A hybrid EM and Monte Carlo EM algorithm and its application to analysis of transmission of infectious diseases

Abstract

In epidemics of infectious diseases such as influenza, an individual may have one of four possible final states: prior immune, escaped from infection, infected with symptoms, and infected asymptomatically. The exact state is often not observed. In addition, the unobserved transmission times of asymptomatic infections further complicate analysis. Under the assumption of missing at random, data-augmentation techniques can be used to integrate out such uncertainties. We adapt an importance-sampling-based Monte Carlo Expectation-Maximization (MCEM) algorithm to the setting of an infectious disease transmitted in close contact groups. Assuming the independence between close contact groups, we propose a hybrid EM-MCEM algorithm that applies the MCEM or the traditional EM algorithms to each close contact group depending on the dimension of missing data in that group, and discuss the variance estimation for this practice. In addition, we propose a bootstrap approach to assess the total Monte Carlo error and factor that error into the variance estimation. The proposed methods are evaluated using simulation studies. We use the hybrid EM-MCEM algorithm to analyze two influenza epidemics in the late 1970s to assess the effects of age and preseason antibody levels on the transmissibility and pathogenicity of the viruses.

Figure 1

Epidemic curves of illness onsets of symptomatic cases in the Seattle influenza seasons. The dashed lines denote the relative infectivity levels of the unknown source over time, calculated using rescaled 7-day moving averages of daily case numbers. (a) 1977–1978 season, H3N2; (b) 1978–1979 season, H1N1.

Figure 2

Comparison of the computational efficiency between the traditional MCEM algorithm and the hybrid EM-MCEM algorithm for disease transmission in independent households. Panel (a) shows the distribution of δ_h, the numbers of possible realizations of U_h, across all households over 100 simulated epidemics. Black diamonds along the horizontal axis indicate the values of J used in other panels, Panels (b), (c) and (d) plot the average computational times of importance sampling (black), EM iterations (white) and variance calculation over simulated epidemics versus J, the minimum value of δ_h to apply the MCEM algorithm.