CS 7280代做、代写Python编程语言

日期：2024-11-04 09:33

GT CS 7280: Network Science

Assignment 4: Modeling Epidemics

Fall 2024

Overview

The objective of this assignment is to experiment with the concepts we covered in Module-4

about network epidemics, and see how the theoretical results that were derived in class

compare to simulation results.

Submission

Please submit your Jupyter Notebook A4-YOURGTUSERNAME.ipynb with requirements.txt

so that we may be able to replicate your Python dependencies to run your code as needed.

With Anaconda, you can do this by running:

conda list -e > requirements.txt

Ensure all graphs and plots are properly labeled with unit labels and titles for x & y axes.

Producing readable, interpretable graphics is part of the grade as it indicates understanding of

the content – there may be point deductions if plots are not properly labeled.

Getting Started

Assignment 4 requires Epidemics on Network python module available here. You can download

this module and run the examples given in the documentation to become familiar with it.

You can install the library using: pip install EoN

This homework, especially parts 2 and 3, might take several minutes to run. Be aware of

this and plan to complete it accordingly.

**IMPORTANT** As with prior assignments the structure has been designed

to have several subsections in each part. The first few subsections are

meant to just define useful functions and the final subsection of each part

is where the functions are called and the analysis is done. If you are

confused about how a function is meant to be used, check the final

subsection in each part to see how they are being called. This should clear

up a lot of potential points of confusion early on.

GT CS 7280: Network Science

Part 1: Outbreak Modeling [40 Points]

The file “fludata.txt” has the list of students, teachers and staff at a school. The interaction

between them was measured based on the proximity of the sensors that people were carrying

on them. The data file has three columns, the first two columns are the IDs of the people and

the third column is the number of interactions.

Construct an undirected graph from that text file using functions in the networkX module. For

the purpose of this assignment, we consider only the unweighted graph (i.e., you can ignore

the third column).

1. [10 points] Suppose there is a pathogen with transmission rate of 0.01 and recovery

rate of 0.5. Suppose that an outbreak started at node 325 (“patient-0”). Complete the

simulate_outbreak function to simulate an outbreak under the SIS model using the

provided parameters. The function should return a list of length n_iter containing

simulation runs where n_iter is an argument to the function.

Important: When running your simulations, you will want to discard the outbreaks that

died out stochastically. To do this, check whether the number of infected nodes at the

last time step is 0 and replace them with a simulation that does not die out. In total you

should have n_iter simulations.

Additionally, complete the plot_outbreaks function to visualize the results of the

simulate_outbreak function. Show the results for each of the simulations on a single

plot and break each simulation into 2 lines, one for the number of infected and the other

for number of susceptible over time. Make sure to properly label these lines and to

create a legend identifying which lines are which.

2. [10 points] In the lecture we modeled the initial exponential increase of the number of

infected nodes as 𝐼(𝑡) ≈ 𝐼 , where 𝜏 is a time constant. Note that here as only

0

𝑒

𝑡/τ

𝐼

0 = 1

one node was infected initially. Now, complete the get_exponent function to fit an

exponent to the curve of the number of infections. Choose only the initial portion of the

outbreak, say for 𝐼(𝑡) ≤ 100 (the exponential region of the outbreak, where the number of

infected is less than or equal to 100) and return the estimated time constant 𝜏.

Hint: scipy.optimize.curve_fit is a helpful function to fit the exponent to a curve.

Additionally, complete the plot_curve_fit function to plot both the actual number of

infected and the theoretical curve given a value of 𝜏 (for values of Infected < 100). This

function should also compute the r-squared between the two curves and print the value

for 𝜏 and r-squared in the title of the plot. Again, make sure to label both curves and

create a legend identifying which is which.

3. [5 points] In the lecture and textbook we discussed theoretical values for 𝜏 that can be

calculated from properties of the graph and the dynamics of the infection spread.

Complete the calculate_theoretical_taus function to compute:

GT CS 7280: Network Science

○ The random distribution shown in the Lesson 9 Canvas lecture “SIS Model”.

○ The arbitrary distribution from the Canvas lectures shown in the Lesson 9

Canvas lecture “Summary of SI, SIS, SIR Models with Arbitrary Degree

Distribution”.

○ The arbitrary distribution from the textbook found in Ch. 10, Equation 10.21.

Additionally, complete the compare_taus function to show a boxplot of the distribution of

sample 𝜏’s calculated from simulation runs (see 1.5 to understand where these come

from). Visualize the theoretical calculations as dots on the box plot. Again, label each of

these dots with the calculation used to generate them.

4. [10 points] Complete the calculate_theoretical_endemic_size function to compute the

size of the population that remains infected at the endemic state.

