June 5, 2019

# Development of a UV-transparent Lens Array for Enlarging the Effective Area
of Multichannel SiPMs

Akira Okumura Akira Asano Kazuhiro Furuta Naoya Hidaka Yuki Nakamura Hiroyasu
Tajima and Anatolii Zenin Institute for Space–Earth Environmental
ResearchInstitute for Space–Earth Environmental Research Nagoya University
Nagoya University Furo-cho Furo-cho Chikusa-ku Chikusa-ku Nagoya 464-8601
Nagoya 464-8601 Japan Japan<EMAIL_ADDRESS>(A.O.)

###### Abstract

We developed a UV-transparent lens array that can increase the photon
detection efficiency of a silicon photomultiplier (SiPM) array comprising of
64 pixels ($3\times 3$ mm2 each) and 0.2-mm gaps. Through the plano-convex
spherical lens on each $3.2\times 3.2$ mm2 region, we showed that the loss of
photon detection efficiency due to the pixel gaps could be recovered as the
incident photons get concentrated on the sensitive regions of the SiPM array.
By using a prototype lens array, we achieved approximately 10%–30% relative
increase in photon detection efficiency in our target angles of incidence of
30–60 deg.

Silicon photomultipliers (SiPMs), lens array, ray-tracing simulation,
Cherenkov telescopes

##  1 Introduction

Silicon photomultipliers (SiPMs) are widely used in various physics
experiments owing to their compactness, low operation voltage, and high photon
detection efficiency (PDE). PDE is defined as a product of quantum efficiency
(QE) and other factors, namely, the geometrical fill factor and probability of
the avalanche process of individual Geiger-mode avalanche-photodiode (G-APD)
cells for SiPMs and the collection efficiency of 90%–95% for conventional
photomultiplier tubes (PMTs).

SiPM PDE reaches as high as 50% when a G-APD cell size of 75-$\mu$m is chosen
(e.g., Hamamatsu Photonics S13360 series with a fill factor of 82%), whereas
that of PMTs with an ultrabialkali photocathode (maximum QE of typically
nearly 43%) can achieve approximately 40% at most. Thus, the use of SiPMs
could be a more preferable option than PMTs in low photon luminosity and
compact multichannel applications, such as small focal-plane ($<50$ cm)
cameras of ground-based very-high-energy (100 GeV–10 TeV) gamma-ray telescopes
(also referred to as Cherenkov telescopes).

Covering the entire focal-plane area with a large number of SiPMs with the
smallest possible dead area is ideally desirable. However, non-zero gaps
between two adjacent channels usually exist in multichannel SiPM products
(e.g., $0.2$ mm for the Hamamatsu Photonics S13361 series), as shown in Figs.
1 and 2, thereby resulting in a non-negligible effective PDE loss amounting to
relatively $12.1$% ($=1-3.0^{2}/3.2^{2}$) for $3\times 3$-mm2 pixels with
0.2-mm gaps. In relation to one of the advantages of SiPMs, the high PDE
becomes less significant, unless this PDE loss is recovered using some
techniques.

Figure 1: $8\times 8$ pixel SiPM (Hamamatsu Photonics S13361-3050AS) (left)
and the same product with a lens array (right).

Figure 2: (a) Close-up view of one of the corner pixels of the SiPM array, and
3584 G-APD cells and the central metal pad are presented. (b) The same pixel
with a lens array. The G-APD cells and the pad are apparently magnified
because of the convex lens shape. The circular central shadow with a radius of
approximately 10 G-APD cells is a reflected image of a camera lens. Small red
squares in both photos point the identical G-APD cell.

On the contrary, intentionally having wider pixel gaps can reduce the
production cost of a large SiPM array, as the SiPM unit price under mass
production is roughly proportional to the total area of the silicon substrate.
On such basis, having a higher effective PDE and building a less expensive
SiPM array come as trade-offs.

Installing solid light concentrators called compound parabolic concentrators
(CPCs, also known as Winston cones) on individual single-channel SiPMs is a
feasible method for identifying a less effective PDE loss, considering a
smaller silicon area [1]. However, the absorption of ultraviolet (UV: 300–400
nm) photon inside long solid cones (a few cm) is not negligible; thus, it is
not extremely suitable for UV–visible Cherenkov photon detectors. The use of
hollow Winston cones with an optimized Bézier curve profile has been proposed
and demonstrated [2, 3]. However, the minimum plastic thickness required in
injection molding is approximately 0.2 mm, a size that produces an unavoidable
pixel gap of 0.4 mm in total.

In the present study, we explored another solution to reduce the effective PDE
loss caused by the pixel gaps by using a lens-array molding technology and a
UV-transparent glass material. A discussion of lens array design, simulation,
and measurements and a comparison of the effective PDE between SiPMs with and
without a lens array were carried out.

##  2 Lens-array Design

A micro lens array (roughly 10 $\mu$m) on an imaging sensor is quite commonly
used in commercial digital camera products with complementary metal oxide
semiconductor (CMOS) or charge-coupled device (CCD) imaging sensors with a
typical pixel size of $5$–$10$ $\mu$m. Convex micro lenses aligned on
individual imaging pixels concentrate incident photons onto the active area of
the pixels to achieve higher sensitivity.

Accordingly, it is natural to apply the same idea to multichannel SiPMs to
achieve a higher effective PDE with the silicon substrate area being kept
small. Figures 2 and 2 clearly show that the lens-array concept can optically
hide the 0.2-mm gaps between 3-mm pixels, where a lens array comprising of
$8\times 8$ plano-convex spherical lenses is placed and coupled to the SiPM
array with optical grease (Saint-Gobain BC-630, refractive index $n$ of
approximately 1.5).

Each lens has a $3.2\times 3.2$ mm2 footprint to cover a single SiPM pixel.
The lens height and the radius of curvature are 2.0 mm and 2.3 mm,
respectively, which are the chosen values to maximize the effective PDE for
the Cherenkov photon spectrum at the ground level. Moreover, a flat
distribution of angles of incidence from 30 to 60 deg was assumed to consider
a future Cherenkov camera application with the Schwarzschild–Couder optical
system[4]. As illustrated in Fig. 2, the radius of curvature is too short for
normal incidence photons, and the outer G-APD cells are totally hidden. On the
contrary, a large curvature is required to concentrate photons with large
angles of incidence of up to 60 deg onto the cells.

##  3 Simulation

During design optimization, we simulated the performance of the lens array
assuming the PDE curve of Hamamatsu Photonics S12571-050C (50-$\mu$m cell
size, silicone resin coating). The wavelength-dependent photon absorption
length, $\tau_{\mathrm{abs}}(\lambda)$, and refractive indices, $n(\lambda)$,
of the silicone resin, optical grease, and UV-transparent lens-array material
(OMG UVC-200B) were measured and considered in the simulation. For simplicity,
the Fresnel reflection and the interference at the media boundary on the
silicon substrate were ignored. The effective PDE increase for a flat
distribution of angles of incidence (30–60 deg) for a Schwarzschild–Couder
Cherenkov camera under these assumptions, as simulated by the ROBAST ray-
tracing software[5], was 13.5%. For example, if the PDE without the lens array
was 40%, then the effective PDE with the lens array would be $40\times
1.135=45.4$%. Consequently, the effective PDE loss of $12.1$% due to the pixel
gaps would be almost compensated by the lens array
($\frac{1}{1-0.121}=1.138$).

