2 Phase portrait & Linear system Flashcards
What is the reproduction number?
R0 = rate or infection / rate of recovery
What is the SI model?
One infected = lifelong infection
S > I
Logistic model
What is the equation for the SI model?
Change in infection = βIS/N
S = N-I
Write it in terms of I = βI(N-I)/N
What is N in the SI model?
N = S + I
What do you talk about when analysing the phase portrait?
Dimensionality
Linearity
Trajectory
Fixed points = stable/unstable
How do we calculate fixed points?
When dx/dt = 0
What changes β?
Probability of vial transmission is effected by safety measures = vaccines, thickness of masks, time spent together, proximity
What type of values are β and Υ?
Probabilities
Have no units and range from 0-1
What is a variable vs parameter?***
A variable represents a model state, and may change during simulation.
A parameter is commonly used to describe agents statically. A parameter is normally a constant in a single simulation, and is changed only when you need to adjust your model behavior.
What is the relationship of I dot and S dot?
I dot = -S dot
S dot = -I dot
Define stable and unstable fixed point slopes in phase portraits
Unstable fixed point = positive slope
—When x decreases a bit x dot also decreases
—When x increases a bit x dot also increases
Stable fixed point = negative slope
What is the SIS model?
Infected individuals may rcover and get infected again (example: common cold)
What is the difference in equation between SI and SIS model?
Subtract the recovery rate (Υ) x Infection (I)
-ΥI
What are the fixed points in SIS model?
I* = 0 (unstable)
I* = (1-Υ/β) N (stable)
To get these numbers, we solve when I dot = 0
How do we write the equation for SIS model in terms of R0? ***
How does R0 control the system?
When R0 is greater than 1 = infection will stabilize at the stable fixed point
This can lead to an epidemic outbreak and persistent endemic
When R0 is less than 1 = recovery is larger than infection
There will be no outbreak and the disease will subside
We cannot have a negative number of patients so effectively there is only 1 stable fixed point at 0
What does it mean to plot the SIS model in 2D?***
What are the rules in the 2D phase portrait?
Each pint in the phase portrait has one and ONLY ONE state, depicted by adding all the vectors (so only 1 vector)
There is no intersecting of any trajectory (no crossing of any vectors)
Non-crossing trajectories make up the VECTOR FIELD (the current that carries the point in the 2D (ocean)
What is a nullcline?
Nullcline is found in 2D phase portraits
A nullcline is a curve in the phase plane where the vector field points in a purely horizontal or vertical direction
Nullcline has a 0 vector in one direction either
(x-dot,0) or (0, y-dot)
What are eigenvectors in 2D phase portraits?
Set of vectors where the trajectories run straight in a 2D phase portrait
Since trajectories cannot intersect, eigenvectors form boundaries where ‘flow’ cannot cross from one region to another
How are eigenvectors described?
Linearly independent
Do all models have eigenvectors?
No, some model have imaginary eigenvalues
What is an example of a true second-order system?
SIR model
Example: dengi
What is the equation for the characteristic time of recovery?
Tau = 1 / gamma
What does Tau describe?***
The characteristic time of recovery
What type of system is the SIR model?
True second-order system = in which the highest-order derivative is a second derivative
Describe I dot on a population change over time
When I dot increases = more people are getting infected
When I dot hits 0 = no more infected at that time
When below 0 = infection rate decreasing
What is the equation for S dot?
S dot = - beta/N x susceptible x infection
What is the equation for I dot?
I dot = beta/N x SI - (gamma x infection)
What is the equation for R dot?
R dot = gamma x infection
How do we non-dimensionalise the SIR model?
Divide all populations by N
Dividing all rates by gamma
Why do we non-dimensionalize?
Model works with any population size and rate of dynamics
Populations are not percentages between 0-1
Dimensionless number R0 = beta/gamma controls the model
SIR model phase portrait, what does the line represent?
S + I = 1
R = 0
SIR model phase portrait, what does the below line represent? (inside the triangle)
S + I < 1
R > 0
How do we find the nullclines?
When S dot = 0 and I dot = 0
Solve S dot = 0 no movement in the left to right, so nullcline is vertical (up-down line)
Solve I dot = 0
What does the phase portrait show?
System dynamics
What does 1/R0 represent?
The threshold = predicts if epidemic starts
Threshold crosses trajectories at the maximal infected population = predicts when epidemic starts to subside
What happens when S0 is greater than 1/R0?
Then the infected population will increase in the beginning = leading to an epidemic outbreak
What happens when S0 is less than 1/R0?
The infected population will decrease, so no epidemic occurs
Do phase portraits have dimensions?***
NO
