We start Chapter 3 by considering the dynamics of an SIS-type infection in a population that can be structured into a high-risk and a low-risk group. Focussing initially on the behavior of the high-risk group, and denote the number of susceptible and infectious individuals within this group by X_H and Y_H , and the total number in the high-risk group by N_H (=X_H +Y_H ). Alternatively, it is often simpler to use a frequentist approach, such that S_H and I_H refer to the proportion of the entire population that are susceptible or infectious and also in the high-risk group, in which case n_H is the proportion of the population in the high-risk group: S_H = X_H/N, I_H = Y_H/N, n_H = N_H/N.
The dynamics of either group is derived from two basic events, infection and recovery.  We initially focus on the dynamics of the high-risk group. Recovery, or the loss of infectious cases, can occur only through treatment and, following the unstructured formulation, we assume this occurs at a constant rate γ . New infectious cases within the high-risk group occur when a high-risk susceptible is infected by someone in either the high- or low-risk group. These two distinct transmission types require different transmission parameters: We let β_HH denote transmission to high risk from high-risk and β_HL represent transmission to high risk from low risk. (Note throughout this book we use the same ordering of subscripts such that transmission is always β_{to from}) Putting these elements together, we arrive at the following differential equations: