Birth and Death Processes

1
Birth and Death Processes

• Stochastic Process
• By
• TMJA Cooray

2
Introduction

• Markov processes in continuous time was
  considered in the previous section.
• There we considered the Poisson processes for
  which the new events occur entirely at random and
  quite independently of the (current) present
  state of the system.
• However, there are processes where the
  occurrence/non-occurrence of the new events
  depend in some degree to the present state of the
  system.

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Examples

• Appearance of new individuals births,
• disappearance of existing individuals deaths

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The simple birth and death process

• Usual notations
• X(t)-population size of individuals at time t
  notation pn(t)PrX(t)n
• Assume that
• All individuals are capable of giving birth to a
  new individual
• Chance that any individual giving birth to a
  (producing )new one in time ?t, ??t o(?t) .
  where ? is the chance of a new birth per
  individual per unit time. and all the other
  assumptions as in the Poisson process.

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• Also assume that
• All individuals are capable of dying
• Chance that any individual dying (in time ?t,is
  ??t o(?t) . where ? is the chance of a death
  of an individual per unit time, and all the other
  assumptions as in the Poisson process.
• The chance of no reproduction or no death (no
  change ) in time ?t of the individual can be
  written as 1- (??)?t o(?t) ,
• The chance of more than one of these events ( a
  birth and a death in time ?t) is o(?t)

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• Hence the corresponding probability that the
  whole population of size X(t) will produce a
  birth is ? X(t)?t to first order in ?t and
  probability of one death for the whole
  population of size X(t) is ? X(t)?t to first
  order in ?t and
• probability that the whole population of size
  X(t) will not change is
• 1-(? ?)X(t)?t to first order in ?t

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• If the population size is 0,i.e.. no births or
  deaths can occur. thus we have to start the
  process ,with a non zero population size.
  X(0)n?0. say a.
• then pa(0)1 and pa-1(0)0

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• Now by writing the probability of a population
  size X(t ?t) n at time t ?t is

9

• we use the method of generating functions as
  mentioned in the Poisson processes

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The subsidiary equations take the form
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• This expression and integration is bit tedious .
• Reference
• (Stochastic Processes -Norman T J Bailey , page
  91-95)
• Now let us consider the two cases
• separately
• ?0 when ?gt0 (simple birth process)
• and ?0 when ?gt0 (simple death process)

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The pure birth process

• Assume that no deaths are possible . ?0
• All individuals are capable of giving birth to a
  new individual
• Chance that any individual giving birth to a
  (producing )new one in time ?t, ? ?t . where ?
  is average number births per individual. and
  all the other assumptions as in the Poisson
  process.
• The chance of no reproduction in time ?t of the
  individual can be written as 1- ? ?t ,

13

• Now by writing the probability of a population
  size n at time t ?t is
• if the population size is 0,i.e.. no births can
  occur. thus we have to start the process ,with a
  non zero population size. X(0)n?0. say a.
• then pa(0)1 and pa-1(0)0----(9)

14

• now (8) becomes at t0
• this can be solved in succession starting with
  the above.
• we use the method of generating functions as
  mentioned in the Poisson processes

15

• this is a partial differential equation of simple
  linear type already shown in Poisson processes.

• the subsidiary equations take the form

16

• the first and the last term gives
• Mconstant----(13)
• first and second gives
• this is integrated as

17

• this is now in the form shown in solution of
  partial differential equations
• the general solution can now be written as
• using the two independent integrals (13) and (14)
  ,where is an arbitrary function to be
  determined by the initial conditions.
• thus at t0,

18

• thus by writing
• we get
• hence the arbitrary function is
• applying (17) to ( 15) we get
• this is the generating function of a negative
  binomial distribution function?

19
It is equivalent to
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negative binomial distribution

• picking out the xn terms we get

21

• The mean and the variance for this distribution
  are given as
• Mean value is precisely equal to the value
  obtained for a deterministically growing
  population of size n at time t ,in which the
  increment in time ?t is ??n?t

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The simple (pure) death process

• With all the assumptions and assuming ?0
• We get

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Then the subsidiary equations are
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• the first and the last term gives
• Mconstant----(25)
• first and second gives
• this is integrated as

25

• M(?,0)ea? (at to , na and pa(0)1)
• the general solution can now be written as
• using the two independent integrals (25) and (26)
  ,where ?? is an arbitrary function to be
  determined by the initial conditions.
• thus at t0,

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• thus by writing
• we get
• hence the arbitrary function is
• applying (29) to ( 27) we get
• this is the generating function of a binomial
  distribution function?

27
Now (30 and (31) can be written as
Let pe - ?t and q1- e - ?t
28

• (34) and (35) can be easily identified as the
  generating function of the binomial distribution.

29

• The population ?0(vanishes) as t?infinity.

30

• The mean and the variance of the of this
  distribution are
• The stochastic mean decreases according to a
  negative exponential law.