Séminaire AFRIMath de Probabilités et Statistiques

In this talk, we consider a linear birth and death process with immigration to model infectious disease propagation when contamination stems from both person-to-person contact and the environment. Our aim is to estimate the parameters
of the process. The main originality and difficulty comes from the observation framework. Indeed, counts of infected population are hidden. The performance of our estimators is illustrated both on synthetic data and real data of typhoid fever in Mayotte.

Time homogeneous Markov model has been successfully used to extend the
clas sical survival analysis to the multi-states analysis. This model assumes
that the evolution of the process is independent to the waiting time in the state.
In our clinical problem, this constraint is restrictive. The semi-Markov can be
used to extend the time-homogeneous Markov model with discrete states and
continuous time, because waiting time distributions are considered. We
propose a paramet ric semi-Markovian model applied to the malaria
serological data. We develop a semi-Markov model with time-varying
regression coefficients adapted to medical context. The purposes are
threefold. The firt is to introduce a modified Weibull distribution on the
semi-Markovian process class offering some flexibilities than those often used
as Weibull and exponential Weibull. The second is to discuss a simple
approach based on using appropriate time-dependent covariates effect in a
homogeneous semi-Markov model and to propose a non-parametric
estimation for the time-varying regression coefficients using spline functions.
The third purpose is to discuss the application of these results on semi-Markov
models to malaria serology.

Keyword : Multi-state model, Semi-Markov process, Flexible Weibull distribu
tion, Hazard function, Malaria serology, longitudinal analysis