The average number of new virus particles (virions) produced by a virally-infected

Question

cell during its lifetime is called the burst size. We will call the burst size N. Under reasonable
assumptions, N can be approximated by N = p/u�(p) where p is the rate of production of new
viruses and u�(p) is the cell death rate. Over many generations, the virus population can evolve so
that its production rate p lies anywhere on the range 0 < p < pmax. The upper bound pmax is set by
biological constraints (e.g. nutrient supply to the cells). We expect the virus to evolve to maximize
N.

(a) Suppose that the cell death rate is a constant, u�(p) = �0 > 0. What production rate would
you expect the virus to have after a long period of evolutionary change?

(b) Suppose the cell death rate is a linear function of the production rate, �u(p) = ap+�0, where
a and �0 are positive constants. What production rate would you expect the virus to have
after a long period of evolutionary change? Does this depend on a? If so, how?

(c) Finally, suppose the cell death rate is a quadratic function of the production rate, �u(p) =
bp^2 + u0, where b and �u0 are positive constants. What production rate would you expect the virus to have after a long period of evolutionary change? Does this depend on b? If so, how?