quad\quad\cdot\ldots\cdot\\\ &\quad\quad\cdot
f_{I_{\varphi_{i}^{n-1}}^{n-2}}\big{(}t_{(i)}^{n-1}-t_{(i)}^{n-2}\,\big{|}\,\mathbf{x},\mathbf{h}^{i}_{(\varphi_{i}^{2})},\mathbf{l}\big{)}\\\
&\quad=\lambda_{\varphi_{i}^{1}}(t_{(i)}^{1}|\mathbf{x},\mathbf{h}^{i}_{(\varphi_{i}^{1})},\mathbf{l})\exp\left(-\int_{0}^{t_{(i)}^{1}}\lambda_{\varphi_{i}^{1}}(s|\mathbf{x},\mathbf{h}^{i}_{(\varphi_{i}^{1})},\mathbf{l})\text{d}s\right)\cdot\prod_{k=2}^{n-1}\exp\left(-\int_{0}^{t_{(i)}^{1}}\lambda_{\varphi_{i}^{k}}(s|\mathbf{x},\mathbf{h}^{i}_{(\varphi_{i}^{k})},\mathbf{l})\text{d}s\right)\\\
&\quad\quad\cdot\lambda_{\varphi_{i}^{2}}(t_{(i)}^{2}|\mathbf{x},\mathbf{h}^{i}_{(\varphi_{i}^{2})},\mathbf{l})\exp\left(-\int_{t_{(i)}^{1}}^{t_{(i)}^{2}}\lambda_{\varphi_{i}^{2}}(s|\mathbf{x},\mathbf{h}^{i}_{(\varphi_{i}^{2})},\mathbf{l})\text{d}s\right)\cdot\prod_{k=3}^{n-1}\exp\left(-\int_{t_{(i)}^{1}}^{t_{(i)}^{2}}\lambda_{\varphi_{i}^{k}}(s|\mathbf{x},\mathbf{h}^{i}_{(\varphi_{i}^{k})},\mathbf{l})\text{d}s\right)\\\
&\quad\quad\cdot\ldots\cdot\\\
&\quad\quad\cdot\lambda_{\varphi_{i}^{n-1}}(t_{(i)}^{n-1}|\mathbf{x},\mathbf{h}^{i}_{(\varphi_{i}^{1})},\mathbf{l})\exp\left(-\int_{t_{(i)}^{n-2}}^{t_{(i)}^{n-1}}\lambda_{\varphi_{i}^{n-1}}(s|\mathbf{x},\mathbf{h}^{i}_{(\varphi_{i}^{n-1})},\mathbf{l})\text{d}s\right)\\\
&\quad=\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{]}\\\
&\quad=\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{)}\\\
\intertext{by grouping terms by $\varphi_{i}^{k}$}\end{split}$

∎

######  Corollary 3.

Under hazard model (10), the exposure-controlled contagion effect
$CE_{i}(t,\mathbf{x},\mathbf{h}_{(i)},\mathbf{h}_{(i)}^{\prime})>0$ for
$\mathbf{h}_{(i)}\prec\mathbf{h}_{(i)}^{\prime}$ as long as $\gamma(t)>0$ for
any $t>0$.

###### Proof of Corollary 3.

$\begin{split}CE_{i}(t,\mathbf{x},\mathbf{h}_{(i)},\mathbf{h}_{(i)}^{\prime})&=\mathbb{E}[Y_{i}(t;\mathbf{x},\mathbf{h}_{(i)})-Y_{i}(t;\mathbf{x},\mathbf{h}_{(i)}^{\prime})]\\\
&=1-\exp\big{[}-\int_{0}^{t}\Big{(}\lambda_{0i}(s|x_{i},\mathbf{l}_{i})+\sum_{j\neq
i}y_{j}(s)\lambda_{ji}(s-t_{j}|x_{i},x_{j},\mathbf{l}_{i},\mathbf{l}_{j})\Big{)}\mathrm{d}s\big{]}\\\
&\quad-\Big{(}1-\exp\big{[}-\int_{0}^{t}\Big{(}\lambda_{0i}(s|x_{i},\mathbf{l}_{i})+\sum_{j\neq
i}y_{j}(s)\lambda_{ji}(s-t^{\prime}_{j}|x_{i},x_{j},\mathbf{l}_{i},\mathbf{l}_{j})\Big{)}\mathrm{d}s\big{]}\Big{)}\\\
&=\exp\big{[}-\int_{0}^{t}\Big{(}\lambda_{0i}(s|x_{i},\mathbf{l}_{i})\mathrm{d}s\big{]}\\\
&\quad\times\Big{(}e^{-\sum_{j\neq
i}\int_{0}^{t}y_{j}(s)\lambda_{ji}(s-t^{\prime}_{j}|x_{i},x_{j},\mathbf{l}_{i},\mathbf{l}_{j})\mathrm{d}s}-e^{-\sum_{j\neq
i}\int_{0}^{t}y_{j}(s)\lambda_{ji}(s-t_{j}|x_{i},x_{j},\mathbf{l}_{i},\mathbf{l}_{j})\mathrm{d}s}\Big{)}\end{split}$
(19)

For the term inside the round brackets, plugging in model (10), we have

$\begin{split}&\exp\Big{[}-{\sum}_{j\neq
i}\int_{0}^{t}y_{j}(s)\lambda_{ji}(s-t^{\prime}_{j}|x_{i},x_{j},\mathbf{l}_{i},\mathbf{l}_{j})\mathrm{d}s\Big{]}\\\
&=\exp\Big{[}-{\sum}_{j\neq
i}\int_{0}^{t}y_{j}(s)\gamma(s-t^{\prime}_{j})\exp(\beta_{1}x_{i}+\beta_{2}x_{j}+\theta_{1}\mathbf{l}_{i}+\theta_{2}\mathbf{l}_{j})\mathrm{d}s\Big{]}\text{
by \eqref{eq:haz}}\\\ &=\exp\Big{[}-{\sum}_{j\neq
i}\exp(\beta_{1}x_{i}+\beta_{2}x_{j}+\theta_{1}\mathbf{l}_{i}+\theta_{2}\mathbf{l}_{j})\cdot\int_{0}^{t-t^{\prime}_{j}}\gamma(s)\mathrm{d}s\Big{]}\\\
&>\exp\Big{[}-{\sum}_{j\neq
i}\exp(\beta_{1}x_{i}+\beta_{2}x_{j}+\theta_{1}\mathbf{l}_{i}+\theta_{2}\mathbf{l}_{j})\cdot\int_{0}^{t-t_{j}}\gamma(s)\mathrm{d}s\Big{]}\text{
since $t_{j}<t^{\prime}_{j}$ and $\gamma(t)>0$}\\\ &=\exp\Big{[}-{\sum}_{j\neq
i}\int_{0}^{t}y_{j}(s)\lambda_{ji}(s-t_{j}|x_{i},x_{j},\mathbf{l}_{i},\mathbf{l}_{j})\mathrm{d}s\Big{]}\\\
\end{split}$ (20)

