IMM Extra Credit
常微分方程代写 The form of the fifinished product should be a Python notebook, including both the analysis of the SIR model, as well as the analysis
In this project, you will analyze the SIR epidemic model, and apply it to real Covid-19 epi demiological data, for at least two regions. For the analysis of data for two regions, the Python
notebook CovidDataExample.ipynb demonstrates how to read in data, extract time series for various regions, and make plots of the variables that enter the SIR model. The notebook SIR_data_fit.ipynb provides a lengthy discussion of the SIR model parameter and analysis, as well as code to fifind numerical solutions its deterministic form, as well as a fifitting example for
data for Japan.
Analysis of the SIR model 常微分方程代写
For the analysis of the SIR model, provide the following:
- What are the major weaknesses of the SIR model for modeling Covid, or any epidemic? In what sort of situations is it particularly well-suited?
- How could the SIR model be modifified to include separate compartments for dead and recovered populations? How about for vaccinated people? Write down the deterministic
version of such a model and explain the new parameters.
- News and government agencies often show graphs of “daily new cases”, which is the number of new positive cases per day (I’ll call it C(t)). Using variables from the model (S, I and
R), what is this quantity? 常微分方程代写
- How might you compute the replacement number rt without knowing all of the underlying model parameters? For example, there is a form that involves only the removal rate b. Alternatively, how might you simply calculate the exponential growth rate of I for some short period of time during which an outbreak is starting?
- Draw the phase plane of the solution space for the SIR model, using just the equations for S and I. You may want to use the nondimensionalized population fractions s = S/N and
ı = I/N.
- Derive the peak number of infectives Imax, and if possible, estimate the time at which this peak occur. This may be done, for example, using the approximate solution derived in chapter 10 of Murray (2002, provided in the Resources folder).
- Show that the peak in the daily new cases C(t) occurs before the peak in infections.
Analysis of data for two regions
For the two regions, you will choose:
- A region that controlled the virus and exhibits a classic SIR-like curve for number of infected individuals (e.g. places like Hubei, or a cruise ship like the Diamond Princess)
- A region with an uneven curve or recent outbreak, and hence an unresolved near future (e.g. Thailand). 常微分方程代写
For the simpler region (1), or a time period over which the infections reach a peak and then decline in the more challenging region (2), estimate at a minimum the following:
- The parameters a (average number of adequate contacts per person, per unit time) and b (the inverse average removal time). These can be found using the Scipy optimization/parameter
fifitting method curve_fit, as I I recommend that you leave N as a free variable in the fifit.
- The basic reproduction number r0 (compare this to the estimate r0 = 2.2 (e.g. here)
- The population fraction that needs to be immuned in order to have herd immunity
- If possible, consider including the timeseries of the vaccinated population in the model, based on your answer to Analysis question 2 above.
For the more challenging region (2), in a period during which there is a rapidly growing outbreak (e.g. for India in the last few weeks), estimate the exponential growth rate for I(t), and thereby estimate the time period over which the number of infected people is expected to double. 常微分方程代写
The form of the fifinished product should be a Python notebook, including both the analysis of the SIR model, as well as the analysis of the two regions. [However, if you prefer, you may provide
the math of the analysis of the model in a separate PDF scan of your very neatly handwritten work. The text-based questions should be included in the notebook.]
Remember that JupyterHub cells can be used for text by selecting Markdown instead of Code; LaTeX can and should be used in these blocks for math. For helpusing Markdown in Jupyter, see
this page, and for help with using LaTeX in general, here is a good cheatsheet. And of course, look at the many examples provided in all the shared notebooks. 常微分方程代写
The fifinal product should include:
- A title and project description: regions you are analyzing and why, a description of the data, caveats about this particular data (e.g. is the disease on the rise in these regions, or has the curve been flflattened?), etc...
- Analysis of the SIR model, listed above
- Read-in of the data, with plots of I and R (and S, possibly on a separate plot, if it is much larger than the former), with labels and description.
- Perform fifits for your data, using either the Gauss-Newton method, or another optimization package. Provide comparison plots for the fifit and the data, for each of the regions you’ve
- Compute the quanatities and analysis listed above.
- Conclude your work by interpreting the results. Why did difffferent regions give difffferent r0 values? What does this say about containment strategies? Do you believe the results? Pro
vide any references (including basic news articles) that you use to understand and interpret your results. 常微分方程代写