Direct and indirect effects of influenza vaccination

Background After vaccination, vaccinees acquire some protection against infection and/or disease. Vaccination, therefore, reduces the number of infections in the population. Due to this herd protection, not everybody needs to be vaccinated to prevent infections from spreading. Methods We quantify direct and indirect effects of influenza vaccination examining the standard Susceptible-Infected-Recovered (SIR) and Susceptible-Infected-Recovered-Susceptible (SIRS) model as well as simulation results of a sophisticated simulation tool which allows for seasonal transmission of four influenza strains in a population with realistic demography and age-dependent contact patterns. Results As shown analytically for the simple SIR and SIRS transmission models, indirect vaccination effects are bigger than direct ones if the effective reproduction number of disease transmission is close to the critical value of 1. Simulation results for 20–60% vaccination with live influenza vaccine of 2–17 year old children in Germany, averaged over 10 years (2017–26), confirm this result: four to seven times as many influenza cases are prevented among non-vaccinated individuals as among vaccinees. For complications like death due to influenza which occur much more frequently in the unvaccinated elderly than in the vaccination target group of children, indirect benefits can surpass direct ones by a factor of 20 or even more than 30. Conclusions The true effect of vaccination can be much bigger than what would be expected by only looking at vaccination coverage and vaccine efficacy. Electronic supplementary material The online version of this article (doi:10.1186/s12879-017-2399-4) contains supplementary material, which is available to authorized users.

, is the indirect effect, and the ratio of indirect/direct effect is . Indirect effects surpass direct ones for  Exploring direct and indirect vaccination effects in the SIRS model As the immunity against influenza is not permanent and as vaccinations have to be given repeatedly, the simple SIR model does not adequately model the transmission of influenza. The so-called SIRS model improves some of the shortcomings of the SIR model: immunity is lost over time and vaccination is modeled as a continuous process which goes on throughout the life. As in the previous section, we develop a second SIRS model where vaccination does not prevent infection in order to separate direct and indirect vaccination effects (parameters: per capita birth and death rate µ, contact rate β, recovery rate γ, vaccination rate φ, loss rate of naturally acquired immunity ρ, loss rate of vaccination-derived immunity τ, population size N). The SIRS models can be described by the following sets of differential equations: Similarly, the modified SIR model with non-protective vaccination is given by The main purpose of the second model is to provide the infection incidence for unvaccinated individuals which is not influenced by indirect effects to determine how many non-vaccinees would be infected in the absence of indirect effects. Comparing the results of only the first model with and without vaccination, the incidence of infections which is prevented by vaccination totals µ τ µ ρ µ ρ γ µ γ φ The incidence among vaccinees in the second model, , denotes infections that would have occurred among vaccinees (i.e. infections which are directly prevented by a protective vaccine). The difference between the total effect and the direct effect is the indirect effect. The ratio of indirect/direct effects is Figure 4 of the paper).
( ) The endemic equilibrium of this system is given by The endemic equilibrium of this system is given by

Exploring direct and indirect vaccination effects with the simulation tool Q-LAIV-Sim
The simulation tool Q-LAIV-Sim describes the spread of influenza in a population with realistic demographic structure, age-dependent contacts, annual vaccination and loss of immunity. Previous versions of the simulation tool have been published elsewhere (1,2). The new version of the simulation tool differs in several aspects from the previously published one: (a) instead of calculating the demographic development, the demography in the new version of the simulation tool is solely based on demographic data and on official demographic predictions (for details, see paragraph "Birth ν, death µ and ageing α"). (b) Instead of using a single "birthday" at the beginning of the simulation year at which the age of every individual in the whole population increases by one year, ageing of the population now occurs continuously throughout the simulation year (for details, see paragraph "Birth ν, death µ and ageing α"). (c) The POLYMOD matrix which describes the age-dependent frequency of contacts was extended to 1-year age groups and made symmetric by averaging "incoming" and outgoing" contacts (for details, see paragraph "Contact matrix"). (d) Whereas the previous version of the simulation tool only distinguished between influenza A and B, the extended version allows for the independent transmission of four influenza viruses (A(H1N1), A(H3N2), B/Yamagata, B/Victoria; for details, see paragraph "Influenza strains"). (e) This also allows to consider that trivalent influenza vaccine (TIV) only protects against the B lineage contained in the vaccine (for details, see paragraph "Vaccination rate φ"). (f) Vaccination in the new version not only depends on the age and risk-status of the vaccinees, but also considers whether or not they were vaccinated in the previous season, allowing for preferential re-vaccination (the calculation of the revaccination factor from data will be given below). (g) Vaccination-derived immunity, which was assumed to be lost exponentially in the previous version, is now assumed to be lost completely at the end of one or two years (for details, see paragraph "Loss of vaccine-derived immunity"). (h) Introduction of infections from outside of the population (which was already considered in the previous version of the simulation tool) was modified to mainly occur during the main transmission season, assuming that neighboring countries (from which infections are most likely introduced) have similar seasonal waves (for details, see paragraph "Force of infection λ"). (i) Newer parameter values (e.g. those describing the vaccination coverage, the vaccine efficacy and the percentage of at-risk individuals) have become available and are now used in the simulations (Tables S1 and S2).