We performed a more realistic simulation after the production and performance
test of the lens array to reproduce the measurement result. Thus, the G-APD
cell structure, thin $\mathrm{SiO}_{2}$ and $\mathrm{Si}_{3}\mathrm{N}_{4}$
layers on the silicon substrate, and $n(\lambda)$ of the layers and the
substrate were additionally considered, whereas the optical interference on
the layers remained ignored, as ROBAST does not support quantum optics.

The ROBAST simulation for different LED colors (310, 375, 465, and 635 nm) are
presented in Fig. 3(d). The measurement results are discussed in Section 4.
Simulated photons were beamed onto the central $3.2\times
3.2$-$\mathrm{mm}^{2}$ region of the $8\times 8$ SiPM array from outside of
the lens array. The relative PDE increase in 0–40 deg reached nearly 10%–12%,
13%–17%, 14%–16%, and 14%–15% for 310, 375, 465, and 635 nm, respectively,
where all the photons detected by any SiPM channel were counted (Total, in
black color in Fig. 3(d)). With the high refractive index of silicon
(approximately 3.9–6.9 in 300–600 nm), a large fraction of photons got
reflected on the silicon surface, and a part of them was totally reflected at
the media boundary between the lens array and air. Thus, photons with large
angles of incidence exhibited a higher probability of reaching the silicon
surface multiple times, resulting in higher PDE increase than that in 0–40
deg.

(a) 310 nm

(b) 375 nm

(c) 465 nm

(d) 635 nm

Figure 3: Relative PDE increase by lens array vs. angle of incidence for four
LED colors. ROBAST simulations for different pixel groupings are shown as
total (black), central four pixels only (green crosses), central four pixels
and two left neighbor pixels only (orange circles), and central four pixels
and two right neighbor pixels only (small magenta circles). See also Fig.
4(a). Red and blue open squares show the measured increase for positive and
negative rotations, respectively.

Simulated count maps of detected photons and their tracks for 50 and 70 deg
are shown in Fig. 4(d). Here, due to the total reflection, incident photons
beamed toward the central four pixels were detected by not only these pixels
but also the neighbor ones. In the case of 50 deg, photons were focused onto
the two left neighbor pixels. In the case of 70 deg, a part of the beamed
photons first entered the lenses of the two right neighbor pixels and the
refracted tracks reached the silicone surface of the same pixels. These
complex reflection and refraction processes resulted in a non-flat PDE
increase at 40–70 deg.

(a) Count map (50 deg)

(b) Count map (70 deg)

(c) Photon tracks (50 deg)

(d) Photon tracks (70 deg)

Figure 4: (a) A simulated count map of detected photons in individual SiPM
channels for 465-nm photons with an angle of incidence of 50 deg. (b) Same as
(a) but for 70 deg. (c) Bird’s eye view (top) and side view (bottom) of the
simulated photon tracks of an angle of incidence of 50 deg. The latter clearly
shows the tracks of photons that were reflected on the silicon surface and
then totally reflected on the lens array surface. (d) Same as (c) but for 70
deg.

##  4 Measurement

We measured the actual performance of the lens array via uniform illumination
of parallel LED beam flashes on the SiPM surface. Comparing the numbers of
detected photons by one of the 64 channels of the SiPM array with the lens
array being coupled and removed, we calculated the relative PDE increase from
their ratio for various angles of incidence. Fig. 3(d) shows the measurement
results from $-75$ to $+75$ deg. Note that a uniform $3.2\times
3.2$-$\mathrm{mm}^{2}$ beam was assumed in the ROBAST simulation, although
parallel LED flashes were illuminated on the whole area of the SiPM array in
our measurement. Thus, in Fig. 3(d), we can only provide a direct comparison
between “Sim. (Total),” “Measured (Pos. Rot.)” (0–$+75$ deg), and “Measured
(Neg. Rot.)” ($-75$–0 deg).

PDE increase measured in the all LED colors was roughly 10%–20% in 0–40 deg,
which is consistent with the simulation result. Nonetheless, the expected
large relative increase of more than 40% in 50–70 deg was not necessarily as
high as the simulation, whereas there was the occurrence of two peak
structures at approximately 52 and 63 deg in the measurement. This result
implied that reflectance at the media boundary between the silicon substrate
and the protection silicone resin was significantly smaller than a simple
assumption of Fresnel reflection. Moreover, the thin $\mathrm{SiO}_{2}$ and
$\mathrm{Si}_{3}\mathrm{N}_{4}$ layers existing between these two media were
expected to work as antireflection coating to improve the PDE; on the
contrary, the quantum optical property of the layers was not considered in the
simulation.

##  5 Conclusion

Through a ray-tracing simulation and a measurement, we showed the idea of
coupling a spherical lens array on a SiPM array to increase their effective
active area or to recover their physical 0.2-mm pixel gap. Based on our
measurements, the relative PDE increase in the target angle regions of 30–60
deg was approximately 10%–30% for 310, 375, 465, and 635 nm of LED colors.
Nonetheless, the simulation did not reproduce the measurement, particularly
for large angles of incidence, probably due to imperfection in our optical
treatment of the thin multi layers. This could be resolved by better modeling
of the layers for further assessment and lens shape optimization. Degradation
of focused image on the SiPM array due to the photon propagation onto the
neighboring pixels as well as possible increase of optical crosstalk[6] should
also be evaluated.

## References

  * [1] H. Anderhub, et al., J. Instrum. 8 P06008 (2013).
  * [2] A. Okumura, Astropart. Phys. 38 18–24 (2012).
  * [3] J. Aguilar, et al., Astropart. Phys. 60 32–40 (2015).
  * [4] V. Vassiliev, S. Fegan, and P. Brousseau. Astropart. Phys. 28 10–27 (2007).
  * [5] A. Okumura, K. Noda, and C. Rulten, Astropart. Phys. 76 38–47 (2016).
  * [6] Y. Nakamura, A. Okumura, H. Tajima, N. Yamane, and A. Zenin These Proceedings (2019).


Modeling the Effect of Observational Social Learning on Parental Decision-
Making for Childhood Vaccination and Diseases Spread over Household Networks

Tamer Oraby∗, Andras Balogh

School of Mathematical and Statistical Sciences, The University of Texas Rio
Grande Valley,

1201 W. University Dr., Edinburg, Texas, USA,

∗Corresponding author<EMAIL_ADDRESS>
## Abstract

In this paper, we introduce a new model of parental decision-making concerning
vaccines against a childhood disease that spreads over a contact network. We
consider a bilayer network composed of two overlapping networks which are
either Erdős-Rényi (random) networks or Barabási-Albert networks. The new
model uses a Bayesian aggregation rule for observational social learning,
occurring over a social network, of which other decision models, like voting
and DeGroot models, are special cases. Using our new model, we show how some
levels of social learning about vaccination preferences can lead to the
convergence of opinions and affect levels of vaccine uptake and so disease
spread. In addition, we study the effect of the existence of two cultures of
social learning on the establishment of social norms of vaccination and levels
of vaccine uptake. In all cases, the mutual influence between the dynamics of
observational social learning and disease spread is dependent on the network’s
topology and vaccine safety and availability.

Keywords: Social Learning; Children Vaccination; Networks; Disease Model;
Social Norms.

