A recently proposed consistent approach for elastically scattering gas mixtures, of the type introduced by Bhatnagar, Gross, and Krook (BGK), has been extended to deal with a four species gas undergoing reversible bimolecular chemical reactions. The single BGK collision operator introduced for each species must take into account also transfer of mass and of energy of chemical bond. Suitable auxiliary fields have then to be introduced not only for temperatures and velocities, but also for densities, in order to fulfill correctly balance equations for mass, momentum, and total energy. The exact collision equilibrium, satisfying the mass action law of chemistry, is also recovered, and the proper choice of collision frequencies is discussed. Preliminary numerical results for the relaxation problem in space-homogeneous conditions are reported and briefly commented on.

An ellipsoidal BGK model is proposed for a binary mixture of rarefied gases in the frame of kinetic theory. It fulfils the crucial properties of the actual Boltzmann equation (collision invariants, equilibria, entropy dissipation), and introduces a further constraint on velocity equalization of the two species. The model features two disposable relaxation parameters which can be used to fit exactly, in the continuum limit, Fick's law for diffusion velocities and Newton's law for the viscous stress in the relevant set of Navier-Stokes equations. Positivity of temperature fields is guaranteed by a physically meaningful restriction on the parameters themselves.

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.

We propose a kinetic model of BGK type for a gas mixture of an arbitrary number of species with arbitrary collision law. The model features the same structure of the corresponding Boltzmann equations and fulfils all consistency requirements concerning conservation laws, equilibria, and Htheorem. Comparison is made to existing BGK models for mixtures, and the achieved improvements are commented on. Finally, possible application to the case of Coulomb interaction is briefly discussed.

A new class of high-order accuracy numerical methods for the BGK model of the Boltzmann equation is presented. The schemes are based on a semi-lagrangian formulation of the BGK equation; time integration is dealt with DIRK (Diagonally Implicit Runge Kutta) and BDF methods; the latter turn out to be accurate and computationally less expensive than the former. Numerical results and examples show that the schemes are reliable and efficient for the investigation of both rarefied and fluid regimes in gasdynamics. arXiv:1411.7929v1 [math.NA] 28 Nov 2014 1 More precisely, from the BGK model, to zero-th order in ε, one obtains the compressible Euler equations in the fluid-dynamic limit, while to first order in ε, the moments satisfy equations of compressible Navier-Stokes type, but with the wrong value for the Prandtl number. This problem can be fixed by resorting to the so-called ES-BGK model [4], but in the present paper we shall restrict to the classical BGK model.

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A recently proposed consistent Bhatnagar-Gross-Krook-type approach for reversible bimolecular chemical reactions, well suited to deal with collision dominated gas mixtures in which mechanical and chemical relaxation times are of the same order of magnitude (fast reactions), is discussed. The model recovers essential features of the chemical process such as mass action law at equilibrium and reactive H theorem. The hydrodynamic limit, at both Euler and Navier-Stokes levels, is derived by a Chapman-Enskog procedure, in terms of the relevant hydrodynamic variables, and compared to the corresponding limits holding in the nonreactive and in the slowly reactive cases. In particular, results show that reactive corrections to transport coefficients cannot be neglected for fast reactions.

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