Branching Processes History And Examples Biology Essay
Ramifying Procedures have traditionally been studied as portion of Markov Processes and Renewal Theory. Traditionally they have been considered as a tool for applications in natural scientific disciplines and more peculiarly in the Fieldss of Biology, Microbiology and Epidemiology. One can see the correlativity as a Branching Procedure can depict populations harmonizing to the birth and/or decease rates. In add-on, a great involvement exists in the uninterrupted Branching Processes, peculiarly when they can be written in the signifier of Stochastic Differential equations. Another facet of the Branching Processes that we are interested in is the designation of households of calculators for their parametric quantitiesA Branching Process ( BP ) is a stochastic procedure, and more specifically a Markov procedure, that theoretical accounts the size of the population at a given clip T, based on premises on the length of the life of any single and the ensuing coevals that is produced by that peculiar person.
It comes as no surprise that the procedure was foremost described, in its simplest signifier, in an effort to analyze the endurance of household names. The first recorded bookman to work on that job was the Gallic statistician Irenee-Jules Bienayme in 1845 [ Wiley, 1975 & A ; Heyde and Seneta, 1873 ] . His success on the topic was that he attempted to stand for mathematically the extinction of baronial household names based on the average figure of males ( offspring ) for every male ascendant of the anterior coevals. Though he lacked the cogent evidence, he did pull off to put up a first preparation.
The procedure is more known as the Galton-Watson Process based on the work of English Lord and scientific discipline bookman Francis Galton and the mathematician Rev. Henry W. Watson. Galton was interested in look intoing the survival/extinction of English blue family names. He proceeded to print the inquiry in the 1873 Educational Times. Watson answered his enquiry with a solution, which led to the publication of [ 2 ] .
As such, the procedure is either referred to as Galton-Watson ( GW ) procedure or Bienayme-Galton-Watson ( BGW ) procedure. The procedure is normally denoted as the household with the size of the population at clip.Definition 3.1 ( Branching Process ) ( Kimmel and Axelrod, 2002 )Let be a household of non-negative random variables defined on with elements I‰ , the figure of bing members of the settlement at clip T and I‰ the index of the coevals. Let besides the birth of the original ascendant to happen at, the life anticipation of the ascendant at coevals I‰ and the count of the posterities occurred at decease of the ascendant. Then:Since it has a self perennial belongings we havein distributionReplacing ( 3.2 ) in ( 3.
1 ) we get:Next, it is of import to discourse the chance bring forthing map ( p.g.f ) of the procedure. We use M. Kimmel and D.
Axelrod ‘s definition as it appears in [ 3 ] :
Definition 3.2 The Probability Generating Function
The pgf of a -valued random variable Ten is a map
So, in kernel we can follow their notation. In add-on, the undermentioned theorem is a aggregation of belongingss of the p.g.
f. as besides provided in [ 3 ]Theorem 3.1 Properties ( Kimmel and Axelrod, 2002 )Suppose is a valued random variable with pgf which may non be proper. Besides assume the non-triviality status:, so:is non-negative and uninterrupted with all derived functions on [ 0,1 ) .
Under ( 2.6 ) is increasing and convex.If is proper, ; otherwise.If is proper, the kth factorial minute of, , is finite if and merely if is finite. In such instance.If and are independent valued random variables soIf is a valued random variable and is a sequence of iid valued random variables independent of, so has the pgf.Suppose that is a sequence of valued random variables,exists for each if and merely if the sequence converges in distribution to a random variable. Then is the pgf of the bound of.
By the relationship ( 3.3 ) and theorem 3.1, one can infer that:As we have discussed the general belongingss of Branching procedures, we would wish to continue to a categorization that depends on the lifetime of the members of the settlement.Definition 2.3 ( BGW or GW ) The Galton Watson ProcessLet be Branching procedure and presume the ascendant generates progenies at clip of decease, where. Furthermore, assume that the lifetime is indistinguishable for each member and it is equal to 1.
Then the ramification Process is a GW procedure.Definition 2.4 Markov Branching ProcessLet be a Branching procedure whose persons have uninterrupted life-times that are exponentially distributed.
Then the procedure is a Markov Branching procedure.Definition 2.5 Bellman-HarrisLet be a Branching procedure where the life-time of the members follows a non-negative random variable. Then the procedure is a Bellman-Harris Branching procedure.Figure. Example of a Galton-Watson Branching ProcessNext we want to discourse the mean of the procedure, i.e.
. It will be stated for the GW procedure, though it is true for all instances. Based on the belongingss from Theorem 3.1 and Definition 3.3 we have that, andIn add-on, we have that, composed T times, and we can therefore acquire thatBecause of ( 3.10 ) we can discourse the criticalness of the procedure based on the value m, i.e.
the trichotomy:For, , hence supercriticalFor, , hence criticalFor, , hence subcriticalIn the last two instances one can easy see that the procedure does non hold adequate energy, i.e. the members of the settlement are non bring forthing plenty offspring to replace the original members.
In fact, population will extinct about certainly.Next we would wish to demo the chance bring forthing map for the uninterrupted Markov Branching Process ( Kimmel and Axelrod, 2002 ) . By definition, the life-time of an single follows an exponential distribution with parametric quantity I» .
Then the cumulative distribution for Y, the life-time of the person, takes the signifier, which gives the p.d.f.Any ascendant, during its life-time, will bring forth progeny harmonizing to the p.
Specifying as the entire population size at clip T, so is a uninterrupted clip Markov Process with initial status. Our intent here is to deduce a differential equation for the p.g.f for the population size. We foremost use the fact thatTaking and little plenty we getAs such we can composeNow, we divide both sides by and take the bound as goes to zero so that, which by using L’Hospital regulation we get:, with initial status Q ( s,0 ) =s and alone solution ifExample 2.1 Drug Resistance in Cancer Cells ( Kimmel and Axelrod, 2002 ) & A ; ( Allen 2003 )As mentioned earlier, Ramifying Procedures can be applied in assorted Fieldss, including but non limited to Biology.
What we found peculiarly attractive was its application to malignant neoplastic disease and malignant neoplastic disease therapy. An first-class illustration is in chemotherapy and drug opposition found in assorted books ( Kimmel and Axelrod, 2002 ) & A ; ( Allen 2003 ) . In the scenario presented here the being of two different types of malignant neoplastic disease cells is assumed: Type A being the drug sensitive cells in a tumour and Type B being the drug immune cells.
Define Y as the clip it takes for a cell to split ( split ) , which y is traveling to be exponentially distributed with parametric quantity. Besides lets define p the chance that out of the two generated cells from a Type A cell one of them will be a Type B cell. Another sensible premise is that every Type B cell will generated ever two type B cells. In such a instance a multi-type ramification procedure is necessary.
Therefore the p.g.f.
s satisfy:, andBesides, the set satisfying ( 3.16 )Then, by separation of variables we have the set of differential equations, with initial statusOnce ( 3.20 ) and replace in ( 3.19 ) to acquireThe most interesting consequence for this illustration comes when the chance of no immune cells at clip T is evaluated, i.e:It is easy to see that the bound of ( 3.23 ) as, is equal to nothing.
This is a really interesting consequence as it indicates that without any outside influence, the chance of holding no immune cells is zero, except when. If P is really equal to zero, it would connote that the chance of a drug sensitive cell to bring forth a drug resistant cell is itself nothing ( Allen 2003 ) .R.
W. Brown, Estimation of Branching Processes with Immigration by Adaptive Control, Master ‘s Thesis supervised by B. Pasik-Duncan, 2000.H.
W. Watson and F. Galton, On the Probability of the Extinction of Families, Journal of the Anthropological Institute of Great Britain, volume 4, pages 138-144, 1875.
M. Kimmel and D.E. Axelrod, Branching Processes in Biology, Springer, 2002.L.J.S Allen, An Application to Stochastic Processes with Applications to Biology, Prentice Hall, 2003B. Bercu, Weighted appraisal and tracking for ramifying procedures with in-migration, IEEE Transations on Automatic Control, 46, pp.
43-50, 2001.B. Bercu, Weighted Estimation and Tracking for ARMAX Models, SIAM J. Control and Optimization, Vol. 33, No.
1, pp. 89-106, January 1995.J. Winnicki, Estimation of the Discrepancies in the Branching Process with Immigration, Prob. Th. Rel. William claude dukenfields, vol 88, pp.
77-106, 1991.A. J. Gao and B. Pasik-Duncan, Stochastic Linear Quadratic Adaptive Control for continuous-time First Order Systems, Systems & A ; Control Letters 31, pp. 149-154, 1997P. Jagers, Branching Processes with Biological Applications, Wiley, 1975C.
c. Heyde and E. Seneta, The simple ramification procedure, a turning point trial and a cardinal individuality: a historical note on I.J. Bienayme, Biometrika, 59, 680-683, 1972
As we mentioned in the anterior chapter, we were attracted to ramifying procedures due to the biological applications and more peculiarly the applications in malignant neoplastic disease therapy and chemotherapy. What attracted us to look into Branching Process ‘s calculators for its parametric quantities is the work of Bernard Bercu.
