SquIP

A model for the distribution of the multiplicity of infection (MoI) in a cohort of humans as it ages. The model extends a basic queuing model that has been used in malaria epidemiology before called $M/M/\infty$: it adds parameters to model treatment with anti-malarial drugs, which cures infections and is followed by a short period of chemo-protection. The model is named after its states:

SquIP $=$

S - Susceptible $+$

quI - a queuing process for Infected $+$

P - chemo-Protected

The master equations are presented below. The software implementation of SquIP has two parts:

dSquIP computes the derivatives
solve_SquIP is a wrapper around dSquIP that
- sets the value of a truncation parameter
- sets up the initial values, times, and parameters
- calls deSolve::ode to solve the system of ordinary differential equations
- parses the outputs

This software implementation truncates the infinite system of dIifferential equations. To check the accuracy of the truncated system, we have derived hybrid variables.

The Dynamical System

Variables

The independent variable is age, $a.$

The dependent variables are:

$I_i$ is the expected number infected with $i$ clones
$S$ is the expected number uninfected and susceptible to infection $(S=I_0)$
$P$ is the expected number uninfected and chemo-protected
$H = S+P+\sum_i I_i$ is expected number in the cohort

The model assumes 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 — $h(a)$

Let $h(a)$ denote the force of infection (FoI). This model assumes that each incident infection increases the MoI by exactly one. The FoI is passed as a trace function.

Natural Clearance — $r$

The model assumes that each clone clears independently of the others at rate $r,$ so if a person has $i$ clones, the MoI goes down by one at the rate $ri.$

Treatment and Chemoprotection — $\xi, \rho, \sigma, \eta$

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

Everyone in the population who is not chemoprotected takes drugs at a background rate $\xi$
Some incident infections cause disease, and a fraction $\rho$ gets treated
In addition to these 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 $S.$

Demography — $\mu$

The model assumes that all individuals die at the rate $\mu(a)$, such that

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

Master Equations

The dynamics are described by an infinite set of differential equations: $$ \begin{array}{rl} dS/da &= \eta P + r I_1 - (h + \xi + \mu) S \\ dP/da &= \left(h \rho + \xi \right) \left(H-P \right) + \sigma \sum_i I_i - \left(\eta + \mu \right) P \\ dI_1/ da &= h(1-\rho)S + 2 r I_2 - (r + h + \xi + \sigma + \mu) I_1 \\ dI_i/da &= h(1-\rho)I_{i-1} + (i+1) r I_{i+1} - (i r + h + \xi + \sigma + \mu) I_i\\ \end{array} $$

In this software implementation, the infinite system of equations is truncated after $i=N$ and $$dI_N/da = h(1-\rho)I_{N-1} - (N r + h + \xi + \sigma + \mu) I_N$$

Hybrid Variables

To check the accuracy of the truncated system, we have derived hybrid variables for the first two moments of the distribution of the MoI.

Let $m_n$ denote the $n^{th}$ moment of the MoI: $$m_n = \frac{\textstyle{1}}{\textstyle{H}}\sum_i i^n I_i$$

This software implementation computes the dynamics of the hybrid variables $m_1$ and $m_2$: $$ \begin{array}{rl} d m_1 /da &= h(1-\rho)\left(1-P/H\right) - \left(r + h \rho +\sigma + \xi\right) m_1 \\ d m_2 /da &= h(1-\rho)\left( 1- P/H + 2 m_1\right) - \left(2r + h \rho + \sigma + \xi \right) m_2 + r m_1 \\ \end{array} $$

After solving, the output is parsed and the first two moments of the distribution of the MoI are computed from $I_i$. The empirical moments should match $m_1$ and $m_2$ unless the value of the truncation parameter $N$ is set too low.

SquIP \(=\)

The Dynamical System

Master Equations

Hybrid Variables

See also