Mitigating effects of vaccination on influenza outbreaks given constraints in stockpile size and daily administration capacity

Abstract

Background: Influenza viruses are a major cause of morbidity and mortality worldwide. Vaccination remains a powerful tool for preventing or mitigating influenza outbreaks. Yet, vaccine supplies and daily administration capacities are limited, even in developed countries. Understanding how such constraints can alter the mitigating effects of vaccination is a crucial part of influenza preparedness plans. Mathematical models provide tools for government and medical officials to assess the impact of different vaccination strategies and plan accordingly. However, many existing models of vaccination employ several questionable assumptions, including a rate of vaccination proportional to the population at each point in time.

Methods: We present a SIR-like model that explicitly takes into account vaccine supply and the number of vaccines administered per day and places data-informed limits on these parameters. We refer to this as the non-proportional model of vaccination and compare it to the proportional scheme typically found in the literature.

Results: The proportional and non-proportional models behave similarly for a few different vaccination scenarios. However, there are parameter regimes involving the vaccination campaign duration and daily supply limit for which the non-proportional model predicts smaller epidemics that peak later, but may last longer, than those of the proportional model. We also use the non-proportional model to predict the mitigating effects of variably timed vaccination campaigns for different levels of vaccination coverage, using specific constraints on daily administration capacity.

Conclusions: The non-proportional model of vaccination is a theoretical improvement that provides more accurate predictions of the mitigating effects of vaccination on influenza outbreaks than the proportional model. In addition, parameters such as vaccine supply and daily administration limit can be easily adjusted to simulate conditions in developed and developing nations with a wide variety of financial and medical resources. Finally, the model can be used by government and medical officials to create customized pandemic preparedness plans based on the supply and administration constraints of specific communities.

Figure 1

Proportional and non-proportional decay of the vaccinable population. Proportional decay (dashed line) is given by x(t) = x₀e^-kt, for k = 0.1. Non-proportional decay (solid line) is given by , where .

Figure 2

Effects of vaccination in the proportional and non-proportional models for different campaign starts. The proportional model is represented by dashed black lines and the non-proportional model by solid black lines. The graphs in the left column (a, c, e) show the proportion of infected people as a function of time. The graphs in the right column (b, d, f) show the proportion of the population vaccinated, and those still eligible for vaccination (vaccinable), over time. The initial population size is 108 people. Infected individuals are inserted into the susceptible population with a pulse on day 10 (t₀ = 10; solid vertical gray line). The vaccination campaign is initiated on day 20 (a, b), 50 (c, d), or 80 (e, f), and lasts 28 days. Start (t_a)and stop (t_b) times of the campaign are indicated by dashed vertical lines. Vaccination occurs at a rate of 1% of the eligible population per day (proportional; k = 0.01), or at a maximum of 10⁶ vaccines per day (non-proportional, ).

Figure 3

Effects of vaccination in the two models for different administration rates and campaign durations. Epidemic measures are shown for proportional (open circles) and non-proportional (filled dots) models. Final size, peak size, peak time, and epidemic duration are plotted as a function of the difference between the vaccination start time (t_a) and the onset of the initial outbreak (t₀; solid gray line). The vaccination campaign durations and daily administration rates are as follows: (1) 56 day campaign with k = 0.001 (proportional) or (non-proportional) (a1-a4), (2) 28 day campaign with k = 0.01 or (b1-b4), and (3) 3 day campaign with k = 0.1 or (c1-c4).

Figure 4

Effects of vaccination in the non-proportional model given different levels of population coverage. Simulations were performed using the non-proportional model of vaccination with . The pulse inserting infected individual(s) into the susceptible population occurs at t₀ = 10 (gray solid vertical lines). The proportion of people infected over time is plotted for vaccination start times, t_a = 20 (a-d), t_a = 50 (e-h), and t_a = 80 (i-l). Start times are indicated with dashed lines in each panel. The target level of vaccination coverage in the total population varies between 20% and 80%, as indicated. Dotted lines mark the end of the vaccination campaign when the target coverage level is reached. The probability of being confirmed, p, is set at either 0.20 (thick gray lines) or 0.65 (thin black lines), and b adjusted accordingly such that R₀ = 2.0 for all simulations. Note that the gray and black lines overlap. The inset in panel (i) illustrates the change in the epidemic dynamics at the time vaccination ends.