Modelling the effect of travel-related policies on disease control in a meta-population structure

Abstract

Travel restrictions, while delaying the spread of an emerging disease from the source, could inflict substantial socioeconomic burden. Travel-related policies, such as quarantine and testing of travelers, may be considered as alternative strategies to mitigate the negative impact of travel bans. We developed a meta-population, delay-differential model to evaluate a strategy that combines testing of travelers prior to departure from the source of infection with quarantine and testing at exit from quarantine in the destination population. Our results, based on early parameter estimates of SARS-CoV-2 infection, indicate that testing travelers at exit from quarantine is more effective in delaying case importation than testing them before departure or upon arrival. We show that a 1-day quarantine with an exit test could outperform a longer, 3-day quarantine without testing in delaying the outbreak peak. Rapid, large-scale testing capacities with short turnaround times provide important means of detecting infectious cases and reducing case importation, while shortening quarantine duration for travelers at destination.

Appendix

1.1 Reproduction numbers

To set a baseline for our analysis, we consider the two populations without any control measures, such as testing, isolation, or quarantine, and assume the same travel rate between $P_A$ and $P_B$. This simplifies our model into two populations with a susceptible-exposed-infectious-recovered structure, connected by dispersal. At the disease-free equilibrium (without any infection), all variables except $S_A$ and $S_B$ are zeros with $\bar{S}_{A}=\bar{S}_{B}=N:= (N_A+N_B)/2$ when $\delta _{_A}=\delta _{_B}$. The basic reproduction number is given by $\mathcal {R}_0=\beta N/\gamma$.

When control measures are applied, $\mathcal {R}_0$ may be reduced, which is characterized by the control reproduction number $\mathcal {R}_c$. To calculate $\mathcal {R}_c$ for model (6), we use the Next Generation Matrix (NGM) approach introduced in Xu and Zhao (2012). Let $x_1, \ldots , x_{11}$ be the number of individuals in the $E_A, E_{iA}, I_A, I_{iA}$, and $E_B, E_{iB}, I_B, I_{iB}, I_{q1B}, E_{qB}, I_{q2B}$ classes, respectively. It then follows from the model (6) that the distribution of the remaining individuals at time $t>0$ is

$$\begin{aligned} \begin{aligned} \bigg (&e^{-(\alpha +\delta _{_A}) t}x_1, e^{-\alpha t}x_2, e^{-(\gamma +\delta _{_A}) t}x_3, e^{-\gamma t}x_4, e^{-(\alpha +\delta _{_B}) t}x_5, \\&e^{-\alpha t}x_6, e^{-(\gamma +\delta _{_B}) t}x_{7}, e^{-\gamma t}x_{8}, e^{-\gamma t}x_{9}, e^{-\alpha t}x_{10}, e^{-\gamma t}x_{11} \bigg ) \end{aligned} \end{aligned}$$

Thus, the total numbers of newly infected individuals in each class are

$$\begin{aligned} \bar{x}_1&= \int _0^\infty \beta \bar{S}_A e^{-(\gamma +\delta _{_A}) t}x_3 dt + \int _{0}^\infty \delta _{_B}e^{-(\alpha + \delta _{_B})} x_5dt = \frac{\beta \bar{S}_A}{\gamma +\delta _{_A}}x_3 + \frac{\delta _{_B}}{\alpha +\delta _{_B}}x_5. \end{aligned}$$

Similarly, we have

$$\begin{aligned} \bar{x}_2&= \frac{\sigma _1 \psi _{_E} \delta _{_A}}{\alpha +\delta _{_A}}x_1, \qquad \bar{x}_3 = \frac{(1-\eta \psi _{_I})\alpha }{\alpha +\delta _{_A}} x_1 + \frac{\delta _{_B}}{\gamma +\delta _{_B}} x_6, \\ \bar{x}_4&= \frac{\eta \psi _{_I}\alpha }{\alpha +\delta _{_A}} x_1 + x_2 + \frac{\sigma _1\psi _{_I}\delta _{_A}}{\gamma +\delta _{_A}}x_3,\\ \bar{x}_5&= \frac{\beta \bar{S}_B}{\gamma +\delta _{_B}}x_{7}+ \frac{(1-\sigma _2 \psi _{_E})(1-\sigma _1 \psi _{_E})\delta _{_A}e^{-\alpha \tau }}{\alpha +\delta _{_A}}x_1.\\ \bar{x}_6&= \frac{\sigma _2\psi _{_E}\delta _{_A}(1-\sigma _1\psi _{_E}) e^{-\alpha \tau }}{\alpha +\delta _{_A}}x_1, \\ \bar{x}_{7}&= \frac{(1-\eta \psi _{_I})\alpha }{\alpha +\delta _{_B}}x_5 + \frac{(1-\sigma _2\psi _{_I})\delta _{_A}(1-\sigma _1\psi _{_I})e^{-\gamma \tau }}{\gamma +\delta _{_A}}x_3 \\&\quad + \frac{(1-\sigma _2\psi _{_I})\alpha (e^{(\alpha -\gamma )\tau } - e^{-\alpha \tau })}{(e^{\alpha \tau }-1)(2\alpha -\gamma )}x_{10}\\ \bar{x}_{8}&= \frac{\eta \psi _{_I}\alpha }{\alpha +\delta _{_B}} x_5 + x_6 + \frac{\sigma _2\psi _{_I}\delta _{_A}(1-\sigma _1\psi _{_I})e^{-\gamma \tau }}{\gamma +\delta _{_A}}x_3 \\&\quad + \frac{\sigma _2\psi _{_I}\alpha (e^{(\alpha -\gamma )\tau } - e^{-\alpha \tau })}{(e^{\alpha \tau }-1)(2\alpha -\gamma )}x_{10} \\ \bar{x}_{9}&= \frac{(1-\sigma _1\psi _{_I})\delta _{_A}}{\gamma +\delta _{_A}}x_{3}, \qquad \bar{x}_{10} = \frac{(1-\sigma _1\psi _{_E})\delta _{_A}}{\alpha +\delta _{_A}}x_{1}, \qquad \bar{x}_{11} = x_{10}. \end{aligned}$$

