We stratify the population into two sexes (
g ∈ {
f,
m}), 76 1-year age groups (
a = 10,11,12,…,85), and three sexual activity levels (
s ∈ {
none,
low,
high} that denote no, one, and multiple sexual partners during the past 6 months, respectively). Let
Ng, a, u(
t) be the number of individuals in stratum (
g,
a,
u) at time
t, and
cg, a, u be the rate at which these individuals form new sexual partnerships. The age-specific distribution of individuals with different sexual activity levels are based on the sexuality study results published by the Family Planning Association of Hong Kong (FPAHK) [
14]; see Additional file
1 for details. We model assortativity of sexual mixing by age and sexual activity based on the formulation in Walker et al. [
15]. Specifically, given that an individual in stratum (
g,
a,
u) forms a sexual partnership at time
t, the probability that their partner belongs to stratum (
g',
b,
v),
g ≠
g', is
$$ {\displaystyle \begin{array}{l}{\rho}_{g,a,u,b,v}(t)={\varepsilon}_A{\varepsilon}_S\underset{\begin{array}{c}\mathrm{assortative}\ \mathrm{mixing}\ \mathrm{for}\ \mathrm{both}\ \\ {}\mathrm{age}\ \mathrm{and}\ \mathrm{sexual}\ \mathrm{activity}\ \mathrm{level}\end{array}}{\underbrace{\Phi \left(\frac{a-b}{\sigma_g}\right){\delta}_{uv}}}+{\varepsilon}_A\left(1-{\varepsilon}_S\right)\underset{\begin{array}{c}\mathrm{assortative}\ \mathrm{mixing}\ \mathrm{for}\ \mathrm{age}\\ {}\ \mathrm{proportionate}\ \mathrm{mixing}\ \mathrm{for}\ \mathrm{sexual}\ \mathrm{activity}\ \mathrm{level}\end{array}}{\underbrace{\Phi \left(\frac{a-b}{\sigma_g}\right)\frac{c_{g^{\prime },b,v}{N}_{g^{\prime },b,v}(t)}{\sum \limits_l{c}_{g^{\prime },b,l}{N}_{g^{\prime },b,l}(t)}}}\\ {}+\left(1-{\varepsilon}_A\right){\varepsilon}_S\underset{\begin{array}{c}\mathrm{proportionate}\ \mathrm{mixing}\ \mathrm{for}\ \mathrm{age}\\ {}\ \mathrm{assortative}\ \mathrm{mixing}\ \mathrm{for}\ \mathrm{sexual}\ \mathrm{activity}\ \mathrm{level}\end{array}}{\underbrace{\frac{c_{g^{\prime },b,v}{N}_{g^{\prime },b,v}(t)}{\sum \limits_k{c}_{g^{\prime },k,v}{N}_{g^{\prime },k,v}(t)}{\delta}_{uv}}}+\left(1-{\varepsilon}_A\right)\left(1-{\varepsilon}_S\right)\underset{\begin{array}{c}\mathrm{proportionate}\ \mathrm{mixing}\ \mathrm{for}\ \mathrm{both}\\ {}\mathrm{age}\ \mathrm{and}\ \mathrm{sexual}\ \mathrm{activity}\ \mathrm{level}\end{array}}{\underbrace{\frac{c_{g^{\prime },b,v}{N}_{g^{\prime },b,v}(t)}{\sum \limits_k{\sum}_l{c}_{g^{\prime },k,l}{N}_{g^{\prime },k,l}(t)}}}\end{array}} $$
where
δuv has value 1 when
u =
v and 0 otherwise, and Φ(⋅) is the Gaussian kernel. We use the Gaussian kernel to model age assortativity because its shape conforms with intuition as well as the patterns empirically observed in sexual activity surveys from the UK, Australia, and the US [
16‐
18]. In this formulation, age assortativity is controlled by
εA and
σg whereas risk assortativity is controlled by
εS. For simplicity, we assume that
σg is the same for males and females.
Let
Ig, a, u, h(
t) be the prevalence of HPV class
h among individuals in stratum (
g,
a,
u) at time
t. The force of infection from HPV class
h for individuals in stratum (
g,
a,
u) at time
t is
$$ {\lambda}_{g,a,u,h}(t)=\sum \limits_b\sum \limits_v\left[{\alpha}_a{\alpha}_b{\beta}_h{c}_{g,a,u,b,v}^{\ast }(t){\rho}_{g,a,u,b,v}(t)\frac{I_{g^{\prime },b,v,h}(t)}{N_{g^{\prime },b,v}(t)}\right] $$
where
βh is the class-specific baseline probability of transmission per sexual partnership,
\( {c}_{g,a,u,b,v}^{\ast }(t) \) is the adjusted contact rate between stratum (
g,
a,
u) and (
g',
b,
v) (see Additional file
1 for details), and
$$ {\alpha}_a=\left\{\begin{array}{l}1\\ {}1+\frac{\mu -1}{W_2-{W}_1}\left(a-{W}_1\right)\\ {}\mu \end{array}\kern0.5em \begin{array}{l}\mathrm{if}\ a<{W}_1\\ {}\mathrm{if}\ {W}_1\le a\le {W}_2\\ {}\mathrm{if}\ a>{W}_2\end{array}\right. $$
for modeling the effect of age on susceptibility and infectiousness (e.g., to reflect the age dependence of condom use, which increased from 26% in the 15–19 age group to 70% in the 18–27 age group according to the Youth Sexuality Study 2011 by the FPAHK [
19]). The model assumes that susceptibility and infectiousness is (1) highest (
αa = 1) for individuals aged below
W1; (2) linearly decreases from 1 to
μ as individuals age from
W1 to
W2; and (3) remains at
αa =
μ for individuals aged above
W2. The parameters
μ,
W1, and
W2 are inferred during model parameterization.