Plugging the result of (20) in (19), we have
$CE_{i}(t,\mathbf{x},\mathbf{h}_{(i)},\mathbf{h}_{(i)}^{\prime})>0$.

Similar conclusion applies for the exposure-marginalized contagion effect, and
the proof is omitted here. ∎

######  Corollary 4.

Under hazard model (10), the exposure-controlled susceptibility effect
$SE_{i}(t,\mathbf{x}_{(i)},\mathbf{h}_{(i)})$ has the same sign as
$e^{\beta_{1}}$.

###### Proof of Corollary 4.

$\begin{split}&SE_{i}(t,\mathbf{x}_{(i)},\mathbf{h}_{(i)})=\mathbb{E}[Y_{i}(t;1,\mathbf{x}_{(i)},\mathbf{h}_{(i)})-Y_{i}(t;0,\mathbf{x}_{(i)},\mathbf{h}_{(i)})]\\\
&=1-\exp\big{[}-\int_{0}^{t}\Big{(}\lambda_{0i}(s|1,\mathbf{l}_{i})+\sum_{j\neq
i}y_{j}(s)\lambda_{ji}(s-t_{j}|1,x_{j},\mathbf{l}_{i},\mathbf{l}_{j})\Big{)}\mathrm{d}s\big{]}\\\
&\quad-\Big{(}1-\exp\big{[}-\int_{0}^{t}\Big{(}\lambda_{0i}(s|0,\mathbf{l}_{i})+\sum_{j\neq
i}y_{j}(s)\lambda_{ji}(s-t^{\prime}_{j}|0,x_{j},\mathbf{l}_{i},\mathbf{l}_{j})\Big{)}\mathrm{d}s\big{]}\Big{)}\\\
&=\exp\big{[}-\int_{0}^{t}\alpha(t)\exp(\theta_{1}\mathbf{l}_{i})+\sum_{j\neq
i}y_{j}(s)\gamma(t-t_{j})\exp(\beta_{2}x_{j}+\theta_{1}\mathbf{l}_{i}+\theta_{2}\mathbf{l}_{j})\mathrm{d}s\big{]}\\\
&\quad-\exp\big{[}-e^{\beta_{1}}\int_{0}^{t}\alpha(t)\exp(\theta_{1}\mathbf{l}_{i})+\sum_{j\neq
i}y_{j}(s)\gamma(t-t_{j})\exp(\beta_{2}x_{j}+\theta_{1}\mathbf{l}_{i}+\theta_{2}\mathbf{l}_{j})\mathrm{d}s\big{]}\\\
\end{split}$

Therefore, if $\beta_{1}<0$ thereby $e^{\beta_{1}}<1$, then
$SE_{i}(t,\mathbf{x}_{(i)},\mathbf{h}_{(i)})<0$; if $\beta_{1}=0$ thereby
$e^{\beta_{1}}=1$, then $SE_{i}(t,\mathbf{x}_{(i)},\mathbf{h}_{(i)})=0$; if
$\beta_{1}>0$ thereby $e^{\beta_{1}}>1$, then
$SE_{i}(t,\mathbf{x}_{(i)},\mathbf{h}_{(i)})>0$.

Similar conclusions apply for the exposure-marginalized susceptibility effect
and the exposure-and-treatment-marginalized susceptibility effect, and the
proofs are omitted here. ∎

######  Corollary 5.

Under hazard model (10), the exposure-controlled infectiousness effect
$IE_{i}(t,x_{i},\mathbf{h}_{(i)})$ has the same sign as $e^{\beta_{2}}$.

###### Proof of Corollary 5.

$\begin{split}&IE_{i}(t,x_{i},\mathbf{h}_{(i)})=\mathbb{E}[Y_{i}(t;x_{i},\mathbf{1},\mathbf{h}_{(i)})-Y_{i}(t;x_{i},\mathbf{0},\mathbf{h}_{(i)})]\\\
&=1-\exp\big{[}-\int_{0}^{t}\Big{(}\lambda_{0i}(s|x_{i},\mathbf{l}_{i})+\sum_{j\neq
i}y_{j}(s)\lambda_{ji}(s-t_{j}|x_{i},1,\mathbf{l}_{i},\mathbf{l}_{j})\Big{)}\mathrm{d}s\big{]}\\\
&\quad-\Big{(}1-\exp\big{[}-\int_{0}^{t}\Big{(}\lambda_{0i}(s|x_{i},\mathbf{l}_{i})+\sum_{j\neq
i}y_{j}(s)\lambda_{ji}(s-t^{\prime}_{j}|x_{i},0,\mathbf{l}_{i},\mathbf{l}_{j})\Big{)}\mathrm{d}s\big{]}\Big{)}\\\
&=\exp\big{[}-\int_{0}^{t}\alpha(t)\exp(\beta_{1}x_{i}+\theta_{1}\mathbf{l}_{i})+\sum_{j\neq
i}y_{j}(s)\gamma(t-t_{j})\exp(\beta_{1}x_{i}+\theta_{1}\mathbf{l}_{i}+\theta_{2}\mathbf{l}_{j})\mathrm{d}s\big{]}\\\
&\quad-\exp\big{[}-\int_{0}^{t}\alpha(t)\exp(\beta_{1}x_{i}+\theta_{1}\mathbf{l}_{i})+e^{\beta_{2}}\sum_{j\neq
i}y_{j}(s)\gamma(t-t_{j})\exp(\beta_{1}x_{i}+\theta_{1}\mathbf{l}_{i}+\theta_{2}\mathbf{l}_{j})\mathrm{d}s\big{]}\\\
\end{split}$

Therefore, if $\beta_{2}<0$ thereby $e^{\beta_{2}}<1$, then
$IE_{i}(t,x_{i},\mathbf{h}_{(i)})<0$; if $\beta_{2}=0$ thereby
$e^{\beta_{2}}=1$, then $IE_{i}(t,x_{i},\mathbf{h}_{(i)})=0$; if $\beta_{2}>0$
thereby $e^{\beta_{2}}>1$, then $IE_{i}(t,x_{i},\mathbf{h}_{(i)})>0$.

