We start Chapter 3 by
considering the dynamics of an SIStype infection in a population that
can be structured into a highrisk and a lowrisk group. Focussing
initially on the behavior of the highrisk 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 highrisk 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 highrisk group, in which case n_{H}
is the proportion of the population in the highrisk 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
highrisk 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 highrisk group occur when a highrisk susceptible is infected by
someone in either the high or lowrisk group. These two distinct
transmission types require different transmission parameters: We let β_{HH}
denote transmission to high risk from highrisk 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:
Parameters
β 
is the matrix of transmission
rates and incorporates the encounter rate between susceptible and
infectious individuals together with the probability of transmission. 
γ 
is
called the removal
or recovery rate, though often we are more interested in its reciprocal
(1/γ) which determines the average infectious period. 
n_{H} 
is
the
proportion of the population that are in the high risk group

I_{H}(0) 
is
the initial
proportion of the population that are both infectious and in the high
risk group. 
I_{L}(0) 
is
the initial
proportion of the population that are both infectious and in the low
risk group.

All rates are
specified
in years.
Requirements.
All parameters must be positive, and n_{H} ≤ 1, I_{H}(0)≤
n_{H}, I_{L}(0)≤
1n_{H},
Files
C++ Program, Python Program, Fortran Program, Parameters, MATLAB Code.
