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How to Minimize the Attack Rate during Multiple Influenza Outbreaks in a Heterogeneous Population

Figure 1

Illustration of the concept of optimal control for multiple outbreaks.

We assume that multiple outbreaks can occur, with the intervention only being feasible during the first outbreak. If the intervention is weak (or absent), the first outbreak will be large enough to deplete the number of susceptible people below a critical threshold level (the herd immunity level below which effective reproduction number <1), such that if the infection is re-introduced, its effective reproductive number would be too low to cause a second outbreak (black and cyan lines). If the intervention is very strong, it is possible that after the first outbreak, the number of susceptible people remaining is large enough to support a second (uncontrolled) outbreak upon re-introduction of the pathogen, leading to an overall number of people infected that might be the same as that reached during just one outbreak (red line). In both the “too much” and “too little” intervention scenarios, the number of susceptible people drops below the critical threshold level, which defines the level of herd immunity. The excess drop is termed ‘overshoot’. The optimal intervention is one that minimizes the overshoot by allowing the susceptible population to drop to the critical threshold level during the first outbreak, such that a second outbreak cannot occur (green line). The solid lines represent the susceptible people and the broken lines represent the infected people.

Figure 1