Similar conclusions apply for the exposure-marginalized infectiousness effect
and the exposure-and-treatment-marginalized infectiousness effect, and the
proofs are omitted here. ∎

###### Proof of Equations (12)–(14).

First, we prove the result in Equation (12).

$\begin{split}HSE^{C}(t,\mathbf{x}_{(i)},\mathbf{h}_{(i)},\mathbf{l})&=\frac{\lambda_{i}(t|x_{i}=1,\mathbf{x}_{(i)},\mathbf{h}_{(i)},\mathbf{l})}{\lambda_{i}(t|x_{i}=0,\mathbf{x}_{(i)},\mathbf{h}_{(i)},\mathbf{l})}\\\
&=\frac{e^{\beta_{1}+\theta_{1}\mathbf{l}_{i}}\left(\alpha(t)+\sum_{j\neq
i}y_{j}(t)\gamma(t-t_{j})e^{\beta_{2}X_{j}+\theta_{2}\mathbf{l}_{j}}\right)}{e^{\theta_{1}\mathbf{l}_{i}}\left(\alpha(t)+\sum_{j\neq
i}y_{j}(t)\gamma(t-t_{j})e^{\beta_{2}X_{j}+\theta_{2}\mathbf{l}_{j}}\right)}\\\
&=e^{\beta_{1}}\\\ \end{split}$

Second, we prove the result in Equation (13). For $\mathbf{h}_{(i,j)}$ with
$\mathbf{t}_{(i,j)}$, then

$\begin{split}&\lambda_{i}(t|x_{j}=1,\mathbf{x}_{(j)},h^{\prime}_{j},\mathbf{h}_{(i,j)},\mathbf{l})-\lambda_{i}(t|x_{j}=1,\mathbf{x}_{(j)},h_{j},\mathbf{h}_{(i,j)},\mathbf{l})\\\
&\quad=\lambda_{i}(t|x_{j}=1,\mathbf{x}_{(j)},h^{\prime}_{j},\mathbf{h}_{(i,j)},\mathbf{l})-\lambda_{i}(t|x_{j}=1,\mathbf{x}_{(j)},h_{j},\mathbf{h}_{(i,j)},\mathbf{l})\\\
&\quad=e^{\beta_{1}X_{i}+\theta_{1}\mathbf{l}_{i}}\Big{(}\alpha(t)+\sum_{k\neq
i,j}y_{k}(t)\gamma(t-t_{k})e^{\beta_{2}X_{k}+\theta_{2}\mathbf{l}_{k}}+y_{j}(t)\gamma(t-t^{\prime}_{j})e^{\beta_{2}+\theta_{2}\mathbf{l}_{j}}\Big{)}\\\
&\quad\quad-e^{\beta_{1}X_{i}+\theta_{1}\mathbf{l}_{i}}\Big{(}\alpha(t)+\sum_{k\neq
i,j}y_{k}(t)\gamma(t-t_{k})e^{\beta_{2}X_{k}+\theta_{2}\mathbf{l}_{k}}\Big{)}\\\
&\quad=e^{\beta_{1}X_{i}+\theta_{1}\mathbf{l}_{i}}y_{j}(t)\gamma(t-t^{\prime}_{j})e^{\beta_{2}+\theta_{2}\mathbf{l}_{j}}\\\
\intertext{Similarly,}&\lambda_{i}(t|x_{j}=0,\mathbf{x}_{(j)},h^{\prime}_{j},\mathbf{h}_{(i,j)},\mathbf{l})-\lambda_{i}(t|x_{j}=0,\mathbf{x}_{(j)},h_{j},\mathbf{h}_{(i,j)},\mathbf{l})=e^{\beta_{1}X_{i}+\theta_{1}\mathbf{l}_{i}}y_{j}(t)\gamma(t-t^{\prime}_{j})e^{\theta_{2}\mathbf{l}_{j}}\end{split}$

Thus,

$HIE^{C}(t;h_{j},h^{\prime}_{j},\mathbf{x}_{(j)},\mathbf{h}_{(i,j)})=\frac{e^{\beta_{1}X_{i}+\theta_{1}\mathbf{l}_{i}}\gamma(t-t^{\prime}_{j})e^{\beta_{2}+\theta_{2}\mathbf{l}_{j}}}{e^{\beta_{1}X_{i}+\theta_{1}\mathbf{l}_{i}}\gamma(t-t^{\prime}_{j})e^{\theta_{2}\mathbf{l}_{j}}}=e^{\beta_{2}}$

Third, we prove the result in Equation (14).

$\begin{split}&\int_{0}^{t}\big{[}\lambda_{i}(u|\mathbf{x}=\mathbf{0},h^{\prime}_{j},\mathbf{h}_{(i,j)},\mathbf{l})-\lambda_{i}(u;\mathbf{x}=\mathbf{0},h_{j},\mathbf{h}_{(i,j)},\mathbf{l})\big{]}du\\\
&=\int_{0}^{t}\Big{\\{}e^{\theta_{1}\mathbf{l}_{i}}\big{[}\alpha(u)+\sum_{k\neq
i,j}y_{k}(u)\gamma(u-t_{k})e^{\rho_{k}\mathbf{l}_{k}}+y_{j}(t)\gamma(u-t^{\prime}_{j})e^{\theta_{2}\mathbf{l}_{j}}]\\\
&\quad\quad\quad\quad-e^{\theta_{1}\mathbf{l}_{i}}\big{[}\alpha(u)+\sum_{k\neq
i,j}y_{k}(u)\gamma(u-t_{k})e^{\rho_{k}\mathbf{l}_{k}}]\Big{\\}}du\\\
&=\int_{0}^{t}e^{\theta_{1}\mathbf{l}_{i}}y_{j}(u)\gamma(u-t^{\prime}_{j})e^{\theta_{2}\mathbf{l}_{j}}du=e^{\theta_{1}\mathbf{l}_{i}+\theta_{2}\mathbf{l}_{j}}\cdot\int_{0}^{t}y_{j}(u)\gamma(u-t^{\prime}_{j})du\\\
\intertext{Similarly,}&\int_{0}^{t}\big{[}\lambda_{i}(u;\mathbf{x}=\mathbf{0},h^{\prime\prime}_{j},\mathbf{h}_{(i,j)})-\lambda_{i}(u;\mathbf{x}=\mathbf{0},h_{j},\mathbf{h}_{(i,j)})\big{]}du=e^{\theta_{1}\mathbf{l}_{i}+\theta_{2}\mathbf{l}_{j}}\int_{0}^{t}y_{j}(u)\gamma(u-t^{\prime\prime}_{j})du\end{split}$

