Direct and indirect effects of influenza vaccination

The classic susceptible-infected-recovered (SIR) model forms the backbone of most infectious disease transmission models (Fig. 1a). Vaccinated newborns are assumed to become fully immune (V) whereas unvaccinated ones are susceptible (S). These can become infected (I) after contact with infected individuals; they finally recover, and become permanently immune (R). Although for influenza, neither vaccination-derived nor naturally acquired immunity lasts lifelong (as is assumed in the SIR model), we start with analyzing this basic model for the sake of simplicity. In order to quantify direct and indirect effects of vaccinations (which in this model only occur shortly after birth), we have slightly re-structured the SIR model: Fig. 1a shows a version of the standard model (model A) where vaccinated individuals (V) are fully immune. Figure 1b shows a version where vaccinees are fully susceptible to infection (i.e. vaccination in Fig. 1b is assumed to be completely useless); model B, therefore, represents the SIR model without vaccination effects. It shows how many vaccinees and how many non-vaccinees would have been infected if the vaccination either had never occurred or if it were completely useless. In the standard SIR model A, infection incidence is reduced because vaccinees are immune: they can neither be infected nor can they cause secondary infections. As a consequence, fewer susceptible individuals (S) are infected in model A than in the modified model B where the vaccination is without effect. In a first step, we only use model A to calculate the infection incidence for two vaccination scenarios: I₀ is the infection incidence without any vaccinations and I_S is the incidence with vaccinations. The total effect of vaccination (comprising both, direct and indirect effects) is given by the difference I₀-I_S. In a second step, we take a look at model (b) where vaccination is regarded to be non-protective: as vaccinees can be infected, we can calculate their infection incidence I_V, too. As these infections will not occur if the vaccine is fully protective, I_V is the direct effect of vaccination. The indirect effect is finally given as difference between total and direct effect (I₀-I_S-I_V). For the SIR model, the evaluation can be done by looking at the equilibrium state of the corresponding set of differential equations (as given in the Additional file 1). A key parameter in the calculation is the so-called basic reproduction number R₀ which describes how many secondary infections are caused by a single initial case in a completely non-immune population without interventions. Calculating the ratio of indirect/direct protection, we obtain the expression 1/(R₀–1) which is displayed in Fig. 2. If R₀ is larger than 2 (as is the case for e.g. measles), indirect effects are smaller than direct ones (the ratio becomes less than 1); if R₀ is below 2 (as is the case for influenza), indirect effects exceed direct ones.

The SIRS extension (Fig. 3a) of the SIR model allows for loss of immunity and for repeated vaccinations which not necessarily need to occur shortly after birth. Thus, it represents the situation of influenza much better than the SIR model. Individuals are born susceptible (S); they can either be vaccinated (V) or they can become infected (I) after contact with infected individuals and finally recover, and become immune (R). Immune individuals can lose their immunity and become susceptible again (S) whereby the immunity loss rate can be different for naturally acquired (R) and vaccination-derived immunity (V). We again modified the model: Fig. 3a shows the standard SIRS model where vaccinated individuals (V) are immune; Fig. 3b shows the version where vaccinees remain fully susceptible. Describing the systems by differential equations again allows calculating the ratio of indirect/direct vaccination effects as outlined in the previous section (see Additional file 1 for details). As shown in Fig. 4, the ratio becomes big if the basic reproduction number R0 is small.