Model description of Q-LAIV-Sim
The core of the mathematical model which underlies the computer simulations is given below. In order to reduce the complexity of the model which is described by 32,330 differential equations, only the basic features are shown in the equations; further explanations will be given below.
a a a n a a n a a n a a a a n a a n a a n a n a n a a n a n a n a S r a a n a a n a a n a n a n a a n a E r r t a a n a a n a a n a n a n a I r r t a a n a a n a a n a a n a n a n a V r r t a a n a a n a a n a n a n a R r r t

Model parameters of Q-LAIV-Sim
Indexing a age group (in 6 month steps for the first two groups; in years thereafter) n category "no risk" r category "at risk"

Influenza strains
As influenza can be caused by any one of four different viral strains (A(H1N1), A(H3N2), B/Yamagata, B/Victoria), we assume that these four strains are transmitted independently, causing strain-specific immunity; i.e. the differential equations shown above have to be applied separately to each one of them. As trivalent vaccination (TIV) only contains one of the two B lineages, simulation results differ between strains.
Birth ν, death µ and ageing α Using official demographic data until 2012 (3) and predicted data (using "Main scenario (proj_13npms)" from 2013 to 2026 (4), the simulation tool is constructed such that the exact size and age-distribution of the German population is reproduced for each simulation year. The numbers of individuals in the lowest age group are translated into birth rates ν and transitions from the numbers of individuals of age a in one year to age a+1 in the following year allow to calculate age-specific mortality rates µ a . Unlike in other simulation tools (8,17), births and deaths not only occur once a year, but are spread evenly over the simulation year. Whereas ageing in dynamic models frequently occurs only once a year (1,8,17), aging is implemented in Q-LAIV-Sim as a continuous process. Although the description of the differential equations seems to allow for individuals to continuously age, the simulation has been implemented such that each individual can only age once a year, using additional indexing of the differential equations which was omitted from the description. The only exception are the two age groups below 1 year of age: as influenza vaccination can only be performed for children of at least 6 months, newborn individuals also "celebrate" their 6-months anniversary instead of only having an annual "birthday" (i.e. they pass through two aging steps within one simulation year). As the percentage of individuals who are "at risk" when being infected (status r) increases over age, some individuals with "no risk" status (n) are added into the "at risk" category when they grow older. Individuals who reach the highest age group (100+ years) remain in that age-group until they die.

Maternal protection (fraction m, loss rate ω)
A fraction m=30% of newborn individuals is protected by maternal antibodies, the remaining fraction (1-m) is born susceptible. Maternal protection is lost at rate ω and the individuals become susceptible. As the average duration of maternal protection is assumed to be 4 months, only 0.25% of those newborns who initially are protected by maternal antibodies are still protected after 2 years. The model is implemented such that these few remaining individuals lose their maternal protection abruptly at their second birthday (this transition is not shown in the equations).

Contact matrix
Age-dependent mixing of contacts in Q-LAIV-Sim is based on the POLYMOD matrix for Germany (6). As Q-LAIV-Sim distinguishes annual cohorts (see above, paragraph "Birth ν, death µ and ageing α"), the original 5-year age group structure of the POLYMOD matrix was first extended into a 1-year structure, using a straight-forward extension algo-rithm, and afterwards the resulting abrupt changes at the transitions between the former 5-year age groups were smoothed by rearranging contacts between cohorts such that the sum of contacts within the original 5-years age groups remained unchanged. As the POLYMOD matrix is by nature asymmetric, individuals of cohort a1 may be predicted to have a total number of x contacts with individuals of cohort a2, yet -viewed from the other side -individuals of cohort a2 may be predicted to have a completely different number of contacts with cohort a1. Such kinds of asymmetric contact structures may be appropriate for infectious diseases where the transmission is easier passed on from group a1 to group a2 than from a2 to a1 (as may be the case for the transmission of a venereal disease via bisexual contacts), yet such asymmetries should not play a major role in influenza transmission. In order to create a symmetric contact structure which is based on the POLYMOD study, we calculate an average of the (a1,a2) and the (a2,a1) contact rates which is weighted by the (time-varying) population sizes of the two cohorts. Performing this calculation for all 101 x 101 cohort combinations, we obtain a symmetric matrix ( ) t a a 2 1 , β which we use to calculate the force of infection (cf. next paragraph).

