Andrew J Black
email: andrew.black@postgrad.manchester.ac.uk

Alan J Mckane

Stochastic amplification in an epidemic model with seasonal forcing

Keywords: non-linear dynamics, period doubling, measles

We study the stochastic susceptible-infected-recovered (SIR) model with time-dependent forcing using analytic techniques which allow us to disentangle the interaction of stochasticity and external forcing. The model is formulated as a continuous time Markov process, which is decomposed into a deterministic dynamics together with stochastic corrections, by using an expansion in inverse system size. The forcing induces a limit cycle in the deterministic dynamics, and a complete analysis of the fluctuations about this time-dependent solution is given. This analysis is applied when the limit cycle is annual, and after a period-doubling when it is biennial. The comprehensive nature of our approach allows us to give a coherent picture of the dynamics which unifies past work, but which also provides a systematic method for predicting the periods of oscillations seen in whooping cough and measles epidemics.

Introduction

The availability of extensive time-series data for childhood diseases is often the reason given for the amount of attention that this subject receives.

However, the possibility that relatively simple models can capture the essence of the disease dynamics also makes the topic an attractive one for modellers.

Mathematical epidemiologists are especially intrigued by the rich variety of oscillatory dynamics seen in this data [START_REF] Earn | A simple model for complex dynamical transitions in epidemics[END_REF][START_REF] Grenfell | Dynamics of measles epidemics: scaling noise, determinism and predictability with the TSIR model[END_REF]. Within the literature there is a broad consensus that there are two main elements needed to model these oscillations: firstly stochasticity, due to the individual nature of the population [START_REF] Bartlett | Stochastic population models in ecology and epidemiology[END_REF][START_REF] Durrett | The importance of being discrete and spatial[END_REF]; and secondly, seasonal forcing, arising from the term-time aggregation of children in schools, which is deterministic [START_REF] London | Recurrent outbreaks of measles, chickenpox and mumps I: Seasonal variation in contact rates[END_REF][START_REF] Schenzle | An age-structured model of pre-and post-vaccination measles transmission[END_REF][START_REF] Altizer | Seasonality and the dynamics of infectious diseases[END_REF]. Independently these two factors are well understood, but how they interact when both included in the same model is still an open question [START_REF] Keeling | Seasonally forced dynamics explored as switching between attractors[END_REF][START_REF] Rohani | The interplay between determinism and stochasticity in childhood diseases[END_REF][START_REF] Coulson | Skeletons, noise and population growth: the end of an old debate?[END_REF]. Measles is the canonical example of a disease which displays recurrent epidemic behaviour. In larger cities regular periodic oscillations, usually annual or biennial, are observed, whereas smaller cities display more irregular dynamics [START_REF] Grenfell | Dynamics of measles epidemics: scaling noise, determinism and predictability with the TSIR model[END_REF][START_REF] Lloyd | Spatiotemporal dynamics of measles: synchrony and persistence in a disease metapopulation[END_REF]. The introduction of mass vaccination in the 1960s provides a 'natural experiment' after which the dynamics become much more irregular [START_REF] Grenfell | Meta)population dynamics of infectious diseases[END_REF].

One of the early successes in the field was a simple deterministic model with seasonal forcing which could recreate the regular dynamics of measles [START_REF] Dietz | The incidence of infectious diseases under the influence of seasonal fluctuations[END_REF][START_REF] Schwartz | Infinite subharmonic bifurcation in an SEIR epidemic model[END_REF]. Where simple models such as this fail, is in capturing the more irregular dynamics seen in smaller populations [START_REF] Grenfell | Dynamics of measles epidemics: scaling noise, determinism and predictability with the TSIR model[END_REF], after vaccination [START_REF] Rohani | Opposite patterns of synchrony in sympatric disease metapopulations[END_REF], and in other diseases such as whooping cough [START_REF] Nguyen | Noise, nonlinearity and seasonality: the epidemics of whooping cough revisited[END_REF]. These aspects can only be captured by fully stochastic models.

Stochastic models without external forcing show large oscillations caused by the stochasticity exciting the system's natural frequency [START_REF] Bartlett | Measles periodicity and community size[END_REF][START_REF] Alonso | Stochastic amplification in epidemics[END_REF]. When forcing is included, it is less clear how the stochasticity interacts with the cyclic solutions that are produced. It could act passively to kick the system between different deterministic states [START_REF] Schwartz | Multiple stable recurrent outbreaks and predictability in seasonally forced nonlinear epidemic models[END_REF], as well as interacting with the non-linearity to excite the transients.

Power spectra [START_REF] Priestley | Spectral analysis and time series[END_REF][START_REF] Anderson | Oscillatory fluctuations in the incidence of infectious disease and the impact of vaccination: time series analysis[END_REF] have proved a useful tool in investigating these factors. They can help distinguish various components in the time-series and classify them as essentially seasonal, stochastic or an interaction of the two [START_REF] Benton | Revealing the ghost in the machine: using spectral analysis to understand the influence of noise on population dynamics[END_REF]. The most successful synthesis, by Bauch and Earn (2003b), showed that a simple mechanistic model can accurately predict the position of peaks in the power spectrum of a number of different disease time series.

We approach this problem by starting with an individual based model, which is inherently stochastic. We can then both simulate it and derive the emergent population level dynamics. The novel aspect of this work is that we calculate the power spectrum analytically for explicitly time-dependent external forcing, and compare the results with stochastic simulations. We do this by formulating the model as a master equation which can then be studied using van [START_REF] Van Kampen | Stochastic processes in physics and chemistry[END_REF] expansion in the inverse system size. The macroscopic dynamics can then be viewed as a sum of a deterministic and a stochastic part. The value of the analytic approach is that we can more easily deduce the mechanisms behind the dynamics and better understand the interplay between deterministic and stochastic forces.

The theory we develop in this paper unifies much of the previous work on these models. It encompasses the influential work of [START_REF] Earn | A simple model for complex dynamical transitions in epidemics[END_REF] in understanding the transitions in measles epidemics, the later work of Bauch and Earn (2003b) relating to the transient fluctuations close to cyclic attractors for different diseases and the more recent work on stochastic amplification in epidemic models [START_REF] Alonso | Stochastic amplification in epidemics[END_REF][START_REF] Black | Stochasticity in staged models of epidemics: quantifying the dynamics of whooping cough[END_REF]. The picture that emerges is close to that proposed by Bauch and Earn (2003b), but goes beyond it in two important respects. Firstly, we calculate the exact power spectrum for the forced model. Secondly, we show how the forcing changes the form of the fluctuations, and how in a stochastic model these are intimately related to the period doubling bifurcation, which is vital for explaining the dynamics of measles.

The rest of this paper is as follows. In section 2 we introduce the seasonallyforced version of the stochastic susceptible-infected-recovered (SIR) model and the system-size expansion of the master equation. Section 3 provides a discussion of the results for the simple case when the deterministic dynamics are described by an annual limit cycle. In Section 4 we apply our method to elucidate the dynamics investigated by [START_REF] Earn | A simple model for complex dynamical transitions in epidemics[END_REF], which can account for the transitions in measles epidemics. This is an interesting parameter regime, as the deterministic theory predicts a period doubling bifurcation.

Finally, in section 5 a broad discussion of our results is given, describing how this approach can account for the different dynamics of measles and whoop-ing cough. There are two appendices giving technical details relating to the system-size expansion and Floquet theory.

