We first examine the effect of disease mortality in the I1 class (parameter ρ). As relative disease mortality in I1 increases, the ESS virulence decreases. The degree of decrease in ESS virulence depends on the parameter scaling virulence relative to transmission (ψ, a). As the amount of transmission increases per unit of virulence, the ESS virulence decreases. The decrease in ESS virulence is greatest when there is no disease mortality in I1 (ρ = 0; a).
Figure 1. (a) The relationship between evolutionarily stable virulence (α) and the relative virulence in the first infected class (ρ) for three levels of infection hazard (solid line, ψ = 0.1; dashed line, ψ = 0.2; dotted line, (more ...)
The relative amount of transmission in I2 (ϕ) also has a large effect on ESS virulence (b). As the amount of transmission in I2 increases, the ESS virulence decreases, and the rate of decrease depends on the level of mortality that occurs in the I1 class. As the level of transmission in I2 and the disease mortality rate in the I1 class (ρ) approach zero, the ESS virulence goes to infinity (b). These results can be understood by realizing that for any fixed level of virulence (α), decreases in the transmission parameter ϕ reduce the fitness benefit of reaching the second class (I2), while increases in ρ both decrease the probability of reaching the second class and decreases the infectious period in the first class. Therefore, as both parameters reach zero, there is no benefit in reaching the second class and no cost to virulence in the first class. Thus, ESS virulence is very high and virulence will have a greater tendency to increase after introduction.
The effect of the transition rate (γ) between the classes on ESS virulence depends strongly on the relative amount of disease mortality in I1 (ρ) and on the recovery rate (σ; ). When disease mortality in the classes is similar (ρ = 1) and there is recovery from the second class (σ = 10), ESS virulence increases with increases in transition rate between the classes (a). When there is no recovery, however, there is no change in ESS virulence with transition rate (b). This result is because of the pathogen, on average, spending relatively more time in the I1 class as γ decreases. In the limit as γ goes to zero, the pathogen spends its whole life in I1 and has an infectious period of 1/(μ + α). Thus, pathogen virulence adapts to I1, and reductions in virulence translate directly to a longer infectious period. As γ increases, however, the pathogen spends little time in I1 and, in the limit, experiences an infectious period of 1/(μ + α+ σ). Thus, the pathogen adapts to the conditions of the I2 class and to a higher virulence as long as σ > 0.
Figure 2. The relationship between evolutionarily stable virulence (α) and the transition rate between the infected classes (γ) for three levels of relative virulence (ρ) with and without recovery (σ) from the second infected class. (more ...)
With less disease mortality in I1 (ρ < 1), ESS virulence first increases and then decreases as transition rate (γ) increases from zero (a,b). The degree of increase, and the inflection point where ESS virulence begins to decrease, depends on the amount of disease mortality in I1 (ρ, ). At the extremes of γ (0, ∞), the ESS virulence converges to that of a model with one infectious stage, just as described above, but here the ESS virulence is always higher for low values of ρ and γ ().