We applied optimal control theory to an SI epidemic model to identify optimal culling strategies for diseases management in wildlife. We focused on different forms of the objective function, including linear control, quadratic control, and control with limited amount of resources. Moreover, we identified optimal solutions under different assumptions on disease-free host dynamics, namely: self-regulating logistic growth, Malthusian growth, and the case of negligible demography. We showed that the correct characterization of the disease-free host growth is crucial for defining optimal disease control strategies. By analytical investigations of the model with negligible demography, we demonstrated that the optimal strategy for the linear control can be either to cull at the maximum rate at the very beginning of the epidemic (reactive culling) when the culling cost is low, or never to cull, when culling cost is high. On the other hand, in the cases of quadratic control or limited resources, we demonstrated that the optimal strategy is always reactive. Numerical analyses for hosts with logistic growth showed that, in the case of linear control, the optimal strategy is always reactive when culling cost is low. In contrast, if the culling cost is high, the optimal strategy is to delay control, i.e. not to cull at the onset of the epidemic. Finally, we showed that for diseases with the same basic reproduction number delayed control can be optimal for acute infections, i.e. characterized by high disease-induced mortality and fast dynamics, while reactive control can be optimal for chronic ones.
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