On emergence times in competition processes
Résumé
The estimation of the time by which mutants adapted to control measures emerge is a new challenge in modern epidemiology. Here, we use competition processes, the extension of birth-death processes in higher dimensions, to study the time $S$ by which a particular type emerges in a controlled population through a spontaneous birth or mutation. We give some bounds for the tail of the probability distribution of $S$. Then using properties of quasi-stationary distributions, we give conditions for the existence of an initial law under which $S$ is exponentially distributed. Finally, we provide some examples of our model use for the estimation of the emergence time of pathogen mutants overcoming host resistance in a host-pathogen system.
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