Introduction to

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• Introduction to
• Mathematical and
• Computational
• Biology
• Lecture 7B
• Dr. Eduardo Mendoza
• Mathematics Department Physics
  Department
• University of the Philippines
  Ludwig-Maximilians-University
• Diliman Munich, Germany
• eduardom_at_math.upd.edu.ph
  Eduardo.Mendoza_at_physik.uni-muenchen.de
• Department of Computer Science
• Munich University of Applied Science

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Topics to be covered

• Delay Models
• Concepts
• Applications
• Australian sheep blow-fly
• Lemming population in Canada
• Delay Models in Physiology Periodic Dynamic
  Diseases

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1.3 Delay models

• Previous models birth rate considered to act
  instantaneously, no time delay for reaching
  maturity
• Delay differential equation models
• dN/dt f (N(t), N(t T))
• Logistic delay differential equation
• dN/dt rN(t)(1 N(t T)/K)

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Solution heuristics (1)

• Suppose for t t1, N(t1 ) K and that for some
  t lt t1 , N (t T) lt K. Then
• dN/dt gt 0 (since 1 N (t T)/K gt 0)
• For t t1 T, N(t T) N(t1 ) K, hence dN/dt
  0.
• Analogously, for t with t1 T lt t lt t2, dN/dt lt
  0, and t t2, dN/dt 0.
• ? Possibility of oscillatory behavior

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Solution heuristics (2)

• Example dN/dt p/2T (N (t T)) ?
• N(t) A cos pt/2T
• Period 2T

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Applications (1)

• Australian sheep-blowfly (May, 1975)
• Pest of considerable importance in Australian
  sheep farming
• Data from Nicholsons 1957 experiments (over two
  years) regular basic periodic oscillation of
  35-40 days
• In LDDE K food level available
• T delay time need for larva to mature
  
• r unknown parameter

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Applications (2)

• Lack of change with K observed (explained)
• Good fit between model and data
• Encourages use of logistic delay differential
  equations for populations, but care is needed in
  doing this!
• Further applications
• lemming population in Churchill, Canada (May,
  1981) 4 year-period, gestation time T 0.75
  year
• Vole population in Scottish highlands (Stirzaker,
  1975) similar values as with lemmings
• Rodent populations (Myers and Krebs, 1974) 3-4
  year cycles

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Linear Analysis of Delay Population Models
Periodic Solutions (Excerpts from 1.4)

• N 0 unstable with exponential growth
• Need to consider only N K
• Non-dimensionalization
• N ? N(t)/K, t ? rt, T ? tT
• Strictly speaking.N,r,T
• Equation is now
• dN(t)/dt N(t)(1-N(t-T))
• Discussion s. blackboard

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1.5 Periodic dynamic diseases

• Cheyne-Stokes Respiration
• human respiratory ailment manifsted in altered
  breathing pattern
• Amplitude (directly related to breath volume or
  ventilation V) regularly waxes and wanes,
  separated by periods of apnea (very low volume)
• S. blackboard for illustration

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Physiological facts for model

• Level c(t) of arterial carbon dioxide (CO2) is
  monitored by receptors (believed to be in the
  brainstem) ? timelag of T in overall control of
  breathing levels
• Empirical data show ventilation response to CO2
  is sigmodial (s-form) ? use Hill function for
  the model

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Model equation (1)

• V Vmax cm(t T) / (am cm(t T) )
• with
• Vmax maximum possible volume
• a, m determined by experimental data
• (m is called the Hill constant)
• Note we also assume that CO2 removal from blood
  ventilation x level of CO2

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Model equation (2)

• Model equation
• dc(t)/dt p bVc(t)
• with
• p constant CO2 production rate
  in the body
• b empirically determined constant
• For extensive steady-state analysis, cf. pp.22-26
  
• of Murray
• Experimental parameter values estimated
• Co 40 mmHg, p0 6 mmHg/min, V0 7
  liter/min,
• T 0.25 min

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Regulation of Haematopoiesis

• Haematopoiesis formation of blood cell elements
  in the body
• Physiological background
• White and red blood cells, placelets and so on
  are produced in the bone marrow from where they
  enter the blood stream
• When the level of oxygen in the blood decreases,
  this leads to a release of a substance, which in
  turn leads to an increase of blood elements from
  the marrow
• Abnormalities in this feedback system are among
  major causes of haematological diseases

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Model equation (1)

• Let c(t) concentration of cells (the population
  species) in the circulating blood (in units of
  cells/mm3)
• Assume cells are lost at a rate proportional to
  their concentration, say gc, where g has
  dimension (day)-1
• After cell reduction, there is a 6-day delay
  before the marrow releases further cells to
  replenish
• Assume flux ? of cells intro the bloodstream
  depends on cell concentration at an earlier time,
  namely c(t T), where T is the dely

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Model equation (2)

• Model equation
• dc(t)/dt ? (c(t T)) gc(t)
• Possible form for ? (c(t T)) due to Mackey
  and Glass, 1977
• dc/dt ? am c(t T)/ am cm (t-T)
• where ? , a, m, g and T are positive constants

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1.6 Harvesting a single natural population