The seasonally forced SIR model

We first summarise the individual-based stochastic SIR model. We emphasise only the aspects which are relevant to this paper; a more general discussion can be found in textbooks on the subject [START_REF] Anderson | Infectious diseases of humans: dynamics and control[END_REF][START_REF] Keeling | Modelling infectious diseases in humans and animals[END_REF]. The population is split into three classes: susceptibles, infected and recovered. Birth and death rates are set equal to μ and these events are linked, even in the stochastic model, so that the total population, N, remains constant. Recovery happens at a constant rate γ, so that the average infectious time is 1/γ; once recovered, an individual is immune for life. Seasonal forcing is included by assuming that the transmission rate β(t) follows a term-time pattern [START_REF] Schenzle | An age-structured model of pre-and post-vaccination measles transmission[END_REF],

β(t) = β 0 (1 + β 1 term(t)), (1)
where β 0 is the baseline contact rate, β 1 the magnitude of forcing and term(t) is a periodic function which switches between 1 during school terms and -1 during holidays. In this paper we use the England and Wales term dates set down by [START_REF] Keeling | Seasonally forced dynamics explored as switching between attractors[END_REF]. The reproductive ratio is determined by

R 0 = β /γ
, where β is the effective (time-averaged) transmission rate:

β = β 0 [p s (1 + β 1 ) + (1 -p s )(1 -β 1 )], (2)
and p s is the proportion of time spent in school. For our choice of terms p s = 0.75. We also include a small immigration term, η, to account for infectious imports. We use a commuter formulation, where susceptibles are in contact with a pool of infectives outside the main population [START_REF] Engbert | Chance and chaos in population biologymodels of recurrent epidemics and food chain dynamics[END_REF][START_REF] Alonso | Stochastic amplification in epidemics[END_REF]. Since N = S + I + R, we can use this constraint to eliminate the variable R from the rate equations.

The model is then defined by the processes through which it evolves. If we write the state of the system as σ ≡ {S, I}, we can specify the following transition rates, T (σ|σ ), between an initial state σ and a final state σ: Since birth and death are coupled, the processes 3 and 4 also imply the birth of a susceptible individual. An important point is that changes in vaccination can be mapped onto the effective transmission rate β using [START_REF] Earn | A simple model for complex dynamical transitions in epidemics[END_REF])

1. Infection: S + I β(t) --→ I + I and S η -→ I. T (S -1, I + 1|S, I) = β(t) S N I + ηS . ( 3
β → β (1 -p), ( 7
)
where p is the proportion of individuals vaccinated at birth. We can also approximately map a change in birth rates onto β , but this is not exact in this model because births and deaths are linked. In this paper we are primarily interested in the parameter range of childhood diseases. These are characterised by μ γ, i.e. the average life expectancy of an individual is orders of magnitude longer than the mean infectious period [START_REF] Anderson | Infectious diseases of humans: dynamics and control[END_REF].

Methods of analysis

We use two methods to investigate the dynamics of this system. Firstly we simulate the system using [START_REF] Gillespie | A general method for numerically simulating the stochastic time evolution of coupled chemical reactions[END_REF] algorithm with the appropriate time-dependent extensions [START_REF] Anderson | A modified next reaction method for simulating chemical systems with time dependent propensities and delays[END_REF]. This method generates exact realisations from which statistical quantities, such as power spectra and moments, can be computed. The second method is analytic, through the construction of a master equation. This cannot be solved exactly so we instead use van Kampen's (1992) expansion in the inverse system size to derive approximate analytic solutions.

This involves making the substitutions,

S = Nφ + N 1/2 x, I = Nψ + N 1/2 y, (9)
and expanding the master equation in powers of N -1/2 . This technique and similar ones have been documented at length in the literature, but almost exclusively for time-independent models [START_REF] Alonso | Stochastic amplification in epidemics[END_REF]. The novel aspect of this paper is that we analyse the full time-dependent system [START_REF] Boland | Limit cycles, complex Floquet multipliers and intrinsic noise[END_REF], whereas in a previous paper we used the approximation which replaced β(t) by β [START_REF] Black | Stochasticity in staged models of epidemics: quantifying the dynamics of whooping cough[END_REF].

The details of the system-size expansion for this model are given in Appendix A. At leading order we find a pair of deterministic equations, describing the mean behaviour, which scale with the system size N,

φ = -β(t)φψ -ηφ + μ(1 -φ), ψ = β(t)φψ + ηφ -(μ + γ)ψ. ( 10
)
These are the same as the equations that are found from a purely phenomenological treatment of the SIR system. At next-to-leading order we obtain a pair of Langevin equations for the stochastic corrections to the deterministic equations ( 10),

ẋ = K(t)x(t) + f(t), (11)
where x ≡ {x, y}, and f(t) are Gaussian white-noise terms with correlation function f(t)f(t ) T = G(t)δ(tt ). The matrices K(t) and G(t) are determined from carrying out the expansion and are given by

K(t) = ⎛ ⎝ -β ψ -η -μ -β φ β ψ + η β φ -γ -μ ⎞ ⎠ , (12)
and

G 11 = β φ ψ + η φ + μ(1 -φ), G 22 = β φ ψ + η φ + (γ + μ) ψ, G 12 = G 21 = -β φ ψ -η φ -μ ψ, ( 13
)
where a bar indicates that the solutions are evaluated on the limit cycle.

These are essentially the same as are found in the non-forced model [START_REF] Alonso | Stochastic amplification in epidemics[END_REF], except now β, φ and ψ are all functions of time.

Stochastic amplification about a limit cycle

The mean behaviour is found by integrating the deterministic equations (10). When β 1 = 0, solutions show damped oscillations tending to a fixed point [START_REF] Anderson | Infectious diseases of humans: dynamics and control[END_REF]. For non-zero β 1 , this model can display a rich set of dynamics including chaos [START_REF] Olsen | Oscillations and chaos in epidemics: a nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark[END_REF][START_REF] Rand | Chaotic stochasticity -a ubiquitous source of unpredictability in epidemics[END_REF], but for realistic parameter values the most common long-time solution is a limit cycle with a period that is an integer multiple, n, of a year [START_REF] Dietz | The incidence of infectious diseases under the influence of seasonal fluctuations[END_REF][START_REF] Schwartz | Infinite subharmonic bifurcation in an SEIR epidemic model[END_REF]. As the forcing is a step function in time, we can visualise this as the system alternately switching between two spiral fixed-points [START_REF] Keeling | Seasonally forced dynamics explored as switching between attractors[END_REF]) resulting in a piecewise continuous limit cycle, illustrated in figure 1. Any other periodic forcing function, for instance a sinusoidally varying one, could be used without more difficulty, and would typically lead to a limit cycle which is smooth. As β 1 is increased, the limit cycle grows (although typically not linearly with β 1 ) and at critical values bifurcations are induced to longer period solutions [START_REF] Aron | Seasonality and period-doubling bifurcations in an epidemic model[END_REF][START_REF] Kuznetsov | Bifurcation analysis of periodic SEIR and SIR epidemic models[END_REF].

In this section we present results where there is only an annual limitcycle (n = 1). The case where we also have a period doubling is examined in section 4. The stability of these limit-cycle solutions can be investigated with the use of Floquet theory [START_REF] Kuznetsov | Elements of applied bifurcation theory[END_REF][START_REF] Boland | Limit cycles, complex Floquet multipliers and intrinsic noise[END_REF]. This quantifies how perturbations to the trajectory of the limit cycle behave and is analogous to linear stability analysis about a fixed point [START_REF] Grimshaw | Nonlinear ordinary differential equations[END_REF].

