SIR Model and Influenza at an English Boarding School 1978

In January and February 1978, an epidemic of influenza occurred in a boarding school in the north of England. The boarding school housed a total of 763 boys, all of whom were at risk during the epidemic. The spring term began January 10. The boys returned from their Christmas vacation spent at many different locations in the world. On average A boy returning from Hong Kong exhibited elevated temperature during the period 15-18 January. On January 22, three boys were sick and on average each boy spend 2 days as infectious.The following data set gives the number of boys ill on the n-th day beginning January 22 (n=1).

Data={{3,25},{4,75},{5,227},{6,296},{7,258},{8,236},{9,192},{10,126},{11,71},{12,28},{13,11},{14,7}};
plot1=ListPlot[Data,PlotStyle->{Red,PointSize[Large]},AxesLabel->{"n","Infected"},AxesStyle->Directive[Black, 20],PlotRange->{0,763}]

If the number of boys who escaped influenza was 19, calculate the value of R_0, α, and β.

We first calculate R_0 using the value of S∞.

SInf=19;
NMax=763;
I0=25;
S0=NMax-I0;
R0=Log[SInf/NMax]/((-1+SInf/NMax));
N[R0]

Now, lets see how well this matches our the numerical value of 296 for Imax.

N[NMax*(1-1/R0*(1+Log[R0]))]

α=1/2;
β=α*R0/NMax;

sol=NDSolve[{S'[t]==-β*S[t]*II[t],II'[t]==β*S[t]*II[t]-α*II[t],R'[t]==α*II[t],S[3]==NMax-I0,II[3]==I0,R[3]==0},{S,II,R},{t,3,14}];
plot2=Plot[Evaluate[{S[t],II[t],R[t]}/.sol],{t,3,14},PlotStyle->Automatic];

Now, lets put the two plots together.

Show[plot1,plot2]

Hot damn! That actually fits really well!

Eyam Plague

Find data on the Eyam Plague and fit an SIR model to the data.