Introduction to Stochastic Models

Shlomo Ta’asanCarnegie Mellon University

Download StochasticModels.zip

tsb.mssm.edu/summerschool/index.php/Introduction_to_stochastic_modeling

How to work with Probabilities?

Simplest model: Tossing a fair coinP (Head) = 1/2 ; P (Tail) = 1/2

How to toss a coin using matlab? rand - generates a random number between 0 and 1

rand(100,1) generate 100 random numbers100*rand generate a random number between 0 and 100 (decimal numbers)

We also need to know how to do something (kill the cell) with probability P

r = randif r < P cell diesend

The models

Reactions

Random Walks

Chemotaxis

Macrophage Looking for Bacteria

Stochastic Models for Reactions

Modeling Reactions

A -> 0per event A count goes down by 1.

A -> Bper event A count goes down by 1, B count goes up by 1

A + B -> Cper event: A,B count go down by 1, C count goes up by 1

Model in two ways:

1. ABM: modeling of each molecule, 1 it exists, 0 is it deleted. for each agent we perform an action with some

probability

2. probabilistic model for total number of molecules in each group.

Na, Nb, etc the total number of molecules of type A,B, …

we change Na, Nb etc by one according to some probability

Our first real ABM: A-> 0

A cell dies with probability p during time interval dt. P(A-> 0; dt) = p * dt

For simulation we need a counter for time, and a variable to monitor if the cell isdead or alive. Use a variable x for this.

x = 1 cell is alivex = 0 cell is dead

What if we have many cells? What are the questions we can ask about this model?

population size as a function of time, average time to extinction, fluctuation in time to extinction Matlab code: realABM_A.m

Efficient Probabilistic Model: A-> 0

A cell dies with probability p during time interval dt. P(A-> 0; dt) = p * dt

We model the total number of agents, Na, instead of each agent. We pick dt small enough such that only one cell can die (i.e the probability that two die is very small, so we ignore it)

The only event in this model: Na -> Na -1 (when one of the agents dies)

P( Na -> Na – 1) = p*Na*dt

This is much faster than the real ABM model. Sometimes it is enough to do this type of models. Sometimes we need a full ABM model

Lets do the simulation! Matlab Code ABM_A.m

Stochastic-SIR

Population has three groups: Susceptible, Infected and Recovered

The dynamics is expressed in the reactions S + I -> I + I (probability: r*dt) event 1I -> R (probability: a*dt) event 2

When to expect epidemic? A relation between parameters

Ns, Ni, Nr population size for Susceptible, Infected and Recovered.We pick dt small enough such that only one event will happen, The

probability of events

P( S+I -> I + I) = r*Ns*Ni*dt event 1 happens just once during dtP(I -> R) = a*dt*Ni event 2 happens just once during dt

Since probabilities are < 1, dt should be small enough. More precisely,

We want the sum of all probabilities to be less than 1. !!!

Stochastic-SIR cont.

Each event changes the variables of the model Ns, Ni, Nr.

Event 1: S + I -> I + I Ns -> Ns -1 and Ni -> Ni + 1Event 2: I -> R Ni -> Ni – 1 and Nr -> Nr + 1

P( Ns -> Ns -1 & Ni -> Ni + 1) = r*Ns*Ni*dtP( Ni -> Ni -1 & Nr -> Nr + 1) = a*dt*Ni

Now lets simulate it. Matlab Code: ABM_SIR.m

How does it compare with the ODE model? Notice the finite time to eradicate infection in this model. Compare

to the ODE.

Stochastic vs. ODE

Reaction: A + 2 X 3XA X

Multiple steady states!

ODE

Stochastic model

Stochastic models do more than just adding noise to results of an ODE!!

Modeling cell movement

Random Walks

A basic random walk in 2D:

P( x-> x+1, y->y) = ¼ P( x-> x-1, y->y) = ¼ P( x-> x, y->y-1) = ¼ P( x-> x, y->y+1) = ¼

Pick a random number, r. if r between 0 and ¼ go rightIf r between ¼ and ½ go left, if r is between ½ and ¾ go down, if r

between ¾ and 1 go up.

Questions: If we start at (0,0), where will we be after n steps? in different places each time

How to describe this? Probability distribution

Lets simulate it: Matlab code randomWalk.m - ABM for bacterial movement

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Stochastic Modeling of Chemotaxis

1st Ingredient: Bacterium performs a biased random walk based on the gradient (gX,gY) of the chemoattractant molecules.

P( x-> x+1, y->y) = ¼ + gXP( x-> x-1, y->y) = ¼ - gXP( x-> x, y->y+1) = ¼ + gYP( x-> x, y->y-1) = ¼ - gY

Simulation similar to previous random walk..

2nd Ingredient: Chemoattractant molecules diffuse (spread) we model them using concentration at every lattice point C(I,j)

Evolution: an average between value at (I,j) and values at nbrs

Cnew(i,j) = 0.6*Cold(i,j) + 0.1*(Cold(i+1,j)+Cold(i-1,j)+Cold(I,j+1)+Cold(i,j-1))

Matlab code: Chemotaxis.m

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¼ +gX

¼ -gY

¼ -gX

A Macrophage looking for Bacteria

1st Ingredient: Bacterium performs a simple random walk and releases chemoattractant molecules.

2nd Ingredient: Chemoattractant molecules diffuse

Cnew(i,j) = 0.6*Cold(i,j) + 0.1*(Cold(i+1,j)+Cold(i-1,j)+Cold(I,j+1)+Cold(i,j-1))

3rd Ingredient: Macrophage performs a random walk based on chemoattractant molecules and kills bacteria as they are reached.

Matlab code: Phagocytosis.m

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¼ +gX

¼ -gY

¼ -gX

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The End