An optimal control theory approach to non-pharmaceutical interventions

Lin, Feng; Muthuraman, Kumar; Lawley, Mark

Effect of NPI policies on the epidemic NPIs reduce and delay the spread of pandemic by moderating social contact between susceptible and infectious people. Because NPIs disrupt daily societal functions, it is important that they be implemented judiciously. This requires identifying effective initiating triggers, which is a challenging research task. Implementing NPIs will impede influenza spread; on the other hand, normal societal functions will be interrupted. The optimal control method takes both aspects into account and tries to find the best balance between them, given the decision makers relative weighting of the two. Based on Figures 2 and 3 , early implementation for moderate and severe pandemic is very important for NPIs to have impact on the outbreak and the impact is effective only if NPIs are implemented early. Late NPI implementation might still be optimal, but the impact is much less. For a severe pandemic, it is optimal to trigger NPIs at the beginning stage when susceptible population is large and infectious population is small. If we miss the beginning stage, it is not optimal to implement NPIs until the outbreak is significantly progressed. This is because once the pathogen achieves a certain level of infection, NPIs are not effective against it, and thus are not worth the cost. That is, the benefit of NPIs at a stage when the disease has progressed significantly is less than the cost of NPI implementation. This finding supports the CDC pandemic mitigation guidelines, which state that when the pandemic is Category 4 or 5, all NPIs are recommended for early implementation [7] . Furthermore, earlier NPI implementation reduces and delays the peak of the outbreak as illustrated Figures 3(a) and 3(b), which allows additional time for vaccine development. If a severe pandemic occurs, hospitals will experience an overwhelming influx of patients and need to operate at their surge capacities. Earlier NPI implementation can reduce the magnitude of infectious at the peak, which relieves some of the burden on hospitals and other health care infrastructures. In contrast, NPIs are not nearly as effective if disease has already spread into the community as the cases shown in Figures 3(c) and 3 (d). In both cases, NPIs are triggered after the peak of the outbreak, where hospitals might have already been operating at their surge capacities for a few weeks. Both cases start at states falling on the control thresholds recommended in Figures 2(c) and 2(d) [37] . This finding indicates that the additional complexity of our model is warranted when compared with the SI model used in [37] . Timely and sensitive surveillance systems are key to successful application of the optimal control method as knowledge of both the pathogen characteristics and the community state are assumed. The surveillance systems should be able to identify the virus quickly and provide accurate estimates for parameters which characterize the severity of an influenza. The effectiveness of the control policy depends on the accuracy of these estimates, which include infection rate b, death rate Ï„ and recovery rate g. Once the control policy is computed, we also need to track the community state to determine if NPIs should be triggered. As early NPI implementation is found to be much more effective, we do not want to miss the beginning stage of the outbreak. Thus, the surveillance system should also estimate the community state, including the size of the infectious and sus