Floquet theory states that for any periodic solution of Eq. ( 10) there exists a matrix B which satisfies the relation, where X(t) is the fundamental matrix [START_REF] Grimshaw | Nonlinear ordinary differential equations[END_REF] and T n is the period of the limit cycle. The eigenvalues of B are called the Floquet multipliers, ρ i ; a related set of quantities are the Floquet exponents λ i = ln(ρ i )/T n (in this paper, since we will be discussing frequencies rather than angular frequencies, these exponents will be divided by a factor of 2π). Another way to think of this is as linear stability analysis of the fixed points of the n-cycle Poincare map of the system (Bauch and Earn, 2003b;[START_REF] Kuznetsov | Elements of applied bifurcation theory[END_REF]. A limit-cycle solution will be stable if |ρ i | < 1. When the multipliers are complex, perturbations to the trajectories return to the limit-cycle in a damped oscillatory manner, analogous to a stable spiral fixed point [START_REF] Grimshaw | Nonlinear ordinary differential equations[END_REF]. Similar ideas have been used to investigate the transients in forced epidemic systems in the past, but only in a deterministic setting (Bauch and Earn, 2003b;[START_REF] He | Epidemiological effects of seasonal oscillations in birth rates[END_REF]. Here we will explore how the nature of the fluctuations can be quantified using Floquet theory. Figure 2 shows a simulation of the full stochastic system together with the deterministic limit-cycle solution. We can see that even for large populations the stochastic corrections to the deterministic solution are important. The noise due to demographic stochasticity (noise at the individual level due to chance events; Nisbet and Gurney 1982) excites the natural oscillatory modes about the limit cycle, creating a resonance and giving rise to large scale coherent oscillations-an effect known as stochastic amplification [START_REF] Mckane | Predator-prey cycles from resonant amplification of demographic stochasticity[END_REF][START_REF] Alonso | Stochastic amplification in epidemics[END_REF]. As described in Appendix B, by solving Eq. ( 11), and invoking aspects of Floquet theory, we can express the auto-correlation function,

X(t + T n ) = X(t)B, ( 14
C(τ ) = 1 T n Tn 0 x(t + τ )x T (t) dt, x ≡ {x, y}, (15)
as an integral without further approximation [START_REF] Boland | Limit cycles, complex Floquet multipliers and intrinsic noise[END_REF]. Taking the Fourier transform of this expression then gives an exact expression for power spectrum of these stochastic oscillations.

Figure 3 shows simulated and analytic power spectra for the system shown in figure 2. We observe a sharp peak at 1 year due to the deterministic annual limit-cycle and a number of broader peaks due to the stochastic amplification of the transients. We would expect on general grounds that the stochastic peaks would be observed at frequencies,

m/T n ± Im(λ), ( 16
)
where m is an integer and λ is the Floquet exponent (Wisenfeld, 1985b;[START_REF] Boland | Limit cycles, complex Floquet multipliers and intrinsic noise[END_REF], and this is indeed what is seen. For the annual limitcycle the dominant peak is at 0 + Im(λ), with the other peaks being much smaller. Near to bifurcations these minor peaks become important and are treated in more detail in the following section.

The area under the peaks in the power spectrum is proportional to the root-mean-square amplitude of the oscillations. Away from any deterministic bifurcation points the amplitude is proportional to Re(λ), as in the unforced model. Thus the spectrum is close in form to that predicted from the unforced model by substituting β for the time-independent transmission rate (Bauch and Earn, 2003a;[START_REF] Black | Stochasticity in staged models of epidemics: quantifying the dynamics of whooping cough[END_REF]. There is good agreement between analytical calculations and simulations.

1 Im Λ 1 Im Λ Im Λ 0.2 0.4 0.6 0.8 1.0 1.2 1.
Although calculations give the power spectrum as an integral, it must be evaluated numerically because the deterministic equations ( 10) cannot be solved in closed form; this is all carried out using the symbolic package Mathematica (Wolfram Research, 2008). This analysis about an annual limit cycle corresponds to that of Bauch and Earn (2003b) except that we can derive the full power spectrum. They term the 'resonant peak' what we describe as the deterministic or annual peak, and the 'non-resonant peak' what we describe as the stochastic peaks. Their terminology is somewhat misleading, as the stochastic peak is generated by a resonance phenomena whereas the macroscopic peak is not.

Period doubling and measles transitions

We can use our analytic methods to help understand the dynamics and large-scale temporal transitions in measles epidemic patterns, first investigated by [START_REF] Earn | A simple model for complex dynamical transitions in epidemics[END_REF]. The main force in driving these transitions is changes in the susceptible recruitment (a mixture of changes in birth rates and vaccination), which can be mapped onto R 0 . Thus a knowledge of the model dynamics as a function of R 0 can be used to explain the changes in epidemic patterns. Although the analysis of [START_REF] Earn | A simple model for complex dynamical transitions in epidemics[END_REF] is in good qualitative agreement with time-series data, there are a number of outstanding questions with regard to the interpretation of the mechanisms for the dynamics. We first provide a brief review of the original analysis and then go on to show how the stochastic dynamics of this model can be understood within the framework we have laid out in the previous section.

Review of original analysis

It is acknowledged that stochasticity plays a role in the dynamics of measles, which can only be captured through simulation of the individualbased model. Fundamentally though, the analysis of these mechanisms by derived from the SIR equations ( 10), as a function of R 0 , with parameters corresponding to measles and no immigration (η = 0). This shows the incidence sampled annually on the 1st of January each year, thus stable limit cycles are shown by different numbers of (colour coded) curves. The single curve, beginning at small R 0 , shows an annual cycle which bifurcates at R 0 = 15.5 into two curves giving a biennial cycle. For values of R 0 lying between about 5 and 15, there are several sets of n curves representing n-year cycles.

For large R 0 (e.g. R 0 ≈ 30) only an annual limit cycle exists. As R 0 is reduced a biennial limit-cycle is found; before vaccination was introduced in England and Wales, most cities would be in this region. Higher birth-rates might move the system back into the region with only an annual attractor, whereas vaccination would act to reduce R 0 , moving it into the region with multiple co-existing longer period attractors. The interpretation put forward by Earn et al, is that stochasticity will then cause the system to jump between these different deterministic states [START_REF] Schwartz | Multiple stable recurrent outbreaks and predictability in seasonally forced nonlinear epidemic models[END_REF], giving rise to irregular patterns. Thus, in this description, noise plays a passive role [START_REF] Coulson | Skeletons, noise and population growth: the end of an old debate?[END_REF].

Although peaks were seen in power spectra from simulations, which appear to confirm this view, there are a number of problems with this interpretation. The crucial aspect that is neglected is that there are no infectious imports included in the deterministic analysis (although presumably they are included in simulations). When this factor is introduced (η = 0) then most of the additional structure disappears, see figure 5a; we are left with an annual limit cycle and a period doubling [START_REF] Engbert | Chance and chaos in population biologymodels of recurrent epidemics and food chain dynamics[END_REF][START_REF] Ferguson | Mass vaccination to control chickenpox: The influence of zoster[END_REF][START_REF] Alonso | Stochastic amplification in epidemics[END_REF].

When η = 10 -6 , there is only a small region in the range 24 < R 0 < 25 where there are coexisting annual and biennial limit-cycles. As the immigration parameter is reduced some of the additional structure reappears; for example at η = 10 -7 some of the period 3 attractors can be found in the range 9 < R 0 < 11. As η is decreased further still, more of the structure is found [START_REF] Bolker | Space, persistence and dynamics of measles epidemics[END_REF][START_REF] Nasell | Measles outbreaks are not chaotic, in: Mathematical approaches for emerging and reemerging infectious diseases: models, methods and theory[END_REF].

