Section 8.1 Mathematical Modeling with Differential Equations.
Ordinary Differential Equations as Macroscopic Modeling Tool · 2008. 11. 21. · Modeling the...
Transcript of Ordinary Differential Equations as Macroscopic Modeling Tool · 2008. 11. 21. · Modeling the...
1
Modeling the Immune System – W9
Ordinary Differential Equations as Macroscopic Modeling Tool
2
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
3
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
4
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
5
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
6
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 ∞
7
Equilibrium or Steady State AnalysisDefinition: dN/dt = 0
8
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)
9
Half -Life
Q: Compute the half-life of one individualA: defined as median life assuming exponential lifetime
= ln(2) / d
10
A Non-Replicating Population Model
d = dead rateExample: s = production from thymus of anergic self reacting T cellsCheck the unitsDoubling time ?R0 ?Half-life ?
11
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)
12
Density Dependent Death
Density Dependent Birth
Interpretation of k ?
“Fratricide” term = Homeostasis
Linear or nonlinear model?
13
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
14
The Logistic Growth ModelBoth previous models are can be combinedThey all are special cases of the “Logistic Growth Model”
15
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 ?
16
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)
17
Unstable equilibrium stable equilibrium
Phase Plot Analysis
18
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
19
Chapter 3.Interacting Populations
What does the model ignore ?
Immune reactionHomeostasis
20
Healthy Steady-StateDefined as equilibrium when there is no infectionCompute it !
21
Other Equilibrium Values
Set second side of equations to 0 and obtain:
22
Fitness/Reproductive Number R0
Defined here as the number of infected cells reproduced by an infected cell, in the worst case
23
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
24
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
25
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)
26
Immune Reaction Model
27
Example of Simulation
28
Equilibria
How do we get them ?
29
30
Stability of Equilibrium Point
31
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
32
Here only x3* is a stable equilibrium
33
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
34
Parameter Influence on Hill/Threshold Functions
f(x), n= 1 f(x), n=2
Red: exponentialfunctions
changing h, n = 10 h = 50, changing n
35
Infection Model
Rate of infection per infected cell is a saturating function of T
Also called Michaelis-Menten
36
Simulations of This Model
T
I
T
I
37
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
38
Elimination of the Fastest Time Scale
Not constant: function of I(t)!
39
Compare to Immune Reaction Model
Conclusion ?The models are the same, with proper parameter settings
40
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*
41
Elimination of the Fastest AND Slowest Time Scale
3-equations system -> 2-equations system:
42
Is this a familiar model ?
43
Added Value of the Model vs. Raw Data
44
Self-Study AssignmentResponsible: IrinaSee distributed assignment (available on Moodle as well)