An individual-level probabilistic model and solution for control of infectious diseases

We present an individual-level probabilistic model to evaluate the effectiveness of two traditional control measures for infectious diseases: the isolation of symptomatic individuals and contact tracing (plus subsequent quarantine). The model allows us to calculate the reproduction number and the generation-time distribution under the two control measures. The model is related to the work of Fraser et al. on the same topic [1], which provides a population-level model using a combination of differential equations and probabilistic arguments. We show that our individual-level model has certain advantages. In particular, we are able to provide more precise results for a disease that has two classes of infected individuals - the individuals who will remain asymptomatic throughout and the individuals who will eventually become symptomatic. Using the properties of integral operators with positive kernels, we also resolve the important theoretical issue as to why the density function of the steady-state generation time is the eigenfunction associated with the largest eigenvalue of the underlying integral operator. Moreover, the same theoretical result shows why the simple algorithm of repeated integration can find numerical solutions for virtually all initial conditions. We discuss the model's implications, especially how it enhances our understanding about the impact of asymptomatic individuals. For instance, in the special case where the infectiousness of the two classes is proportional to each other, the effects of the asymptomatic individuals can be understood by supposing that all individuals will be symptomatic but with modified infectiousness and modified efficacy of the isolation measure. The numerical results show that, out of the two measures, isolation is the more decisive one, at least for the COVID-19 parameters used in the numerical experiments.