Thus,

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

∎


$\displaystyle\bigg{[}\log\big{(}1-\frac{wx_{i^{\prime},j^{\prime},k^{\prime}}}{vx_{ijk}}\big{)}-\log\big{(}1-\frac{wq_{2}x_{i^{\prime},j^{\prime},k^{\prime}}}{vx_{ijk}}\big{)}-\log\big{(}1-\frac{wq_{1}x_{i^{\prime},j^{\prime},k^{\prime}}}{vx_{ijk}}\big{)}+\log\big{(}1-\frac{wq_{1}q_{2}x_{i^{\prime},j^{\prime},k^{\prime}}}{vx_{ijk}}\big{)}\bigg{]}.$
(7.13)

Since $\kappa_{r}=(q^{r}_{1}-1)(q^{r}_{2}-1)(q^{r}_{3}-1)$, the second term is

$\displaystyle\mathrm{(II)}=\sum_{r=1}^{\infty}\Big{(}\frac{w}{v}\Big{)}^{r}\frac{1}{r}\cdot\frac{1-K^{-r}_{1}}{q^{r}_{3}-1}\sum_{(i,j,k)\in\Lambda^{\prime}}x^{r}_{ijk}=\sum_{r=1}^{\infty}\Big{(}\frac{w}{vK_{1}}\Big{)}^{r}\frac{1}{r}\cdot\frac{K^{r}_{1}-1}{q^{r}_{3}-1}\sum_{(i,j,k)\in\Lambda^{\prime}}x^{r}_{ijk}.$
(7.14)

Using the identity (1.12), we see that Eq. (7.14) becomes

$\displaystyle\sum_{r=1}^{\infty}$
$\displaystyle\Big{(}\frac{w}{vK_{1}}\Big{)}^{r}\frac{1}{r}\bigg{[}\frac{K^{r}_{1}}{q^{r}_{3}-1}-\frac{1}{q^{r}_{3}-1}\bigg{]}\sum_{(i,j,k)\in\Lambda^{\prime}}x^{r}_{ijk}$
$\displaystyle=-\sum_{(i,j,k)\in\Lambda^{\prime}}\sum_{r=1}^{\infty}\frac{1}{r(1-q^{r}_{3})}\Big{(}\frac{w}{v}x_{ijk}\Big{)}^{r}+\sum_{(i,j,k)\in\Lambda^{\prime}}\sum_{r=1}^{\infty}\frac{1}{r(1-q^{r}_{3})}\Big{(}\frac{w}{vK_{1}}x_{ijk}\Big{)}^{r}$
$\displaystyle=\sum_{(i,j,k)\in\Lambda^{\prime}}\log\Big{(}\frac{w}{v}x_{ijk};q_{3}\Big{)}_{\infty}-\sum_{(i,j,k)\in\Lambda^{\prime}}\log\Big{(}\frac{w}{vK_{1}}x_{ijk};q_{3}\Big{)}_{\infty}.$
(7.15)

Similarly the third term becomes

$\displaystyle\mathrm{(III)}=\sum_{r=1}^{\infty}$
$\displaystyle\Big{(}\frac{w}{v}\Big{)}^{r}\frac{1}{r}\frac{1-K^{r}_{2}}{1-q^{r}_{3}}\sum_{(i,j,k)\in\Lambda}x^{-r}_{ijk}$
$\displaystyle=\sum_{(i,j,k)\in\Lambda}\sum_{r=1}^{\infty}\frac{1}{r(1-q^{r}_{3})}\Big{(}\frac{w}{vx_{ijk}}\Big{)}^{r}-\sum_{(i,j,k)\in\Lambda}\sum_{r=1}^{\infty}\frac{1}{r(1-q^{r}_{3})}\Big{(}\frac{wK_{2}}{vx_{ijk}}\Big{)}^{r}$
$\displaystyle=-\sum_{(i,j,k)\in\Lambda}\log\big{(}\frac{w}{vx_{ijk}};q_{3}\big{)}_{\infty}+\sum_{(i,j,k)\in\Lambda}\log\big{(}\frac{wK_{2}}{vx_{ijk}};q_{3}\big{)}_{\infty}.$
(7.16)

Now merging all the three terms, we obtain

$\displaystyle\tilde{A}^{K_{1}K_{2}}_{\Lambda\Lambda^{\prime}}(v,w)=$
$\displaystyle z_{\Lambda}(K_{1};v)z_{\Lambda^{\prime}}(K_{2};w)$
$\displaystyle\cdot\bigg{[}\prod_{(i^{\prime},j^{\prime},k^{\prime})\in\Lambda^{\prime}}\prod_{(i,j,k)\in\Lambda}\frac{\big{(}1-\frac{wx_{i^{\prime}j^{\prime}k^{\prime}}}{vx_{ijk}}\big{)}\big{(}1-\frac{wq_{1}q_{2}x_{i^{\prime},j^{\prime},k^{\prime}}}{vx_{ijk}}\big{)}}{\big{(}1-\frac{wq_{2}x_{i^{\prime},j^{\prime},k^{\prime}}}{vx_{ijk}}\big{)}\big{(}1-\frac{wq_{1}x_{i^{\prime},j^{\prime},k^{\prime}}}{vx_{ijk}}\big{)}}\bigg{]}\cdot\bigg{[}\prod_{(i,j,k)\in\Lambda^{\prime}}\frac{\Big{(}\frac{w}{v}x_{ijk};q_{3}\Big{)}_{\infty}}{\Big{(}\frac{w}{vK_{1}}x_{ijk};q_{3}\Big{)}_{\infty}}\bigg{]}$
$\displaystyle\cdot\bigg{[}\prod_{(i,j,k)\in\Lambda}\frac{\big{(}\frac{wK_{2}}{vx_{ijk}};q_{3}\big{)}_{\infty}}{\big{(}\frac{w}{vx_{ijk}};q_{3}\big{)}_{\infty}}\bigg{]}.$
(7.17)

