Variations on the Luria-Delbruck model

The Luria-Delbruck model essentially deals with an intertwining of atwo-types process (sensitive against mutants), whereby individual resistantmutants collectively emerge randomly and grow or die at birth eventssustained by an exponentially growing sensitive population. Werevisit this classical problem and investigate new scenarii involving(sub-)critical branching process opportunities for the mutant sub-familiesgrowth. In such models, the production of mutants appears highly dependenton the way the sensitive population grows. For comparison, thecase of a linearly growing sensitive population is also investigated; inthis context, we derive some results on the structure of the random subsetof times free of mutants where the whole population lacks immunity. Thisrandom set is shown of particular interest would the sub-families growaccording to a (sub-)critical binary branching process.In both scenarii, we give some insight on the ultimate number of mutationsexplaining the total mutant population size whenever an equilibriumdistribution is attained.