A dynamic model of bovine tuberculosis spread and control in Great Britain

In this paper, we consider a simple mathematical description of bovine TB transmission between cattle and badgers. The motivation for this work came from an exam question I wrote which asked students to calculate R₀ (the basic reproduction number) for a two-by-two next-generation matrix: 

Here R₁ is the reproduction number in host 1, R₂ is the reproduction number in host 2, R₁₂ is the number of secondary cases in host 2 due to host 1 and R₂₁ is the number of secondary cases in host 1 due to host 2. 

I wanted to get the students to show that none of the entries had to be greater than 1 for R₀ to be greater than 1. 

Then I realised that this could be exactly the situation for bovine TB in cattle and badgers. Perhaps transmission is close to control in each population, but when considered together, the situation is far from under control. 

We set up the model with four reproduction numbers, one for cattle-to-cattle transmission (R_CC), one for badger-to-badger transmission (R_BB), one for cattle-to-badger transmission (R_CB) and one for badger-to-cattle transmission (R_BC): 

So the question was how to determine the magnitude of each of these terms. We used published estimates of the reproduction number in badger populations and in the cattle population as well as evidence from the Randomised Badger Culling Trial. 

Using this framework leads to the interesting conclusion that you can get amplification in the system by transmitting the infection to the other host. 

We looked at how reducing the reproduction numbers affected infection levels in the cattle population. Because any amount of amplification in the badger population is bad, you have to reduce badger-to-badger transmission by a lot to reduce infection in cattle. This figure from the paper illustrates why:

As you reduce R_BB then you're actually moving along a contour line so the impact on (in this case) time to eradication is relatively modest - although not zero. The black arrow on the figure shows this. 

The full paper is here and you can also listen to me talking about it on Farming Today on Radio 4. 

Other coverage on the Countryfile.com website and here.