To show the relation of
$\tilde{A}^{K_{1}K_{2}}_{\Lambda\Lambda^{\prime}}(v,w)$ and the Nekrasov
function, we show that under a certain appropriate limit the inverse of the
normalized two-point correlators
$\tilde{A}^{K_{1}K_{2}}_{\Lambda\Lambda^{\prime}}(v,w)$ reduces to the
Nekrasov function up to multiplicative constant141414Note that in the ordinary
case, the Nekrasov functions appear in the denominator of the two-point
correlators.. Using
$x_{ijk}=q^{i-1}_{1}q^{j-1}_{2}q^{k-1}_{3}=q^{i}_{1}q^{j}_{2}q^{k}_{3}$, we
see that under the conditions $K_{1}=K_{2}=q_{3}$ and $k=1$,

$\displaystyle\tilde{A}^{q_{3},q_{3}}_{\lambda\lambda^{\prime}}(v,w)^{-1}=$
$\displaystyle z_{\lambda}(q_{3};v)^{-1}z_{\lambda^{\prime}}(q_{3};w)^{-1}$
$\displaystyle\cdot\bigg{[}\prod_{(i^{\prime},j^{\prime})\in\lambda^{\prime}}\prod_{(i,j)\in\lambda}\frac{\big{(}1-\frac{w}{v}q_{2}q^{i^{\prime}-i}_{1}q^{j^{\prime}-j}_{2}\big{)}\big{(}1-\frac{w}{v}q_{1}q^{i^{\prime}-i}_{1}q^{j^{\prime}-j}_{2}\big{)}}{\big{(}1-\frac{w}{v}q^{i^{\prime}-i}_{1}q^{j^{\prime}-j}_{2}\big{)}\big{(}1-\frac{w}{v}q_{1}q_{2}q^{i^{\prime}-i}_{1}q^{j^{\prime}-j}_{2}\big{)}}\bigg{]}\cdot\bigg{[}\prod_{(i^{\prime},j^{\prime})\in\lambda^{\prime}}\big{(}1-\frac{w}{v}q^{i^{\prime}}_{1}q^{j^{\prime}}_{2}\big{)}\bigg{]}$
$\displaystyle\cdot\bigg{[}\prod_{(i,j)\in\lambda}\big{(}1-q_{1}q_{2}\frac{w}{vq^{i}_{1}q^{j}_{2}}\big{)}\bigg{]}.$
(7.18)

Comparing with the Nekrasov function (1.2), we see that

$\displaystyle\tilde{A}^{q_{3},q_{3}}_{\lambda\lambda^{\prime}}(v,w)^{-1}=z_{\lambda}(q_{3};v)^{-1}z_{\lambda^{\prime}}(q_{3};w)^{-1}\cdot
N_{\lambda\lambda^{\prime}}(w,v).$ (7.19)

This justifies that
$\tilde{A}^{K_{1}K_{2}}_{\Lambda\Lambda^{\prime}}(v,w)^{-1}$ can be regarded
as a generalized Nekrasov function.

It is an interesting challenge to understand the solution (7.17) as some kind
of the partition function of supersymmetric gauge theory and/or generalization
of Macdonald function [50] [36]. For example, it is tempting to relate it to
six dimensional gauge theory. However the direction $q_{3}$ appears as
preferred in (7.17) and it seems to suggest the existence of an additional
defect in the theory151515If we adopt a suggestion in [52] that the MacMahon
representation may be associated with a system of $D7$ and anti-$D7$ branes,
the defect would be five branes attached to it.. Though the MacMahon
representation is symmetric in $(q_{1},q_{2},q_{3})$, the horizontal Fock
representation breaks it by the commutation relation with
$\mathfrak{q}=q_{3}^{1/2}$. The appearance of the infinite product is another
issue against an interpretation in terms of the gauge theory.

Acknowledgements

We would like to thank H. Awata, A. Mironov, A. Morozov and Y. Zenkevich for
useful discussions. Our work is supported in part by Grants-in-Aid for
Scientific Research (# 18K03274) (H.K.). The work of P.C. is supported by the
MEXT Scholarship.

##  Appendix A Determine the proportional factor

Here we determine the proportional factor of (6.27) explicitly. From (4.18) we
get that

$\displaystyle\Xi_{\Lambda}(K;pv)=\Big{[}\frac{z_{\lambda}(K;pv)}{z_{\lambda}(K;v)}\Big{]}z_{\lambda}(K;v)\mathcal{M}^{[n]}(K)\tilde{\Phi}^{[n]}_{\Lambda}(pv)\Gamma_{n}(K;pv).$
(A.1)

Now we can write

$\tilde{\Phi}^{[n]}_{\Lambda}(pv)\Gamma_{n}(K;pv)=A_{\Lambda}^{(-)}(v;p)\tilde{\Phi}^{[n]}_{\Lambda}(v)A_{\Lambda}^{(+)}(v;p)\cdot
B_{n}^{(-)}(v,p)\Gamma_{n}(K;v)B_{n}^{(+)}(v,p),$ (A.2)

where

$\displaystyle
A_{\Lambda}^{(\pm)}(v;p)=\exp\bigg{[}\mp\sum_{r=1}^{\infty}\frac{(p^{\mp
r}-1)H_{\pm
r}}{\mathfrak{q}^{r}-\mathfrak{q}^{-r}}\sum_{k=1}^{n}\mathfrak{q}^{-r/2}(q^{k-1}_{3}v)^{\mp
r}\chi_{\mp r}^{(k)}(q_{1},q_{2})\bigg{]},$

