## Introduction and the problem

The SIR model for spread of disease was first proposed in 1927 in a collection of three articles in the Proceedings of the Royal Society by Anderson Gray McKendrick and William Ogilvy Kermack; the resulting theory is known as KermackâMcKendrick theory; now considered a subclass of a more general theory known as compartmental models in epidemiology. The three original articles were republished in 1991, in a special issue of the Bulletin of Mathematical Biology.

The SIR model is so named because it assumes a static population, so no births
or deaths, divided into three mutually exclusive classes: those *susceptible* to
the disease; those *infected* with the disease, and those *recovered with
immunity* from the disease. This model is clearly not applicable to all
possible epidemics: there may be births and deaths, people may be re-infected, and so
on. More complex models take these and other factors into account.

The SIR model consists of three non-linear ordinary differential equations, parameterized by two growth factors \(\beta\) and \(\gamma\):

\begin{eqnarray*}
\frac{dS}{dt}&=&-\frac{\beta IS}{N}\\

\frac{dI}{dt}&=&\frac{\beta IS}{N}-\gamma I\\

\frac{dR}{dt}&=&\gamma I
\end{eqnarray*}

Here \(N\) is the population, and since each of \(S\), \(I\) and \(R\) represent the number of people in mutually exclusive sets, we should have \(S+I+R=N\). Note that the right hand sides of the equations sum to zero, hence \[ \frac{dS}{dt}+\frac{dI}{dt}+\frac{dR}{dt}=\frac{dN}{dt}=0 \] which indicates that the population is constant.

The problem here is to see if values of \(\beta\) and \(\gamma\) can be found which will provide a close fit to a well-known epidemiological case study: that of influenza in a British boarding school. This was described in a 1978 issue of the British Medical Journal.

This should provide a good test of the SIR model, as it satisfies all of the criteria. In this case there was a total population of 763, and an outbreak of 14 days, with infected numbers as:

Day: | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | ||

Infections: | 1 | 3 | 6 | 25 | 73 | 222 | 294 | 258 | 237 | 191 | 125 | 69 | 27 | 11 | 4 |

Some years ago, on my old (and now taken off line) blog, I explored using Julia for this task, which was managed easily (once I got the hang of Julia syntax). However, that was five years ago, and when I tried to recreate it I found that Julia has changed over the last five years, and my original comments and code no longer worked. So I decided to experiment with Python instead, in which I have more expertise, or at least, experience.

## Using Python

### Setup and some initial computation

We start by importing some of the modules and functions we need, and define the data from the table above:

```
import matplotlib.pyplot as plt
import numpy as np
from scipy.integrate import solve_ivp
data = [1, 3, 6, 25, 73, 222, 294, 258, 237, 191, 125, 69, 27, 11, 4]
```

Now we enter the SIR system with some (randomly chosen) values of \(\beta\) and
\(\gamma\), using syntax conformable with the solver `solve_ivp`

:

```
beta,gamma = [0.01,0.1]
def SIR(t,y):
S = y[0]
I = y[1]
R = y[2]
return([-beta*S*I, beta*S*I-gamma*I, gamma*I])
```

We can now solve this system using `solve\_ivp`

and plot the results:

```
sol = solve_ivp(SIR,[0,14],[762,1,0],t_eval=np.arange(0,14.2,0.2))
fig = plt.figure(figsize=(12,4))
plt.plot(sol.t,sol.y[0])
plt.plot(sol.t,sol.y[1])
plt.plot(sol.t,sol.y[2])
plt.plot(np.arange(0,15),data,"k*:")
plt.grid("True")
plt.legend(["Susceptible","Infected","Removed","Original Data"])
```

Note that we have used the `t_eval`

argument in our call to `solve_ivp`

which
allows us to exactly specify the points at which the solution will be given.
This will allow us to align points in the computed values of \(I\) with the
original data.

The output is this plot:

There is a nicer plot, with more attention paid to setup and colours,
here.
This use of scipy to solve the SIR equations uses the `odeint`

tool. This
works perfectly well, but I believe it is being deprecated in favour of
`solve_ivp`

.

### Fitting the data

To find values \(\beta\) and \(\gamma\) with a better fit to the data, we start by defining a function which gives the sum of squared differences between the data points, and the corresponding values of \(I\). The problem will then be minimize that function.

```
def sumsq(p):
beta, gamma = p
def SIR(t,y):
S = y[0]
I = y[1]
R = y[2]
return([-beta*S*I, beta*S*I-gamma*I, gamma*I])
sol = solve_ivp(SIR,[0,14],[762,1,0],t_eval=np.arange(0,14.2,0.2))
return(sum((sol.y[1][::5]-data)**2))
```

As you see, we have just wrapped the SIR definition and its solution inside a calling function whose variables are the parameters of the SIR equations.

To minimize this function we need to import another python function first:

```
from scipy.optimize import minimize
msol = minimize(sumsq,[0.001,1],method='Nelder-Mead')
msol.x
```

with output:

`array([0.00218035, 0.44553886])`

To see if this does provide a better fit, we can simply run the solver with these values, and plot them, as we did at the beginning:

```
beta,gamma = msol.x
def SIR(t,y):
S = y[0]
I = y[1]
R = y[2]
return([-beta*S*I, beta*S*I-gamma*I, gamma*I])
```

`sol = solve_ivp(SIR,[0,14],[762,1,0],t_eval=np.arange(0,14.2,0.2))`

```
fig = plt.figure(figsize=(10,4))
plt.plot(sol.t,sol.y[0],"b-")
plt.plot(sol.t,sol.y[1],"r-")
plt.plot(sol.t,sol.y[2],"g-")
plt.plot(np.arange(0,15),data,"k*:")
plt.legend(["Susceptible","Infected","Removed","Original Data"])
```

with output:

and as you see, a remarkably close fit!

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