Delay in booster schedule as a control parameter in vaccination dynamics

Abstract

The use of multiple vaccine doses has proven to be essential in providing high levels of protection against a number of vaccine-preventable diseases at the individual level. However, the effectiveness of vaccination at the population level depends on several key factors, including the dose-dependent protection efficacy of vaccine, coverage of primary and booster doses, and in particular, the timing of a booster dose. For vaccines that provide transient protection, the optimal scheduling of a booster dose remains an important component of immunization programs and could significantly affect the long-term disease dynamics. In this study, we developed a vaccination model as a system of delay differential equations to investigate the effect of booster schedule using a control parameter represented by a fixed time-delay. By exploring the stability analysis of the model based on its reproduction number, we show the disease persistence in scenarios where the booster dose is sub-optimally scheduled. The findings indicate that, depending on the protection efficacy of primary vaccine series and the coverage of booster vaccination, the time-delay in a booster schedule can be a determining factor in disease persistence or elimination. We present model results with simulations for a vaccine-preventable bacterial disease, Heamophilus influenzae serotype b, using parameter estimates from the previous literature. Our study highlights the importance of timelines for multiple-dose vaccination in order to enhance the population-wide benefits of herd immunity.

Keywords: Booster schedule; Delay equations; Persistence; Reproduction number; Vaccination.

Fig. 1

Schematic diagram for the basic structure of the model in the absence of vaccination

Fig. 2

Schematic diagram for timelines of primary and booster vaccination with delay, and durations of vaccine-induced and naturally acquired protection

Fig. 3

The coverage of booster vaccination ($\rho$) as a function of time delay (${\tau }_p$) following primary vaccination. Solid line corresponds to a fixed coverage; dashed line represents the exponentially declining booster coverage; and dotted line illustrates an inverted logistic coverage of booster vaccination declining over time

Fig. 4

Reproduction number (${\mathcal{R}}_c$) as a function of protection efficacy of the primary vaccination ($1-\eta$, $x-$axis) and delay in the booster dose schedule (${\tau }_p$, $y-$axis). The coverage of booster dose is: a$\rho =0.95$ fixed; b$\rho =0.95e^{-0.001{\tau }_p}$; and c$\rho =14.2307e^{-0.006{\tau }_p}/(0.1+e^{-0.006({\tau }_p-450)})$. The white curve corresponds to $R_c=1$

Fig. 5

Prevalence of disease with: a$\eta =0.8$ and fixed $\rho =0.95$; b$\eta =0.95$, $\rho =0.89$ (for ${\tau }_p=2$ months) and $\rho =0.46$ (for ${\tau }_p=24$ months); c$\eta =0.88$, $\rho =0.95$ (for ${\tau }_p=1$ month), $\rho =0.92$ (for ${\tau }_p=9$ months), and $\rho =0.62$ (for ${\tau }_p=24$ months). The threshold of ${\tau }_p$ (for ${\mathcal{R}}_c=1$ in Fig. 4) is approximately (a) 19 months; b 7 months; and c 7 or 22 months

Fig. 6

Prevalence of disease in varying populations. Dashed (black) curve corresponds to a constant population size (i.e., simulated dashed curve in Fig. 5b). Colour curves represent the disease prevalence over time with changing population size. Parameter values are the same as Fig. 5b with ${\tau }_p=24$ months, while the birth rate changes by $\Delta$