Co-operation and defection: playing the field in virus dynamics.

A previous model (Szathmáry, 1992) is further developed for the dynamics of standard (V) and defective interfering (DI) viruses. The crucial retained element is the incorporation of population structure in the form of a complete distribution of cells infected by particles differing in number. New elements are: the non-linear shared benefit from the contribution of Vs to the group of viruses infecting the same cell (synergistic at low numbers, diminishing returns at high numbers, respectively); a dynamics for the total number of particles (V and DI); and the possibility of extinction if the frequency of Vs is small enough. In evolutionary genetical terms this is a frequency- and density-dependent evolutionary game. A crucial result is retained: coexistence of Vs and DIs is possible provided the multiplicity of infection (hence the size of the coinfection group) is large enough. Phase portraits and vector-field plots for a continuous-time and numerical solutions for a corresponding discrete-time case are presented. The latter reflect the basic features of serial, undiluted passage. The causes for two possible means of extinction (low initial frequency of Vs, and very high vigour of Vs) are revealed: the appearance of high amplitude fluctuations. Coexistence can apparently be ensured by stable points, periodic behaviour, or strange attractors. The paper clarifies the dynamical background of cycles found experimentally in V-DI systems.