Pandemic influenza: risk of multiple introductions and the need to prepare for them

Abstract

Containing an emerging influenza H5N1 pandemic in its earliest stages may be feasible, but containing multiple introductions of a pandemic-capable strain would be more difficult. Mills and colleagues argue that multiple introductions are likely, especially if risk of a pandemic is high.

Figure 1. Probabilities of Zero, One, or Two or More Events (Introductions of a Pandemic-Capable Strain) When the Expected Number of Events Ranges from Zero to Ten, under a Constant Hazard (Poisson Distribution)

Note that if the expected number of events (v) is more than 1.26, then it is more likely that two or more events will occur than that exactly one event will occur.

Figure 2. Expected Gain in Time to a Pandemic (G) of a Containment Policy as a Function of the Expected Time to Introduction (T) of a Pandemic-Capable Strain

Either infinite containment attempts (blue) are possible or only a single containment attempt (red) is possible, and we assume an optimistic (80%, solid) or pessimistic (50%, dashed) probability of success for each containment. G is defined as the difference in the expected time to a pandemic under the status quo and the containment policy. The Poisson hazard of introduction of pandemic-capable strains (λ) is the reciprocal of the expected time to introduction (T). The results are identical if the hazard of introduction is fixed but unknown.

Figure 3. Schematic Diagram of the Escalating Hazard of Introduction Model with Time on the Horizontal Axis and Hazard of Introduction of a Pandemic-Capable Strain on the Vertical Axis

The initial hazard of introduction (λ ₀) increases to a higher hazard of introduction (λ ₁) at a rate determined by the hazard of escalation (λ _E). The expected time of escalation is the reciprocal of the hazard of escalation. The effect of escalation on the hazard of introduction is measured by the ratio of the final to the initial hazards of introduction (ε = λ ₁/λ ₀).

Figure 4. Relative Gain in Time to a Pandemic (G/T) as a Function of the Effect of Escalation on the Hazard of Introduction (ε = λ ₁/λ ₀) , Assuming Infinite Containment Attempts with 50% Success Probability Are Possible

Each curve corresponds to a 10-fold increase in the magnitude of the escalation hazard relative to the initial hazard of introduction (ρ = λ _E/λ ₀) , ranging from 0.1 to 10 (A, short-dashed, long-dashed, and solid lines), and from 100 to 100,000 (B, dotted, short-dashed, long-dashed, and solid lines). When the relative magnitude of the escalation hazard is zero (A, dotted line), the relative gain in time to a pandemic is identical to the simple Poisson process in Figure 2.