How hosts and pathogens choose the strengths of defense and counter-defense. A game-theoretical view

Host-pathogen interactions consist of an attack by the pathogen, frequently a defense by the host and possibly a counter-defense by the pathogen. Here, we present a game-theoretical approach to describing such interactions. We consider a game where the host and pathogen are players and they can choose between the strategies of defense (or counter-defense) and no response. Specifically, they may or may not produce a toxin and an enzyme degrading the toxin, respectively. We consider that the host and pathogen must also incur a cost for toxin or enzyme production. We highlight both the sequential and non-sequential versions of the game and determine the Nash equilibria. Further, we resolve a paradox occurring in that interplay. If the inactivating enzyme is very efficient, producing the toxin becomes useless, leading to the enzyme being no longer required. Then, production of the defense becomes useful again. In game theory, such situations can be described by a generalized matching pennies game. As a novel result, we find under which conditions the defense cycle leads to a steady state or to an oscillation. We obtain, for saturating dose-response kinetics and considering monotonic cost functions, 'partial (counter-)defense' strategies as pure Nash equilibria. This implies that producing a moderate amount of toxin and enzyme is the best choice.