2 Phase portrait & Linear system Flashcards

1
Q

What is the reproduction number?

A

R0 = rate or infection / rate of recovery

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2
Q

What is the SI model?

A

One infected = lifelong infection
S > I
Logistic model

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3
Q

What is the equation for the SI model?

A

Change in infection = βIS/N
S = N-I
Write it in terms of I = βI(N-I)/N

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4
Q

What is N in the SI model?

A

N = S + I

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5
Q

What do you talk about when analysing the phase portrait?

A

Dimensionality
Linearity
Trajectory
Fixed points = stable/unstable

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6
Q

How do we calculate fixed points?

A

When dx/dt = 0

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7
Q

What changes β?

A

Probability of vial transmission is effected by safety measures = vaccines, thickness of masks, time spent together, proximity

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8
Q

What type of values are β and Υ?

A

Probabilities
Have no units and range from 0-1

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9
Q

What is a variable vs parameter?***

A

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.

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10
Q

What is the relationship of I dot and S dot?

A

I dot = -S dot
S dot = -I dot

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11
Q

Define stable and unstable fixed point slopes in phase portraits

A

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

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12
Q

What is the SIS model?

A

Infected individuals may rcover and get infected again (example: common cold)

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13
Q

What is the difference in equation between SI and SIS model?

A

Subtract the recovery rate (Υ) x Infection (I)
-ΥI

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14
Q

What are the fixed points in SIS model?

A

I* = 0 (unstable)
I* = (1-Υ/β) N (stable)

To get these numbers, we solve when I dot = 0

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15
Q

How do we write the equation for SIS model in terms of R0? ***

A
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16
Q

How does R0 control the system?

A

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

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17
Q

What does it mean to plot the SIS model in 2D?***

18
Q

What are the rules in the 2D phase portrait?

A

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)

19
Q

What is a nullcline?

A

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)

20
Q

What are eigenvectors in 2D phase portraits?

A

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

21
Q

How are eigenvectors described?

A

Linearly independent

22
Q

Do all models have eigenvectors?

A

No, some model have imaginary eigenvalues

23
Q

What is an example of a true second-order system?

A

SIR model
Example: dengi

24
Q

What is the equation for the characteristic time of recovery?

A

Tau = 1 / gamma

25
What does Tau describe?***
The characteristic time of recovery
26
What type of system is the SIR model?
True second-order system = in which the highest-order derivative is a second derivative
27
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
28
What is the equation for S dot?
S dot = - beta/N x susceptible x infection
29
What is the equation for I dot?
I dot = beta/N x SI - (gamma x infection)
30
What is the equation for R dot?
R dot = gamma x infection
31
How do we non-dimensionalise the SIR model?
Divide all populations by N Dividing all rates by gamma
32
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
33
SIR model phase portrait, what does the line represent?
S + I = 1 R = 0
34
SIR model phase portrait, what does the below line represent? (inside the triangle)
S + I < 1 R > 0
35
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
36
What does the phase portrait show?
System dynamics
37
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
38
What happens when S0 is greater than 1/R0?
Then the infected population will increase in the beginning = leading to an epidemic outbreak
39
What happens when S0 is less than 1/R0?
The infected population will decrease, so no epidemic occurs
40
Do phase portraits have dimensions?***
NO