##  1 Introduction

Human herding behavior or cascading is a convergence in opinion driven by
social learning [1, 2]. Such herding could also be induced by payoff
externalities, sanctions, preference interaction, direct communication, and/or
observational influence [2]. Observational influence results from integrating
learned behavior or perception of others’ opinions with one’s own opinion.
Observational social learning causes a subtle pressure on people to conform
[3, 4, 5]. That social learning takes place through different channels of
information sharing, and through observation and/or perception of public
choices [6]. However, boundedly rational observational learning occurs when
there is incomplete or insufficient information on the behavior of others,
[7].

Social learning has been shown to play a vital role in people’s decision-
making even in the presence of information [8]. Parents learn about the
vaccination choices/attitudes of other parents and look for consensus
vaccination signals [9]. When parents share their feelings about the
vaccination of their children while others observe these opinions, pressure is
placed on other parents [10, 11]. Mixed messages and signals of parents’
choices can cause problems for other parents when interpreting them.
Incomplete information on the vaccine and the opinions of friends on social
networks, creating boundedly social agents, can cause problems in vaccine
acceptance and therefore in vaccine uptake and disease spread [12].

People and households are connected to each other through different types of
networks. From a graph theory point of view, there are several types of
network models, e.g. Erdős-Rényi (random) network model (ERN) and Barabási-
Albert network model (BAN), see [13]. Different network models represent
various real-life systems. Many real-life networks are scale-free networks
(SFN) in which the degree distribution is a power law with exponent
$2<\gamma\leq 3$, see [14].

Many studies modeled the spread of diseases on networks, see, e.g., [15, 16,
17, 18], and on multiplex network models, see, e.g., [19, 20, 21, 22, 23, 24].
Some studies also considered the structure of the home [25, 26, 27, 28]. Few
models of spread of diseases considered vaccine decision making on those
networks through voter model [23], through DeGroot model of selection [14],
through social norms [29], and through social learning [30]. Moreover, most of
the mathematical models in those articles consider rational agents making
decisions to vaccine children with complete information, e.g. [15, 10, 31,
32], except for a few like [24].

Here, we consider bilayer networks in which two same-type overlapping networks
and their mutual influence are of interest. The first network is the physical
network through which face-to-face contacts take place and pediatric disease
transmission occurs. The second network is a social (bidirectional) weighted
network through which information and opinions about the vaccine are shared,
shaping parents’ decisions. Pediatric disease spreads on the physical network
within and between households, while information and opinion sharing and
perception transpire on the social network of households, parents in that
case. We assume that the two networks overlap with parents linked to a larger
number of other parents, relatives, coworkers, and distant friends, whose
children might or might not be connected on the physical network. We introduce
some adjustments to network models to reflect some important aspects of
household networks. First, household networks should account for the number of
children in households. Second, households without children must not have
physical connections. That is, the expected degree of a household should
increase with the number of occupying children, on average.

In this paper, we present a new model of parental decision-making to protect
their children against a measles-like disease that spreads over household
networks. This model considers boundedly rational observational social
learning using a Bayesian aggregation formula. This Bayesian formula is
different from that of Muller-Frank and Neri [7], who presented a quasi-
Bayesian model of boundedly rational observational learning in a general
context. We show that our model can give rise to social norms and at the same
time encompasses other selection models like voting and DeGroot. Our new model
considers socially bounded agents (parents who value children the most) who
possess imperfect information about the vaccination choices of their network’s
neighbors.

We postulate that those agents can only perceive a correct or wrong message,
probably, because of fear of retribution or confusion. There is a chance that
the agents will send a correct message of their opinion with probability $q$
and a message with probability $1-q$, see [33]. Using that model, we study the
cascading opinion of vaccination in a boundedly rational observational social
learning and compare it to other models of social pressure [29, 34], see also
[12]. Moreover, we study the influence of those types of signal games on the
spread of vaccine opinion and disease in social and physical networks in the
presence of resource limitations represented in vaccine efficacy and
accessibility, as well as vaccine safety. Finally, we study the effect of the
presence of two cultures of social learning on the establishment of a social
norm and so on vaccine uptake.

##  2 Model and Methods

#### Networks.

To model the spread of the disease, we use an agent-based network model whose
nodes are households $N$. Households are occupied by a number of children
($C_{i}$, such that $0\leq C_{i}\leq n_{C}$, for $i=1,\ldots,N$) who are
connected through a physical network. The number $n_{c}$ is the maximum number
of children per household. Parents are connected through a different (social,
internet and physical) bi-directional weighted network via which they exchange
opinions and information, and observe choices. Parents in a household without
children could still be connected to other parents and shape their opinions.
We use two types of networks: Erdős-Rényi (random) network model (ERN) and
Barabási-Albert network model (BAN), see [13]. In physical networks, we
postulate the degree of nodes to be proportional to the number of children in
the households. Meanwhile, the parent’s network overlaps with the children’s
network via random rewiring of the children’s network but with wiring
probabilities that are greater than severance probabilities. We assume that
the parent’s network is weighted bidirectional with weights given by the
(learning) probabilities $q_{j,i}$, for $i,j=1,\ldots,N$ and $(i,j)$ is a
social network link. See supplementary material (SI. Model) for complete
information about the networks. An instance of a histogram for each random
network is shown in Fig. S1.

#### Birth process.

A birth process is postulated to depend on the number of children occupying a
household. The probability of a new pregnancy is modeled using a logistic
function with a median of $C^{*}$ such that it decreases as the number of
children in a household approaches $n_{c}$. See supplementary material (SI.
Model). A pregnancy lasts for a period of 280 days. The delivery of a newborn
updates the number of children, but not the number of links in the children’s
network. We consider miscarriages and children’s deaths to be rare and so are
not included.

#### Disease spread.

We assume that a new measles-like disease (vaccine-preventable, pediatric) is
spreading between children within and between households. We assume a mean
length $m_{p}$ days of incubation period with a maximum of $\ell$ days. A new
infection in a household occurs with a probability
$1-(1-\beta_{h})^{I(i)}\cdot(1-\beta)^{n_{I}(i)/C_{i}}$; where $\beta$ is the
probability of infecting a child in another household (through the physical
network), $\beta_{h}$ is the probability of infecting a sibling within the
same household, and $I(i)$ is the total number of infected siblings in the
same household. The number $n_{I}(i)$ is the number of infected children
connected to the household $i$ through the network. The number $n_{I}(i)$ is
divided by $C_{i}$ to approximate the probability of infection formula based
on the assumption that the children in a household have the same number of
friends on average. We assume that the epidemic starts with $I_{0}$ infected
children in different households and is randomly chosen.

#### Vaccination decision-making.

Parents are randomly allocated into three types: never-vaccinators, who oppose
vaccination at all times, non-vaccinators, and vaccinators. We assume a small
percentage of parents are never-vaccinators who are still sharing their
opinions. Parents in the household $i$ shape their subjective decision to
vaccinate their children based on the reward of vaccination, given by
$\pi_{i}=\alpha_{i}I-\gamma_{i}A$, where $I$ and $A$ are the total number of
adverse events infected and vaccine, respectively, up to the decision-making.
Parents who experience an adverse event due to vaccinating their children are
switched to never-vaccinators in the sequel. The parameter $\alpha_{i}$ is the
degree of relevance of disease infectiousness and $\gamma_{i}$ is the degree
of relevance of vaccine adverse events to the subjective opinion of the
household $i$. The probability of accepting vaccination against the disease is
given by $pr_{i}=1/(1+\exp(-\pi_{i}))$ for the household $i$ and is equal to
zero for never-vaccinators. Probabilities
$pr_{i}=1/(1+\exp(-\alpha_{i}I_{0}))$ are used to generate the initial stance
of parents in the household $i$ toward vaccination.

