I have a problem which I cant get my head around.

I have the following equations:

Differential Equations problem-biology.jpg

Where:
S(t) is Susceptibles (never had disease)
I(t) is Infectious (have the disease and can infect susceptibles)
R(t) is Removed (become immune, or have died and or neither S nor I.

A plague lasted for 14 months.

In that period around 270 people out of an inital population of 350 died.

However a few of the survivors were probably already immune, having previously recovered from the disease.

But most including the Queen, who buried her husband and six of her children within one eight-day span —seem never to have contracted it.

How can I use these data to get an estimate for R0 in this outbreak.
Any help is much appreciated.