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Modeling the Immune System – W9

Ordinary Differential Equations as Macroscopic Modeling Tool

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Lecture Notes for ODE ModelsWe use the lecture notes “Theoretical Fysiology” 2006 by Rob de Boer, U. Utrecht

available online at

http://theory.bio.uu.nl/rdb/books/tf.pdf

We use a modified version of the slides produced by Jean-Yves Le Boudec for the MIS course during the AY 2006-2007

We will study Chapters 1, 2, 3, 4, and 7

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Goal of Lectures on ODE ModelsKnow the method and limitations of ODE modelsKnow the following concepts

Logistic equations

Saturation functionsLotka Volterra predator prey modelSeparation of timescales

Phase planes NullclinesSteady state analysis

Asympotic stabilityDoubling timeHalf life

Know how to simulate an ODE model

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Chapter 1 and 2Population Growth

Replicating Population Model

Where N = the total number of individuals in a populationb = birth rated = death rate

Convention: state variables (e.g. N) upper case, parameters (e.g., b,d) and independent variable (e.g. t) lower case; all italic (note the difference between differential operator d/d and d); model parameters are strictly positive

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Birth and Death ParametersWhat do they mean ? How can you measure them ?

Birth rate b = normalized number of births per time unitMeasure births every hour, plot ratio births / population

Death rate: d = normalized number of deaths per time unit Measure deaths every hour, plot ratio deaths / population

Issue: assume one can measure only the net growth rate b-d

Inverse of death rate: 1/d = expected life spanMeasure lifetime of each cell

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What does the model tell us ?

(also called expected “fitness” of an individual, reproductive number)

If R0 < 1 N goes to 0If R0 > 1 N goes to ∞

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Equilibrium or Steady State AnalysisDefinition: dN/dt = 0

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Doubling Time

What is the doubling time ?

Deduce another way to measure (b-d)Plot the growth in log log scales; the slope is (b-d)

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Half -Life

Q: Compute the half-life of one individualA: defined as median life assuming exponential lifetime

= ln(2) / d

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A Non-Replicating Population Model

d = dead rateExample: s = production from thymus of anergic self reacting T cellsCheck the unitsDoubling time ?R0 ?Half-life ?

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Non-replicating vs. Replicating Population Models

Non-replicating population (saturation: independent external input balanced by death, with casualties proportional to the population size)

Replicating population (continuous, exponential growing)

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Density Dependent Death

Density Dependent Birth

Interpretation of k ?

“Fratricide” term = Homeostasis

Linear or nonlinear model?

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Steady State Analysis

Density dependent death:

Carrying capacity proportional to fitness (reproductive ratio) R0

Density dependent birth:

Also N* (equilibrium point and in ecology “carrying capacity”)

Carrying capacity little dependent on fitness (reproductive ratio) R0

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The Logistic Growth ModelBoth previous models are can be combinedThey all are special cases of the “Logistic Growth Model”

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Equilibrium Values

For r >0 : growth, asymptotically going to KN= K is the only stable equilibrium

For r <0: decay to 0N=0 is the only stable equilibrium

How do we know ?

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Stability Analysis

Assume case r>0 Two equilibria: N= 0 and N= K from dN/dt = f(N) = 0N= K is the only stable equilibrium

Lyapunov exponents:Nonlinear ODEIdea: linearize f(N) at equilibria (e.g. Taylor expansion)Calculate

As a function of the sign of λ around the equilibrium point we can check whether a small perturbation h will be damped or amplified

Return time (for stable points)

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Unstable equilibrium stable equilibrium

Phase Plot Analysis

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What are the Limitations of these ODE Models ?

Note: all these assumption are NOT characteristic of ODE models in general and most of them can be partially relaxed. For instance:

1. different individuals can be considered at the price of larger ODE systems (i.e. each caste of individual represented by an explicit state variable)

2. Spatial models usually require PDEs but crude spatiality (i.e. individuals placed in a given zone) can be captured with ODEs at the price of additional state variables (i.e. the same individual in a different zone is characterized by a different state variable)

3. Population can be small (but characterized by a lot of interactions) if the model try to reproduce the average behavior over several runs of the same experiment

4. Parameters can vary as a function of independent and state variables; ODEsbecome nonlinear and more difficult to solve with close form solutions

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Chapter 3.Interacting Populations

What does the model ignore ?

Immune reactionHomeostasis

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Healthy Steady-StateDefined as equilibrium when there is no infectionCompute it !

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Other Equilibrium Values

Set second side of equations to 0 and obtain:

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Fitness/Reproductive Number R0

Defined here as the number of infected cells reproduced by an infected cell, in the worst case

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Study Stability by the Method of Nullclines

1. Draw the lines in (I,T) space given steady state conditions.Equilibrium point is at intersection

2. Analyze the direction of vector field and see if system tends to beattracted or not by equilibrium point

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Case δI/β > σ/δT

f1(T,I) f2(T,I)

f1(T,I)=0

f2(T,I)=0

f1(T,I)<0f2(T,I)<0

Only one equilibrium(healthy state)

Healthy state is stable

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f1(T,I)<0f2(T,I)>0

We will see a more systematic method (eigenvalues) later

Two equilibria(healthy state + chronical infection)

Unstable (saddle point)

stable

Case δI/β < σ/δT

f2(T,I)f1(T,I)

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Immune Reaction Model

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Example of Simulation

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Equilibria

How do we get them ?

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Stability of Equilibrium Point

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f1(T,I,E) = 0f2(T,I,E) = 0f3(T,I,E) = 0

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

Ef

If

Tf

Ef

If

Tf

Ef

If

Tf

333

222

111

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Here only x3* is a stable equilibrium

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Chapter 4 / Section 16.3Saturation Functions

Problem: find rate functions that saturate

Hill functions are also called « threshold functions »; h : saturation constant or threshold f(h)=0.5n : degree of nonlinearity, determine curve shape

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Parameter Influence on Hill/Threshold Functions

f(x), n= 1 f(x), n=2

Red: exponentialfunctions

changing h, n = 10 h = 50, changing n

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Infection Model

Rate of infection per infected cell is a saturating function of T

Also called Michaelis-Menten

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Simulations of This Model

T

I

T

I

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Chapter 7Simplification of ODE Model by Separation of

Time Scales

What do we add to previous model ?

Viral population dynamics; viral load = steady state of virus population

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Elimination of the Fastest Time Scale

Not constant: function of I(t)!

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Compare to Immune Reaction Model

Conclusion ?The models are the same, with proper parameter settings

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Elimination of the Slowest Time ScaleDuring therapy of chronically infected patients:

From steady state analysis of the original

4-equations system

Replace E(t) with steady state value E*

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Elimination of the Fastest AND Slowest Time Scale

3-equations system -> 2-equations system:

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Is this a familiar model ?

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Added Value of the Model vs. Raw Data

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Self-Study AssignmentResponsible: IrinaSee distributed assignment (available on Moodle as well)