EPIDEMIOLOGICAL CONSEQUENCES OF IMPERFECT VACCINES FOR IMMUNIZING INFECTIONS

Abstract

The control of some childhood diseases has proven to be difficult even in countries that maintain high vaccination coverage. This may be due to the use of imperfect vaccines and there has been much discussion on the different modes by which vaccines might fail. To understand the epidemiological implications of some of these different modes, we performed a systematic analysis of a model based on the standard SIR equations with a vaccinated component that permits vaccine failure in degree ("leakiness"), take ("all-or-nothingness") and duration (waning of vaccine-derived immunity). The model was first considered as a system of ordinary differential equations, then extended to a system of partial differential equations to accommodate age structure. We derived analytic expressions for the steady states of the system and the final age distributions in the case of homogenous contact rates. The stability of these equilibria are determined by a threshold parameter R_p , a function of the vaccine failure parameters and the coverage p. The value of p for which R_p = 1 yields the critical vaccination ratio, a measure of herd immunity. Using this concept we can compare vaccines that confer the same level of herd immunity to the population but may fail at the individual level in different ways. For any fixed R_p > 1, the leaky model results in the highest prevalence of infection, while the all-or-nothing and waning models have the same steady state prevalence. The actual composition of a vaccine cannot be determined on the basis of steady state levels alone, however the distinctions can be made by looking at transient dynamics (such as after the onset of vaccination), the mean age of infection, the age distributions at steady state of the infected class, and the effect of age-specific contact rates.

Keywords: age-structured population models; imperfect vaccines; infectious disease dynamics.

Fig. 2.1

A model of imperfect vaccine immunity incorporating three possible modes of failure: degree (blue), take (green) and duration (red).

Fig. 3.1

Bifurcation of the I steady state of the leaky (ε_L, ε_A, ε_W ) = (ε, 0, 0), all-or-nothing (ε_L, ε_A, ε_W ) = (ε, 0, 0) and waning (ε_L, ε_A, ε_W ) = (0, 0, ε) models as parameters are varied. We vary p ∈ [0, 1] in (a), ε ∈ [0, 1) in (b) β and ∈ [0, 1600] in (c). Default values of the parameters that are not being varied are given in Table 2.1. The values of the steady state for the all-or-nothing model overlap with those for waning immunity model. The imaginary part of the complex conjugate eigenvalue pair of the Jacobian matrix from part of the bifurcation diagram in (c) is shown in (d). Here the values for the leaky model is slightly below the overlapping plots for the all-or-nothing and waning models.

Fig. 3.2

Simulations showing what happens when the system is initially at the zero vaccination endemic steady state, then vaccination is gradually increased from p = 0 at t = 15 to p = 0.85 at t = 30. In (a) the waning model displays a honeymoon period of about 30 years. Afterward the solution oscillates as it approaches equilibrium yielding initially large outbreaks every 15 years. When is increased in (b), the oscillations get smaller.

Fig. 4.1

The value of λ* and the mean age of the infective class at steady state for the leaky, all-or-nothing, waning and combined models (refer to Table 2.2). For this figure we set b(t, a, x) = β with ∈ [0, 500]. The value of λ* for the all-or-nothing and waning models overlap.

Fig. 4.2

Illustration of the proportions of the infected class within the age ranges [0, 1), [1, 7), [7, 11), [11, 20) and [20, ∞) years at the endemic steady state of (4.1) for homogenous transmission β = 400 and age-specific transmission based on the contact structure described in [15].