Introduction to ODE Modeling

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Introduction to ODE Modeling

• Shlomo Taasan
• Carnegie Mellon University

3nd Biodefense Summer School, August 2008
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Plan
We will learn in this tutorial translating
biological knowledge to differential equations
models In the afternoon Lab we use matlab to
simulate models generate graphs, make
predictions, ... In particular we will
model Reactions, Trafficking, Simple
infections Participants will also have an
opportunity to play with more complex
models Lotka-Volterra periodic
solutions Lorenz model chaos

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Modeling
Biology Driven Models Biology is understood and
is translated into equations, reactions,
graphs, Data Driven Modeling Use experimental
data only to construct models Main ingredients
Objects molecules (cytokines/chemokines/...),
cells (Macrophages, Neutrophils,
...), organs( lymph node, spleen, .., lung,.. )
Actions trafficking/migration, interaction
(activation/inhibition), proliferation
Differential equations are about rate of change
of quantities

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Ordinary Differential Equations (ODE)
a some quantity examples cell count, receptor
expression level, cell damage, ... We write ODE
as da/dt f where f may be a complex
formula We interpret this ODE as da f dt
- the change in a during a short time interval
dt is equal to f times dt f may depend on
time or may depend on a or both.

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Basic Examples

• da/dt 0
• - This means da 0 dt 0 ? change in a is
  0, ? a does not change
• da/dt 1
• - This means da dt ? a changes by dt
• 3. da/dt - a da - a dt ? a changes by
  a dt
• - This means that a decreases, and the reduction
  is large when a is large and getting small when a
  is getting smaller.
• Lets look at the results of simulation of these
  equations.

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Want to model

• - interactions in the immune system
• cell-cell, virus/bacterium cell,
• molecule (cytokine/chemokine)-cell, etc
• - trafficking
• - Natural killer trafficking between organs in
  the body
• - Dendritic cell migration from tissue to
  lymph-node
• Spread of disease in a population (in a given
  location)
• HIV, Influenza A
• ?? Pandemic worldwide spread of an infection
• focus on the spatial aspect spread between
  countries, continents

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Preparation for Modeling Some Syntax
Syntax A ? 0 Meaning A dies, Neutrophil
goes apoptosis, bacteria die etc. Syntax
A ? B Meaning A changes into B
for example A macrophage, B activated
macrophage Syntax A B ? C Meaning A and
B interact to give C or If A meets B then C is
produced.

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Modeling Reactions - The Law of Mass Action

• The rate of change of products is proportional to
  the product of reactants concentration
• A ? 0
• The only reactant (left side) is a
• ? rate of change is proportional to a,
• ODE da/dt -ka (minus sign since we
  loose a)
• A ? B
• Similar to the previous case but here one B is
  created per each A that disappear
• ODE
• da/dt -ka as before but we also have
• db/dt ka here the sign is

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Modeling Reactions cont.

• A B ? C
• Here the reactants (left side) is A and B, the
  product (right side) is C.
• dc/dt k ab C is created at a rate
  proportional to the product of the concentration
  of A and B
• da/dt - kab The rate of change of A is
  tha same as the rate of change of C per each C
  that is created one A is lost
• db/dt - kab, similar to A.

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Modeling Trafficking
An example Macrophages are trafficking between
lung to Lymph node and back Want to know the
number of macrophages in lung ad Lymph node as
time progress. L Number of Macrophages in the
Lung LN Number of Macrophages in the Lymph
Node Assumption When a macrophage leaves the
lung it ends at the lymph node and vice versa.
The rate of trafficking is proportional to the
number of cell. This sounds a lot like our
reactions before.

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Trafficking cont.

• We use our syntax
• L ? LN and LN ? L written also as L ??
  LN
• The rate at which cells arrive to the lymph node
  from the lung is proportional to the number of
  cells in the lung.
• Similarly, rate at which cells arrive to the lung
  from the lymph node is proportional to the number
  of cells in the lymph node.
• The ODE
• dL/dt -k1L k2 LN loss gain
• dLN/dt k2LN k1L loss gain

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Modeling Infection-The SIR model
Population has three groups Susceptible (S),
Infected (I) and Recovered (R) The dynamics is
expressed in the reactions S I -gt I I
(rate r) I -gt R (rate a) A difficulty
more reactions/ I is affected form two different
equations. How to construct the reaction? -
each reaction is independent of the other - they
appear simultaneously - the rate of change of a
product is a sum of change coming from
all reactions

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SIR Model

dS/dt - rSI dI/dt rSI - aI dR/dt aI
Complete ODE Model SUM of contributions from
all reactions
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SIR model

The differential equations dS/dt - rSI
dI/dt rSI aI dR/dt aI This model is
more interesting. We change the parameters a, r/
We can also change the initial values for S,R,I
and see what happens. When to expect epidemic? A
relation between parameters Such questions can
be answered using some mathematical analysis. In
this lectures we do it by simulation. -- we will
do it in the lab
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An HIV model
The HIV virus targets specific cells, the CD4 T
cells. These cells may get infected and serve as
a virus producing factory. In HIV infection the
main problem is the decline in the number of CD4
T cells that are essential for protecting the
body form different pathogens. It is important
to understand the dynamics of the CD4 cell count
as a function of time. In this simplified model
(Perelson) we consider three populations T -
Target cells (CD4 T cells) I - Infected
cells V - Virus

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HIV model cont.

• Model assumptions
• -gt T (lambda) target cells production
• T -gt 0 (d) target cells natural death
• T V -gt I V (k) target cell becomes
  infected by virus
• I -gt 0 (delta) infected cells death
• I -gt I V (p) virus replication in
  infected cells
• V -gt 0 (c) virus clearance
• We construct the equations similar to the SIR
  model.
• Each reaction contribute to changes in several of
  the variables. We add all the changes together
  for each variable separately

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HIV Model
Complete ODE Model is SUM of contributions
from all reactions
dT/dt lambda d T k V T dI/dt k V
T delta I dV/dt p I c V
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HIV model cont.

• The ODE
• dT/dt lambda d T k V T
• dI/dt k V T delta I
• dV/dt p I c V
• This model has 6 parameters that may affect the
  behavior.
• We will study this in the lab

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Quick Manual From Reactions To ODE
Complete ODE Model is SUM of contributions
from all reactions
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• Now You are Ready to Do Your Own ODE Models