Force of infection λ
To consider the seasonality of transmission, the force of infection depends on the time t, using the cosine function

Vaccination rate φ
Vaccinations are implemented such that, in a given simulation year approximately the same number of vaccinations takes place on every day during October and November, and that finally the given age-and risk-specific vaccination coverage is reached (for a mathematical description, see (1)). Vaccinations depend on the age and the risk status of the individuals, but not on their immunity status. As vaccination of immune individuals and infected individuals is assumed not to change their immunologic status, these vaccinations are not shown in the equations above, but they are recorded during the simulation, using additional indexing of the differential equations. This is necessary for two reasons: (a) although the description of the differential equations above would allow for individuals to receive multiple vaccinations per year, the system is implemented such that each individual can only be vaccinated once a year. (b) individuals who were vaccinated in the previous year are assumed to preferentially be re-vaccinated in the next year (an estimation of the re-vaccination factor for Germany is given below). Only the vaccination of susceptible individuals can change their immunity status, depending on the vaccine efficacy (which depends on the age and risk status of the

Protection P of vaccinees against infection
In the standard version of the simulation tool Q-LAIV-Sim, successfully vaccinated individuals are fully protected against infection and disease until they lose their immunity (loss of vaccination-derived immunity will be described in the next section). In order to reproduce the reasoning which is given above for the SIR and SIRS models, a new parameter P was introduced which describes the protection of vaccinees against infection. With the default value P=1, vaccinees cannot be infected until they lose their immunity; with the experimental value P=0, vaccinees remain fully susceptible and the number of vaccinated individuals who are infected while they are still in category "V" can be calculated (in order not to change anything during the initialization procedure or in the reference scenario with QIV, the setting P=0 is only used for QLAIV vaccination).

Loss rate of vaccine-derived immunity (not shown in equations)
Vaccinations with the inactivated vaccines (TIV and QIV) are assumed to protect the vaccinees for one season. Individuals who have become immune due to TIV or QIV vaccination lose their immunity at the end of the simulation year (i.e. on 31 August) in which they were vaccinated. For the live vaccine, it has been shown that a percentage of individuals are still protected in the second season: the vaccine efficacy in the first season was 80% (12,13); it dropped to 56% in the second year (18), i.e. 70% of the individuals who are protected in the first year, are also protected in the second year. The model is implemented such that 30% of individuals lose their Q-LAIV immunity at the end of the simulation year in which they are vaccinated, and the remaining ones lose their immunity at the end of the following year. Technically, this demands further indexing of the differential equations which is not shown in the model description.

Simulation results of Q-LAIV-Sim
Simulation results are given in Table S3 and in Table 1 and Figure 5 of the paper. In a sensitivity analysis, we have used the QIV vaccine efficacy and the duration of QIV immune protection of QLAIV, too (Table  S4). Indirectly prevented infections (total) i T i C + i A Indirect/direct ratio (total) r T i T /d C Table S3. Annual number of directly and indirectly prevented influenza-related events caused by pediatric QLAIV vaccination in Germany. Simulations are initialized from 2010 to 2016 using TIV with the baseline vaccination coverage shown in Table S2. In the reference scenario, TIV is replaced by QIV in 2017, but the vaccination coverage remains unchanged. In the evaluated scenario, the same QIV vaccinations are performed, except for 2-17 year old children who receive QLAIV; in the first evaluation year their QLAIV coverage is identical to the baseline coverage (around 5%), then it is increased in three equal annual steps to reach the final coverage as shown below. In the columns "Prevented cases", 10-year cumulative numbers of directly (dC) and indirectly prevented cases among children (iC), as well as indirectly prevented cases among adults (iA) and in the total population (iT) are given. In the "Ratio" columns, ratios of indirectly prevented cases divided by directly prevented cases are given for the sub-group of children (rC) and for the total population (rT). For a graphical display of the results see Figure 5 of the paper.   Table S4. Annual number of directly and indirectly prevented influenza-related events caused by pediatric QLAIV vaccination in Germany. In these simulations, the vaccine efficacy of QLAIV was set to the QIV value of 59% and the duration of QLAIV immunity was set to one year (as for QIV); all other details are described in Table S3. Estimation of the re-vaccination factor