Then, complete the compare_endemic_sizes function to plot the distribution of

endemic sizes across several simulation runs as a boxplot, and compare it with the

theoretical calculation for endemic size as a single dot, similarly to the previous

subsection.

5. [5 points] Run the code provided in cell 1.5 and look at the resulting figures. How good of

a fit is the exponential curve in section 1.2? Explain how the theoretical estimates in 1.3

& 1.4 compare to the empirical distribution and indicate which you would consider a

reasonable fit for the data.

Part 2: Transmission Rate [25 Points]

Next, let us vary the transmission rate and see how it affects the spread of infection. Since we

know that only the ratio of the transmission rate and the recovery rate matters, let us keep the

recovery rate constant at 0.5 and vary only the transmission rate.

1. [10 points] Complete the simulate_beta_sweep function to vary the transmission rate

over a range of beta values between beta_min, beta_max with beta_samples number of

points. For each value of the transmission rate, compute 5 simulations to avoid outliers.

You can reuse your simulate_outbreak function from Part 1 in this function.

Next, complete the extract_average_tau function to return a list of the average 𝜏 value

calculated over the five simulation runs for EACH beta value. You may reuse the

get_exponent function from Part 1.

Finally, complete the plot_beta_tau_curves function to show the exponential curve

given by the 𝜏 values for each beta value. The x-axis is time and y-axis is the number of

infected people. Use a log scale on the y-axis and make sure that each line has its own

color. This function should be similar to the plot_curve_fit function in part 1.2, but you

GT CS 7280: Network Science

will be showing a series of exponentials instead of comparing an experimental with a

theoretical curve.

2. [10 points] Complete the extract_average_endemic_size function to return a list of the

average endemic size calculated over the five simulation runs for EACH beta value.

Next, complete the calculate_theoretical_endemic function to find the minimum

theoretical beta values of the transmission rate for an epidemic to occur. Calculate this

minimum based on the equations derived in lecture for both the random distribution and

the arbitrary distribution. Also, calculate the theoretical endemic size for each value of

beta under the assumption of random distribution.

Finally, complete the compare_endemic_sizes_vs_beta function to plot the average

endemic sizes and theoretical endemic sizes as a curve vs beta. Additionally, plot the

minimum values for beta to start an epidemic as vertical lines. Make sure to label each

line and provide a legend.

3. [5 points] Run the code provided in cell 2.3 and look at the resulting figures. How similar

is the theoretical to experimental endemic sizes? How closely do the minimum beta

values provide a reasonable lower bound for the start of an endemic?

Part 3: Patient-0 Centrality & 𝜏 [30 Points]

Now, let us see how the choice of “patient-0” affects the spread of an outbreak. Consider every

node of the network as patient-0, and run the SIS model using the parameters in Part 1 to

compute . Run the simulation with each node in the simulation as patient-0. Hint: You can skip

cases where the infection quickly diminishes to 0.

1. [10 points] Complete the sweep_initial_infected function to complete a single

simulation run for each node in the graph as the initial infected. Check for runs that

stochastically die out and do not save those. Return the list of simulation run results and

a list of nodes (integer IDs) where the simulation was successful.

Additionally, complete the compute_centrality function to calculate the: degree

centrality, closeness centrality (with wf_improved=false), betweenness centrality, and

eigenvector centrality of the graph. Remember to use the unweighted centrality metrics.

Return the centralities for each node where the simulation was kept in the previous

function.

Hint: We provide “nodes” as an argument which is meant to represent the second output

of the previous function. You can use this to filter for centralities of only these nodes

before you return them. Check the cell for 3.3 to see exactly how this is used.

GT CS 7280: Network Science

2. [15 points] Complete the calculate_pearson_correlation to compute the Pearson

correlation coefficient between each centrality metric and 𝜏, along with a p-value for that

correlation.

Additionally, complete the plot_centrality_vs_tau function to plot a scatter plot between

the 𝜏 value that corresponds to each node, and different centrality metrics of that node:

degree centrality, closeness centrality, betweenness centrality, and eigenvector

centrality. Do this all as one figure with four subfigures. Include the Pearson correlation

values as well as the corresponding p-values in the title for each scatter plot. Remember

to use the unweighted centrality metrics.

3. [5 points] Rank these centrality metrics based on Pearson’s correlation coefficient, and

determine which metrics can be a better predictor of how fast an outbreak will spread

from the initial node. Analyze your results. That is, do the results match your intuition? If

they differ, why might that be?

Part 4: Knowledge Question [5 Points]

Answer the following food for thought question from Lesson 10 – Submodularity of Objective

Function:

Prove that a non-negative linear combination of a set of submodular functions is also a

submodular function.

Hint: Make sure you understand the definition of linearity.