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.

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A Network Landscape Model (NLM) for the evaluation of the ecological trend of an environmental system is here presented and investigated. The model consists in a network of dynamical systems, where each node represents a single Landscape Unit (LU), endowed by a system of ODEs for two variables relevant to the production of bio-energy and to the percentage of green areas, respectively. The main goal of the paper consists in testing the relevance of connectivity between the LUs. For this purpose we consider first the Single LU Model (SLM) and investigate its equilibria and their stability, in terms of two bifurcation parameters. Then the network dynamics is theoretically investigated by means of a bifurcation analysis of a proper simplified differential system, that allows to understand how the coupling between different LUs modifies the asymptotic scenarios for the single LU model. Numerical simulations of NLM are performed, with reference to an environmental system in Northern Italy, and results are discussed in connection with SLM

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