2 Phase portrait & Linear system

Question 1

What is the reproduction number?

Answer 1

R0 = rate or infection / rate of recovery

Question 2

What is the SI model?

Answer 2

One infected = lifelong infection
S > I
Logistic model

Question 3

What is the equation for the SI model?

Answer 3

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

Question 4

What is N in the SI model?

Answer 4

N = S + I

Question 5

What do you talk about when analysing the phase portrait?

Answer 5

Dimensionality
Linearity
Trajectory
Fixed points = stable/unstable

Question 6

How do we calculate fixed points?

Answer 6

When dx/dt = 0

Question 7

What changes β?

Answer 7

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

Question 8

What type of values are β and Υ?

Answer 8

Probabilities
Have no units and range from 0-1

Question 9

What is a variable vs parameter?***

Answer 9

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.

Question 10

What is the relationship of I dot and S dot?

Answer 10

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

Question 11

Define stable and unstable fixed point slopes in phase portraits

Answer 11

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

Question 12

What is the SIS model?

Answer 12

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

Question 13

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

Answer 13

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

Question 14

What are the fixed points in SIS model?

Answer 14

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

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

Question 15

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

Question 16

How does R0 control the system?

Answer 16

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

Question 17

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

Question 18

What are the rules in the 2D phase portrait?

Answer 18

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)

Question 19

What is a nullcline?

Answer 19

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)

Question 20

What are eigenvectors in 2D phase portraits?

Answer 20

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

Question 21

How are eigenvectors described?

Answer 21

Linearly independent

Question 22

Do all models have eigenvectors?

Answer 22

No, some model have imaginary eigenvalues

Question 23

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

Answer 23

SIR model
Example: dengi

Question 24

What is the equation for the characteristic time of recovery?

Answer 24

Tau = 1 / gamma

Question 25

What does Tau describe?***

Answer 25

The characteristic time of recovery

Question 26

What type of system is the SIR model?

Answer 26

True second-order system = in which the highest-order derivative is a second derivative

Question 27

Describe I dot on a population change over time

Answer 27

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

Question 28

What is the equation for S dot?

Answer 28

S dot = - beta/N x susceptible x infection

Question 29

What is the equation for I dot?

Answer 29

I dot = beta/N x SI - (gamma x infection)

Question 30

What is the equation for R dot?

Answer 30

R dot = gamma x infection

Question 31

How do we non-dimensionalise the SIR model?

Answer 31

Divide all populations by N
Dividing all rates by gamma

Question 32

Why do we non-dimensionalize?

Answer 32

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

Question 33

SIR model phase portrait, what does the line represent?

Answer 33

S + I = 1
R = 0

Question 34

SIR model phase portrait, what does the below line represent? (inside the triangle)

Answer 34

S + I < 1
R > 0

Question 35

How do we find the nullclines?

Answer 35

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

Question 36

What does the phase portrait show?

Answer 36

System dynamics

Question 37

What does 1/R0 represent?

Answer 37

The threshold = predicts if epidemic starts

Threshold crosses trajectories at the maximal infected population = predicts when epidemic starts to subside

Question 38

What happens when S0 is greater than 1/R0?

Answer 38

Then the infected population will increase in the beginning = leading to an epidemic outbreak

Question 39

What happens when S0 is less than 1/R0?

Answer 39

The infected population will decrease, so no epidemic occurs

Question 40

Do phase portraits have dimensions?***

Answer 40

NO