SARS-CoV-2 Transmission in Italy and Vaccination Status: A Novel Approximate Bayesian Computation Approach for Inferring Compartmental Models From Multiple Data Sources

Abstract

INTRODUCTION
Infectious disease spread is commonly modelled using stochastic compartmental models. However, often model parameters cannot be fully identified from a single data source. Therefore, reliable inference requires the integration of multiple epidemiological datasets, each providing partial and complementary information. The joint analysis of heterogeneous and interdependent data sources, combined with the complexity of realistic compartmental models, typically leads to an analytically intractable likelihood function, making standard Bayesian inference infeasible. We therefore resort to likelihood-free methods, specifically Approximate Bayesian Computation (ABC).
OBJECTIVES
We study SARS-CoV-2 transmission dynamics in Italy during May–October 2022, when the vaccination campaign was in its final phase and vaccination coverage in the population was stable. The aim is to estimate the basic reproduction number stratified by vaccination status, exploiting two complementary data sources: daily COVID-19 deaths from the Protezione Civile (PC) and COVID-19 mortality records stratified by vaccination status from the Istituto Superiore di Sanità (ISS). To this end, we develop a (SIRD)² compartmental model and an ABC algorithm that jointly assimilates both data sources. 
METHODS
The (SIRD)² compartmental model partitions the Italian population into vaccinated and unvaccinated sub-populations with distinct transition rates. Specifically, infection transmission is governed by different reproduction numbers, depending on the vaccination status of the person who is infected. The estimation methodology relies on an ABC algorithm based on a Double Nested Simulator (DNS). The DNS employs a macro-simulator to generate the whole latent data-generating process, i.e. the trajectories of all model compartments, including the daily time series of COVID-19 deaths, and a second simulator that, using the output of the macro-simulator, produces the number of COVID-19 deaths occurring in different time windows, stratified by vaccination status. While the macro-simulator mimics daily data made available by PC, the second simulator mimics contingency tables reported in periodic reports issued by ISS. The ABC algorithm returns samples from an approximate posterior distribution of model parameters conditioned on both data sources simultaneously.
RESULTS
Pooling the PC and ISS sources within the DNS-ABC framework substantially improves parameter identifiability relative to single-source analyses, reducing posterior uncertainty on the vaccination-status-specific reproduction numbers. The stratified (SIRD)² model successfully disentangles the transmission dynamics of vaccinated and unvaccinated individuals over the study period. To ensure parameter identifiability, we fix mortality and recovery rates and estimate R_(0,V) and R_(0,N), the basic reproduction numbers among susceptible vaccinated and unvaccinated individuals, respectively. The posterior mean of R_(0,V) is 2.567 (95% credible interval: 2.358–2.778), whereas of R_(0,N) is 1.213 (95% credible interval: 0.876–1.551). Predictive performance further confirms the advantage of multi-source integration. 
CONCLUSIONS
The proposed DNS-ABC framework, together with the (SIRD)² model, enables the estimation of group-specific reproduction numbers by stratifying transmission according to vaccination status. The multi-source ABC approach is well suited to settings lacking a single comprehensive data source, where heterogeneous partial data can be integrated to enable inference. Since the basic reproduction numbers capture both biological transmissibility and behavioural factors, the results suggest that the latter may have contributed to the higher reproduction number estimated among vaccinated than among unvaccinated individuals.