Immigration is an important aspect in the simulation because without it the disease would fade out as the minimum number of infections can go far below a single individual [START_REF] Bartlett | Measles periodicity and community size[END_REF][START_REF] Bolker | Space, persistence and dynamics of measles epidemics[END_REF][START_REF] Conlan | Resolving the impact of waiting time distributions on the persistence of measles[END_REF]. In a deterministic analysis this term is easily omitted because the variables are continuous and therefore fadeout cannot happen [START_REF] Nasell | On the time to extinction in recurrent epdemics[END_REF]. This raises the question: do these longer period solutions have an effect on the stochastic dynamics? If not, how can we describe the nature of the stochastic dynamics? We can use our analytic method to help clarify these questions. The power spectrum is especially useful as it can show up anomalous peaks from simulations. exponents. These parameter values will be used for the rest of this section.

Analytic predictions

Figure 6 shows the Floquet multipliers on a larger scale near the period doubling bifurcation point and figure 7 shows the analytical and numerical power spectra for various values of R 0 with N = 5 × 10 6 . Away from any bifurcation points there is good agreement between the analytic and the simulated spectra.

As we approach the period-doubling bifurcation point from below, the stochastic oscillations follow a virtual-Hopf pattern (Wisenfeld, 1985a,b).

This is where the oscillations first show the precursor characteristics of a Hopf bifurcation before changing into the precursor characteristics of a perioddoubling. This is clearly seen in the power spectra shown in figure 7. In the Hopf-like regime (R 0 < 14.94), the Floquet multipliers are a complex conjugate pair, giving rise to two peaks in the spectrum: a major one at frequency Im(λ) and a minor one at 1-Im(λ), as in section 3. Therefore in figure 7a the two peaks are most widely separated for R 0 = 4.

As we increase R 0 , Im(λ) also increases, and the major and minor peaks move closer together, converging at 0.5 y -1 when the multipliers become real and negative; this marks the onset of the period doubling regime, see figure 6. In this regime (14.94 < R 0 < 15.34), as the multipliers are negative, so their phase is ±π and so the imaginary part of the Floquet exponents is ±π/2πT 1 = ±0.5. Therefore the peak stays fixed at 0.5 years -1 as we increase R 0 further within this range, but the amplitude increases quickly.

At R bif 0 = 15.34 one of the multipliers reaches -1 and we see a deterministic period doubling [START_REF] Kuznetsov | Elements of applied bifurcation theory[END_REF], and the size of the fluctuations grows to order N. Figure 8 shows how in this way the oscillations smoothly turn into the macroscopic biennial limit cycle. The same pattern is seen if we hold R 0 fixed and increase β 1 to induce a period doubling.

When the system is in the biennial regime we can still calculate the fluctuations about the limit cycle and get a good correspondence with analytic predictions (figure 7b). The positions of the peaks are now at m/2 ± Im(λ) and the spectrum changes little within this parameter range. The peaks at m/2 + Im(λ) are barely visible, as compared to the prominent peaks at m/2 -Im(λ). In the annual regime after the doubling (R 0 > 25), the analytic results are again very accurate, with stochastic peaks at frequencies m ± Im(λ) (figure 7c). Here as well, the set of peaks at m + Im(λ) are much smaller. Note that in both of these regions the time-series will be dominated by the deterministic signal as the stochastic oscillations are much smaller than in the pre-bifurcation region (R 0 < 15).

Near the bifurcation point

For values of R 0 near the bifurcation point, the deviations between the analytic and simulated spectra become larger (see for example figure 7a; R 0 = 14). This is expected: the analysis developed here is essentially linear and thus predicts an unbounded increase in the fluctuations as we approach the bifurcation point [START_REF] Greenman | The impact of environmental fluctuations on structured discrete time population models: resonance, synchrony and threshold behaviour[END_REF]. As the fluctuations become larger the linear approximation breaks down and non-linear effects become important and act to bound the fluctuations. Going to larger system sizes can result in better agreement between analytic results and simulation, but this will always break down at some point.

Although the analytic approximation breaks down near the bifurcation point, the structure we have uncovered is still visible. Figure 8 shows stochastic power spectra from simulations for 14 < R 0 < 18, as we move though the bifurcation point. The virtual-Hopf pattern is still clear, as predicted by the analysis, but the fluctuations remain bounded, growing to the same order as the system size [START_REF] Van Kampen | Stochastic processes in physics and chemistry[END_REF][START_REF] Kravtsov | Nonlinear saturation of prebifurcation noise amplification[END_REF]. Within this region the macroscopic dynamics cannot be split into a deterministic and stochastic part and it is not in general possible to reconstruct the deterministic part by averaging over many realisations. Thus, determining exactly where the bifurcation takes place is difficult (Wisenfeld, 1985b). At R 0 = 16 the deterministic biennial peak should be observed, but is not clearly visible until R 0 = 18. It is possible that the bifurcation point is shifted in the stochastic system, but more analysis is required to determine that this is so.

Smaller populations

The results presented in the previous sections were for N = 5 × 10 6 , which roughly corresponds to the largest populations we would be interested ations from the analytic calculations and results are sensitive to N, η and β 1 . The forcing pushes the system close to the fade-out boundary (I = 0), where fluctuations are non-Gaussian, and so large deviations from the theory are expected. Figure 9 shows the stochastic power spectra from simulations, within the range 4 ≤ R 0 ≤ 30 and with N = 5 × 10 6 , 10 6 and 5 × 10 5 .

For smaller values of R 0 we still clearly observe the virtual-Hopf pattern, but a visual inspection of the time-series shows much more irregular dynam-ics. This is due to the increased stochasticity in the smaller systems, but also the closeness of the fade-out boundary, where extinction and re-colonisation events start to have an impact on the dynamics [START_REF] Griffiths | The effect of measles vaccination on the incidence of measles in the community[END_REF]. This has an effect on the power spectra in two ways: firstly as a broadening of the power spectra, showing a greater range or amplified frequencies and thus a more irregular dynamics. Secondly the endogenous period is systematically shifted higher, as in unforced versions of this model [START_REF] Simoes | Stochastic fluctuations in epidemics on networks[END_REF].

This reflects the fact that the period of oscillations also depends on the reintroduction of the disease after fade-out [START_REF] Bartlett | Measles periodicity and community size[END_REF].

The most important effect is on the fluctuations in the biennial regime after the period doubling. For N = 5 × 10 6 the peaks are sharp, indicating a deterministic limit cycle, and the stochastic oscillations are much smaller (figure 7b), hence the good predictability of these larger systems. For the two smaller populations this is not the case. We do not observe the deterministic biennial limit cycle, but instead see an enhanced stochastic peak and a broadening of the spectrum. The range of this enhanced region is also reduced.

Although there are large deviations, having an analytical description still helps us interpret the dynamics at smaller N. Taking the average of Eq. ( 9)

we obtain

I = Nψ(t) + N 1 2 y . ( 17
)
In the linear noise approximation, which we have used in this paper, the fluctuations are Gaussian and therefore y = 0. At some point this will break down and we must include the next order corrections, which will be of the order N 1/2 , to the macroscopic equations. It will no longer be true that the mean is equal to the average [START_REF] Van Kampen | Stochastic processes in physics and chemistry[END_REF][START_REF] Grima | Noise-induced breakdown of the Michaelis-Menten equation in steady-state conditions[END_REF]. This effect of the fluctuations on the deterministic dynamics could be enough to retard the onset of the biennial limit cycle and is the subject of further research.

Switching between attractors

As seen from the bifurcation diagram in figure 5, where η = 10 -6 , the only region where deterministically there are predicted to be two coexisting states is when 24 < R 0 < 25. This can be detected in simulations and the period of switching depends strongly on the system size. If the system is large it will tend to stay in the state it started in, because the fluctuations are not large enough compared to the mean to kick the system into the other state. Decreasing the system size makes this possible, and we see periods of annual dynamics followed by biennial and back to annual, where the period of switching depends on the system size.