#### Observational social learning.

Let the learning probability $q_{i,j}$ that the household $i$ has the correct
perception or learning about the household’s $j$ opinion or stance about
vaccination. Meanwhile, $1-q_{i,j}$ is the probability that the household $i$
will make the wrong perception of the household opinion $j$’. Such social
learning does not have to be symmetric, that is, $q_{i,j}\neq q_{j,i}$. For
instance, followers of a celebrity learn from the celebrity more than in the
other way around. Reciprocal or symmetric social learning involves
$q_{j,i}=q_{i,j}$. Let household $i$ have a set of vaccinator neighbors in the
social network $N_{V}(i)$ with cardinality $n_{V}(i)$ and have a set of non-
vaccinator’s social network neighbors $N_{N}(i)$ with cardinality $n_{N}(i)$.
Let also the total number of neighbors be $n_{S}(i)=n_{V}(i)+n_{N}(i)$, where
$N_{S}(i)=N_{V}(i)\cup N_{N}(i)$. Then parents in the household $i$ make the
decision to vaccinate their children based on the following posterior
probability:

$P_{S}(i)=\dfrac{pr_{i}\cdot\prod_{j\in N_{V}(i)}q_{j,i}\cdot\prod_{k\in
N_{N}(i)}(1-q_{k,i})}{pr_{i}\cdot\prod_{j\in N_{V}(i)}q_{j,i}\cdot\prod_{k\in
N_{N}(i)}(1-q_{k,i})+(1-pr_{i})\cdot\prod_{j\in
N_{V}(i)}(1-q_{j,i})\cdot\prod_{k\in N_{N}(i)}q_{k,i}}$ (2.1)

or what we call the Bayesian aggregation rule in observational social
learning. The rationale of the formula in equation (2.1) is that the prior
probability of vaccination $pr_{i}$ is updated by independent information
collected or perceived from neighbors in the network. A vaccinator is
perceived to have that opinion with probability $q_{i,j}$, and a non-
vaccinator is perceived to have the opinion to vaccinate with probability
$1-q_{i,j}$.

Nondirectional social learning, in the sense of outward uniformity, means
$q_{j,i}=q_{j,k}=:q_{j}$ for all $i\neq k$ and for all $j$.

In case that $q_{j}=q$ for all $j$, then

$P_{S}(i)=\dfrac{pr_{i}\cdot q^{n_{V}(i)}\cdot(1-q)^{n_{N}(i)}}{pr_{i}\cdot
q^{n_{V}(i)}\cdot(1-q)^{n_{N}(i)}+(1-pr_{i})\cdot(1-q)^{n_{V}(i)}\cdot
q^{n_{N}(i)}}$ (2.2)

for $0<q<1$. An uninformative probability $q=.5$ results in no social
influence on the parent, since $P_{S}(i)=pr_{i}$.

The model in equation (2.1) can be re-written as

$P_{S}(i)=\dfrac{1}{1+\exp\left(-\Delta_{i}-\pi_{i}\right)}$ (2.3)

where $\pi_{i}=\text{logit}(pr_{i})$ and $\Delta_{i}=\sum_{j\in
N_{V}(i)}\text{logit}(q_{j,i})-\sum_{k\in N_{N}(i)}\text{logit}(q_{k,i})$.
(Note: $\text{logit}(p)=\log(\dfrac{p}{1-p})$.)

It is also a model of bounded rationality. Putting $PS(i)$ in the Boltzmann
distribution form

$P_{S}\left(i\right)=\frac{\exp{\left(.5{(\Delta}_{i}+\pi_{i})\right)}}{\exp{\left(.5{(\Delta}_{i}+\pi_{i})\right)}+\exp{\left(.5{(\bar{\Delta}}_{i}+{\bar{\pi}}_{i})\right)}}$

with temperature $\tau=.5$, reveals a bounded rational model supported by
physics [35]. In that case, the reward for vaccination is given by
$\pi_{i}=\alpha_{i}I-\gamma_{i}A=-{\bar{\pi}}_{i}$. Meanwhile,
$\Delta_{i}=\sum_{j\in
N_{V}\left(i\right)}\text{logit}\left(q_{j,i}\right)-\sum_{k\in
N_{N}\left(i\right)}\text{logit}\left(q_{k,i}\right)={-\bar{\Delta}}_{i}$ is
the induced social pressure to vaccinate.

DeGroot model of selection [14], can also be seen in the term $\Delta_{i}$ if
the neighbor $j\in N_{S}(i)$ is given a weight of $\text{logit}(q_{j,i})$ and
using the discrete opinions to be valued as $+1$ for vaccination and $-1$ for
no vaccination. This makes a stochastic DeGroot model of selection a special
case of the Bayesian aggregation rule in equation (2.1).

Similarly, when $q_{i}=q$ and $pr_{i}=pr$ for all $i$ as in the model of
equation (2.2), the model can be re-written as

$P_{S}(i)=\dfrac{1}{1+\exp\left(-\delta_{i}\dfrac{n_{V}(i)-n_{N}(i)}{n_{S}(i)}-\pi\right)}$
(2.4)

where $\pi=\text{logit}(pr)$ and $\delta_{i}=n_{S}(i)\,\text{logit}(q)$ is the
degree of injunctive social norm practiced by household $i$. See [29, 34, 12]
for the case $\delta_{i}=:\delta$ for all $i$, in which case

$P_{S}(i)=\dfrac{1}{1+\exp\left(-\delta(2\frac{n_{V}(i)}{n_{S}(i)}-1)-\pi\right)}$
(2.5)

which leads to the voting model of selection using
$G:=2\frac{n_{V}(i)}{n_{S}(i)}-1$ to decide the winner strategy when $G>0$ and
loser when $G<0$. In other words, the latter model is also a special case of
(2.1) in case of homogeneous observational learning. But since $\delta$ is
nonnegative, the equation (2.5) can only reveal the human behavior when $q\geq
0.5$ and so the model in (2.5) is less attractive than that in (2.1).

On each day, the parental position on vaccination is updated randomly based on
the probabilities $P_{S}$ and vaccinators are chosen with the probability
$\rho$ of vaccinating all their children where $\rho$ is the probability of
getting access to vaccination based on the resources.

#### Epidemiological measures.

To analyze the effect of the probability of learning $q_{j,i}$ on the opinion
of parents and on the spread of disease, we use a number of epidemiological
measures: the size of epidemics, the peak of the epidemic, the uptake of
vaccines, the number of vaccinators, and the basic reproduction number $R_{0}$
that we use only for the calibration of ERN. We use $R_{0}$ as an
epidemiological measure and not as a threshold, see [36]. The size of the
epidemic is the total number of children infected at the end of the epidemic.
The uptake of the vaccine is the total number of children who are vaccinated.
The ending number of vaccinators is used to measure whether the opinion of
vaccination becomes a consensus at the end of the epidemic. The basic
reproduction number $R_{0}$ is defined to be the average number of secondary
cases in a completely susceptible population. We use that definition to build
an algorithm to estimate the value of $R_{0}$ (see Algorithm 1 in
Supplementary Material SIII, Methods). In that algorithm, we use Bayes’
theorem to calculate the probability that infection happens due to contact
with the index case, which is then used to calculate the mean number of
infections. To find the grand mean, we averaged over several simulations of
disease transmission and then over the $N$ households, which can include the
index case, then finally over simulations of the various networks.

#### Model simulation.

The model is implemented using stochastic simulation for 100 runs to examine
the effect of the probability of social learning $q_{j,i}$ on the uptake of
vaccines and the spread of pediatric diseases. We assume that $q_{j,i}$ are
uniformly distributed on $q\pm 0.05$ for a prespecified value of $q$ where
$0<q<1$. The stance towards vaccination and the disease states of infected
children are updated at the beginning of each time step (day). Multiple
infections can occur on the same day in the network and the numbers $n_{I}(i)$
and $I(i)$ are updated every day for all $i$, $i=1,2,\ldots,N$. An infected
child on the $j^{th}$ day after infection jumps either to the end of the
incubation period (recovers) or remains infected with a transition to the
following day with probabilities given by the truncated exponential
distribution (see supplementary material SI, Model).

Our simulation codes rely on the NumPy-compatible CuPy Python library [37],
accelerated with NVIDIA CUDA [38] for parallel calculations in Graphical
Processing Units (GPUs). Most of the calculations were performed on a CentOS
workstation with $8$ NVIDIA Tesla (Kepler) K80 GPU cards, each of which has
$2496$ CUDA cores and 12GB memory. For more information, see Supplementary
Material SV, Codes of Simulations.

The eight GPU cards were used to distribute the 100 runs of stochastic
simulations, with each card running one network simulation at a time. While
the epidemic process can be run immediately after the network generations, we
typically run the two processes separately. After network generation, the
networks were saved to a hard drive, and at the beginning of the epidemic
process, these networks were read from the hard drive.

  * 1.

Average generation time of one network, including the savings of the networks
to hard drive:

    * (a)

Erdő s-R’enyi Network (ERN): $21$ seconds of wall clock time.

    * (b)

Barab\’asi-Albert Network (BAN): $116$ seconds of wall clock time. This
network has more serial steps than the ERN, since the households are added one
by one to the network.

  * 2.

Average epidemic run time over one network:

    * (a)

Erdő s-R’enyi Network (ERN): $13$ seconds of wall clock time.

    * (b)

Barab\Asi-Albert Network (BAN): $11$ seconds of wall clock time. The shorter
time is due to the fact that the sizes of the epidemics are smaller than those
on the ERNs.

#### Parameter values.

Model parameterization is done using the literature, calibration, and
guesstimation. We use a number of $N=100,000$ households in a moderate-sized
city with a random number of children in each household of a mean equal to two
and half children. We assume that the mean degree in the children’s network is
$40$ and the mean degree of the parents’ network is $60$ in the ERN. We assume
that the epidemic starts with $I_{0}=10$ initially infected children that are
randomly dispersed among $N$ households. We postulate a disease of a mean
incubation period of $11$ days and a maximum of $16$ days, [39, p.8]. We find
the values for $\beta$ by calibration using the $R_{0}$ values between 12 and
18, [39, p.8]. We assume that a fraction of $5\%$ of the population will
refuse to vaccinate at all (never-vaccinators), due to medical or ideological
reasons. A full table of the definition of parameters and their values can be
found in the supplementary material SII and Table S1.

##  3 Results

Using voting models of selection, the effect of the degree of the injunctive
social norm or peer pressure $\delta$ on epidemic sizes and their peaks as
well as the vaccine uptake is almost not noticeable for selected values of
$\delta$ in $[0.025,0.225]$, Fig. 1. That is true for the case of Erdős-Rényi
(random) network model (ERN) in Fig. 1 (a), (b) and (c), and clearer for
Barabási-Albert network model in Fig. S1 (d), (e) and (f).The degree of
injunctive social norm works differently on the two types of networks on
vaccine uptake and on the sizes and peaks of epidemics as indicated by the
results of the simulations shown in top panels versus bottom panels of Fig. 1.
Irrelevant to the group pressure, vaccine uptake on Barabási-Albert networks
is larger than on Erdős-Rényi networks.

Figure 3.1: Fig. 1. Simulations of sizes of the epidemic, the total number of
vaccinated children, and the peak of the epidemic for different values of
$\delta$ in equation (2.5). Simulations are done on Erdős-Rényi (random)
network model (ERN) in (a), (b), and (c), and on Barabási-Albert network model
(BAN) in (d), (e) and (f). In all of the simulations $P_{adv}=.0001$ and
$\rho=.01$.

In case of Erdős-Rényi (random) network model (ERN) and using the general
Bayesian aggregation rule in equation (2.1), we find richer dynamical
behaviors than those when voter model with an injunctive social norm or peer
pressure $\delta$ is used. First, let us assume that there are enough vaccines
to vaccinate one child in every 100 children every day. When $P_{adv}=.0001$,
as $q$ increases, the pressure imposed on parents increases and leads to
higher uptake of vaccines and, consequently, to a smaller size and peak of
epidemics, Figs. 2 (a), (b), and (c). But when the probability of an adverse
event increases, and so more adverse events occur, as the probability of
perceiving the correct position increases, vaccine uptake levels drop. When
$P_{adv}=.001$, the uptake of the vaccine will continue to increase with the
values of $q<.5$, a pattern that changes for $q>.5$, Figs. 2 (d), (e), and
(f). A higher probability of adverse events of $P_{adv}=.01$, results in a
decrease in vaccine uptake as the value of $q$ increases beyond $q\sim.2$. In
that case, both the size and the peak of the epidemics increase with the
increase in the probability $q$, Fig. 2 (g), (h) and (i).

Figure 3.2: Fig. 2. Simulations of sizes of the epidemic, the total number of
vaccinated children, and the peak of the epidemic on Erdős-Rényi (random)
network model (ERN) for different values of $q$. Simulations are done using
$P_{adv}=.0001$ in (a), (b), and (c), $P_{adv}=.001$ in (d), (e), and (f), and
$P_{adv}=.01$ in (g), (h) and (i). In all of the simulations $\rho=.01$.

Those patterns change when we assume that vaccines are scarce, for instance,
when it is only possible to vaccinate one child in every 1000 children every
day. First, the uncertainty of the outcomes increases, compare panels of Fig.
2 to Fig. 3. The less vaccine available, the less number of adverse cases will
appear. It takes a higher probability of an adverse event to effectively
motivate parents to refuse vaccination of their children, Fig. 3 (b), (e) and
(h). It is noticeable that the peak of the epidemics in that case do not
change significantly with the change in the learning probability.

Figure 3.3: Fig. 3. Simulations of sizes of the epidemic, the total number of
vaccinated children, and the peak of the epidemic on Erdős-Rényi (random)
network model (ERN) for different values of $q$. Simulations are done using
$P_{adv}=.0001$ in (a), (b), and (c), $P_{adv}=.001$ in (d), (e), and (f), and
$P_{adv}=.01$ in (g), (h) and (i). In all of the simulations $\rho=.001$.

In case of Barabási-Albert network model (BAN) and using the Bayesian updating
rule in equation (2.1), the sizes of the epidemics are smaller than those in
ERNs, compare panels of Fig. 2 to Fig. 4, and panels of Fig. 3 to Fig. 5. That
is against the known fact that disease spreads faster on BANs. But that might
be due to the increase in vaccine uptake that we observe in BANs in contrast
to ERNs. The difference between BANs and ERNs is also true for the values of
$q$. When the probability of adverse event increases to $P_{adv}=.01$, vaccine
uptake levels dropped from those levels when $P_{adv}=.0001$ and
$P_{adv}=.001$, compare panel (h) of Fig. 4 with panels (b) and (e). However,
in that case, the learning probability causes a pressure to increase vaccine
uptake even if it causes more adverse events; compare panel Fig. 4 (h) to
panel Fig. 2 (h).

