Kristan Schneider How often are you hit by an infection ...

KristanSchneider

Howoftenareyouhitbyaninfection?-Alikelihood

approachtodeterminethemultiplicityofinfection

ROeS2013,Sept.12,2013

Whatismalaria?

•Infectiousdiseasecausedbyparasites(genus

ium;eukaryote)

•40%oftheworld’spopulationatmalariarisk

•Worldwide200-300millioninfections&1-3milliondeathsperyear

•Enormouseconomicdamageeveryyear

Transmissioncycle

Malariacontrol

•Goalofmalariacontrol:

◦reducedeceaseburden⇒drugtreatments

Malariacontrol

◦reducetransmission⇒bednets,vectorcontrol,...

Malariacontrol

Q:Howtomeasureefficiencyofcontrolinterventions?

Malariacontrol

A:Multiplicityofinfection=metricfortransmissionintensity

Malariacontrol

→co-infectionsmightincreasediseaseseverity

Malariacontrol

→metrictransmissionintensity

Malariacontrol

→impactstatisticsbasedongeneticdata

Malariacontrol

→impactstatisticsbasedongeneticdata

Multiplicityofinfection-Whyisitimportant?

•Largenumberofco-infections=hightransmission

Multiplicityofinfection-Whyisitimportant?

•Largenumberofco-infections=hightransmission

•Hightransmission=moregeneticvariation

→Multiplicityofinfection=keyquantityingeneticstudies

Approach

•Nbloodsamples

Approach

•Nbloodsamples

•nstrains(versionsofageneticmarkers)

Approach

•Nbloodsamples

•n ii iobservednumbersampleswithconfigurationii i(0-1vector;lengthn)

Approach

•Nbloodsamples

•Qii iexpectedfrequencysampleswithconfigurationii i

Approach

•Nbloodsamples

•Qii iexpectedfrequencysampleswithconfigurationii i

Likelihood:

Qn ii iii i

Log-likelihood:

n ii ilogQii i

HowtoderiveQii i?

•Assumptions:

HowtoderiveQii i?

•Assumptions:

◦pp p=(p1,...,pn)...frequencyvectorofstrain

HowtoderiveQii i?

•Assumptions:

◦Infectionsrare&independentevents

HowtoderiveQii i?

•Assumptions:

•Implications:

◦Numberofinfectingstrains∼positivePoissondistribution(parameterλ)

HowtoderiveQii i?

•Assumptions:

•Implications:

P(X=m)=

1eλ−1λm

HowtoderiveQii i?

•Assumptions:

•Implications:

P(X=m)=

1eλ−1λm

◦InfectionconditionedonX=mstrainsmultinomiallydistributed∼Mult(m,pp p)

HowtoderiveQii i?

•Assumptions:

•Implications:

P(X=m)=

1eλ−1λm

◦Quantityofinterest=

λ1−e−λ(meanofpositivePoissondistr.)

HowtoderiveQii i?

•Assumptions:

•Implications:

P(X=m)=

1eλ−1λm

◦Quantityofinterest=

λ1−e−λ(meanofpositivePoissondistr.)

◦Likelihood:

L=L(λ,pp p)=−Nlog(eλ−1)+

n∑ k=1

Nklog(eλpk−1)

◦Nk...numberofsampleswithstraink

Results •Aims:

◦MLestimateofparametersθ̂θ θ=(λ̂,p̂p p)?

Results •Aims:

→multidimensionalNewtonmethod(nindependentparameters)

Results •Aims:

◦Existence,uniqueness,numericaleffort?

Results •Aims:

•Results:

◦θ̂θ θ=(λ̂,p̂p p)existsandisunique

Results •Aims:

•Results:

◦Eθ̂θ θNk=Nkexpected=observednumberofsamplescontainingstraink

Results •Aims:

•Results:

◦λ̂foundbyiterating

λt+1=λt−

n∑ k=1

1−NkN(1−e−λt))

n∑ k=1

Neλt−Nk(e−λt−1)

convergesfromallstartingvaluesλ0>λ̂

Results •Aims:

•Results:

◦λ̂foundbyiterating

λt+1=λt−

n∑ k=1

1−NkN(1−e−λt))

n∑ k=1

Neλt−Nk(e−λt−1)

convergesfromallstartingvaluesλ0>λ̂

◦p̂k=−1 λ̂log(

1−NkN(1−e−λ̂))

Results

•Cameroon

Results

•Cameroon

•331bloodsample

Results

•Cameroon

•331bloodsample

Results

•Cameroon

•331bloodsample

Results

•Cameroon

•331bloodsample

Results

•Cameroon

•331bloodsample

Results

•Cameroon

•331bloodsample

•8microsatellitemarkers

Results

•Cameroon

•331bloodsample

•8microsatellitemarkers

302kb313kb319kb379kb335kb363kb383kb429kb

microsatellitelocus

Results •Aims:

◦Confidenceintervalls

Results •Aims:

◦Testingtheestimates

Results •Aims:

•Approach:ProfileLikelihood

◦λfixedL(λ,pp p)→max?

Results •Aims:

◦2(maxλ,pp pL(λ,pp p)

︸︷︷

︸max.LH

−max pp pL(λ0,pp p)

︸︷︷

︸profileLH

)∼χ2 1

Results •Aims:

︸︷︷

︸max.LH

︸︷︷

︸profileLH

)∼χ2 1

◦Findλ,λwith:max pp pL(λ,pp p)=max pp pL(λ,pp p)=maxλ,pp pL(λ,pp p)−χ2 1(1−α)/2=ℓ∗

Results •Aims:

︸︷︷

︸max.LH

︸︷︷

︸profileLH

)∼χ2 1

◦Findλ,λwith:max pp pL(λ,pp p)=max pp pL(λ,pp p)=maxλ,pp pL(λ,pp p)−χ2 1(1−α)/2=ℓ∗

◦Trick:maximizeconditionedonL(λ,pp p)=ℓ∗

→Lagrangemultiplies&(n+1)-dimensionalNewtonmethod

Results

Results:

◦1−αconfidenceintervalsexist&uniquelydefined

Results

Results:

◦Boundsfoundbyiterating2-dimrecursion

Results

Results:

◦Approachconverges(locally)quadratically

Results

Results:

◦TestforH0:λ=λ0vs.HA:λ6=λ0

rejectH0ifλ06∈[λ,λ]

p-value:

2(maxλ,pp pL(λ,pp p)−max pp pL(λ0,pp p)))

Results

Results:

p-value:

microsatellitelocus

Results

Results:

p-value:

microsatellitelocus

302kb313kb 319kb379kb335kb363kb 383kb429kb

Results (θ̂θ θ−θθ θ0)∼N(00 0,I−1N(θθ θ0))

microsatellitelocus

302kb313kb 319kb379kb335kb363kb 383kb429kb

Results

microsatellitelocus

Venezuela

Results

microsatellitelocus

Venezuela

Cameroon

Venezuela

mean±sd

Conclusions

•MLestimatefornumberofco-infections

(keyparameterinmalaria)

Conclusions

•ML-approachworkswell

Conclusions

•Robustnessstudy→futurework

Conclusions

•Includingseveralmarkersatthesametime

Conclusions

McCollumetal,2012

MalariaJ

Schneider&Kim,2010,

Theo.Pop.Biol.

Conclusions

McCollumetal,2012

MalariaJ

Schneider&Kim,2010,

Theo.Pop.Biol.

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