There is another intriguing region where we see signs of this type of behaviour. For N = 10 6 and R 0 = 10 (figure 9b), we observe an enhanced stochastic peak in the spectrum with a period of 3 years. Visual inspection of the time-series shows regions of irregular annual oscillations interspersed with very regular triennial oscillations. Note that this is not observed in the larger or smaller systems and the power spectrum is shifted by the proximity to the fade-out boundary from its infinite system size limit. Very similar behaviour is observed for measles data from Baltimore between 1928 and 1935 [START_REF] London | Recurrent outbreaks of measles, chickenpox and mumps I: Seasonal variation in contact rates[END_REF][START_REF] Earn | A simple model for complex dynamical transitions in epidemics[END_REF], which has similar parameter values (Bauch and Earn, 2003b).

Discussion

We have used an analytic approach, as well as simulations, to quantify the effect of stochastic amplification in the forced SIR model. The time dependence has been treated explicitly, instead of by using an approximation as in a previous paper [START_REF] Black | Stochasticity in staged models of epidemics: quantifying the dynamics of whooping cough[END_REF]. Because this model has a finite population, and therefore is inherently stochastic, it can only be Our analysis suggests a simple explanation for the differences seen in the epidemic patterns of measles and whooping cough in England and Wales both before and after vaccination [START_REF] Rohani | Opposite patterns of synchrony in sympatric disease metapopulations[END_REF], and which are representative of the two main parameter regimes for childhood diseases. The generic situation occurs when we are far away from the bifurcation point.

Here we observe a deterministic annual limit cycle with stochastic oscillations, as in Figures 2 and3. In general the form of the spectrum is close to that predicted by the unforced model. As already shown in a previous paper, this situation can account for the dynamics of whooping cough pre-and post-vaccination [START_REF] Black | Stochasticity in staged models of epidemics: quantifying the dynamics of whooping cough[END_REF]. Pre-vaccination the stochastic oscillations are centred on 2-3 years. Vaccination acts to shift the endogenous frequency lower and increases the amplitude of these fluctuations giving large four yearly outbreaks.

Measles epidemics show a contrasting behaviour and represent the second important parameter regime, where the deterministic dynamics are near to a bifurcation point. Pre-vaccination, large cities such as London are in the regime with a deterministic biennial limit-cycle. Vaccination acts to lower R 0 and shift the system into the regime where there is an annual limit cycle with large stochastic oscillations. As vaccination coverage is increased, the endogenous period of these oscillations is also increased [START_REF] Grenfell | Travelling waves and spatial hierarchies in measles epidemics[END_REF].

Measles dynamics show a strong dependence on population size [START_REF] Bartlett | Measles periodicity and community size[END_REF][START_REF] Grenfell | Dynamics of measles epidemics: scaling noise, determinism and predictability with the TSIR model[END_REF]. Our analysis also offers some insight into this:

in large populations the stochastic oscillations are very small compared with the deterministic biennial limit-cycle. This accounts for the regularity and explains why purely deterministic models capture this aspect so well [START_REF] Dietz | The incidence of infectious diseases under the influence of seasonal fluctuations[END_REF]. For smaller populations the deterministic biennial limit-cycle is not observed, just enhanced stochastic oscillations, thus accounting for the more irregular dynamics seen in these smaller populations.

Finite size effects and immigration / imports are closely related in a stochastic individual-based model because the population is finite. Immigration reflects the basic fact that no population is isolated and there must be reintroduction of the disease if it fades out. One advantage of the approach which starts from an individual based model and derives the population level model, is that the immigration terms from the stochastic model are automat-ically included in the deterministic equations on the macro-scale. As we have shown, this is vital, as the longer period solutions are no longer found when they are included, and thus removes some speculation as to the influence of multiple co-existing attractors. These terms are easily omitted in a deterministic analysis because the system size is an innocent parameter [START_REF] Nasell | On the time to extinction in recurrent epdemics[END_REF]. It is interesting to note the similarities between immigration and age-structure in these forced models [START_REF] Schenzle | An age-structured model of pre-and post-vaccination measles transmission[END_REF]. Adding age-structure creates a constant pool of infectives in the infant class which acts to damp the dynamics [START_REF] Bolker | Chaos and biological complexity in measles dynamics[END_REF], exactly analogous to how we model immigration.

Owing to the importance of immigration in these epidemic systems one relevant outstanding question is: what form should the immigration parameter take? As measles dynamics can be highly synchronised it could be argued that the immigration parameter should reflect this [START_REF] Bolker | Space, persistence and dynamics of measles epidemics[END_REF][START_REF] Xia | Measles metapopulation dynamics: a gravity model for epidemiological coupling and dynamics[END_REF][START_REF] Lloyd | Spatiotemporal dynamics of measles: synchrony and persistence in a disease metapopulation[END_REF]. On the other hand for larger cities, this parameter can be viewed as an aggregate of many infectious encounters from varied sources and could be approximated as a constant. Previous work has hinted at the sort of spatial effects that can arise [START_REF] Keeling | Metapopulation moments: coupling, stochasticity and persistence[END_REF][START_REF] Keeling | Estimating spatial coupling in epidemiological systems: a mechanistic approach[END_REF][START_REF] Hagenaars | Spatial heterogeneity and the persistence of infectious diseases[END_REF], but investigation of explicitly spatial stochastic models should be a high priority.

The bifurcation diagrams for SEIR and SIR models are very similar, which justifies our use of the SIR model in this paper. The extension to uncoupled births and deaths would be straightforward, but would offer no further insight [START_REF] Keeling | Modelling infectious diseases in humans and animals[END_REF]. There are technical difficulties in extending the method to the SEIR model because of the difference in time scales between the collapse onto the centre manifold [START_REF] Schwartz | Infinite subharmonic bifurcation in an SEIR epidemic model[END_REF]) and the period of forcing; this creates difficulties in computing the Floquet multipliers.

These could in principle be overcome either by calculating the multipliers by a different method [START_REF] Fairgrieve | Floquet multipliers[END_REF][START_REF] Lust | Improved numerical Floquet multipliers[END_REF], or by carrying out a centre-manifold reduction before doing the van-Kampen expansion [START_REF] Forgoston | Accurate noise projection for reduced stochastic epidemic models[END_REF]. The breakdown of the linear theory near the bifurcation point can be remedied by including next-to-leading order terms from the expansion of the master equation (van Kampen, 1992), but would result in a much more complex calculation. The calculations and discussions we have given here once again highlight the important role that simple models play in the understanding of complex systems. It also makes a novel contribution to the wider debate on the relative importance of stochastic and deterministic forces in ecology [START_REF] Coulson | Skeletons, noise and population growth: the end of an old debate?[END_REF].

expansion in a more straightforward way. These are defined by:

E ±1 I f (S, I, t) = f (S, I ± 1, t), E ±1 S f (S, I, t) = f (S ± 1, I, t). (A.1)
The master equation ( 8) can then be written in full as,

d dt P (S, I, t) = (E S E -1 I -1) β S N I + ηS + (E -1 S -1) [μ(N -S -I)] + (E I E -1 S -1) [μI] + (E I -1) [γI] P (S, I, t), (A.2)
which agrees with [START_REF] Alonso | Stochastic amplification in epidemics[END_REF], up to typographical errors in that paper. The essential step in the expansion is the ansatz (9); we anticipate that the approximate probability distribution has a mean which scales as N and width which scales as N 1/2 (van Kampen, 1992). To expand Eq. (A.2) in a power series in N -1/2 , we write the step operators in terms of the fluctuation variables x and y:

E ±1 S = 1 ± N -1 2 ∂ ∂x + 1 2 N -1 ∂ 2 ∂x 2 ± • • • E ±1 I = 1 ± N -1 2 ∂ ∂y + 1 2 N -1 ∂ 2 ∂y 2 ± • • • . (A.3)
Substituting these and the ansatz (9) into Eq. (A.2) we identify a hierarchy of equations multiplied by different powers of N -1/2 . At leading order we find the deterministic equations (10) for the macroscopic variables, φ(t) and ψ(t). At next-to-leading order we obtain a linear Fokker-Planck equation for the fluctuations variables x ≡ {x, y}, of the form, .4) thus it satisfies the relation, Ẋ(t) = K(t)X(t).