Each simulations was run three times
The preferential re-vaccination factor will be calculated from reported data on people who were vaccinated at least once in three subsequent seasons from 2004/5 to 2006/7 in Germany (Table S4). Vaccination coverage of >60 year olds was 2004/5: p1=45%, 2005/6: p2=50% and 2006/7: p3=49%, respectively; the vaccination coverage p0 below 60 years was not reported (19).  Figure S1. Estimation of the re-vaccination factor f As the data set only contains individuals who were vaccinated at least once, we obtain that -a fraction ( ) ( ) is expected to be vaccinated three times The unknown parameters p0 and f were estimated using the method of least squares, obtaining the following estimates: -p0 = 25.9% of young adults were vaccinated -f = 4.25, meaning that previously vaccinated individuals are more than 4 times as likely to be vaccinated again as previously unvaccinated individuals As can be seen from the two last columns of Table S4, the estimated percentages of individuals with one, two or three vaccinations (using f=4.25) are satisfactorily close to the observed ones.

Exploring direct and indirect vaccination effects in a static model with seasonality
The simulation results in Q-LAIV-Sim are strongly driven by the seasonality of transmission and by the annual vaccination campaigns which precede the transmission seasons ( Figure S2) . As annually only 10.6% of individuals are infected with influenza (7), only about 2-3% of the population is infected with any one of the four influenza strains. In simulations with Q-LAIV-Sim (using the parameter settings shown in Tables S1 and S2), about 30% of the population are immune to a given influenza strain before the transmission season. Because of the low infection rate, this percentage hardly changes during the transmission season (as can be seen from Figure S2, stages 1-3, vaccination and loss of vaccination-derived immunity which take place outside of the seasonal transmission window have much more impact on the number of susceptibles). As the number of immune individuals only slightly changes during the transmission period, seasonality must be a major driver of the dynamics which determines the increase and decrease in the number of cases.  (1), vaccination-derived immunity which was acquired during the last season (inactivated vaccine) or during the last two seasons (live vaccine) is lost (2). As hardly any infectious individuals are present and the transmissibility of the infection is very low at that time, the number of susceptible individuals continues to grow due to births. From 1 October to 30 November, children and adults are vaccinated (3). Soon thereafter, newly introduced infections spread in the population (4) and start a seasonal wave which reaches its peak around March or April (5). After that, the incidence strongly declines, aided by the reduced transmissibility of the infection.
In the summer months when virtually all infections are gone, the susceptible population slightly increases due to loss of naturally acquired immunity and to new births (6), until vaccination-derived immunity is lost again (1) at the beginning of the next simulation year and a new vaccination campaign starts.
In the following simplified model, we assume that the immunity does not change during the seasonal wave at all, but remains at a constant level of 30%. We start with a single infection on 1 December (i.e. t0 = 91, which is immediately after the end of the annual vaccination campaign and which precedes the seasonal wave). At this time, the value of the reproduction number ( ) 0 t R is 1.542, meaning that the index case is expected to infect 1.542 others in a nonimmune population. As 30% of the population is immune, the expected number of secondary infections is reduced to 1.08 which is just above the critical number 1.0 which is needed for an epidemic to evolve. On average, it takes a generation time of 3.5 days until the next generation of cases is ready to infect others. By then, the reproduction number has increased to ( ) 552 . Although this seems to negligibly decrease the slope at which the number of cases grows, the expected annual number of cases drops from 89.0 to 62.5 cases (i.e. by 29.2%). As only 2% of the population were vaccinated, only 2% of the original 89.0 secondary cases (i.e. 1.8 cases) were prevented because the prospective victims were protected (i.e. directly protected). The remaining 24.8 cases which were prevented by vaccinating 2% of the population were obviously an indirect effect of vaccination. The ratio of indirectly/directly prevented cases was, thus 24.8/1.8 = 13,8, meaning that more than nearly 14 times as many cases were prevented indirectly as were prevented directly.