Effect of Movement on the Early Phase of an Epidemic

Abstract

The early phase of an epidemic is characterized by a small number of infected individuals, implying that stochastic effects drive the dynamics of the disease. Mathematically, we define the stochastic phase as the time during which the number of infected individuals remains small and positive. A continuous-time Markov chain model of a simple two-patch epidemic is presented. An algorithm for formalizing what is meant by small is presented, and the effect of movement on the duration of the early stochastic phase of an epidemic is studied.

Keywords: Early spread of infection; Markov chain; Metapopulation.

Fig. 1

SARS-CoV-2 incidence in Campbell County, Wyoming, from the first case on 2020-03-20, to the start of the exponential increase in case counts (last date shown 2020-10-16)(Color figure online)

Fig. 2

Flow diagram of the two population SIS epidemic model. The per capita rates of flow between compartments are shown (Color figure online)

Fig. 3

Mean duration of the stochastic phase in a single population patch A for various values of ${\mathcal{R}}_0^A$ (Color figure online)

Fig. 4

Range of durations (shaded) and mean duration (line) of the stochastic phase versus $m^{\mathit{AB}}$ with disease introduced in patch A (red) or B (blue) for ${\mathcal{R}}_0^A=1.2$, ${\mathcal{R}}_0^B=2.5$ and $k=0.9$ (Color figure online)

Fig. 5

Mean duration (in days) of the stochastic phase for varying values of $m^{\mathit{AB}}$ and ${\mathcal{R}}_0^A$ with ${\mathcal{R}}_0^B=2.5$ where $k=0.1$ on the left and $k=0.9$ on the right. Blue: value of the reproduction number for the two-patch model as given by (3) (Color figure online)

Fig. 6

Left: Mean duration (in days) of the stochastic phase for varying values of $m^{\mathit{AB}}$ and ${\mathcal{R}}_0^B$ with $k=0.9$ and ${\mathcal{R}}_0^A=2.5$. Right: Mean duration of the stochastic phase for varying values of $m^{\mathit{AB}}$ and ${\mathcal{R}}_0^B$ with $k=0.9$ and ${\mathcal{R}}_0^A=1.2$ (Color figure online)

Fig. 7

Mean duration (days) of the stochastic phase as k varies with ${\mathcal{R}}_0^A$ (left) and ${\mathcal{R}}_0^B$ (right)(Color figure online)

Fig. 8

Mean duration (days) of the stochastic phase as a function of $m^{\mathit{AB}}$ and k (Color figure online)

Fig. 9

Absolute value of the difference between the MTBP and stochastic phase approximations of ${\mathbb{P}}_0$ as a function of the threshold $\widehat{I}$ and $m^{\mathit{AB}}$ (Color figure online)

Fig. 10

Absolute value of the difference between the MTBP and stochastic phase approximations of ${\mathbb{P}}_0$ as a function of the threshold $\widehat{I}$ and ${\mathcal{R}}_0^A$. In alphabetical order ${\mathcal{R}}_0^B$ = 0.8, 1.2, 2.5 (Color figure online)