Modeling Infectious Diseases in Humans and Animals
Matt J. Keeling & Pejman Rohani

SIS model with 2 risk groups (page 58)

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 XH and YH , and the total number in the high-risk group by NH (=XH +YH ). Alternatively, it is often simpler to use a frequentist approach, such that SH and IH refer to the proportion of the entire population that are susceptible or infectious and also in the high-risk group, in which case nH is the proportion of the population in the high-risk group: SH = XH/N, IH = YH/N, nH = NH/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 βHH denote transmission to high risk from high-risk and βHL represent transmission to high risk from low risk. (Note throughout this book we use the same ordering of subscripts such that transmission is always βto from) Putting these elements together, we arrive at the following differential equations:

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.
nH is the proportion of the population that are in the high risk group
IH(0) is the initial proportion of the population that are both infectious and in the high risk group.
IL(0) is the initial proportion of the population that are both infectious and in the low risk group.
All rates are specified in years.

All parameters must be positive, and  nH ≤ 1, IH(0)≤ nH, IL(0)≤ 1-nH,

C++ ProgramPython ProgramFortran ProgramParametersMATLAB Code.

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