2birthdeath - Birth Processes Birth-Death Processes...

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Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Birth-death processes Jorge J´ulvez University of Zaragoza 1 / 47
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Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Outline 1 Birth Processes 2 Birth-Death Processes 3 Relationship to Markov Chains 4 Linear Birth-Death Processes 5 Examples 2 / 47
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Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Outline 1 Birth Processes 2 Birth-Death Processes 3 Relationship to Markov Chains 4 Linear Birth-Death Processes 5 Examples 3 / 47
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Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process) Example: Consider cells which reproduce according to the following rules: A cell present at time t has probability λ h + o ( h ) of splitting in two in the interval ( t , t + h ) This probability is independent of age Events betweeen different cells are independent Time > 4 / 47
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Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process) Example: Consider cells which reproduce according to the following rules: A cell present at time t has probability λ h + o ( h ) of splitting in two in the interval ( t , t + h ) This probability is independent of age Events betweeen different cells are independent Time > What is the time evolution of the system? 4 / 47
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Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process) Non-Probabilistic Analysis Let n ( t ) = number of cells at time t Let λ be the birth rate per single cell Thus λ n ( t )Δ( t ) births occur in ( t , t + Δ t ) Then: n ( t + Δ t ) = n ( t ) + n ( t ) λ Δ t n ( t + Δ t ) - n ( t ) Δ t = n ( t ) λ dn dt = n 0 ( t ) = n ( t ) λ The solution of this differential equation is: n ( t ) = Ke λ t If n ( 0 ) = n 0 then n ( t ) = n 0 e λ t 5 / 47
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Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process) Probabilistic Analysis Notation: N ( t ) = number of cells at time t P { N ( t ) = n } = P n ( t ) Assumptions: A cell present at time t has probability λ h + o ( h ) of splitting in two in the interval ( t , t + h ) The probability of more than one birth occurring in time interval ( t , t + h ) is o ( h ) All states are transient 6 / 47
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Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process) Assumptions: Probability of splitting in ( t , t + h ) : λ h + o ( h ) Probability of more than one split in ( t , t + h ) : o ( h ) The probability of birth in ( t , t + h ) if N ( t ) = n is n λ h + o ( h ) .
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