Fig. 4

Ratio of indirect/direct vaccination effects in the SIRS model for different vaccination rates (from top to bottom: ϕ = 0.2, 0.1, 0.01 per year). As the whole population is eligible for vaccination in the SIRS model, transmission can go to extinction for moderately high annual vaccination coverage if R₀ is small (thus, the lines cannot be drawn for the whole R₀ range). Parameters: life expectancy μ⁻¹ = 70 years, duration of naturally acquired immunity ρ ⁻¹ = 6 years, duration of vaccination-derived immunity τ⁻¹ = 2 years, duration of contagiousness γ⁻¹ = 5 days. The mathematical description of the curves is given in the Additional file 1