Vaccination and the dynamics of immune evasion

Abstract

Vaccines exert strong selective pressures on pathogens, favouring the spread of antigenic variants. We propose a simple mathematical model to investigate the dynamics of a novel pathogenic strain that emerges in a population where a previous strain is maintained at low endemic level by a vaccine. We compare three methods to assess the ability of the novel strain to invade and persist: algebraic rate of invasion; deterministic dynamics; and stochastic dynamics. These three techniques provide complementary predictions on the fate of the system. In particular, we emphasize the importance of stochastic simulations, which account for the possibility of extinctions of either strain. More specifically, our model suggests that the probability of persistence of an invasive strain (i) can be minimized for intermediate levels of vaccine cross-protection (i.e. immune protection against the novel strain) and (ii) is lower if cross-immunity acts through a reduced infectious period rather than through reduced susceptibility.

Figure 1

Compartmental sketch of the model used in this study. Births and deaths at rate μ per capita are not shown, nor vaccine uptake p.

Figure 2

The greyscale represents the initial growth rate of strain 2 (obtained by solving equation (3.4) for x), as a function of cross-protection on susceptibility τ on the horizontal axis, and (a) vaccine coverage p or (b) reduction in infectious period η on the vertical axis. In (a), the left-hand side of the vertical dashed line corresponds to a vaccine conferring narrow antigenic protection, τ<θ. Parameter values: R₀=17; θ=0.5; average infectious period 1/γ=21 days; average immune periods 1/σ=1/σ_V=20 years; average lifespan 1/μ=50 years; (a) no reduction in infectious period through cross-immunity (ν=η=0); and (b) vaccine coverage p=1 and ν=0.5.

Figure 3

Deterministic trajectories, obtained by the numerical integration of equations (2.1), following the introduction of strain 2 with an incidence of 10⁻⁶. Black lines show the incidence (infected fraction of the population) of strain 1, darker grey lines that of strain 2 and lighter grey lines the fraction in the vaccinated compartment V. Solid lines, vaccine cross-protection τ=0.7; dashed lines, τ=0.9. Arrows point at post-epidemic troughs, as recorded in figures 5 and . Other parameter values are as in figure 2, except vaccine immune period 1/σ_V=50 years.

Figure 4

Trough depth of strain 2 plotted against the reduction in susceptibility through vaccine cross-immunity (τ) on the horizontal and the vertical axes: (a) vaccine coverage p and (b) reduction in infectious period through vaccine cross-immunity η. Other parameter values are as in figure 2, except (b) ν=0.5.

Figure 5

Series of 100 stochastic simulations, grouped by outcome (see §2) in (a)–(e), showing the number of individuals infected with strains 1 (black) and 2 (grey). Parameter values: initial population size, 1 million; average life span 1/μ=50 years; R₀=17; vaccine coverage p=1; infectious period 21 days; immune periods 1/σ =1/σ_V=20 years; natural cross-protection θ=0.6; vaccine cross-protection τ=0.2; and no reduction in infectious period ν=η=0.

Figure 6

Summary of 10 series of 1000 stochastic simulations, for two values of vaccine cross-protection ((a) τ=0, (b) τ=0.7) and five values of vaccine coverage (horizontal axis). In each series, we recorded the frequencies (vertical scale) of the five possible outcomes (pattern coded) and the corresponding number of cases (the diameter of bubbles represents the average number for each outcome) in the 10 years that follow the introduction of strain 2. Other parameters are as in figure 5, except natural cross-protection θ=0.5.

Figure 7

(a) Frequencies of replacement of strain 1 by strain 2 and (b) frequencies of eradication of both strains, estimated from 36 series of 1000 stochastic simulations for a range of vaccine cross-protection parameters (τ and η) from 0 to 0.95. Other parameter values are as in figure 5, except natural cross-protection θ=0.6 and ν=0.7. Only three outcomes were obtained in these series: initial extinction of strain 2 (not shown); (a) extinction of strain 1 only; (b) or extinction of both strains.