Seasonality is neither considered in the basic SIR nor in the SIRS model, yet it plays a major role in the transmission of influenza. More realistic simulation models frequently implement a seasonality module which compresses the main influenza transmission period to a few winter months [7, 9‐13] (for a closer look at the specific effects of seasonality on direct and indirect effects, see Additional file 1). In Germany, influenza vaccination is mainly recommended for the elderly and for other individuals at increased risk whereby mostly trivalent inactivated vaccine is used. Following a general discussion on pediatric influenza vaccination, a survey initiated by the Robert-Koch-Institute in 2015 revealed that over 50% of parents are willing to have their children vaccinated against influenza if it is officially recommended [14]. In the following, we use an extension of the previously published influenza transmission model QLAIV-Sim [11, 12, 15] to look at direct and indirect effects of pediatric influenza vaccination in Germany. QLAIV-Sim is based on a model which extends the SIRS model to a system of 32,330 differential equations: based on realistic demographic data [16], the population is structured in 101 age groups, further distinguished into “at risk” and “no risk” groups, who are connected by an age-dependent contact matrix [17]. Vaccinations, using reported age- and risk-dependent vaccination coverage [18, 19], occur annually in October and November whereby individuals who were vaccinated in the previous season are preferentially vaccinated again (see Additional file 1 for details [20]). Vaccine efficacy of the inactivated vaccine is assumed to be 45% (0.5–1 year old children), 59% (2–17), 60% (18–59 no risk individuals) and 58% (at risk individuals and elderly 60+) [21‐24]; the vaccine efficacy of the live vaccine for 2–17 year old children is assumed to be 80% in the season following vaccination and 56% in the subsequent season [24, 25]. In order to create realistic age-dependent immunity patterns in the population, the model is run from 2000 to 2016 with trivalent influenza vaccine (TIV), using the recorded vaccine composition and allowing for the independent transmission of the four influenza strains A(H1N1), A(H3N2), B/Yamagata and B/Victoria. During the following 10-year evaluation period (starting in 2017), only quadrivalent influenza vaccines are used (i.e. vaccines which contain both Influenza B lineages). In a first simulation, we calculate how many infections occur (“baseline” result) if vaccinations are performed as in the initialization period with the only exception, that quadrivalent inactivated vaccine (QIV) is used instead of TIV. In a second step, we run the same simulation with the identical vaccination coverage, except that an increased percentage of children (2–17 years) receive quadrivalent live vaccine (QLAIV) instead of QIV. The resulting difference in infection incidence is the total effect of the additional QLAIV vaccinations. In order to separate this total effect into direct and indirect effects, we use the same strategy as described above: a third simulation is run where we assume that QLAIV vaccination is completely without effect, and the cumulative incidence is recorded separately for individuals who are in stage S and in stage V, respectively.

Table 1 shows the simulation results for 20% annual QLAIV vaccination coverage; infections are translated into symptomatic cases, assuming that 66.9% of infections lead to clinical disease [26]. Without additional vaccinations, 40.9 million symptomatic cases are expected to occur in the 10-year evaluation period. QLAIV vaccination of 20% of children reduces the cumulative incidence to 31.9 million cases. The prevented 9 million cases can be split into 1.1 million directly prevented cases among vaccinees, 2.4 million indirectly prevented cases among non-vaccinated children and 5.5 million indirectly prevented cases among adults. Thus, more than twice as many cases among children are indirectly prevented than directly. Combining all indirectly prevented cases among children and adults (7.9 million), we can see that more than 7 times as many cases are indirectly prevented as are directly prevented. Fig. 5a shows the results for 20 to 60% annual QLAIV coverage: the number of directly prevented symptomatic cases (white) and the numbers of indirectly prevented ones (light and dark grey) increase with increasing QLAIV coverage; due to mathematical reasons, the ratio of indirectly/directly prevented cases declines from 7.4 to 4.6 with increasing coverage. The numbers of symptomatic cases are finally translated into cases of acute otitis media (AOM) and to deaths due to influenza (Fig. 5b-c). As AOM cases predominantly occur among children, more cases are (directly and indirectly) prevented among children, leading to lower ratios of indirectly/directly prevented cases (ranging from 2.0 to 4.3; Fig. 5b). Considering, on the other hand, influenza deaths which predominantly occur in the elderly, the ratios of indirectly/directly prevented cases become much higher (ranging from 22.2 to 35.9; Fig. 5c), meaning that for every directly prevented death among vaccinees up to 36 deaths are indirectly prevented. In a sensitivity analysis, the vaccine efficacy of QLAIV is reduced to the QIV value of 59% and the duration of QLAIV immunity is reduced to 1 year (as for QIV), the absolute vaccination effects are reduced accordingly, but the ratio of indirect / direct effects grows: for 20% to 60% vaccination coverage, the ranges of ratios are 5.1–10.1 (symptomatic cases), 2.5–8.6 (AOM) and 23.4–63.0 (deaths), respectively (full results are shown in Additional file 1: Table S4).