SquIPz
SquIPz.RdThis is a model for the dIistribution of the multiplicity of infection (MoI) in a cohort of humans as it ages. It extends the queuing model SquIP by adding a Tweedie process to model heterogeneous exposure: each infection event increases the MoI by the multiplicity of exposure (MoE), which is given by a probability mass function \(F_\mbox{MoE}.\) The model is named after its states and processes:
SquIPz \(=\)
S - Susceptible \(+\)
quI - a queuing process for Infected \(+\)
P - chemo-Protected
z - the Tweedie process distribution
This software implementation of SquIP has two parts:
dSquIPz computes the derivatives
solve_SquIPz is a wrapper around dSquIPz 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 Model
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 (\(h\)) —
Let \(h\) denote the force of infection (FoI). This model assumes that each incident infection increases the MoI by \(i\) with probability $$F_\mbox{MoE}(X=i) = z_i.$$
Natural Clearance — 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 — These equations assume that individuals are treated and cured for several different reasons:
Everyone in the population takes drugs at a background rate \(\xi\)
Incident infections cause disease and a fraction \(\rho\) gets treated
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 \(S.\)
Demography – The model assumes that all individuals die at the rate \(\mu(a)\), such that
$$\frac{\textstyle{dH}}{\textstyle{da}} = -\mu H$$
Dynamics — The dynamics are described by an infinite set of differential equations: $$ \begin{array}{rl} dS/da &= \eta P + r I_1 - (h + \xi + \mu) S \\ dP/da &= \left(h \rho + \xi \right) \left(H-P \right) + \sigma \sum_i I_i - \left(\eta + \mu \right) P \\ dI_i / da &=h (1-\rho) \left( S z_i + \sum_j I_{i-j} z_j \right) + (i+1) r I_{i+1} - \left(h + \sigma + r i + \xi + \mu \right) I_i\\ \end{array} $$
In this software implementation, the infinite system of equations is truncated at \(N\) and $$dI_N/da = h (1-\rho) \sum_{i \geq N} \left( S z_i + \sum_j I_{i-j} z_j \right) - \left(h + \sigma + r N + \xi + \mu \right) I_N$$
Hybrid Variables
Let \(m_n\) denote the \(n^{th}\) moment of the MoI: $$m_n = \frac{\textstyle{1}}{\textstyle{H}}\sum_i i^n I_i$$ and let \(\tilde z_n\) denote the moments of \(F_\mbox{MoE}:\) $$\tilde z_n = \sum_i i^n z_i$$
This software implementation computes the dynamics of the hybrid variables \(m_1\) and \(m_2\): $$ \begin{array}{rl} d m_1 /da &= h(1-\rho)\tilde z_1 \left(1-P/H \right) - \left(r + h \rho +\sigma + \xi\right) m_1 \\ d m_2 /da &= h(1-\rho)\hat z_2 \left( 1-P/H\right) + (r+2 h(1-\rho)\hat z) m - \left(2r + h \rho + \sigma + \xi \right) m_2 \\ \end{array} $$ 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.