Effects of Internal Border Control on Spread of Pandemic Influenza

Population size, travel rates, and residence of travelers can aid in determining travel restrictions as a control policy.

where i(t) denotes the number of new infected persons on day t and is determined by the Poisson random variable X(t) with mean μ = S(t)p(t). Here, is the probability on day t of a given susceptible person becoming infected. This equation corresponds to an illness with total infection period of D days and a varying degree of infectivity ρ(τ) over the course of each infection. Note that τ = 0, 1,…, D-1, and Σρ(τ) = 1.
Multiple cities are accommodated by introducing a pair of labels to indicate a person's city of origin and destination as well as a rate of travel A kl from city k to city l per day. The situations considered in this article involve only 2 cities, with just 2 rates of travel, A 12 and A 21 . We set A 12 ≡ A 1 and A 21 ≡ A 2 , for convenience. The epidemic equations for city Here, X 1 (t) is a Poisson random variable with mean μ 1 where p 1 (t) is given by ( ) Note that the equations for city 2 and p 2 (t) can be obtained by replacing the label 1 with 2 and vice versa in the previous 2 equations.
The sensitivity of the results to the assumption that all infected persons travel can be tested by making an optimistic assumption that two thirds of infected persons are symptomatic and do not travel. This leads to an additional median delay of between 2 days (reproduction number R 0 = 3.5, peaked infectivity) and 11 days (R 0 = 1.5, flat infectivity) if applied to infections acquired while a person is at home or traveling. If, however, one assumes that persons infected while traveling would return home, then the delay is less significant, 1-4 days in scenario 1 and 0-1 days in scenario 2.

(B) Simple Model in Continuous Time
A simpler deterministic model for the infected persons is used to analyze differences in rates of travel on a single route. If we consider only the early stages of the epidemic, well before the peak, then the number of susceptible persons is approximately equal to the total population. The epidemic equations then reduce to the pair of coupled, linear ordinary differential equations that describe the change in the number of infected persons over time: where β is the effective contact rate, 1/γ is the average duration of infection, η is the population of city 1 divided by the population in city 2, A 1 and A 2 are the travel rates from city 1 to city 2 and vice versa, and φ + and φ − are population modifiers due to travel, with definitions ).
Symbolically, the solution to these equations can be written as When susceptible persons are prevented from traveling, this matrix simplifies to the form

(D) Analytic Approximations
The effect of travel restrictions can be approximated in terms of the initial growth rate, r, of the epidemic as T p = 0.99 = log(100)/r, for 99% restrictions and similarly for other levels of travel restriction. The time taken to reach 20 cases can be used to estimate r, and in fact, we can re-express T p by using this time: T p = 0.99 = T 20 *log(100)/log(20). In practice, estimates based on T 20 can be somewhat inaccurate for low values of R 0 (Figure panel D).
The probability of an outbreak spreading from city 1 to city 2, as described by the simulation model above, can be approximated by using extinction probabilities for branching processes (1). If, over the course of an outbreak, an attack rate, AR, occurs in the source region, then with N visitors in town A per day and N citizens of town A visiting the source region, the probability of an outbreak occurring is where q is the extinction probability for a Poisson branching process with mean R 0 . As is seen in Figure panel C, where AR ≈ (1-q), this approximates the simulation model very well.