Bolzoni Luca, Bonacini Elena, Soresina Cinzia, Groppi Maria
Risk Analysis Unit, Istituto Zooprofilattico Sperimentale della Lombardia e dell'Emilia Romagna, Via dei Mercati 13, Parma 43126, Italy.
Department of Mathematics and Computer Science, University of Parma, Parco Area delle Scienze 53/A, Parma 43124, Italy.
Math Biosci. 2017 Oct;292:86-96. doi: 10.1016/j.mbs.2017.07.011. Epub 2017 Aug 8.
We investigate the time-optimal control problem in SIR (Susceptible-Infected-Recovered) epidemic models, focusing on different control policies: vaccination, isolation, culling, and reduction of transmission. Applying the Pontryagin's Minimum Principle (PMP) to the unconstrained control problems (i.e. without costs of control or resource limitations), we prove that, for all the policies investigated, only bang-bang controls with at most one switch are admitted. When a switch occurs, the optimal strategy is to delay the control action some amount of time and then apply the control at the maximum rate for the remainder of the outbreak. This result is in contrast with previous findings on the unconstrained problems of minimizing the total infectious burden over an outbreak, where the optimal strategy is to use the maximal control for the entire epidemic. Then, the critical consequence of our results is that, in a wide range of epidemiological circumstances, it may be impossible to minimize the total infectious burden while minimizing the epidemic duration, and vice versa. Moreover, numerical simulations highlighted additional unexpected results, showing that the optimal control can be delayed also when the control reproduction number is lower than one and that the switching time from no control to maximum control can even occur after the peak of infection has been reached. Our results are especially important for livestock diseases where the minimization of outbreaks duration is a priority due to sanitary restrictions imposed to farms during ongoing epidemics, such as animal movements and export bans.
我们研究了SIR（易感-感染-康复）传染病模型中的时间最优控制问题，重点关注不同的控制策略：疫苗接种、隔离、扑杀以及降低传播率。将庞特里亚金最小值原理（PMP）应用于无约束控制问题（即无控制成本或资源限制），我们证明，对于所研究的所有策略，只允许最多有一次切换的bang-bang控制。当发生切换时，最优策略是延迟一定时间后再采取控制行动，然后在疫情剩余阶段以最大速率应用控制。这一结果与之前关于在疫情期间最小化总感染负担的无约束问题的研究结果形成对比，在那些研究中最优策略是在整个疫情期间都使用最大控制。那么，我们结果的关键影响在于，在广泛的流行病学情况下，可能无法在最小化疫情持续时间的同时最小化总感染负担，反之亦然。此外，数值模拟突出了其他意外结果，表明当控制再生数低于1时最优控制也可能被延迟，甚至在感染峰值出现后才会从无控制切换到最大控制。我们的结果对于牲畜疾病尤为重要，因为在持续疫情期间，由于对农场实施的卫生限制措施（如动物流动和出口禁令），将疫情持续时间最小化是首要任务。