Figure 3.4: Fig. 4. Simulations of sizes of the epidemic, the total number of
vaccinated children, and the peak of the epidemic on Barabási-Albert network
model (BAN) for different values of $q$. Simulations are done using
$P_{adv}=.0001$ in (a), (b), and (c), $P_{adv}=.001$ in (d), (e), and (f), and
$P_{adv}=.01$ in (g), (h) and (i). In all of the simulations $\rho=.01$.

Again, those patterns disappear when we assume that vaccines are scarce, and
they are only available for one child in every 1000 children every day. Like
in the ERNs, the outcome uncertainty increases, compare panels of Fig. 4 to
Fig. 5. But in contrast to the ERNs, the learning probability does not have a
significant influence over the vaccine uptake nor the size and peak of
epidemics; compare panels of Fig. 3 to Fig. 5.

Figure 3.5: Fig. 5. Simulations of sizes of the epidemic, the total number of
vaccinated children, and the peak of the epidemic on Barabási-Albert network
model (BAN) for different values of $q$. Simulations are done using
$P_{adv}=.0001$ in (a), (b), and (c), $P_{adv}=.001$ in (d), (e), and (f), and
$P_{adv}=.01$ in (g), (h) and (i). In all of the simulations $\rho=.001$.

The parameter planes in Fig. 7 for the relevance of the disease $\alpha$ and
the relevance of the adverse event of the vaccine $\gamma$ to the rational
choice component and for different values of $q$ show consistent patterns with
the simulations in Fig. 2. That means that for diseases and opinions spreading
on ERN, the learning effect effectively suppresses the rational perception of
the parent about the reward of vaccination. Parameter planes in Figs. S2 and
S3 show consistent patterns as well.

Figure 3.6: Fig. 6. Parameter planes of $\alpha$ against $q$ and $\gamma$
against $q$ for epidemic sizes, the total number of vaccinated children and
the peak of the epidemic on Erdős-Rényi (random) network model (ERN). In all
simulations, the median value of the simulations is used to plot the parameter
planes that are performed at $P_{adv}=.0001$ and $\rho=.01$.

Similarly, the parameter planes in Fig. 8 for the relevance of the disease
$\alpha$ and the relevance of the vaccine’s adverse event $\gamma$ to the
rational choice component and for different values of $q$ show consistent
patterns with the simulations in Fig. 4. Qualitatively, for diseases and
opinion spreading on BAN, vaccine uptake and epidemics can have a symmetric
reaction to high and low learning probability. Quantitatively, the parents’
rational perception of the payoff of vaccination show influence over vaccine
uptake and epidemics. As perceived disease risk increases, vaccine uptake
increases, and epidemic sizes and peaks decrease, Fig. 8 (a), (b), and (c).
And as the perceived risk of adverse events from vaccine increases, vaccine
uptake decreases, and epidemic sizes and peaks increase, Fig. 8 (d), (e), and
(f). The parameter planes in Figs. S4 and S5 also show consistent patterns.

Figure 3.7: Fig. 7. Parameter planes of $\alpha$ against $q$ and $\gamma$
against $q$ for epidemic sizes, total number of vaccinated children and the
peak of the epidemic on Barabási-Albert network model (BAN). In all
simulations, the median value of the simulations is used to plot the parameter
planes that are performed in $P_{adv}=.0001$ and $\rho=.01$.

The probability of learning of parents can lead to a vaccination consensus in
the case of Erdős-Rényi (random) network model (ERN) and Barabási-Albert
network model (BAN) shown in Figs. 8 (a) and (b) as the number of vaccinators
on the last day of the epidemic. However, a moderate to high learning
probability is required in the case of ERN. The pattern stays exactly the same
as in Fig. 8 (a) for ERN when $P_{adv}=.01$, data are not shown to avoid
redundancy. It changes drastically, however, for BAN when $P_{adv}=.01$, Fig.
8 (c). In the latter, fewer parents end up accepting to vaccinate their
children with a decline over the increase in the learning probability.

Figure 3.8: Fig. 8. Simulations of the final total number of vaccinators on
Erdős-Rényi (random) network model (ERN) and Barabási-Albert network model
(BAN) for different values of $q$. Simulations are done on Erdős-Rényi
(random) network model (ERN) using $P_{adv}=.0001$ in (a). It was done on
Barabási-Albert network model (BAN) when using $P_{adv}=.0001$ in (b) and
using $P_{adv}=.01$ in (c). In all of the simulations $\rho=.01$.

Up to this point, the simulations of the model assumed that the population had
a homogeneous culture. That is, parents show their actual preference or
strategy with probabilities $q_{i,j}\in(q-.05,q+.05)$ for one fixed value of
$q$. In the next part, we examine the effect of the population having a
cultural attribute. That attribute gives rise to a heterogeneous population,
in which the population consists of two groups that use probabilities in two
different regions: $.1\pm.05$ and $.9\pm.05$. We choose $.1$ and $.9$ as
extreme case. We call the group of the latter type, subpopulation with
attribute 1. The proportion of those with attribute 1 can affect the fate of
vaccination uptake and the epidemic.

In the case of an epidemic spreading on an ERN, having a small proportion of
parents with attribute 1, results in a large epidemic size and its peak, as
well as less vaccine uptake. This is apparent in the simulations shown in Fig.
9. In the case of an epidemic spreading on a BAN, heterogeneity in the
population’s culture does not show a significant effect on the epidemics or
vaccine uptake; see simulations in Fig. 10.

Figure 3.9: Fig. 9. Simulations of sizes of epidemic, total number of
vaccinated children, and the peak of the epidemic on Erdős-Rényi (random)
network model (ERN) for different proportions of attribute 1. Simulations are
done using $\rho=.01$ in (a), (b) and (c), and $\rho=.001$ in (d), (e) and
(f). In all of the simulations $P_{adv}=.0001$.

Figure 3.10: Fig. 10. Simulations of sizes of the epidemic, the total number
of vaccinated children, and the peak of the epidemic on Barabási-Albert
network model (BAN) for different proportions of attribute 1. Simulations are
done using $\rho=.01$ in (a), (b) and (c), and $\rho=.001$ in (d), (e) and
(f). In all of the simulations $P_{adv}=.0001$.

If the proportion of parents with attribute 1 is more than one-half,
vaccination becomes a consensus in the case of ERN; see Fig. 11 (a).
Vaccination becomes a consensus in all cases for BAN; see Fig. 11 (b).

Figure 3.11: Fig. 11. Simulations of final total number of vaccinators on
Erdős-Rényi (random) network model (ERN) and Barabási-Albert network model
(BAN) for different proportions of attribute 1. Simulations are done on Erdős-
Rényi (random) network model (ERN) in (a) and Barabási-Albert network model
(BAN) in (b). In all of the simulations are done using $P_{adv}=.0001$ and
$\rho=.01$.