From the relationship between $(\bar{x}_1, \ldots , \bar{x}_{11})$ and $x_1, \ldots , x_{11}$, we have the next generation operator $\mathcal {M}$ (Matrix (9)), where

$$\begin{aligned} m = \frac{\alpha (e^{(\alpha -\gamma )\tau } - e^{-\alpha \tau })}{(e^{\alpha \tau }-1)(2\alpha -\gamma )}. \end{aligned}$$

Since the reproduction number is defined as the spectral radius of $\mathcal {M}$ (Xu and Zhao 2012), we obtain $\mathcal {R}_c = (\rho (\mathcal {M}))^2$.

We note that the interventions of testing before departure from $P_A$, quarantine and its duration upon arrival in $P_B$, and testing at exit quarantine, have no significant effects on the control reproduction number (Fig. 5). However, the proportion of infectious individuals isolated within the population ($\eta$) reduces the reproduction number, leading to further delay in case importation and the outbreak within $P_B$ with a lower peak and cumulative incidence.

$$\begin{aligned} \mathcal {M}=\left[ \begin{array}{ccccccccccc} 0 &{} 0 &{} \frac{\beta \bar{S}_A}{\gamma +\delta _{_A}} &{} 0 &{} \frac{\delta _{_B}}{\alpha +\delta _{_B}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ \frac{\sigma _1 \psi _{_E} \delta _{_A}}{\alpha +\delta _{_A}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ \frac{(1-\eta \psi _{_I})\alpha }{\alpha +\delta _{_A}} &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{\delta _{_B}}{\gamma +\delta _{_B}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ \frac{\eta \psi _{_I}\alpha }{\alpha +\delta _{_A}} &{} 1 &{} \frac{\sigma _1\psi _{_I}\delta _{_A}}{\gamma +\delta _{_A}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ \frac{(1-\sigma _2 \psi _{_E})(1-\sigma _1 \psi _{_E})\delta _{_A}e^{-\alpha \tau }}{\alpha +\delta _{_A}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{\beta \bar{S}_B}{\gamma +\delta _{_B}} &{} 0 &{} 0 &{} 0 &{} 0\\ \frac{\sigma _2\psi _{_E} (1-\sigma _1\psi _{_E})\delta _{_A}e^{-\alpha \tau }}{\alpha +\delta _{_A}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} \frac{(1-\sigma _2\psi _{_I})\delta _{_A}(1-\sigma _1\psi _{_I})e^{-\gamma \tau }}{\gamma +\delta _{_A}} &{} 0 &{} \frac{(1-\eta \psi _{_I})\alpha }{\alpha +\delta _{_B}} &{} 0 &{} 0 &{} 0 &{} 0 &{} (1-\sigma _2\psi _{_I})m &{} 0\\ 0 &{} 0 &{} \frac{\sigma _2\psi _{_I}\delta _{_A}(1-\sigma _1\psi _{_I})e^{-\gamma \tau }}{\gamma +\delta _{_A}} &{} 0 &{} \frac{\eta \psi _{_I}\alpha }{\alpha +\delta _{_B}} &{} 1 &{} 0 &{} 0 &{} 0 &{} \sigma _2\psi _{_I}m &{} 0 \\ 0 &{} 0 &{} \frac{(1-\sigma _1\psi _{_I})\delta _{_A}}{\gamma +\delta _{_A}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ \frac{(1-\sigma _1\psi _{_E})\delta _{_A}}{\alpha +\delta _{_A}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \end{array}\right] \end{aligned}$$

(9)

1.2 Effect of dispersal rate on the reproduction number

See Fig. 5.

Fig. 5

Control reproduction number as a function of dispersal rate. Solid and dashed lines (overlapped) correspond to scenarios without and with testing before departure from $P_A$. The duration of quarantine is A, D 1 day; B, E 2 days; and C, F 3 days, without testing at exit from quarantine (A, B, C), and with testing at exit from quarantine (D, E, F) in $P_B$

1.3 Effect of dispersal rate on peak-time of the outbreak

See Fig. 6.

Fig. 6

Range for the peak time of incidence in $P_B$ when dispersal rate, $\delta$, varies between 0.0001 and 0.001, with quarantine of travelers upon arrival. Solid and dashed lines correspond to scenarios without and with testing before departure from $P_A$. The duration of quarantine is A, D 1 day; B, E 2 days; and C, F 3 days, without testing at exit from quarantine (A, B, C), and with testing at exit from quarantine (D, E, F)

1.4 Cumulative incidence

See Fig. 7.

Fig. 7

Attack rate (i.e., the proportion of the population infected throughout the outbreak) in $P_B$ when quarantine of travelers is implemented upon arrival. Solid and dashed lines correspond to scenarios without and with testing before departure from $P_A$. The duration of quarantine is A, D 1 day; B, E 2 days; and C, F 3 days, without testing at exit from quarantine (A, B, C), and with testing at exit from quarantine (D, E, F)