Epidemic models with heterogeneous mixing and treatment

We consider a two-group epidemic model with treatment and establish a final size relation that gives the extent of the epidemic. This relation can be established with arbitrary mixing between the groups even though it may not be feasible to determine the reproduction number for the model. If the mixing of the two groups is proportionate, there is an explicit expression for the reproductive number and the final size relation is expressible in terms of the components of the reproduction number. We also extend the results to a two-group influenza model with proportionate mixing. Some numerical simulations suggest that (i) the assumption of no disease deaths is a good approximation if the disease death rate is small and (ii) a one-group model is a close approximation to a two-group model but a two-group model is necessary for comparing targeted treatment strategies.