Stochastic age-specific transmission model
We formulate a stochastic age-specific transmission model in the general Susceptible(S)-Exposed(E)-Reported(I)-Unreported(U)-Recovered(R) framework. For a particular age group (i) at time (t-1) ((i=1) corresponding to the 0–17 years, (i=2) to 18–44, (i=3) to 45–64 and (i=4) to 65+), we have
$$begin{array}{l}{S}_{i}(t)= {S}_{i}(t-1)-{n}_{S{E}_{i}}(t) {E}_{i}(t)= {E}_{i}(t-1)+{n}_{S{E}_{i}}(t)- {n}_{E{I}_{i}}(t)-{n}_{E{U}_{i}}(t) {I}_{i}(t)= {I}_{i}(t-1)+{n}_{E{I}_{i}}(t)-{n}_{I{R}_{i}}(t) {U}_{i}(t)= {U}_{i}(t-1)+{n}_{E{U}_{i}}(t)-{n}_{U{R}_{i}}(t) {R}_{i}(t)= {R}_{i}(t-1)+{n}_{I{R}_{i}}(t)+{n}_{U{R}_{i}}(t),end{array}$$
(1)
where ({n}_{{XY}_{i}}(t)) represents number of transitions between a class X and class Y for age group (i) at time (t).
The number of transitions from the susceptible to exposed class for group (i) at time (t) is modelled by
$$begin{aligned}{n}_{S{E}_{i}}(t)&sim Poi({S}_{i}(t-1)times {gamma }_{i}(t)times & quad sum_{j=1}beta (t)times {c}_{j,i}(t)times {{I}_{j}(t-1)+{U}_{j}(t-1)}).end{aligned}$$
(2)
Here, (beta (t)) denotes the average infectiousness of an infectious individual and ({c}_{j,i}(t)) is the average number of contacts per day made by age group (j) to (i). Also note that the product (beta (t)times {c}_{j,i}(t)) may represent age-specific transmissibility (of age group (j)) accounting for contacts. We allow and infer two change points of (beta (t)) (one potentially correlates to changes due to the implementation of lockdown and another one to changes due to the lifting of lockdown), i.e.,
$$beta left(tright)=left{begin{array}{ll}{beta }_{0},&quad if; tle {T}_{1} {beta }_{1}={omega }_{1}times {beta }_{0},&quad if ;{T}_{1}<tle {T}_{2 } {beta }_{2}={omega }_{2}times {beta }_{0},&quad if; t>{T}_{2},end{array}right.$$
(3)
where ({T}_{1}) and ({T}_{2}) are the two change points to be inferred (({T}_{2}ge {T}_{1})). ({gamma }_{i}(t)) denotes the susceptibility of group (i) relative to the oldest age group (i.e., ({gamma }_{4}=1)), which is also allowed to change proportionally after lifting the lockdown. Note that ({gamma }_{i}(t)) implicitly incorporates any behavioral effects (e.g., potential reduction of risk of getting infection due to facemask wearing). Transitions between other classes are modelled as:
$$begin{aligned}{n}_{E{U}_{i}}(t)sim & Bin({n}_{S{E}_{i}}(t-{D}_{EU}),{p}_{{U}_{i}}(t-{D}_{EU})) {n}_{E{I}_{i}}(t)=& {n}_{S{E}_{i}}(t-{D}_{EI})-{n}_{E{U}_{i}}(t) {n}_{I{R}_{i}}(t)=& {n}_{E{I}_{i}}(t-{D}_{IR}) {n}_{U{R}_{i}}(t)=& {n}_{E{U}_{i}}(t-{D}_{UR}),end{aligned}$$
(4)
where ({D}_{EI}), ({D}_{EU}), ({D}_{IR}) and ({D}_{UR}) denote the mean waiting times between the indicated two classes. We assume that ({D}_{EI})= ({D}_{EU})=7 days and ({D}_{IR})= ({D}_{UR})=14 days. ({p}_{{U}_{i}}(t)) represents probability that an infection is unreported at times (t) for age group (i), we assume
$${p}_{{U}_{i}}(t)=1-frac{{e}^{{f}_{i}(t)}}{1+{e}^{{f}_{i}(t)}}.$$
(5)
({f}_{i}(.)) is an increasing function with ({f}_{i}(t)={a}_{i}+{b}_{i}times t), where (-infty <{a}_{i}<infty ) and ({b}_{i}ge 0), which is used to model time-varying average reporting rate in a particular age group (i) (which may be increasing due to, for example, increasing efforts for asymptomatic screening and testing). We provide a schematic overview of our modelling framework in Fig. 5.
A schematic illustration of our modelling framework.
We also explore the sensitivity of the assumption ({D}_{UR})= ({D}_{IR}). Specifically, we also consider the scenario when ({D}_{UR})= ({0.5times D}_{IR}) . Our results show that our main conclusions are largely robust towards the assumption (see Table S1 in SI). In particular, the trend of susceptibility increasing with age (prior to lifting the lockdown) and the homogeneity of susceptibility after lifting the lockdown remain robust. However, we do observe that transmissibility is estimated to be higher in the scenario ({D}_{UR})= ({0.5times D}_{IR}), but maintaining the same trend obtained under the assumption of ({D}_{UR})= ({D}_{IR}).
Bayesian model inference and data-augmentation
We infer ({varvec{Theta}}) (i.e. the parameter vector) in the Bayesian framework by sampling it from the posterior distribution (Pleft({varvec{Theta}}|mathbf{z}right)) where (mathbf{z}) include both observed and unobserved data21,22,23,24. Denoting the likelihood by (L({varvec{Theta}};mathbf{z})), the posterior distribution of ({varvec{Theta}}) is (Pleft({varvec{Theta}}|mathbf{z}right))propto Lleft({varvec{Theta}};mathbf{z}right)pi left({varvec{Theta}}right)), where (pi ({varvec{Theta}})) is prior distribution for ({varvec{Theta}}). Markov chain Monte Carlo (MCMC) techniques are used to obtain samples from the posterior distribution. We assumed that, at time 0, the ratio between observed and unreported cases was assumed to be 1/1025. Mortality and seroprevalence data are used to facilitate the estimation of the number of recovered individuals ({R}_{i}(t)). Specifically, knowing that number of recovered is bounded above by the cumulative incidence, prior distribution of the number of recovered individuals ({R}_{i}(t)) was assumed to follow a Uniform distribution bounded above by the cumulative incidence. The cumulative incidence is estimated from the mortality data and cross-sectional seroprevalence data following the approach previously developed by the authors26. As seroprevalence data were not collected for the 0–17 age group, we conservatively assume that ({R}_{i=1}(t)) was bounded above by the estimated cumulative incidence of the 18–44 group. Non-informative uniform priors for parameters in ({varvec{Theta}}) are used (see Supplementary Information (SI)). More details of the inferential algorithm are referred to SI Text in Supplementary Information (SI). Posterior distributions of parameters are given in Figure S2 in SI Figures.
Imputations of missing contacts
Since no children younger than 18 years were surveyed in our data, we imputed (during-pandemic) contacts made by individuals aged 0–17 years. Specifically, following Jarvis et al.9, we take pre-pandemic contacts and rescale them based on the ratio of the dominant eigenvalue of a during-pandemic matrix to dominant eigenvalue of the pre-pandemic matrix. Also, since there are no publicly available pre-pandemic (0–17 years) contact data for the US, we used pre-pandemic estimates from the UK POLYMOD study as a proxy in the imputation27,28.
Source: Ecology - nature.com