##  4 Discussions and conclusion

In this paper, we introduce a Bayesian aggregation model for boundedly
rational observational social learning as a method of decision-making
concerning vaccination. The learning model is based on social observations of
neighbors on the social network through which information is spreading. Other
models in the literature are shown to be special cases of our model. Some
models, such as the degree of injunctive social norm $\delta$, in contrast,
lack the scale that our model enjoys. Using our new model, we explore the
influence of social learning on the development of consensus on a network. The
model also explores how heterogeneity in the culture of observational social
learning affects vaccine uptake. We used stochastic simulations of disease and
information processes that spread over two overlapping networks. The processes
took part in one of two types of networks: Erdős-Rényi (random) network model
(ERN) and Barabási-Albert network model (BAN). The results of observational
social learning on those different networks and its mutual influence on the
disease spread on the overlapping network were dependent on the network type
and on the vaccine safety and availability. In ERN, as adverse events become
rare due to their small likelihood or inaccessibility to vaccines, the
increase in pressure ensued through declaring the adopted strategy with higher
probability, especially when $q\geq.5$, will result in an increase in vaccine
uptake and eventually vaccination as a social norm. (Notice that $q=.5$ is the
rational agent case.) The vaccine uptake in BAN is relatively higher than in
ERN, and vice versa for epidemic sizes. Furthermore, the higher the
probability of learning when $q\geq.5$ in BAN, the less vaccine uptake will
be.

Based on simulations of epidemic processes on the ERN and BAN, degree
distribution plays a role in the levels of vaccine uptake and parental
acceptance of vaccination. The uptake of the vaccine in the case of BAN is
higher than in ERN. That might be for two reasons: As the disease is known to
spread quickly in BAN, the accumulation of the number of cases will be rapid,
increasing the subjective probability of vaccinating. The second reason is
that the large number of neighbors will increase the inclination toward one of
the two opinions even with small values of $q$.

Vaccine availability and accessibility interact with vaccine adverse events in
a way that could influence parental opinion and thus vaccine uptake. In ERN,
the more the vaccine is available and the less the chances of its adverse
events, it will help to increase the uptake of the vaccine and establish the
vaccination as a consensus. In BAN, that paradigm changes as the degree
distribution has a heavy tail.

Mixed populations, with two different cultures of sharing and perceiving their
opinion about vaccination, could have a significant effect on vaccine uptake.
Having a fraction of the population with low learning probability $q_{1}$ in a
population with $q_{2}>q_{1}$ could decrease the resulting level of vaccine
uptake and increase the size of the epidemic compared to a homogeneous
population with the same learning probability $q_{2}$.

Social studies using surveys and behavioral game experiments might consider
the personal characteristics of each parent that give rise to directional
learning probabilities $q_{i,j}$ and $q_{j,i}$. Population surveys can then be
used to predict vaccine uptake levels. To elicit an increase in vaccine
uptake, it is not enough to consider only the degree of parent link in the
information network to spread awareness. Further efforts to promote social
information exchange in social norm interventions will induce prosocial
decisions about vaccination [3, 4, 5, 40]. According to our model, those
efforts can lead to a consensus on the vaccination opinion and increase
vaccine uptake even in the presence of a fraction of never-vaccinators and
amid the vaccine’s lack of safety and low availability.

##  5 Supplementary Material

A complete description of the model and methods, parameter values, and
auxiliary figures is given in the Supplementary Online Material.

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## S. Supplementary Material

### SI. Model

In this section, we give a complete description of physical and social
networks and the dynamics of disease and information spread.

#### Children physical network

The first network $\mathcal{N}^{C}$ connects children to households, so it is
the network in which disease transmission occurs from child to child. Links
between children represent face-to-face possibilities through school, clubs,
communal activities, etc. These links are assumed to be static; that is, they
do not change over time once they are established. That would be reasonable
since the modeled outbreaks span a few weeks. At this level, we only allow
transmission through the network, and no other transmission routes (such as
environmental ones) are allowed. In that network, we assume that nodes are
formed from a number of $N$ households that are linked through physical
transmission connections for each child (agent) in the household. The set of
neighbors of the household $i$ on this network is denoted by $N_{P}(i)$ and
their size is $n_{P}(i)$.

  1. 1.

Formation. A number of children $C_{i}$ (possibly zero) is randomly assigned
to each household $i$, for $i=1,\ldots,N$, using a binomial distribution with
parameters $n_{c}$ and $p_{c}$. Here, we consider two types of networks:

1) Erdős-Rényi (random) network (ERN): Two households $i$ and $j$ are
physically connected with the probability $p\cdot\sqrt{C_{i}C_{j}}$, which is
proportional to the geometric mean of the number of children in both
households. That probability could be interpreted through Newton’s law of
gravity and deserves future wider investigation. A household with zero
children will not have physical connections. A regular ERN is a special case
where $C_{i}=1$ for all $i$. The parameter $p$ is selected to be less than
$1/n_{c}$.

2) Barabási-Albert Network (BAN): In this network, we connect the first two
households and then connect each of the following households, iteratively,
according to the weighted degree distribution of the preceding households.
That is, the household, say number $j$ ($j=3,4,\ldots,N$), is connected to one
of the households $1,2,\ldots,j-1$ according to their weighted degree
distribution $\dfrac{C_{k}d(k)}{\sum_{i=1}^{j-1}C_{i}d(i)}$,
$k=1,2,\ldots,j-1$, where $d(k)$ is the degree of the household $k$. A
household with zero children will not have physical connections. A regular BAN
is a special case where $C_{k}=1$ for all $k$.

See Fig. S1, for simulation instances of the overlapping ERNs and overlapping
BANs.

  2. 2.

Birth process. We assume that families decide to have a new baby (start a new
pregnancy) with a probability dependent on the number of children per
household. We assume that the probability of a new birth in a household $i$,
if not already in pregnancy, is given by

$\dfrac{\sigma}{1+\exp(k(C_{i}-C^{*}))}.$

After 280 days $C_{i}$ is updated to $C_{i}+1$. The parameter $\sigma$ is the
population-level birth rate. The parameter $k$ measures the sensitivity of the
probability to number of children in the household and $C^{*}$ is the median
for that probability. We estimated $k$ so as to have that probability
approaching zero as $C_{i}$ approaches $n_{C}$.

Humans’ gestation period is assumed to be 280 days. After delivery, parents
either vaccinate or do not vaccinate their new babies. The first 280 days of
the simulation of the model are discarded as a burn-in period.

  3. 3.

Disease transmission within and between households. Initially, a number of
children $I_{0}$ are randomly and uniformly selected to be infected. A new
infection happens in household, say number $i$, on any single day due to
transmission within the household or between households with probability
$1-(1-\beta_{h})^{I(i)}\cdot(1-\beta)^{n_{I}(i)/C_{i}}$; where $\beta$ is the
probability to infect a child in another household (through the physical
network), $\beta_{h}$ is the probability to infect a sibling within the same
household, and $I(i)=\sum_{k=1}^{\ell}I(i,k)$ is the total number of infected
siblings in the same household where $I(i,k)$ is the number of infected
siblings at day/stage $k$ of the incubation period. The number
$n_{I}(i)=\sum_{j\in N_{P}(i)}I(j)$ is the number of infected children
connected to that household $i$ through the network. The number $n_{I}(i)$ is
divided by $C_{i}$ in the probability of infection formula to reflect the
approximating assumption that the children in a household have the same number
of friends on average. Multiple infections can happen on the same day in the
network and the numbers $n_{I}(i)$ and $I(i)$ are updated every day for all
$i$, $i=1,2,\ldots,N$.

  4. 4.

