In thisResearch Project, in the framework of the StochasticProcessesTheory,severalprobabilistic and statisticalproblemsrelated to four important classes of modelswillbestudied:
1) ControlledBranchingProcesses.
2) Two-sex or BisexualBranchingProcesses.
3) EpidemicStochasticModels.
4) StochasticModels in System Biology.
The researchwillbefocused on the theoreticalstudy of these classes of population stochasticprocesses and on the developmentof their applications in the context of System Biology, Genetic and Epidemiology. To developit, we have established four lines of workwith the followingspecific objectives:
L1. ControlledBranchingProcesses:
- To develop the inferentialtheory in controlledprocessfromboth, frequentist (bootstrapmethodology, sequentialmethods, robust estimation based on disparitymeasures) and Bayesian (MCMC, particularfilters, ABC) points of view.
- To introduceand to studythe probabilistictheory ofcontinuous time controlledprocesses and of modifiedcontrolledprocessesallowing to model populations whosenumber of individualstends to become stable when the maximum capacity (carrying-capacity) of the ecosystemwheretheydevelopisreached.
L2. Two-sex or BisexualBranchingProcesses:
- To investigatemodifiedbisexualbranchingprocesses (controlledbisexualprocesses, population-size-dependentbisexualprocesses, continuous timebisexualprocesses, etc.), payingespecial attention to the study of the extinction time and to their applications in Population Dynamics.
- To introduce new two-sexbranchingmodelswithrandommating and to developtheirprobabilistic(conditions for extinction and survival,asymptoticbehaviour) and inferentialtheory.
- To introduce new two-sexmultitypebranchingprocessesdepending on the population density, allowing to model some situations in which the reproductive capacity of the couples isdetermined by the pressure of the environmentwheretheydevelop, and determining the evolution in the number of carriers of somegeneslinked to X and Y chromosomes. To developtheirprobabilistic (conditions for extinction, fixation, coexistence,limitbehaviour) andinferential (frequentistand Bayesiana)theory.
L3. EpidemicStochasticModels:
- To introducegeneralizations of the SIR generalepidemic model, allowing to include network epidemicmodels. To studytheir basic probabilistic and inferentialproperties.
L4. StochasticModels in System Biology:
- To introduce new population stochasticmodelsallowing to model somecomplexbiologicalsystems and to developtheirprobabilistic and inferentialtheory.
From the development of theselinesweexpect to obtainresultsatthreelevels. In a pure mathematicallevel, ourresultswillprovidesatisfactoryanswers to the set up problems, underhypotheses as general as possible.In acomputationallevel, wewilldevelop the programs needed for the simulation of the models and the application of the inferentialmethods.Finally,the theoreticalresults,through the developed software, willbeapplied to the problemsthatmotivated the introduction of the differentmodels,sothat the multidisciplinar vocation will becarried out.