and

$\displaystyle B_{n}^{(\pm)}(v,p)=\exp\bigg{(}\sum_{r=1}^{\infty}\frac{(p^{\mp
r}-1)H_{\pm r}}{\mathfrak{q}^{r}-\mathfrak{q}^{-r}}\frac{q^{\mp nr}_{3}-K^{\mp
r}}{\kappa_{r}}\mathfrak{q}^{-r/2}v^{\mp r}\bigg{)},$ (A.4)

where we have used the notation (6.14). By using (1.14), we get that

$\displaystyle
A_{\Lambda}^{(+)}(v;p)B_{n}^{(-)}(v,p)=\exp\big{(}\clubsuit\big{)}B_{n}^{(-)}(v,p)A_{\Lambda}^{(+)}(v;p),$

where

$\displaystyle\clubsuit=-\sum_{r=1}^{\infty}(p^{-r}-1)(p^{r}-1)\bigg{(}\frac{q^{nr}_{3}-K^{r}}{\kappa_{r}}\bigg{)}\frac{(1-q^{r}_{1})(1-q^{r}_{2})}{r}\bigg{(}\sum_{(i,j,k)\in\Lambda}x^{-r}_{ijk}-\frac{1-q^{-rn}_{3}}{\kappa_{r}}\bigg{)}.$

Next we would like to switch between $A_{\Lambda}^{(+)}(v;p)$ and
$\Gamma_{n}(K;v)$. We know the explicit expression of $\Gamma_{n}(K;v)$ from
Eq. (4.19). So by using (1.14) again, we can show that

$A_{\Lambda}^{(+)}(v;p)\Gamma_{n}(K;v)\\\
=\exp\big{(}\spadesuit\big{)}\Gamma_{n}(K;v)A_{\Lambda}^{(+)}(v;p)$

with

$\displaystyle\spadesuit=-\sum_{r=1}^{\infty}(p^{-r}-1)\bigg{(}\sum_{(i,j,k)\in\Lambda}x^{-r}_{ijk}-\frac{1-q^{-rn}_{3}}{\kappa_{r}}\bigg{)}\frac{q^{nr}_{3}-K^{r}}{\kappa_{r}}\frac{(1-q^{r}_{1})(1-q^{r}_{2})}{r}.$
(A.5)

In the last step we would like to switch between
$\tilde{\Phi}^{[n]}_{\Lambda}(v)$ and $B_{n}^{(-)}(v,p)$. Again since we know
the explicit expression of $\tilde{\Phi}^{[n]}_{\Lambda}(v)$ from Lemma 6.2,
and (1.14) implies

$\tilde{\Phi}^{[n]}_{\Lambda}(v)B_{n}^{(-)}(v,p)=\exp\big{(}\heartsuit\big{)}B_{n}^{(-)}(v,p)\tilde{\Phi}^{[n]}_{\Lambda}(v),$
(A.6)

where

$\displaystyle\heartsuit=-\sum_{r=1}^{\infty}\frac{q^{nr}_{3}-K^{r}}{\kappa_{r}}(p^{r}-1)\frac{(1-q^{r}_{1})(1-q^{r}_{2})}{r}\bigg{(}\sum_{(i,j,k)\in\Lambda}x^{-r}_{ijk}-\frac{1-q^{-rn}_{3}}{\kappa_{r}}\bigg{)}.$

Consequently,

$\displaystyle\tilde{\Phi}^{[n]}_{\Lambda}(pv)\Gamma_{n}(K;pv)=\exp\big{(}\clubsuit\big{)}\exp\big{(}\spadesuit\big{)}\exp\big{(}\heartsuit\big{)}$
$\displaystyle\cdot
A_{\Lambda}^{(-)}(v;p)B_{n}^{(-)}(v,p)\cdot\tilde{\Phi}^{[n]}_{\Lambda}(v)\Gamma_{n}(K;v)\cdot
A_{\Lambda}^{(+)}(v;p)B_{n}^{(+)}(v,p).$ (A.7)

From the explicit expressions of $\clubsuit,\spadesuit,$ and $\heartsuit$ we
get that $\clubsuit+\spadesuit+\heartsuit=0$. In conclusion,

$\displaystyle\Xi_{\Lambda}(K;pv)=\frac{z_{\Lambda}(K;pv)}{z_{\Lambda}(K;v)}$
$\displaystyle\cdot\exp\bigg{(}\sum_{r=1}^{\infty}\frac{(p^{r}-1)H_{-r}}{(\mathfrak{q}^{r}-\mathfrak{q}^{-r})}\big{\\{}(\mathfrak{q}^{-1/2}v)^{r}\big{(}\frac{q^{nr}_{3}-K^{r}}{\kappa_{r}}\big{)}+\sum_{k=1}^{n}(\mathfrak{q}^{-1/2}q^{k-1}_{3}v)^{r}\chi_{r}^{(k)}(q_{1},q_{2})\big{\\}}\bigg{)}\cdot\,\Xi_{\Lambda}(K;v)$
$\displaystyle\cdot\exp\bigg{(}\sum_{r=1}^{\infty}\frac{(p^{-r}-1)H_{r}}{(\mathfrak{q}^{r}-\mathfrak{q}^{-r})}\big{\\{}(\mathfrak{q}^{1/2}v)^{-r}\frac{q^{-nr}_{3}-K^{-r}}{\kappa_{r}}-\sum_{k=1}^{n}(\mathfrak{q}^{1/2}q^{k-1}_{3}v)^{-r}\chi_{-r}^{(k)}(q_{1},q_{2})\big{\\}}\bigg{)}.$
(A.8)

When $p=\mathfrak{q}^{-2}=q_{3}^{-1}$, by (1.13) and (6.13) we can show
$\displaystyle\frac{z_{\Lambda}(K;\mathfrak{q}^{-2}v)}{z_{\Lambda}(K;v)}=K^{|\Lambda|}$,
and hence we arrive at (6.27).

