Modeling the impact of air, sea, and land travel restrictions supplemented by other interventions on the emergence of a new influenza pandemic virus

Background During the early stages of a new influenza pandemic, travel restriction is an immediate and non-pharmaceutical means of retarding incidence growth. It extends the time frame of effective mitigation, especially when the characteristics of the emerging virus are unknown. In the present study, we used the 2009 influenza A pandemic as a case study to evaluate the impact of regulating air, sea, and land transport. Other government strategies, namely, antivirals and hospitalizations, were also evaluated. Methods Hong Kong arrivals from 44 countries via air, sea, and land transports were imported into a discrete stochastic Susceptible, Exposed, Infectious and Recovered (SEIR) host-flow model. The model allowed a number of latent and infectious cases to pass the border, which constitutes a source of local disease transmission. We also modeled antiviral and hospitalization prevention strategies to compare the effectiveness of these control measures. Baseline reproduction rate was estimated from routine surveillance data. Results Regarding air travel, the main route connected to the influenza source area should be targeted for travel restrictions; imposing a 99% air travel restriction delayed the epidemic peak by up to two weeks. Once the pandemic was established in China, the strong land connection between Hong Kong and China rendered Hong Kong vulnerable. Antivirals and hospitalization were found to be more effective on attack rate reductions than travel restrictions. Combined strategies (with 99% restriction on all transport modes) deferred the peak for long enough to establish a vaccination program. Conclusion The findings will assist policy-makers with decisions on handling similar future pandemics. We also suggest regulating the extent of restriction and the transport mode, once restriction has been deemed necessary for pandemic control. Although travel restrictions have yet to gain social acceptance, they allow time for mitigation response when a new and highly intrusive virus emerges.

• Screening sensitivity at entry border points; • Implementation date on travel restrictions

Sensitivity analysis
• Time step of model; • Multivariate sensitivity analysis 1 Mathematical model formulation Basic stochastic SEIR model The model applied the concept of binomial chain process [1,2] and the similar notation as Lekone (2006) [3]. Let ∆t be a time step and (t, t + ∆t] be a time interval, we denote S(t), E(t), I(t), and R(t) as the number of individuals in Susceptible, Exposed, Infected, where the rate of infection is equal to βI(t)/N for a time step, β is the transmission rate, and N is the population size. The α and γ are the constant transition rates from latent state to infectious state and from infectious state to removed state respectively.
And the rates are transformed into probabilities assuming in poisson process. Here are the model compartments of imported cases in latent status,

Arrived and departed cases
and infectious status, The number of latent subjects, E i (t), and the number of infectious subjects, I i (t), at time t of country i-th are generated from discrete-time SEIR model based on the reproduction numbers of the countries,  non-local countries, we will estimate the reproduction numbers by the initial exponential growth rate method [5] employing two months after dates of their first onset cases (which showed in Table 2) daily surveillance data [6] [7] [8], where r is the initial exponential growth rate estimated by the least square fitting to the model, i.e. logarithm(cumulative number of cases at time t) ∝ rt.
At the same time, a number of infected individuals will leave and carry the pathogens away from the local city. Departure statistics are collected from the Census and Statistics Department, Hong Kong [9] and are listed in Table 1. Let m E k be the probability of departure from local area by the mode of transport k, the compartments of exported cases in latent status, EX E (t), and in infectious status, EX I (t), will be 3 k=1 bin(m E k , E(t)) and 3 k=1 bin(m E k , I(t)) respectively. The calculation of m E k is similar to that of m I k,i , which adapts the departure data in Table 1.

Result of estimated R 0 and corresponding confidence interval (CI)
The reproduction numbers of the forty-four non-local countries are estimated by the initial exponential growth rate method. All of the initial growth rates are fitted significantly (p − value < 0.05). Showed in Table 2, the R 0 range from 1.1 to 1.9.

Antiviral and hospitalization
Two new compartments are added into the model, antiviral Treatment T (t) and Hospitalization H(t). Once individuals become infectious, they seek for antiviral treatment and hospitalization with proportions p T and p H respectively. With regard to limited resources, part of them may be untreated as proportions p U . We adapt a ψ fraction reduction of infectiousness for individuals who receive antiviral. Suppose classes M (t) and N (t) are the number of infectious individuals who take antiviral treatment and hospitalization at time t respectively. The P (t) and Q(t) are the number of removed individuals from antiviral treatment and hospitalization with transition rates γ T and γ H to the removed status.

