Further analysis of the model shows that, in the short-term, mutant strains that enlarge their replication rate due to evolving an increased growth rate are more favoured than strains that suffer a lower immune-mediated death rate (‘immune tolerance’), as the latter does not completely evade ongoing immune proliferation due to inter-parasitic competition.

Analysis of this model suggests that, in order to enlarge its emergence probability, it is evolutionary beneficial for a mutated strain to increase its growth rate rather than tolerate immunity by haVing a lower immune-mediated deathrate.

We use Equation 8 in our model by setting R* = R2 — yim-t, which is the rescaled growth rate of the mutated strain, corrected for the fact that the baseline immunity rate will reduce its initial selective advantage.

Standard results from birth-death models states that the mean growth rate is equal to R2 — y, with variance equal to R2 + y [33].

(P1, (P2 Growth rate of initial, mutated infection x1, x2 Size of initial, mutated infection y Size of immune response

K Maximum size of immune response r Unscaled growth rate of immune response

R* ‘Effective’ initial reproductive ratio in the presence of immunity, R — yo p Scaled immune growth rate , r/o1

This is because the tau-leaping algorithm is accurate only if the eXpected number of events per time step is small [37]; since the growth rates of the pathogen strains and the lymphocytes are both large, a small time step is needed to make the simulation valid.