Next we would like to we determine the proportional factor in (6.38)
explicitly From Eq. (4.47) we know that

$\displaystyle\Xi_{\Lambda}^{*}(K;pv)=\frac{z_{\Lambda}^{*}(K;pv)}{z_{\Lambda}^{*}(K;v)}z_{\Lambda}^{*}(K;v)\mathcal{M}^{[n]*}(K)\tilde{\Phi}^{[n]*}_{\Lambda}(pv)\Gamma^{*}_{n}(K;pv).$
(A.9)

We can write

$\tilde{\Phi}^{[n]*}_{\Lambda}(pv)\Gamma^{*}_{n}(K;pv)=C_{\Lambda}^{(-)}(v;p)\tilde{\Phi}^{[n]*}_{\Lambda}(v)C_{\Lambda}^{(+)}(v;p)\cdot
D_{n}^{(-)}(v;p)\Gamma^{*}_{n}(K;v)D_{n}^{(+)}(v;p),$ (A.10)

where

$\displaystyle
C_{\Lambda}^{(\pm)}(v;p)=\exp\bigg{[}\pm\sum_{m=1}^{n}\sum_{r=1}^{\infty}\frac{(p^{\mp
r}-1)H_{\pm
r}}{\mathfrak{q}^{r}-\mathfrak{q}^{-r}}\mathfrak{q}^{r/2}(q^{m-1}_{3}v)^{\mp
r}\chi_{\mp r}^{(m)}(q_{1},q_{2})\bigg{]},$

and

$\displaystyle
D_{n}^{(\pm)}(v;p)=\exp\bigg{(}-\sum_{r=1}^{\infty}\frac{(p^{\mp r}-1)H_{\pm
r}}{\mathfrak{q}^{r}-\mathfrak{q}^{-r}}\frac{q^{\mp nr}_{3}-K^{\mp
r}}{\kappa_{r}}\mathfrak{q}^{r/2}v^{\mp r}\bigg{)}.$ (A.11)

By using (1.14), we get that

$C_{\Lambda}^{(+)}(v;p)D_{n}^{(-)}(v;p)=\exp\big{(}\bigstar\big{)}D_{n}^{(-)}(v;p)C_{\Lambda}^{(+)}(v;p)$
(A.12)

where

$\displaystyle\bigstar=-\sum_{r=1}^{\infty}(p^{-r}-1)(p^{r}-1)\frac{q^{nr}_{3}-K^{r}}{\kappa_{r}}\frac{q^{r}_{3}(q^{r}_{1}-1)(q^{r}_{2}-1)}{r}\cdot\bigg{[}\sum_{(i,j,k)\in\Lambda}x^{-r}_{ijk}-\frac{1-q^{-rn}_{3}}{\kappa_{r}}\bigg{]}.$

Next we would like to switch between $C_{\Lambda}^{(+)}(v;p)$ and
$\Gamma^{*}_{n}(K;v)$. We know the explicit expression of
$\Gamma^{*}_{n}(K;v)$ from (4.48). So by using (1.14) again, we can show that

$\displaystyle
C_{\Lambda}^{(+)}(v;p)\Gamma^{*}_{n}(K;v)=\exp\big{(}\mho\big{)}\Gamma^{*}_{n}(K;v)C_{\Lambda}^{(+)}(v;p),$
(A.13)

where

$\displaystyle\mho=-\sum_{r=1}^{\infty}(p^{-r}-1)\frac{q^{nr}_{3}-K^{r}}{\kappa_{r}}\frac{q^{r}_{3}(q^{r}_{1}-1)(q^{r}_{2}-1)}{r}\bigg{(}\sum_{(i,j,k)\in\Lambda}x^{-r}_{ijk}-\frac{1-q^{-nr}_{3}}{\kappa_{r}}\bigg{)}.$
(A.14)

Finally we would like to switch between $\tilde{\Phi}^{[n]*}_{\Lambda}(v)$ and
$D_{n}^{(-)}(v;p)$. Again since we know the explicit expression of
$\tilde{\Phi}^{[n]*}_{\Lambda}(v)$ from Lemma 6.4, we can use (1.14) to show
that

$\tilde{\Phi}^{[n]*}_{\Lambda}(v)D_{n}^{(-)}(v;p)=\exp\big{(}\natural\big{)}D_{n}^{(-)}(v;p)\tilde{\Phi}^{[n]*}_{\Lambda}(v),$
(A.15)

where

$\displaystyle\natural=-\sum_{r=1}^{\infty}(p^{r}-1)\frac{q^{nr}_{3}-K^{r}}{k_{r}}\frac{q^{r}_{3}(q^{r}_{1}-1)(q^{r}_{2}-1)}{r}\bigg{(}\sum_{(i,j,k)\in\Lambda}x^{-r}_{ijk}-\frac{1-q^{-rn}_{3}}{\kappa_{r}}\bigg{)}.$
(A.16)

Consequently,

$\displaystyle\tilde{\Phi}^{[n]*}_{\Lambda}(pv)\Gamma^{*}_{n}(K;pv)=\exp\big{(}\bigstar\big{)}\exp\big{(}\mho\big{)}\exp\big{(}\natural\big{)}$
$\displaystyle
C_{\Lambda}^{(-)}(v;p)D_{n}^{(-)}(v;p)\cdot\tilde{\Phi}^{[n]*}_{\Lambda}(v)\Gamma^{*}_{n}(K;v)\cdot
C_{\Lambda}^{(+)}(v;p)D_{n}^{(+)}(v;p).$ (A.17)

Substituting (A.17) into (A.9), we see that the proportional factor of (6.38)
is

$\displaystyle\frac{z^{*}_{\Lambda}(K;pv)}{z^{*}_{\Lambda}(K;v)}\exp\big{(}\bigstar+\mho+\natural\big{)}.$

However, it is easy to show that $\bigstar+\mho+\natural=0$. Furthermore, for
$p=\mathfrak{q}^{-2}=q_{3}^{-1}$ (1.13) and (6.33) tell us that
$\displaystyle\frac{z^{*}_{\Omega}(K^{\prime};\mathfrak{q}^{-2}w)}{z^{*}_{\Omega}(K^{\prime};w)}=\frac{1}{K^{\prime\,|\Omega|}}$.
Hence, we finally obtain (6.38).

