README.md
In facilityepimath: Analyze Mathematical Models of Healthcare Facility Transmission

facilityepimath

The goal of facilityepimath is to provide functions to calculate useful quantities for a user-defined differential equation model of infectious disease transmission among individuals in a healthcare facility, including the basic facility reproduction number and model equilibrium. A full description and derivation of the mathematical results implemented in these functions can be found in the following manuscript:

Toth D, Khader K, Mitchell M, Samore M (2025). Transmission thresholds for the spread of infections in healthcare facilities. https://doi.org/10.1101/2025.02.21.25322698.

This work was supported by the Centers for Disease Control and Prevention, Modeling Infectious Diseases in Healthcare Network award U01CK000585.

You can install the development version of facilityepimath from GitHub with:

devtools::install_github("EpiForeSITE/facilityepimath")

You can install the facilitymath from CRAN with:

install.packages("facilityepimath")

The functions in this package analyze compartmental differential equation models, where the compartments are states of individuals who are co-located in a healthcare facility, for example, patients in a hospital or patients/residents of long-term care facility.

Our example here is a hospital model with four compartments: two compartments $C_1$ and $C_2$ representing patients who are colonized with an infectious organism, and two compartments $S_1$ and $S_2$ representing patients who are susceptible to acquiring colonization with that same organism. The two colonized and susceptible states are distinguished by having potentially different infectivity to other patients and vulnerability to acquisition from other patients, respectively.

A system of differential equations with those 4 compartments may take the following general form:

$$ \frac{dS_1}{dt} = -(s_{21}+(a_{11}+a_{21})\alpha+\omega_1+h(t))S_1 + s_{12}S_2 + r_{11}C_1 + r_{12}C_2$$

$$\frac{dS_2}{dt} = s_{21}S_1 - (s_{12}+(a_{12}+a_{22})\alpha+\omega_2+h(t))S_2 + r_{21}C_1 + r_{22}C_2 $$

$$\frac{dC_1}{dt} = a_{11}\alpha S_1 + a_{12}\alpha S_2 - (c_{21}+r_{11}+r_{21}+\omega_3+h(t))C_1 + c_{12}C_2 $$

$$\frac{dC_2}{dt} = a_{21}\alpha S_1 + a_{22}\alpha S_2 + c_{21}C_1 - (c_{12}+r_{12}+r_{22}+\omega_4+h(t))C_2 $$

The acquisition rate $\alpha$ appearing in each equation, and governing the transition rates between the S compartments and the C compartments, is assumed to depend on the number of colonized patients in the facility, as follows:

$$ \alpha = \beta_1 C_1 + \beta_2 C_2 $$

We will demonstrate how to calculate the basic reproduction number $R_0$ of this system using the facilityR0 function. The following components of the system are required as inputs to the function call below.

A matrix S governing the transitions between, and out of, the states $S_1$ and $S_2$ in the absence of any colonized patients:

$$S = \left( \begin{matrix} -s_{21}-\omega_1 & s_{12} \ s_{21} & -s_{12}-\omega_2 \end{matrix}\right) $$

A matrix C governing the transitions between, and out of, the states $C_1$ and $C_2$:

$$C = \left( \begin{matrix} - (c_{21}+r_{11}+r_{21}+\omega_3) & c_{12} \ c_{21} & - (c_{12}+r_{12}+r_{22}+\omega_4) \end{matrix}\right) $$

A matrix A describing the S-to-C state transitions when an acquisition occurs:

$$A = \left( \begin{matrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{matrix}\right) $$

A vector transm containing the $\beta$ coefficients (transmission rates from each colonized compartment) appearing in the $\alpha$ equation: $(\beta_1,\beta_2)$

A vector initS containing the admission state probabilities for the susceptible compartments only (i.e., the pre-invasion system before a colonized patient is introduced): $(\theta_1,1-\theta_1)$

A function mgf(x,deriv) that is the moment-generating function (and its derivatives) of the distribution for which the time-of-stay-dependent removal rate h(t) is the hazard function. This is the length of stay distribution when the state-dependent removal rates $\omega$ are zero.

library(facilityepimath)
S <- rbind(c(-1,2),c(1,-2))
C <- rbind(c(-1.1,0),c(0.1,-0.9))
A <- rbind(c(1,0),c(0,2))
transm <- c(0.4,0.6)
initS <- c(0.9,0.1)

mgf <- function(x, deriv=0) MGFgamma(x, rate=0.01, shape=3.1, deriv)
facilityR0(S,C,A,transm,initS,mgf)
#> [1] 0.7244774