The reinfection threshold regulates pathogen diversity: the case of influenza

Abstract

The awareness that pathogens can adapt and evolve over relatively short time-scales is changing our view of infectious disease epidemiology and control. Research on the transmission dynamics of antigenically diverse pathogens is progressing and there is increasing recognition for the need of new concepts and theories. Mathematical models have been developed considering the modelling unit in two extreme scales: either diversity is not explicitly represented or diversity is represented at the finest scale of single variants. Here, we use an intermediate approach and construct a model at the scale of clusters of variants. The model captures essential properties of more detailed systems and is much more amenable to mathematical treatment. Specificities of pathogen clusters and the overall potential for transmission determine the reinfection rates. These are, in turn, important regulators of cluster dynamics. Ultimately, we detect a reinfection threshold (RT) that separates different behaviours along the transmissibility axis: below RT, levels of infection are low and cluster substitutions are probable; while above RT, levels of infection are high and multiple cluster coexistence is the most probable outcome.

Figure 1

Schematic of clusters of antigenic variants, together with drift and shift phenomena.

Figure 2

Diagram representing the model with two clusters.

Figure 3

Simulations with different cut-offs (a,c,e) and associated replacement windows (b,d,f). (a,c,e) Simulation of model (2.1) with two clusters and R₀=3.5, just below the RT, set at 4 (σ=0.25). Black and grey lines correspond to clusters 1 and 2, respectively. Cut-offs for extinction marked at 10⁻⁶, 10⁻⁷ and 10⁻⁸ represent population sizes P=10⁶, 10⁷ and 10⁸, respectively. (b,d,f) Range of R₀ where replacement is expected as determined from the deterministic model with cut-offs as above (black bars) and the stochastic model with corresponding population sizes (grey bars: range of R₀ where replacement occurs in at least one of 100 simulations). Three values of Δσ are represented: (a,b) Δσ=0.01; (c,d) Δσ=0.1; (e,f) Δσ=0.25.

Figure 4

Detailed results characterizing the probabilities of invasion as three possible outcomes: extinction (white), replacement (black) and coexistence (grey). (a) Δσ=0.01 and (b) Δσ=0.1. f(x) represents the frequency of each event, relative to the total number of simulations performed. The RT is set at 4 (σ=0.25).

Figure 5

Summary of the results from the stochastic simulations for σ=0.25 and 0.5 (RT set at 4 and 2, respectively). The nested curves depict the percentage of replacement events, conditional to invasion of the second cluster, for five values of Δσ (from top to bottom): 0.01, 0.1, 0.25, 0.4 and 0.6. The upper curves (Δσ=0.01) are discontinued at R₀=3.2 (in (a)) and R₀=1.8 (in (b)) as stochastic simulations are unfeasible at low R₀. Deterministic simulations suggested the trend indicated by the dotted segments. Equilibrium curve calculated from the deterministic model (black).