Compute the derivatives for SIS compartmental model:
\(S\) is the density of susceptible humans (or hosts)
\(I\) is the density of infected humans (or hosts)
\(H=S+I\) is human (or host) population density
The parameters and terms are:
\(h\) is the force of infection (from )
\(r\) is the natural clearance rate for infections
The clearance rate for infections is \(r\), and by assumption, individuals are assumed to be susceptible to infection after clearing infections.
The xds implementation computes \(dH/dt\)
rather than \(dS/dt.\) In the functions get_XH_vars
and parse_XH_orbits,
\(S\) is computed as \(S=H-I\) and listed as a variable. The
derivatives computed are:
$$ \begin{array}{rl} dH/dt = & B(t,H) + D \cdot H \\ dI/dt = & h (H-I) - r I + D \cdot I \end{array} $$ where \(S=H-I\);
\(B(t, H)\) is the time-dependent population birth rate; and \(D\) is a linear operator, a matrix describing demographic changes, including mortality, migration, and aging;
Usage
# S3 method for class 'SIS'
dXHdt(t, y, xds_obj, i)Note
The model has a port to model mass treatment. The mass treatment rate, \(\xi(t),\) is a function of time. For mass screen and treat, the treatment rate is lowered by the probability of detection. In this model, mass treatment has the same effect as \(r.\) With mass treatment, infection dynamics are described by:
$$ \begin{array}{rl} dI/dt = & h (H-I) - (r + \xi(t)) I + D \cdot I \end{array} $$
The ports for mda and msat are
handled by computing
r_t = r + mda(t) + d_rdt*msat(t)dI <- foi*(H-I) - r_t*I + ...