(A) The number of reported pertussis cases in Massachusetts. The main plot shows cases aggregated yearly from 1910 to 2008. (Inset) Cases aggregated monthly from 1990 to 2008. (B) The distribution of ages of reported infections in Massachusetts. (Left Cases identified between 1918 and 1921, with cases aggregated into twelve age groups (year long from 0 to 10 years of age, then 10–15, and > 15). (Right) Cases identified between January 1988 and December 2008, with all cases aggregated into year-long age intervals.

(A) The model-predicted age distribution at different levels of vaccine coverage for κ = 0.5 and κ = 20 and n = 10. The y-axes indicate the proportion of the total population infected at age a, averaged over time. (Top) Predicted age distributions for κ = 0.5, with p_vacc = 0 (black) or p_vacc = 0.8 (red). The only age group whose incidence significantly changes from the pre- to postvaccine era is young children who are directly protected by the vaccine. (Bottom) The same results, but for κ = 20. Vaccination results in a decrease in infant cases and an increase in teenage and adult cases, consistent with the data. (B) Mean asymptotic incidence as vaccine coverage increases for κ = 0.5 (red), κ = 20 (black), and κ = 5,000 (green). For low κ disease incidence monotonically decreases as vaccine coverage increases. For the moderate value of κ incidence decreases until a critical vaccine coverage threshold (p_vacc ≈ 0.55 with these parameters), at which point it begins increasing again, thereby predicting failure to control disease in the presence of high vaccine coverage. For extremely high κ, herd immunity is restored and incidence decreases monotonically again. (C) Contour plot of the ratio of secondary to primary cases across vaccine coverage (p_vacc) and varying values of the boosting coefficient (κ). The area of parameter space in which there are more primary than secondary cases is in shades of blue. Areas in which more secondary cases are expected are in shades of purple to red. For low values of κ, the model predicts more secondary than primary cases even in the absence of vaccine coverage (purple in the bottom left corner). For values of κ > 1 the model allows for predominantly primary cases in the prevaccine era (blue on the left), shifting to predominantly secondary cases with higher vaccine coverage (purples and reds on the right). For areas of parameter space in which there are coexisting cyclic and equilibrial attractors, the results from the fluctuating regimes are shown.

Effect of 80% coverage with a teenage booster vaccine given at age 15. The arrows indicate ages at which vaccination can occur (0.5, 2, 5, and 15 years). The white bars show the age distribution without the booster dose at 15 years; the black bars show the distribution with the booster. The y-axis is normalized by the total number of cases in the no-booster regime. Simulations carried out with n = 10.

A flow-diagram of the basic model. Individuals in class R (recovered) are resistant to infection and immune boosting. Those in W (waning) are resistant to infection but their immunity can be boosted at a rate determined by the force of infection, λ, and the value of the boosting coefficient, κ. The duration of immunity to infection is determined by the length of time spent in the circled loop, which depends on the relative magnitude of the competing rates κλ and σ.

(A) Curves indicating the location of bifurcation: supercritical Hopf (dashed line), subcritical Hopf (solid line), and fold of limit cycles (dotted line). (B) Simulated trajectories in each dynamical regime with κ set to 20, and three vaccine coverages. (Top) Coexisting cycle and equilibrium. A sample trajectory from the region with only a stable equilibrium is shown in the middle, and only a limit cycle on the bottom. (C) One-dimensional bifurcation diagram over p_vacc. The maximum and minimum infected proportions of the population are shown from simulations over increasing and decreasing vaccine coverage, thereby capturing both attractors in the area in which they coexist (0.4 ≲ p_vacc ≲ 0.5). (D) Same as (C) but varying κ instead of p_vacc.