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Model $1$: An exponential growth model for an epidemic

The simplest reasonable model for the early stages of the growth of an epidemic is one where the rate of change of the number of individuals with the

disease is proportional to the number of people who already have the disease. We can see this assumes that the people with the disease are mixed in

amongst the people without the disease, such that the number of people a sick person meets is a constant. It also assumes that people do not recover

from the disease, at least over the time period of interest. This is characteristic of the earliest phases of the spread of a disease, when there are no controls

put in place to restrict the motion of people, there are many more people who are healthy compared with those who are sick, and the sick haven't yet

recovered.

The model can be written as a differential equation:

$\frac{dP}{dt}=rP$

where $P$ is the number of people with the disease at time $t,$ and $r$ is a measure of how many healthy people each sick person infects. The solution to this

model is

$P(t)=$Aexp$(rt)$

where $A$ is the number of infected people at time $t=0.$

Model $2$: A sub$-$exponential growth model for an epidemic

There are numerous more sophisticated models for epidemic growth. We shall consider the following modification of the previous exponential growth

model:

$\frac{dP}{dt}=r{P}^q$

where $P(t)=(\frac{r}{m}t+A{)}^{m}m=\frac{1}{1-q}A=(P(0){)}^{(\frac{1}{m})}0$ gives a measure $of$ how the growth rate $of$ new cases depends $on$ the number $of$ infected people. The solution $to$ this model $is$

$P(t)=(\frac{r}{m}t+A{)}^m$

where $m=\frac{1}{1-q}$ and now $A=(P(0){)}^{(\frac{1}{m})}.$

$In$ this lab $we$ will compare model $1$ and model $2$ with early covid data.

Below the lab on Canvas you will find a file COVID$19\_$allcases.txt$.$ This file contains the total number of recorded cases up to that day in the early phase of the pandemic. The first record corresponds to January $22$nd$,$ the second to January $23$rd$,$ etc.

Read the data into a numpy array. You do not need to use pandas to do this. One approach is to set up a list, open the file, read each line in the file and append the result to the list. Once the list is complete you can convert it to a numpy array.

a$)$ Plot the data

Using matplotlib, plot the data versus time. Can we identify by eye any phases in the disease spread?

There appears to be a period of rapid growth $($concave up$),$ followed by a steadily slowing growth $($concave down$).$ There is also a significant jump in the number of cases, which when looking at the data collection, is due to a change in the way cases have been recorded.

b$)$ Looking only at the first five data points $($let$\text{'}$s call this data day $1$ to day $5),$ find a value of r which fits the exponential model to this data. Plot the data and the exponential model together over the first $10$ days. What can we conclude about the exponential model?

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c$)$ Now, using your exponential model, extrapolate backwards in time to find the predicted day when 'patient zero' appeared $($i$.$e$.$ when the number of

infected people first goes below one$).$ The disease is thought to have appeared on December $1$st $2019.$ Identify a plausible reason for a discrepancy

between your model and this date.

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d$)$ Fixing $r=2$ for Model $2,$ determine a value for $q$ by comparing the first $14$ days of data with the model. Plot the data and model out to day $20.$ Can we

conclude anything about the spread of the disease in light of the model dynamics?

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e$)$create a png plot showing the data, and models $1$ and $2,$ out to day $31($the end of the data$).$ Constrain the y axis to a maximum value of P$=200000.$Include a legend in your graph and label your axes.

COVID$19\_$allcases.txt:

$580$

$845$

$1317$

$2015$

$2800$

$4581$

$6058$

$7813$

$9823$

$11950$

$14553$

$17391$

$20630$

$24545$

$2826$

$31439$

$34876$

$37552$

$40553$

$43099$

$45134$

$59287$

$64438$

$67100$

$69197$

$71329$

$73332$

$75184$

$75700$

$76577$

$77673$

$78651$