∂Π ∂t = - i,j K ij (t) ∂[x j Π] ∂x i + 1 2 i,j G ij (t) ∂ 2 Π ∂x i ∂x j , ( A
(B.

2)

The matrix X(t) is not unique and will depend on the initial conditions.

Floquet theory states that if K(t + T n ) = K(t), then there exists a canonical fundamental matrix which can be expressed in the form X 0 (t) = P (t)Y (t) [START_REF] Grimshaw | Nonlinear ordinary differential equations[END_REF]. It has the property that the Floquet matrix, defined by B 0 = X -1 0 (0)X 0 (T n ), is diagonal. These diagonal elements are called Floquet multipliers, ρ i and play a central part in the theory. The matrix P (t) carries the periodicity of the limit cycle, while Y (t) = diag[e λ 1 t , e λ 2 t ], where λ i are the Floquet exponents. Using the periodicity of P (t) it can be seen that B = Y -1 (0)Y (T n ) and so the Floquet multipliers are related to the Floquet exponents by ρ i = e λ i Tn , i = 1, 2. Using the canonical form, we can derive an expression for the power spectrum in terms of the matrices, P (t) and Y (t).

In practice, one obtains a fundamental matrix, X(t) by numerically integrating (B.2) with initial condition X(0) = I. With this choice of initial condition, B = X(T n ). The multipliers and exponents are then calculated from the the eigenvalues of B, which allows the construction of the matrix Y (t), since the eigenvalues are independent of the choice of fundamental matrix. One can then calculate the canonical form, X 0 (t) = X(t)S, where the columns of S are the eigenvectors of B. Finally P (t) is found from P (t) = X(0)Y -1 (t) = X 0 (t)Y (-t).

Having described the basic idea behind Floquet theory, we can now return to the Langevin equation ( 11), which is an inhomogeneous linear equation with periodic coefficients. We can use Floquet theory to construct a solution to this by adding a particular solution to the general solution of the corresponding homogeneous equation (B.1) [START_REF] Grimshaw | Nonlinear ordinary differential equations[END_REF]. This gives, x(t) = X(t)X -1 (t 0 )x 0 + X(t) Using the periodicity of the matrix Γ(ts), this integral can be recast as a finite one over the period of the limit cycle: So we can obtain the correlation matrix as an integral, but this has to be evaluated numerically because the neither the limit-cycle solutions nor P (t) can be obtained in closed form.

Φ ij = 1 1 -ρ i ρ j

  1, I|S, I) = μ(N -S -I). (6)

Figure 1 :

 1 Figure 1: Phase portrait illustrating a deterministic solution of the forced SIR model.The term-time forcing creates a limit cycle (red curve) as the system alternately spirals between the two fixed points defined by β H = β 0 (1 + β 1 ) and β L = β 0 (1 -β 1 ). The light blue solutions show the behaviour if the forcing was switched off, to illustrate the two spiral attractors. The red dot shows the fixed point calculated using the approximation where β(t) is replaced by β .

Figure 2 :

 2 Figure 2: Phase portrait of the stochastic SIR system. A time-series of 100 years duration is shown in light blue. The first two years are highlighted in dark blue, with the dot showing the start point. The macroscopic limit-cycle (red) is also superimposed. Parameters are those relevant for whooping cough[START_REF] Nguyen | Noise, nonlinearity and seasonality: the epidemics of whooping cough revisited[END_REF]: R 0 = 17, γ = 1/22, β 1 = 0.25, μ = 5.5 × 10 -5 , η = 10 -6 and N = 2 × 10 6 .

Figure 3 :

 3 Figure3: Power spectra for the number of infectives from simulation (light blue solid curve) and analytic calculation (black dashed curve). From the simulations, we observe a sharp peak at 1 year from the deterministic annual-limit cycle. The other peaks, marked by the red lines, are from stochastic amplification, with the peak frequencies given by m ± Im(λ), where Im(λ) = 0.36. The dominant stochastic period is therefore 1/0.36 = 2.7 years. Parameters are as in figure2

Figure 4 :

 4 Figure 4: Bifurcation diagram showing the SIR dynamics as a function of R 0 . Fixed parameters: β 1 = 0.29, γ = 1/13, μ = 5.5 × 10 -5 and η = 0. The different period limit cycles are shown in different colours, which are produced by different initial conditions.

Figure 5 :

 5 Figure 5: (a) Bifurcation diagram for the SIR model with β 1 = 0.29 and η = 10 -6 . (b)The Floquet multipliers, which are in general a complex conjugate pair, thus we plot the real (dark blue) and imaginary (light green) parts separately. (c) Imaginary parts of the Floquet exponents. Note that in the region where there are the coexisting limit cycles (24 < R 0 < 25), only the multipliers/exponents for the biennial cycle are shown for clarity.

Figure 5 Figure 6 :

 56 Figure5shows the bifurcation diagram for the model presented in the previous section, but with η = 10 -6 , along with the Floquet multipliers and

Figure 7 :

 7 Figure 7: Analytic (black dashed curves) and numerical (coloured) power spectra for a range of R 0 with N = 5 × 10 6 . In most cases the analytic and numerical spectra are virtually indistinguishable, apart from R 0 = 14. (a) Spectra before the bifurcation, R 0 = 4, 6, 8, 10, 12, 14. (b) Typical biennial regime, R 0 = 20. Note that the stochastic peaks have been made clearer by subtracting the deterministic dynamics before calculating the power spectrum. The spectrum would otherwise be dominated by the peak at 0.5 y -1 .(c) The major and minor peaks in the large R 0 annual regime: R 0 = 26, 30, with the larger peaks corresponding to R 0 = 26 for both the major and minor peaks.

Figure 8 :

 8 Figure 8: Simulation results showing how the power spectrum of the stochastic oscillations changes as the period doubling bifurcation point is crossed. The peaks for R 0 = 17 and 18 have been cropped for clarity.

Figure 9 :

 9 Figure 9: Power spectrum through the bifurcation point for different size populations. (a) N = 5 × 10 6 , (b) N = 10 6 , (c) N = 5 × 10 5 . Some of the peaks are cropped for clarity. Notice the anomalously enhanced peak for N = 10 6 , R 0 = 10, see section 4.5 for discussion of this.

  studied 'exactly' by simulation. The system-size expansion, which we use to derive approximate analytic solutions for this model suggests that we should view the population-level dynamics as being composed of a deterministic part and a stochastic part, where the spectrum of the stochastic fluctuations is intimately related to the stability of the deterministic level dynamics. Power spectra of these models have been known for some time, but it has not always been clear what the mechanisms that generate the peaks are. This is the main advantage of being able to calculate the power spectrum of the stochastic fluctuations analytically; by comparison with the simulations we can gain insight into the mechanisms at work.