The field is non needfully new, and many writers have provided calculators. The invention in Bercu ‘s attack though is that he introduces the procedure in its ARMAX signifier and by taking appropriate signifier of the control map he can pull strings the procedure with fewer limitations on the mean and discrepancy. As such, we present his attack every bit good as some of his theorems that we want to retroflex in the uninterrupted instance. The undermentioned two sections cover foremost the scenario with no Immigration and so the scenario with Immigration.
BGW ( No Immigration )
As we discussed earlier, the value of is of great significance.
Estimating, every bit good as estimating, is a disputing undertaking that requires rigorous premises. As the LSE does non needfully supply strong convergence for the household of calculators, in [ 7 ] , B. Bercu offers an attack by using Weighted Least Squares ( WLS ) and appropriate weights. Though his ultimate undertaking is to gauge the parametric quantities in the BGWI ( Bienayme-Galton-Watson with Immigration ) scenario, it is merely sensible to ab initio look into the instance where Immigration is zero ( nothing ) i.e. :, where is the size of the settlement for coevals N, is the progeny count of each member of the n-th coevals and is an adaptative control.
The intent of the control is to command X, i.e. hike it when it drastically declines to zero or coerce it lower when it drastically grows. Besides, another of import component of the attack in [ 7 ] is the revising of ( 2.1 ) in ARMAX signifier i.e. by putingand, whereIn order to explicate the calculators the quadratic standard is, which is minimized by, where.
Besides, the discrepancy can be estimated byand the pick of the control for is, where is a sequence of non-negative whole number valued random variables and P the projection operator on.This peculiar pick of control satisfies the intents we stated earlier. By using the projection operator to the Natural Numberss, we are assured that the summing up makes sense. In add-on, for the instance where, i.e. the procedure is running out of energy and hence would decease out, it forces is at least equal to 1. Furthermore, due to the apparatus, the undermentioned theorems and lemmas were obtained ( Bercu 1999 ) :Theorem 1. Assume that has a finite minute O order greater than 2 and that converges a.
s. to an whole number. If we use the adaptative control from ( 2.6 ) , so is a strongly consistent calculator of m.a.s.In add-on if, we have the cardinal bound theorem
the jurisprudence of iterated logarithma.s.
and the quadratic strong jurisprudencea.s.In fact, the above Theorem ( Bercu, 2001 ) is in kernel the base for what we want to turn out in the following chapter for uninterrupted instance as it covers the chief belongingss for appraisal and most significantly it provides the strongest rate of convergence possible for a household of calculators.
In add-on, the same belongingss were shown ( Bercu, 2001 ) for the household of calculators for the discrepancy:Theorem 2. Assume that has a finite minute O order greater than 2 and that converges a.s. to an whole number. If we use the adaptative control from ( 2.6 ) , so is a strongly consistent calculator of.a.s.
In add-on Lashkar-e-Taiba be the 4th minute of and allow, so we have the cardinal bound theorem
the jurisprudence of iterated logarithma.s.and the quadratic strong jurisprudencea.
BGWI ( with Immigration )
Another interesting consequence ( Bercu, 2001 ) is when the same attack is used in the instance where the BGW procedure has non-zero Immigration involved ( therefore the BGWI instance ) given by, in which instance we are besides interested in the parametric quantity of, viz. I» and. So, in similar manner he sets ( Bercu, 2001 )and, hence achieving the stochastic arrested development equationwith andThen the quadratic standard, with, andThis was solved in [ 6 ] to acquirewhere, S deterministic, symmetric and positive definite matrix.
For the discrepancies:
so the suggested household of calculator iswhere and, Q deterministic, symmetric and positive definite matrix. Similarly to the BGW scenario, the adaptative control would be expected to beUsing the above control though, the calculator for I? is non strongly consistent. Therefore an excitement ( is considered, i.e.where ( is an exogenic bounded sequence of i.
i.d. positive whole number valued random variables, with V the nondegenerate distribution of ( .The two Lemmas and two Theorems that we need from ( Bercu, 2001 ) are:Lemma 1. Assume that converges to an whole number By utilizing the adaptative control in ( 3.9 ) , we get a.s.whereTheorem 3.
Assume that converges to an whole number By utilizing the adaptative control in ( 3.9 ) , so is a strongly consistent calculator of I? .a.s.In add-on assume that and hold finite minutes higher than 2, so the cardinal bound theorem is
the jurisprudence of iterated logarithma.
Assume that converges to an whole number and that that and have finite minutes of order 4. If the adaptative control from ( 3.9 ) is used so a.s.
:Finally, the last theorem ( Bercu, 2001 ) we would wish to reference is:Theorem 4. Assume that converges to an whole number and that that and have finite minutes of order 4. By utilizing the adaptative control in ( 3.9 ) , so is a strongly consistent calculator of I· .a.s.The cardinal bound theorem
and the jurisprudence of iterated logarithma.s.where and.