Stochastic SEIR model with interventions
Because infectious individuals include those being treated and hospitalized, the probability of a susceptible person becoming infected is equal to for a time step ∆t. Given ν is the sensitivity of the entry screening board, so only ν proportion of imported infectious individuals are able to be identified as positive cases and be voluntary quarantined. Let f k be the restriction fraction for import transportation k-th, the stochastic system is as follow, The distributions for the classes are The descriptions of the parameters are highlighted in Table 3. A simple schematic flow is showed in Figure 1.

Epidemic evolution
The pandemic is seeded according to the start dates (Table 2)

Computer simulation
The model is implemented in software SAS 9.1.3. Simulation is started by the first global onset case with one day time step. The program generates one hundred realizations for each scenario. The medians, means, and the 95% non-parametric confidence intervals of the incidence, peak times, and the time of imported case arrivals are calculated over the realizations among different scenarios. week delay for the FPTs; but once all means of transport were 90% or 99% restricted, the FPT would have one month more delay when the R 0 s decreased 20% compared to that of the R 0 s with 20% increases. Moreover, the FHPT could be delayed for more than 2.5 months with 20% decreases of the R 0 s, whereas the FHPT was delayed for 1.5 months with 20% increases of the R 0 s for a 99% restriction of all means of transport.
Since the number of imported cases depended on the changes of the R 0 from the nonlocal countries, the growth of the local epidemic was affected by the cases passage times ( Figure 4). When the R 0 s increased by 20%, the five months' cumulative AR attained

Variations of screening sensitivity at entry border points
In the baseline scenarios, we set the screening sensitivity at entry border points as 30%; here, we assess the model's output at extremely high (95%) and low (5%) screening sensitivities. According to Figure 5 and 6, the screening sensitivities at entry border points affected slightly on the times of cases arrival. Amongst most of the travel restriction strategies, a 95% screening sensitivity showed at most one to two weeks additional delay to the FHPTs compared to that of a 5% screening sensitivity ( Figure 5).
The increase of the screening sensitivity at entry border points offered a moderate benefit on slowing down the growths of cumulative ARs. Showed in Figure 6A-D, a 95% screening sensitivity showed only half of five months' cumulative ARs compared to that of a 5% screening sensitivity. The 95% screening sensitivity also decreased the seven months' cumulative ARs by about 10% in most of the restriction strategies whether or not the AH had been imposed.

Variations of implementation date on travel restrictions
We tested the impact of delaying the imposition of travel restrictions to five and three months following the first global import. Showed in Figure 7A and 7B, imposing travel restrictions five months after the first global case arose would be too late obviously. Even if all means of transport had been 99% rescaled, the reduction in the cumulative AR was too small. However, it could still decrease the seven months' cumulative AR by no more than 10% if the growth of the epidemic was slowed down by the use of AH. Showed in Figure 7C, imposing the travel restrictions three months after the first global case arose would be a little bit late; but fractional blockings on all means of transport worked well in deferring the growth of the ARs. The 99% restriction would reduced the five months' and seven months' cumulative ARs more than half of that without intervention. With the use of AH, imposing the 99% restriction of all mean of transport was able to control the cumulative AR by no more than 2% in the first seven months; a 90% restriction could still maintain the average seven months' cumulative AR about 6% to 7% ( Figure 7D).

Sensitivity analysis Time step of model
The simulation results were based on the stochastic models with a time step of ∆t = 1 day. The simulations were repeated with ∆t = 0.5 day. Figure 8 showed the results for comparison. According to the results, the incidence growth curves differed moderately compared with that of ∆t = 1 day (main text, Figure 3); the daily ARs were less severe for scenarios with ∆t = 0.5 day. However, there were only slight differences for the impacts of interventions on the baseline scenario between two kinds of time-step settings. For

Multivariate sensitivity analysis
A multivariate sensitivity analysis which varied the following parameters with their prior distributions: • Length of latent period (days) ∼ Uniform(range from 1 to 2) • Sensitivity of the screening board for infectious subjects ∼ Uniform(range from 0.05 to 0.5) • Length of infectious period reduction (days) by taking antivirals ∼ Uniform(range from 1 to 2) • Length of infectious period reduction (days) by hospitalization ∼ Uniform(range from 1 to 2) • Fraction of infectiousness reduction for antiviral treatment ∼ Uniform(range from 0.3 to 0.9) was performed. Each random set of parameters was simulated before every realization. Figures 9 and 10 showed the results. The incidence curves were moderately sensitive to the variations of parameters. Restrictions on a single mode of transport had less apparent impact due to moderate deviations. The range of peak times from imposing 99% restriction on all modes of transports was wider. However, the central tendency of the intervention effects were quite stable. The impacts of interventions on the baseline scenario did not differ much compared with the study main findings. For example, when a 99% restriction of all transports was imposed, the peak was averagely deferred to the ninth month and the eleventh month respectively in absences and presences of the uses of antiviral and hospitalization.