Y 0 Y 0 Y

 00 x(t 0 ) = x 0 . We are interested in the steady state solutions, when transients have damped down, thus we can ignore the first part of Eq. (B.3) and set the initial time to the infinite past, t 0 → -∞. Taking the case where X(t) is X 0 (t) = P (t)Y (t), one finds using the properties of the diagonal matrix Y (t), thatx(t) = P (t) t -∞ Y (ts)P (s) -1 f(s)ds. (B.4)The correlation matrix is defined as C(t+ τ, t) = x(t+ τ )x T (t) , which using Eq. (B.4) may be written asC(t + τ, t) = P (t + τ ) (t + τs)P (s) -1 G(s) × δ(ss )(P -1 (s )) T Y (ts ) T ds ds P (t) T , (B.5)where f(s)f T (s ) = G(s)δ(ss ). Integrating over the delta function, the result will depend on the sign of τ . If we take τ ≥ 0 then the integration region is -∞ < s < t, givingC(t + τ, t) = P (t + τ ) t -∞ Y (t + τs)Γ(s)Y (ts) T ds P (t) T , (B.6)where we have definedΓ(s) = P (s) -1 G(s)(P -1 (s)) T , (B.7)which will have the periodicity of the limit cycle. Next we make a change of variables, s → ts , which givesC(t + τ, t) = P (t + τ ) ∞ (τ + s )Γ(ts )Y (s ) T ds P (t) T . (B.8)The form of Y means we may write Y (τ +s ) = Y (τ )Y (s ), and so the integral that we need to evaluate is given by Φ(t) ≡ ∞ (s)Γ(ts)Y T (s) ds. (B.9)

Tn 0 Γ

 0 ij (ts)e (λ i +λ j )s ds. (B.10) Therefore, the final expression for the correlation matrix isC(t + τ, t) = P (t + τ )Y (τ )Φ(t)P (t) T .(B.11)

Acknowledgements

AJB thanks the EPSRC for the award of a postgraduate grant.

Appendix A. Expansion of the master equation

In this appendix we review the van Kampen system-size expansion, which approximates the master equation ( 8) by the deterministic equations ( 10) plus stochastic fluctuations given by Eq. ( 11) when N is large. The method has been described by [START_REF] Alonso | Stochastic amplification in epidemics[END_REF] for the SIR model without forcing (i.e. where β is independent of time), where further details are given.

We begin by introducing step operators which allow us to express the master equation in a more compact form and also allow us to carry out the where the matrices K(t) and G(t) depend on time through β(t), φ(t) and ψ(t). The Fokker-Planck equation (A.4) is equivalent to the Langevin equations (11) given in the main text [START_REF] Van Kampen | Stochastic processes in physics and chemistry[END_REF][START_REF] Gardiner | Handbook of stochastic methods[END_REF]. Since we will not be interested in transient behaviour, only fluctuations about the limit cycle, the solutions to (10) will be those of the limit cycle, which we denote by φ and ψ. The explicit forms for K(t) and G(t) are then given by Eqs. ( 12) and ( 13) respectively. Since β(t), φ(t) and ψ(t) are periodic, so are

K(t) and G(t).

Appendix B. Floquet analysis

The analysis of the Langevin equation ( 11) is not difficult when the deterministic system approaches a fixed point at large times, and for the SIR model this is described by [START_REF] Alonso | Stochastic amplification in epidemics[END_REF]. When the attractor of the dynamics is a limit cycle, the analysis is more complicated, but can still be carried out to give the power spectra as integrals over known functions. Here we outline this analysis, following closely [START_REF] Boland | Limit cycles, complex Floquet multipliers and intrinsic noise[END_REF], which may be consulted for further details.

We begin by considering linear perturbations about the limit cycle, that is, Eq. ( 11), but without the noise which originates from the discreteness of the individuals. The equation describing these small perturbations x ≡ {x, y} is ẋ = K(t)x, (B.1)

where K(t) is given by Eq. ( 12). A fundamental matrix, X(t), is constructed from the linearly independent solutions of the homogeneous equation (B.1),


Pertti Aro
email: pertti.aro@fimnet.fi

Nicholas J Talley

Lars Agréus

Sven-Erik Johansson

Elisabeth Bolling- Sternevald

Tom Storskrubb

Jukka Ronkainen

Functional dyspepsia impares quality of life in the adult population Impairment of quality of life in a general population with functional dyspepsia applying the Rome III criteria

Keywords: Functional dyspepsia < Disease-based, Epidemiology < Topics, Stomach and duodenum < Organ-based, Abdominal pain < Topics, Peptic ulcer disease Functional dyspepsia, Epidemiology, Stomach and duodenum, Abdominal pain, Peptic ulcer disease

Background: Data on the impact of functional dyspepsia on health related quality of life in the general adult population are scarce.

Aim: To explored the impact of functional dyspepsia applying the Rome III definition on health related quality of life in the general population.

Method:

A random sample of an adult Swedish population (n=1,001, The Kalixanda study) was invited to undergo an oesophagogastroduodenoscopy. An extended abdominal symptom questionnaire and Short Form 36 (SF-36) questionnaire, which includes eight domains measuring physical, mental and social aspects of quality of life, were completed at the clinic visit just before oesophagogastroduodenoscopy.

Results: 202 (20.2%) individuals reported uninvestigated dyspepsia (UID), 157 (15.7%) functional dyspepsia (FD), 52 (5.2%) epigastric pain syndrome (EPS) and 122 (12.2%) postprandial distress syndrome (PDS). UID, FD and PDS had a clinically meaningful (a ≥5 point) and statistically significant impact (p<0.05) on health related quality of life in all SF-36 domains except for Role Emotional. EPS had a significant impact on Bodily Pain and Vitality. Overlap of FD with irritable bowel syndrome (IBS) had a significant impact on Bodily Pain (p=0.002) and General Health (p=0.02) while FD overlap with gastro-oesophageal reflux symptoms (GORS) had a significant impact on Bodily Pain (p=0.02) compared with FD without any overlap with IBS or GORS.

Conclusions:

Functional dyspepsia impacts all main domains describing physical, mental and social aspects of health related quality of life in the general population. Overlap of FD with IBS or GORS impacts the domain related to bodily pain.

Introduction

In the western world, between 20% and 37% of adult population suffer from chronic or recurrent dyspepsia, although this number is influenced by the definition applied. (1)[START_REF] Koloski | Epidemiology and health care seeking in the functional GI disorders: a population-based study[END_REF][START_REF] Aro | Valid symptom reporting at upper endoscopy in a random sample of the Swedish adult general population. The Kalixanda study[END_REF] The symptoms of dyspepsia also remain common in the elderly. [START_REF] Crane | Chronic gastrointestinal symptoms in the elderly[END_REF] The most frequently applied criteria to diagnose functional dyspepsia (FD) have been the Rome I and Rome II definitions. [START_REF] Drossman | The Functional Gastrointestinal Disorders[END_REF][START_REF] Drossman | ROME II. The Functional Gastrointestinal Disorders[END_REF] Notably, major changes were made in the Rome III criteria for functional dyspepsia (FD) where two distinct syndromes were postulated¸ postprandial distress syndrome (PDS) and epigastric pain syndrome (EPS). [START_REF] Tack | Functional gastroduodenal disorders[END_REF] A diagnosis of FD requires exclusion of structural disease.