Disease progression. The incubation period is assumed to be distributed
according to a discretized exponential distribution with mean length of
$m_{p}$ days and maximum $\ell$ days. An infected child on day $j$ after
infection moves either to the end of the incubation period (recovers) or stays
infected into the following day with probabilities $Q_{j}$ and $1-Q_{j}$,
respectively, such that

$Q_{j}=\dfrac{\int_{j}^{j+1}f(x)dx}{\int_{j}^{\infty}f(x)dx}$

for $j=0,1,\ldots,\ell-1$, and $Q_{\ell}=1$, where $f(x)$ is the probability
density function of the exponential distribution.

#### Parental social network

The second network $\mathcal{N}^{P}$ overlaps with the children’s physical
network and assumes larger social/information links between parents in
households. Through social networks, parents share information and perceive
opinions to and from network neighbors who are not necessarily physical
neighbors but possibly friends, relatives, colleagues, online friends, etc.
The set of neighbors of household $i$ on this network is denoted by $N_{S}(i)$
and their size is $n_{S}(i)$. To form an overlapping social network with the
children’s network; links in the children’s physical network are retained
randomly with a probability of $p_{re}$. Other links are added to any one of
the updated non-connected households with probability $p_{ad}$.

### SII. Parameter Values

The model simulation is run for a large number of times for different sets of
values of selected parameters. A parameterization of the model is given in
Table 1. The parameters were assigned using the literature, calibration and
guesstimation. The time unit is given in days.

Table S1: Model parameters, their description and base values.

Parameter | Description | Base values
---|---|---
$N$ | Number of households (nodes) | $100,000$
$\sigma$ | Birth rate | $.005$
$n_{c}$ | Maximum number of children per household | $7$
$p_{c}$ | Initial probability of having a child in a household | $.4$
$k$ | Sensitivity of pregnancy probability to number of children | $2.5$
$C^{*}$ | Median number of children in pregnancy probability | $2$
$p$ | Physical connection probability | $.00013$
$p_{re}$ | Probability of retaining a physical connection to form the social network | $.6$
$p_{ad}$ | Probability of adding a connection to the social network | $.0004$
$I_{0}$ | Initial number of infected children | $10$
$\beta$ | Human to human transmission probability through network [29] | $.06$
$h$ | Human to human factor of transmission probability within household | $1.5$
$m_{p}$, $i_{p}$ | Mean and maximum length of infectious period |  $11$, $16$ days
$NV_{0}$ | Initial number of all time non-vaccinators | $5\%\cdot N$
$MS_{0}$ | Initial number of mover-stayer parents | $N-NV_{0}$
$P_{S}$ | Probability to vaccinate with social learning | $-$
$e$ | Vaccine efficacy | $.95$
$p_{adv}$ | Probability of vaccine adverse event | $.0001-.01$
$\alpha$ | Degree of relevance of infectiousness to the decision to vaccinate | $.001$
$\gamma$ | Degree of relevance of vaccine adverse events to the decision to vaccinate | $.01$
$q$ | Probability to give a correct perception of vaccination | $0-1$
$\delta$ | Injunctive social norm | $.025-.225$
$\rho$ | Probability to get access to vaccination | $.001-.01$

### SIII. Supplementary Figures

Figure S1: Fig. S1. Degree histogram for a simulation of Erdős-Rényi (random)
network model (ERN) in (a) and Barabási-Albert network model (BAN) in (b).

Figure S2: Fig. S2. Parameter planes of $\alpha$ against $q$ and $\gamma$
against $q$ for sizes of the epidemic, the total number of vaccinated
children, and the peak of the epidemic on Erdős-Rényi (random) network model
(ERN). In all of the simulations, the median value of the simulations are used
to plot the parameter planes that are performed at $P_{adv}=.001$ and
$\rho=.01$.

Figure S3: Fig. S3. Parameter planes of $\alpha$ against $q$ and $\gamma$
against $q$ for sizes of the epidemic, the total number of vaccinated
children, and the peak of the epidemic on Erdős-Rényi (random) network model
(ERN). In all of the simulations, the median value of the simulations are used
to plot the parameter planes that are performed at $P_{adv}=.01$ and
$\rho=.01$.

Figure S4: Fig. S4. Parameter planes of $\alpha$ against $q$ and $\gamma$
against $q$ for sizes of the epidemic, the total number of vaccinated
children, and the peak of the epidemic on Barabási-Albert network model (BAN).
In all of the simulations, the median value of the simulations are used to
plot the parameter planes that are performed at $P_{adv}=.001$ and $\rho=.01$.

Figure S5: Fig. S5. Parameter planes of $\alpha$ against $q$ and $\gamma$
against $q$ for sizes of the epidemic, the total number of vaccinated
children, and the peak of the epidemic on Barabási-Albert network model (BAN).
In all of the simulations, the median value of the simulations are used to
plot the parameter planes that are performed at $P_{adv}=.01$ and $\rho=.01$.

### SIV. Estimating the Basic Reproduction Number $R_{0}$

The basic reproduction number is defined to be the average number of secondary
infections caused by an index case introduced to a completely susceptible
population till its recovery. To calculate the basic reproduction number as it
is defined for the disease process over a network $\mathcal{N}^{C}$ of
households, we carry out algorithm LABEL:alg:alg1. It is based on running the
disease process for many number of repetition for an index case in a household
that is selected uniformly from the network. Then the expected number of cases
caused by the index case is estimated by the mean of binomial distributions
whose probabilities are found using Bayes’ theorem.

In particular, the probabilities $P_{i}(t)$ and $P_{j}(t)$ are the
probabilities that the index case is the cause of infection of a new case
within the same household $i$ and in the neighbors $j$, respectively. Then the
mean number of infections among $IC_{i}(t)$ and $IC_{j}(t)$ that are caused by
the index case are given by $IC_{i}(t)P_{i}(t)$ and $IC_{j}(t)P_{j}(t)$,
respectively.

### SV. Coding of Simulation.

Our numerical simulations rely on the NumPy-comptatible Python library CuPy
[37], accelerated with NVIDIA CUDA [38] for parallel calculations on Graphical
Processing Units (GPUs). Most of the calculations were done on a CentOS
workstation with $8$ NVIDIA Tesla (Kepler) K80 GPU cards, each having $2496$
CUDA Cores and 12GB memory.

We introduce a sparse storage format for the adjacency matrices of the
networks, where we record only the position of the nonzero elements in the
upper triangular part by a single index number (see Fig. S6). With $k$
denoting the index of the upper triangular entries in column-major order and
$(i,j)$ denoting the corresponding row and column indices the following
relationships hold:

$\displaystyle k$
$\displaystyle=i+\frac{\left(j-2\right)\left(j-1\right)}{2},$ (5.1)
$\displaystyle j$
$\displaystyle=\left\lfloor\frac{3+\sqrt{8k-7}}{2}\right\rfloor,$ (5.2)
$\displaystyle i$
$\displaystyle=k-\frac{\left(j-2\right)\left(j-1\right)}{2}.$ (5.3)

The indexing formulas (5.1)–(5.3) provide a fast and efficient storage and
calculations through parallel implementations on the GPUs.

| 12345
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|
$A=\begin{bmatrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}&0&0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1}&0\\\
0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}&0&0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1}\\\
0&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1}&0\\\
1&0&1&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}&0\\\
0&1&0&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}\end{bmatrix}$
---
|
$\begin{bmatrix}.&1&2&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}4}&7\\\
.&.&3&5&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}8}\\\
.&.&.&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}6}&9\\\
.&.&.&.&10\\\ .&.&.&.&.\end{bmatrix}$
---

Fig. S6. One-index representation of network models.