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# Using Multiparametric MRI with Optimized Synthetic Correlated Diffusion
Imaging to Enhance Breast Cancer Pathologic Complete Response Prediction

Chi-en Amy Tai
University of Waterloo
Waterloo, ON
<EMAIL_ADDRESS>Alexander Wong
University of Waterloo
Waterloo, ON
<EMAIL_ADDRESS>
###### Abstract

In 2020, 685,000 deaths across the world were attributed to breast cancer,
underscoring the critical need for innovative and effective breast cancer
treatment. Neoadjuvant chemotherapy has recently gained popularity as a
promising treatment strategy for breast cancer, attributed to its efficacy in
shrinking large tumors and leading to pathologic complete response. However,
the current process to recommend neoadjuvant chemotherapy relies on the
subjective evaluation of medical experts which contain inherent biases and
significant uncertainty. A recent study, utilizing volumetric deep radiomic
features extracted from synthetic correlated diffusion imaging (CDIs),
demonstrated significant potential in noninvasive breast cancer pathologic
complete response prediction. Inspired by the positive outcomes of optimizing
CDIs for prostate cancer delineation, this research investigates the
application of optimized CDIs to enhance breast cancer pathologic complete
response prediction. Using multiparametric MRI that fuses optimized CDIs with
diffusion-weighted imaging (DWI), we obtain a leave-one-out cross-validation
accuracy of 93.28%, over 5.5% higher than that previously reported.

##  1 Introduction

In 2020, 685,000 deaths across the world were attributed to breast cancer,
underscoring the critical need for innovative and effective breast cancer
treatment [10]. Recently, neoadjuvant chemotherapy emerged as a promising
treatment strategy, gaining popularity due to its ability to shrink large
tumours [4] and elicit a pathologic complete response (pCR) for breast cancer
patients [5]. Unfortunately, neoadjuvant chemotherapy presents challenges in
terms of cost, time commitment, and potential risks for patients, including
exposure to radiation and the possibility of encountering significant side
effects such as decreased fertility [6].

The current methodology for recommending neoadjuvant chemotherapy hinges on
the subjective evaluation by medical experts, assessing the patient’s
prognosis and potential treatment benefits [7]. The inherent biases and
significant uncertainty in human judgment introduce the possibility of
misguided recommendations [12], putting some patients at risk of preventable
progression to advanced cancer or unnecessary exposure to radiation. Advances
in computer vision and medical imaging have demonstrated the potential for
precise and noninvasive techniques in predicting pathologic complete response
prediction [13].

Most promising, using a pretrained 34-layer volumetric residual convolutional
neural network architecture with initial weights from MONAI [3] with synthetic
correlated diffusion MRI images achieved a leave-one-out cross-validation
(LOOCV) accuracy of 87.75%. Initially introduced in [14], CDIs is a derived
MRI modality that combines native and synthetic diffusion signal acquisitions.
This combination, along with signal calibration, aims to enhance consistency
for the dynamic range across different machines and protocols. The study in
[14] illustrates the effects of fine-tuning CDIs for prostate cancer.
Conversely, in [13], modifications to CDIs for breast cancer were minimal,
lacking proper tuning of the coefficients. Inspired by the positive outcomes
of optimizing CDIs for prostate cancer delineation [14], this paper
investigates the application of optimized CDIs to enhance the prediction of
breast cancer pathologic complete response.

##  2 Methodology

The data is obtained from the pre-treatment (T0) cohort in the American
College of Radiology Imaging Network (ACRIN) 6698/I-SPY2 study [11, 8, 9, 2]
filtered for non-null pCR values, leading to a total of 253 remaining
patients. As the classes were imbalanced (class 0:1 is 67.6%:32.4%), a
weighted random sampler was added to the training sampler along with an AdamW
optimizer. In addition, a cosine annealing learning rate scheduler was also
implemented during training.

Figure 1: Clinical support workflow copied from [13] that is used in this
study.

To optimize CDIs, the Nelder-Mead simplex optimization strategy was used to
maximize the area under the receiving operating characteristic curve (AUC) for
breast cancer tumour delineation. Then, the CDIs signals were fused with
diffusion-weighted imaging (DWI) to create a multiparametric MRI. Finally, all
patient volumes were then standardized to 224x224x25 volumetric data cubes for
dimensional consistency.

Building on the foundation of a previously developed deep radiomic clinical
support system outlined in [13] (see Figure 1), we adopt a 34-layer pre-
trained volumetric residual convolutional neural network framework from MONAI
to obtain deep radiomic features. The weights from MONAI were obtained from
training on eight different 3D MRI and CT segmentation datasets [1]. The
radiomic features were subsequently inputted into a fully-connected neural
network predictor, designed to classify breast cancer into no pCR or pCR.

For training, a learning rate of 0.001 was also used. All the model layers
were trained with no freezing and leave-one-out cross-validation was used to
compare the performance. Though the previous paper only reported the average
accuracy, we also provide the average sensitivity, specificity, and F1 score.

##  3 Results

Table 3 shows the results from using the unoptimized and optimized CDIs with
DWI to create a multiparametric MRI. As shown, the accuracy using the
optimized CDIs was 93.28%, over 5.5% higher than that previously reported
(87.75%). Though the sensitivity and specificity metrics were not previously
reported, using optimized CDIs achieved over 90% for both sensitivity and
specificity. Notably the F1 score using optimized CDIs was also high at
90.17%. An example highlighting the visual differences between the imaging
modalities of the unoptimized CDIs and optimized CDIs are shown in Figure 2.
Also provided in Figure 2 are the associated tumour mask and DWI for the
specific patient slice.

Table 1: Results using unoptimized and optimized CDIs with the best result
bolded.

l c c c CDIs Version Accuracy Sensitivity Specificity
Unoptimized 87.75% N/A N/A
Optimized 93.28% 95.12% 92.40%

(a)

(b)

(c)

(d)

Figure 2: An example slice illustrating visual differences between (a)
Unoptimized CDIs, (b) Optimized CDIs, (c) the associated DWI image, and (d)
the associated tumour mask at pre-treatment for a patient who obtained pCR.
For this patient, the pCR prediction was correct using the Optimized CDIs
signal fused with DWI.

Given the promising results, this proposed noninvasive method that predicts
pathologic complete response would enhance patient treatment with minimal side
effects (as it uses MRI modalities that are normally obtained in the course of
the diagnosis). In addition, the results highlight the importance of tuning
CDIs for the specific cancer domain as the optimized CDIs modality obtains
superior performance compared to unoptimized CDIs.

## References

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