Chapter17* Case Studies: Vaccination and Isolation
***This chapter remains under construction. The examples are here and may be helpful, and in the future, more detailed instructions will further help students and instructors work from this chapter.***
This supplemental chapter includes three case studies of outbreaks. The first of these gives a new example of modeling using vaccination. The second example provides extra practice modeling isolation. Finally, the third example returns to vaccination, modeling a vaccine campaign that occurs during an outbreak.
Section17.1A Varicella Example with Vaccination
Varicella is also known as chickenpox. Activity 17.1 describes a 2016 outbreak and the biological info about varicella that we need to help build a model. In addition, this activity asks students to use Google Colab or some other form of Python, in order to read in a data set provided externally. Instructions for how to do this are provided.
Activity17.1.
In 2016, a varicella outbreak occurred in a kindergarten in Suzhou, China. The population consisted of 738 children, of whom 664 had no prior history of varicella. Those who had a previous history of varicella were not considered susceptible. Among the susceptible children, 364 had received a single dose of the varicella vaccine before the outbreak. One dose of the vaccine is approximately \(82\%\) effective (CDC).
(a)
Construct an SEIR model, complete with differential equations (including vaccination) and parameter values. You must do research to find sufficient values for \(\mathcal{R}_0\text{,}\)\(\gamma\text{,}\) etc.
, import the actual data from the outbreak and compare the accuracy of your model. The answer below gives instructions on how to import your data. Answer.
Here’s how to import data into Google Colab:
Make sure the “pandas” package is imported at the beginning of the code:
Mount Google Drive to access data files:
Load data from the file located in Google Drive: After /content/drive/MyDrive, continue adding backslashes with the necessary folder names to complete the file path. For example, if your file ‘SmallpoxData.csv’ was located in a folder ‘Modeling’, your file path would be content/drive/MyDrive/Modeling/SmallpoxData.csv
Extract the column from your data representing time, as a list: Replace ‘Day’ with whatever your column is labeled.
Extract the column representing the infected population, as a list: Replace ’infected’ with whatever your column is labeled.
All together, the code looks like this (also importing additional necessary packages):
Describe the similarites and differences between your model and the data of the outbreak. What are some possible changes that you could implement into your model do better fit it to the data?
(d)
How would the model change if the students who had received one shot of vaccination had instead received two? Research the effectiveness of two shots of the varicella vaccine and implement that into the model. How different would the outbreak have gone?
(e)
How might strict isolation policies have decreased the spread of the disease. Talk specifically about the movement between compartments within the model.
Section17.2
Exploration17.2.An Example with Strict Isolation.
This case study takes a look at the impact of strict isolation measures on disease spread. When modeling this outbreak it is important to recognize that the isolation of infected individuals is not in place at day 0 of the outbreak.
On June 25, 2009, 1376 cadets arrived for a 6-week training at the military facility where an H1N1 outbreak would shortly break out. Throughout the outbreak, there were a total of 167 cases (134 confirmed and 33 suspected). A few days after a July 4th superspreader event, cases of influenza skyrocketed. After suspecting that the virus was H1N1, preventative measures started to be implemented. Starting July 7th (day 11 of the outbreak), cadets who met case definitions were isolated together in a separate dorm for at least 7 days from symptom onset, and then until they were symptom-free for 24 hours. Additional information: H1N1 typically has an \(\mathcal{R}_0\)-value of 1.5, and someone with H1N1 is infected for approximately 8 days (Cleveland Clinic).
List two possible ways of creating a compartmental model that incorporates strict isolation partway through the outbreak. Construct these models, create a compartmental diagram, and include their differential equations.
(b)
If an SIR model is used, the model must be adjusted to incorporate the effects of isolation measures. Which parameter(s) need to change and why? Provide and justify your choice for the new value of the parameter.
(c)
Should the transmission rate (\(\beta\)) change when isolation is implemented? What are the arguments for and against changing \(\beta\) on day 11? Simulate the model with and without changing \(\beta\) on day 11. Analyze the differences in the results and discuss the reasoning behind keeping \(\beta\) constant or adjusting it.
(d)
Compare your model to the outbreak data using Google Colab 3
colab.research.google.com/
(instructions on how to import data would be included). How well does this model represent the actual outbreak dynamics? Which parameter could we adjust to improve the model’s fit? How would this adjustment reflect the specific characteristics of this outbreak?
(e)
What might happen if isolation was implemented earlier or later in the outbreak? How would this affect the infection dynamics? Simulate the model with isolation starting on different days (e.g., days 7 and 14) and analyze the outcomes.
(f)
Identify and discuss the possible limitations of this model.
Section17.3
Exploration17.3.An Example with a Vaccine Campaign.
This case study includes a vaccine campaign implemented part-way through the outbreak.
Between February and April 2024, a migrant shelter in Chicago experienced a sizeable measles outbreak. At the time of the index case, there were 1,801 shelter residents. Of the 1,801 residents, 784 (44%) had previous vaccination documentation. As it is unclear from the investigation, we will take this to mean that those residents with previous vaccine documentation had received one shot of the MMR vaccine. A mass vaccination campaign was initiated and completed within three days of the index cases’ confirmation and approximately 18 days of the index patient’s rash onset. The campaign vaccinated 882 residents (49%) which resulted in 1,666 (93%) of 1,801 shelter residents having vaccine documentation. Again, we will take this to mean having one dose of the MMR vaccine. The outbreak lasted ~50 days.
Using your own research of measles, list the parameter values for an SIR model as well as the starting values for each compartment. Additionally, set up the differential equations at the beginning of the outbreak.
(b)
As mentioned, there was a successful vaccination campaign that took place during the outbreak. This means that the value of \(p\) increases throughout the outbreak. We know that \(p = .44\) and the start of the outbreak and \(p = .93\) and the end of the outbreak. Given that the vaccination campaign started on day 18 of the outbreak and the MMR vaccine takes approximately 2 weeks to become fully effective, find an equation that demonstrates \(p\) increasing linearly between the start of the vaccination campaign and the end.
(c)
Attempt to graph this model, including your new equation, using Python.
(d)
Use Google Colab to compare your model to the data from the outbreak. Do this by importing your data: Answer 1.
(e)
Add Residual Sum of Squares (RSS) and Corrected Akaike Information Criterion (AICc) to evaluate the accuracy of the model. Answer.
\begin{equation*}
RSS = \[ \sum_{/text{All data values}}^{} (\text{Model Value - Data Value})^2\]
\end{equation*}
