  The rapid rise of antibiotic resistance is a serious threat to global public
health. Without further incentives, pharmaceutical companies have little
interest in developing antibiotics, since the success probability is low and
development costs are huge. The situation is exacerbated by the "antibiotics
dilemma": Developing narrow-spectrum antibiotics against resistant bacteria is
most beneficial for society, but least attractive for companies since their
usage is more limited than for broad-spectrum drugs and thus sales are low.
Starting from a general mathematical framework for the study of
antibiotic-resistance dynamics with an arbitrary number of antibiotics, we
identify efficient treatment protocols and introduce a market-based refunding
scheme that incentivizes pharmaceutical companies to develop narrow-spectrum
antibiotics: Successful companies can claim a refund from a newly established
antibiotics fund that partially covers their development costs. The proposed
refund involves a fixed and variable part. The latter (i) increases with the
use of the new antibiotic for currently resistant strains in comparison with
other newly developed antibiotics for this purpose---the resistance
premium---and (ii) decreases with the use of this antibiotic for non-resistant
bacteria. We outline how such a refunding scheme can solve the antibiotics
dilemma and cope with various sources of uncertainty inherent in antibiotic
R\&D. Finally, connecting our refunding approach to the recently established
antimicrobial resistance (AMR) action fund, we discuss how the antibiotics fund
can be financed.
