A stochastic SIR model with contact-tracing: large population limits and statistical inference

Stéphan Clémençon (1, 2) , Viet Chi Tran (3) , Hector de Arazoza (4)

1 LTCI - Laboratoire Traitement et Communication de l'Information
2 MET@RISK - Méthodologies d'Analyse de Risque Alimentaire
3 LPP - Laboratoire Paul Painlevé - UMR 8524
4 MATCOM - Matematica y Computacion

This paper is devoted to present and study a specific stochastic epidemic model accounting for the effect of \textit{contact-tracing} on the spread of an infectious disease. Precisely, one considers here the situation in which individuals identified as infected by the public health detection system may contribute to detecting other infectious individuals by providing information related to persons with whom they have had possibly infectious contacts. The control strategy, that consists in examining each individual one has been able to identify on the basis of the information collected within a certain time period, is expected to reinforce efficiently the standard random-screening based detection and slack considerably the epidemic. In the novel modelling of the spread of a communicable infectious disease considered here, the population of interest evolves through demographic, infection and detection processes, in a way that its temporal evolution is described by a stochastic Markov process, of which the component accounting for the contact-tracing feature is assumed to be valued in a space of \textit{point measures}. For adequate scalings of the demographic, infection and detection rates, it is shown to converge to the weak deterministic solution of a PDE system, as a parameter $n$, interpreted as the population size roughly speaking, becomes large. From the perspective of the analysis of infectious disease data, this approximation result may serve as a key tool for exploring the asymptotic properties of standard inference methods such as maximum likelihood estimation. We state preliminary statistical results in this context. Eventually, relation of the model to the available data of the HIV epidemic in Cuba, in which country a contact-tracing detection system has been set up since 1986, is investigated and numerical applications are carried out.

Mots clés: contact-tracing; measure-valued Markov process; HIV; large population approximation; maximum likelihood estimation; mathematical epidemiology; stochastic SIR model

Domaines: Probabilités [math.PR]; Théorie [stat.TH]; Statistiques [math.ST]; Santé publique et épidémiologie

Stéphan Clémençon, Viet Chi Tran, Hector de Arazoza. A stochastic SIR model with contact-tracing: large population limits and statistical inference. 2007. ⟨hal-00171717v2⟩