Modeling the Immune System W9. Ordinary Differential Equations as Macroscopic Modeling Tool

Transcription

1 Modeling the Immune System W9 Ordinary Differential Equations as Macroscopic Modeling Tool 1

2 Lecture Notes for ODE Models We use the lecture notes Theoretical Fysiology 2006 by Rob de Boer, U. Utrecht available online at We use a modified version of the slides produced by Jean-Yves Le Boudec for the MIS course during the AY We will study Chapters 1, 2, 3, 4, and 7 2

3 Goal of Lectures on ODE Models Know the method and limitations of ODE models Know the following concepts Logistic equations Saturation functions Lotka Volterra predator prey model Separation of timescales Phase planes Nullclines Steady state analysis Asympotic stability Doubling time Half life Know how to simulate an ODE model 3

4 Chapter 1 and 2 Population Growth Replicating Population Model Where N = the total number of individuals in a population b = birth rate d = 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 4

5 Birth and Death Parameters What do they mean? How can you measure them? Birth rate b = normalized number of births per time unit Measure 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 span Measure lifetime of each cell 5

6 What does the model tell us? (also called expected fitness of an individual, reproductive number) If R 0 < 1 N goes to 0 If R 0 > 1 N goes to 6

7 Equilibrium or Steady State Analysis Definition: dn/dt = 0 7

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) 8

9 Half -Life Q: Compute the half-life of one individual A: defined as median life assuming exponential lifetime = ln(2) / d 9

10 A Non-Replicating Population Model d = dead rate Example: s = production from thymus of anergic self reacting T cells Check the units Doubling time? R 0? Half-life? 10

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) 11

12 Density Dependent Death Linear or nonlinear model? Interpretation of k? Fratricide term = Homeostasis Density Dependent Birth 12

13 Steady State Analysis Density dependent death: Also N* (equilibrium point and in ecology carrying capacity ) Carrying capacity proportional to fitness (reproductive ratio) R 0 Density dependent birth: Carrying capacity little dependent on fitness (reproductive ratio) R 0 13

14 The Logistic Growth Model Both previous models are can be combined They all are special cases of the Logistic Growth Model 14

15 Equilibrium Values For r >0 : growth, asymptotically going to K N= K is the only stable equilibrium For r <0: decay to 0 N=0 is the only stable equilibrium How do we know? 15

16 Stability Analysis Assume case r>0 Two equilibria: N= 0 and N= K from dn/dt = f(n) = 0 N= K is the only stable equilibrium Lyapunov exponents: Nonlinear ODE Idea: 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) 16

17 Phase Plot Analysis Unstable equilibrium stable equilibrium 17

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; ODEs become nonlinear and more difficult to solve with close form solutions 18

19 Chapter 3. Interacting Populations What does the model ignore? Immune reaction Homeostasis 19

20 Healthy Steady-State Defined as equilibrium when there is no infection Compute it! 20

21 Other Equilibrium Values Set second side of equations to 0 and obtain: 21

22 Fitness/Reproductive Number R 0 Defined here as the number of infected cells reproduced by an infected cell, in the worst case 22

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 be attracted or not by equilibrium point 23

24 f 1 (T,I) f 2 (T,I) f 2 (T,I)=0 f 1 (T,I)<0 f 2 (T,I)<0 f 1 (T,I)=0 Healthy state is stable Only one equilibrium (healthy state) Case δ I /β > σ/δ T 24

25 f 1 (T,I) f 2 (T,I) f 1 (T,I)<0 f 2 (T,I)>0 stable Unstable (saddle point) Case δ I /β < σ/δ T We will see a more systematic method (eigenvalues) later Two equilibria (healthy state + chronical infection) 25

26 Immune Reaction Model 26

27 Example of Simulation 27

28 Equilibria How do we get them? 28

29 29

30 Stability of Equilibrium Point 30

31 31 f 1 (T,I,E) = 0 f 2 (T,I,E) = 0 f 3 (T,I,E) = 0 E f I f T f E f I f T f E f I f T f

32 Here only x 3 * is a stable equilibrium 32

33 Chapter 4 / Section 16.3 Saturation Functions Problem: find rate functions that saturate Hill functions are also called «threshold functions»; h : saturation constant or threshold f(h)=0.5 n : degree of nonlinearity, determine curve shape 33

34 Parameter Influence on Hill/Threshold Functions changing h, n = 10 h = 50, changing n f(x), n= 1 f(x), n=2 Red: exponential functions 34

35 Infection Model Also called Michaelis-Menten Rate of infection per infected cell is a saturating function of T 35

36 Simulations of This Model I T I T 36

37 Chapter 7 Simplification 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 37

38 Elimination of the Fastest Time Scale Not constant: function of I(t)! 38

39 Compare to Immune Reaction Model Conclusion? The models are the same, with proper parameter settings 39

40 Elimination of the Slowest Time Scale During therapy of chronically infected patients: From steady state analysis of the original 4-equations system Replace E(t) with steady state value E* 40

41 Elimination of the Fastest AND Slowest Time Scale 3-equations system -> 2-equations system: 41

42 Is this a familiar model? 42

43 Added Value of the Model vs. Raw Data 43

44 Self-Study Assignment Responsible: Irina See distributed assignment (available on Moodle as well) 44