Delay compartment models from a stochastic process

Abstract

Compartment models with delay terms are widely used across a range of disciplines. The motivation to include delay terms varies across different contexts. In epidemiological and pharmacokinetic models, the delays are often used to represent an incubation period. In this work, we derive a compartment model with delay terms from an underlying non-Markov stochastic process. Delay terms arise when waiting times are drawn from a delay exponential distribution. This stochastic process approach allows us to preserve the physicality of the model, gaining understanding into the conditions under which delay terms can arise. By providing the conditions under which the delay exponential function is a probability distribution, we establish a critical value for the delay terms. An exact stochastic simulation method is introduced for the generalized model, enabling us to utilize the simulation in scenarios where intrinsic stochasticity is significant, such as when the population size is small. We illustrate the applications of the model and validate our simulation algorithm on examples drawn from epidemiology and pharmacokinetics.

Figures (4)

• Figure 1: Examples of delay exponential functions with varying parameter settings are depicted. Blue lines represent cases where $\mu = 1$, while red lines represent $\mu = 0.2$. Solid and dashed lines correspond to different choices of $\mu\tau$: specifically, solid lines correspond to $\mu\tau = 0.3$, while dashed lines correspond to $\mu\tau = 0.7$.
• Figure 2: Stochastic and deterministic solutions of the two-compartment pharmacokinetic model with a delayed clearance. We consider a drug transport rate $k=1$, a clearance rate $C/V = 1$ and three time delays $\tau = 0$ (red), $0.2$ (orange) and $0.35$ (green), with an initial injection $x_0=100$. (a) Representative sample paths of the stochastic simulation (circle--dashed lines) and the corresponding deterministic solutions (solid lines) are shown. (b) $2000$ sample paths are averaged to show the convergence between the stochastic and the deterministic solution.
• Figure 3: Stochastic and deterministic solutions of the SIS model with delayed re-susceptibility. We consider the re-susceptibility process with rate $\gamma = 1$ three time delays $\tau = 0$ (red), $0.2$ (orange) and $0.35$ (green), for a small population with initial populations $s_0=95$ and $i_0=5$. (a) Representative sample paths of the infected population (circle--dashed lines) and the corresponding deterministic solutions (solid lines) are presented as fractions of the total population $P(t) = S(t)+I(t)$. The triangle-dashed line represents a sample path leading to disease extinction. (b) The deterministic solution is shown against a stochastic solution, calculated as an average of $2000$ sample paths.
• Figure 4: Stochastic and deterministic solutions of the SIS model with delayed re-susceptibility. We consider the re-susceptibility process with rate $\gamma = 1$ three time delays $\tau = 0$ (red), $0.2$ (orange) and $0.35$ (green), for a large population with initial populations $s_0=1900$ and $i_0=100$. (a) Representative sample paths of the infected population (circle--dashed lines) and the corresponding deterministic solutions (solid lines) are presented as fractions of the total population $P(t) = S(t)+I(t)$. (b) The deterministic solution is shown against a stochastic solution, calculated as an average of $2000$ sample paths.