SIPm

SIPm is an enhanced compartmental model that approximates the model SquIPz

Variables

The independent variable is age, $a.$

Three variables make up a compartmental model:

$S$ is the expected number uninfected and susceptible to infection
$P$ is the expected number uninfected and chemo-protected
$I$ is the expected number infected

Two variables track the distribution of the MoI

$m_1$ is the first moment of the MoI
$m_2$ is the second moment of the MoI

Demography

We let $$H = S+I+P$$ denote the expected population density of the cohort as it ages. The model assumes that all individuals die at the rate $\mu$, such that

$$\frac{\textstyle{dH}}{\textstyle{da}} = -\mu H$$

In the model, we assume that everyone is born susceptible so $H$ is passed as a parameter, and the initial conditions are set to $S(0) = H$ and $I_i(0) = P(0)=0.$

Exposure

We let $h$ denote the force of infection (FoI), and we let $F_\mbox{MoE}$ denote the distribution of the multiplicity of exposure (MoE).

$\tilde z_1$ is the first moment of $F_\mbox{MoE}$
$\tilde z_2$ is the second moment of $F_\mbox{MoE}$

Clearance

In the queuing model SquIP, infections clear at the rate $r I_1.$ If we think of SIPm as an approximating model, then $$r I_1 \approx F_r(m_1, m_2) I$$

To get $F_r,$ we assume that the distribution of the MoI in these models is approximately a zero-inflated negatively binomial distributed. We assume $$ \frac{I_i} H\sim \mbox{NB}\left(\zeta=i| m, \phi \right)\left(1-\frac PH\right) $$ where mu1=\eqn{m_1} is the mean and size`$=\phi$ (the size parameter) is given by: $$\phi = \frac{m_1^2}{m_2-m_1^2-m_1}$$ so $$ F_r(m, \phi) = \begin{cases} r & \mbox{if } m = 0 \\ r \frac{ \mbox{NB}\left(\zeta=1| m, \phi \right) }{1- \mbox{NB}\left(\zeta=0 | m, \phi \right) } \left(1 - \frac{P}{H} \right) & \mbox{if } m>0 \end{cases} $$

In effect, the approximation assumes $I_1/H$ is close to the probability the MoI is equal to 1 in a zero-inflated negative binomial distribution. The distributions of the MoI are probably not exactly negatively binomially distributed. We use the Kullback-Liebler divergence to evaluate how well a negative binomial approximates the distributions in the queueing models.

Treatment and Chemprotection

These equations assume that individuals are treated and cured for several different reasons:

Incident infections cause disease and a fraction $\rho$ gets treated
Everyone in the population takes drugs at a background rate $\xi$
In addition to other modes of treatment, infected individuals get treated at the higher rate $\sigma$

After being treated and cured, individuals enter the chemoprotected class $P.$ Chemoprotection is lost at the rate $\eta,$ and they enter the susceptible class $S.$

Differential Equations

Hybrid Variables The compartmental dynamics are enhanced by computing a hybrid variable, a non-compartmental state variable describing the mean MoI ($m_1$), with dynamics: $$ \frac{\textstyle{dm_1}}{\textstyle{da}} = h (1-\rho) \hat z \left( 1-\frac{\textstyle{P}}{\textstyle{H}} \right) - \left(r + h \rho + \sigma + \xi \right) m_1$$

We also compute the dynamics of the second moment of the distribution of the MoI: $$ \begin{array}{rl} \frac{\textstyle{dm_2}}{\textstyle{da}} = & h(1-\rho)\hat z_2 \left( 1- \frac{ \textstyle{P}}{\textstyle{H}}\right) + (r+2 h(1-\rho)\hat z) m_1 \\[6pt] &- \left(2r + h \rho + \sigma + \xi \right) m_2 \\ \end{array}$$

Compartmental Variables

The infinite infection states in the family of queuing models specified by SquIPz are compressed down to a finite set of mutually exclusive and collectively exhaustive states – $S$, $I$, and $P$ – with dynamics: $$ \begin{array}{rl} \frac{\textstyle{dS}}{\textstyle{da}} &= F_r(m, \phi) I + \eta P - (h + \xi + \mu) S \\[6pt] \frac{\textstyle{dI}}{\textstyle{da}} &= h \left(1-\rho\right) S \\[6pt] & - \left( \rho h + \xi + \sigma + F_r(m, \phi) + \mu \right) I \\[6pt] \frac{\textstyle{dP}}{\textstyle{da}} &= (h \rho+ \xi)(H-P) + \sigma I - (\eta+\mu) P \\[6pt] \end{array}$$

Parameters

r: clearance rate for a simple infection
rho: the fraction of incident cases that gets treated
sigma: treatment because of prevalent infection
xi: background drug taking
eta: loss of chemoprotection
mu: death rate