Figure 4.
The dynamics of the SPI system with seasonality incorporated into the model. (i) Low priming rate (parameters: a = 2, b = 1, h = 0.1, p = 0.1, q = 0.2, α = 1, β = 2, τ = 0.2, c = 0, fY = 0.2, fP = 0.5). (ii) Intermediate priming rate (parameters: a = 5, b = 2, h = 0.1, p = 0.5, q = 0.7, α = 3.5, β = 2, τ = 0.5, c = 0, fY = 0.2, fP = 0.5), and (iii) high priming rate (parameters: a = 2, b = 1, h = 0.1, p = 0.8, q = 0.7, α = 1, β = 2, τ = 0.5, c = 0, fY = 0.2, fP = 0.5). (a) Original SPI model with fixed transmission and birth rate. Endemic equilibrium occurs at low priming, prolonged damped oscillations to endemic equilibrium at intermediate priming and endemic limit cycles at high priming. (b) SPI model with seasonal birth rate. Small annual cycles occur at low priming, multi-year cycles at intermediate priming and fewer, larger multi-year cycles at high priming. (c) SPI model with seasonal transmission rate. Annual cycles occur at low priming rate, large biennial cycles at intermediate priming rate and quasi-periodic cycles at high priming rate.