There are a number of outpatient studies suggesting that FD impairs health related quality of life (HRQoL) in patients, and the impact seems to be on all major variables of quality of life, namely mental, social and physical functioning. [START_REF] Talley | Functional dyspepsia, delayed gastric emptying, and impaired quality of life[END_REF](9)[START_REF] Welen | Functional dyspepsia affects women more than men in daily life: a case-control study in primary care[END_REF][START_REF] Haag | Impairment of health-related quality of life in functional dyspepsia and chronic liver disease: the influence of depression and anxiety[END_REF][START_REF] Haag | Is there any association between disturbed gastrointestinal visceromotor and sensory function and impaired quality of life in functional dyspepsia?[END_REF][START_REF] Gutierrez | Quality of life in patients with functional dyspepsia: a prospective 1-year follow-up study in Spanish patients[END_REF] A study from Germany reported that HRQoL in FD was more severely impaired than in chronic liver disease, and co-morbid anxiety and/or depression significantly contributed to impairment of HRQoL. [START_REF] Haag | Impairment of health-related quality of life in functional dyspepsia and chronic liver disease: the influence of depression and anxiety[END_REF] A Swedish case control study found that FD had a greater impact on HRQoL in women than men in terms of their daily activities. [START_REF] Welen | Functional dyspepsia affects women more than men in daily life: a case-control study in primary care[END_REF] A Spanish follow up study of patients with FD observed some improvement of HRQoL after 1 year follow up. [START_REF] Gutierrez | Quality of life in patients with functional dyspepsia: a prospective 1-year follow-up study in Spanish patients[END_REF] Other data suggest that impaired HRQoL in gastro-oesophageal reflux disease might depend on concomitant dyspepsia. [START_REF] Vries | Concomitant functional dyspepsia and irritable bowel syndrome decrease health-related quality of life in gastroesophageal reflux disease[END_REF] In a population-based, nested, case-control study, subjects reporting symptoms of either dyspepsia or irritable bowel syndrome and healthy controls were interviewed and completed questionnaires on psychological measures plus the validated Short Form 36 questionnaire (SF-36); an association between lower mental HRQoL and dyspepsia was found after adjusting for psychological covariates, but FD was not confirmed by endoscopy. [START_REF] Halder | Impact of functional gastrointestinal disorders on health-related quality of life: a population-based case-control study[END_REF] There are two recently published studies from Asia applying the Rome III definition of functional dyspepsia and focusing on the impact of overlapping FD-irritable bowel syndrome (FD-IBS) and FD-gastro-oesophageal reflux symptoms (FD-GORS) on HRQoL [START_REF] Kaji | Prevalence of overlaps between GERD, FD and IBS and impact on health-related quality of life[END_REF][START_REF] Lee | Depressive mood and quality of life in functional gastrointestinal disorders: differences between functional dyspepsia, irritable bowel syndrome and overlap syndrome[END_REF] The Korean study was a case-control study excluding all organic reasons of dyspepsia by both upper and lower gastrointestinal endoscopies and concluded that overlapping FD-IBS had more impact on HRQoL than FD or IBS alone had. [START_REF] Lee | Depressive mood and quality of life in functional gastrointestinal disorders: differences between functional dyspepsia, irritable bowel syndrome and overlap syndrome[END_REF] The Japanese cross-sectional study included Japanese workers being routinely screened for gastric cancer, and the definition of FD and organic dyspepsia was based on a a symptom questionnaire. [START_REF] Kaji | Prevalence of overlaps between GERD, FD and IBS and impact on health-related quality of life[END_REF] In the Japanese study FD had a significant impact on HRQoL in all physical and psychological domains of the Short Form 8 questionnaire and overlap with IBS and GORS was associated with significantly worsened HRQoL and the Korean study concluded that overlapping FD-IBS had more impact on HRQoL than FD or IBS alone had. [START_REF] Kaji | Prevalence of overlaps between GERD, FD and IBS and impact on health-related quality of life[END_REF][START_REF] Lee | Depressive mood and quality of life in functional gastrointestinal disorders: differences between functional dyspepsia, irritable bowel syndrome and overlap syndrome[END_REF] In a Swedish population-based study assessed by upper endoscopy, we have earlier reported the prevalence of uninvestigated dyspepsia (UID) and FD according to the Rome III definition to be 20% and 16%, respectively. [START_REF] Aro | Anxiety Is Associated With Uninvestigated and Functional Dyspepsia (Rome III Criteria) in a Swedish Population-Based Study[END_REF] While we have also reported the impact of gastro-oesophageal reflux symptoms on HRQoL in this population and, [START_REF] Aro | Anxiety Is Associated With Uninvestigated and Functional Dyspepsia (Rome III Criteria) in a Swedish Population-Based Study[END_REF][START_REF] Ronkainen | Gastro-oesophageal reflux symptoms and health-related quality of life in the adult general population--the Kalixanda study[END_REF][START_REF] Talley | Non-ulcer Dyspepsia and Duodenal Eosinophilia: An Adult Endoscopic Population-Based Case-Control Study[END_REF][START_REF] Walker | Duodenla mastocytosis, eosinophilia and intraepithelial lymphocytosis as possible disease markers in the irritable bowel syndrome and functional dyspepsia[END_REF] we have not assessed the impact of uninvestigated dyspepsia or FD. Only 5% of dyspeptics present for medical consultation annually [START_REF] Jones | Dyspeptic symptoms in the community[END_REF] and the lifetime consultation rate is 25-50%. [START_REF] Koloski | Epidemiology and health care seeking in the functional GI disorders: a population-based study[END_REF][START_REF] Agréus | Socio-economic factors, health care consumption and rating of abdominal symptom severity. A report from The Abdominal Symptom Study[END_REF] Therefore, to optimally define the impact of FD on HRQoL requires a

Materials and Methods

Setting and participants

The Kalixanda study setting consisted of two neighboring communities in Northern Sweden (Kalix and Haparanda) with 28,988 inhabitants (as of December 1998). [START_REF] Aro | Valid symptom reporting at upper endoscopy in a random sample of the Swedish adult general population. The Kalixanda study[END_REF] Every seventh adult (n=3,000) from the target population (20-80 years of age, n=21,610 in September 1998) was drawn. The sampled subjects were given an identification number (1-3,000) in random order. [START_REF] Aro | Valid symptom reporting at upper endoscopy in a random sample of the Swedish adult general population. The Kalixanda study[END_REF]

Study design and logistics

The original study population (n=3,000) was invited by mail to take part. The invitation letter included the validated Abdominal Symptom Questionnaire (ASQ) [START_REF] Agréus | Reproducibility and validity of a postal questionnaire. The abdominal symptom study[END_REF] to be returned by mail. Of the original study population 2,860 were eligible for inclusion. The overall response rate was 74.2 % (n=2,122) of the eligible study population. [START_REF] Aro | Valid symptom reporting at upper endoscopy in a random sample of the Swedish adult general population. The Kalixanda study[END_REF] 1,001 responders to ASQ were invited in random order to oesophagogastroduodenoscopy and response rate for those eligible for the oesophagogastroduodenoscopy was 73.3 % as described elsewhere. [START_REF] Aro | Valid symptom reporting at upper endoscopy in a random sample of the Swedish adult general population. The Kalixanda study[END_REF] At the visit for the oesophagogastroduodenoscopy, the participants filled in a more comprehensive ASQ. [START_REF] Aro | Valid symptom reporting at upper endoscopy in a random sample of the Swedish adult general population. The Kalixanda study[END_REF] The study protocol was approved by the Umeå University ethics committee and the study was conducted according to the declaration of Helsinki. Informed consent was obtained from all participants. Assessments ASQ is a questionnaire assessing gastrointestinal symptoms and it has been found to be valid, reproducible and reliable. [START_REF] Agréus | Reproducibility and validity of a postal questionnaire. The abdominal symptom study[END_REF][START_REF] Aro | Validation of the translation and crosscultural adaptation into Finnish of the Abdominal Symptom Questionnaire, the Hospital Anxiety and Depression Scale and the Complaint Score Questionnaire[END_REF] The questionnaire was designed before the Rome III era but it had been updated to reflect Rome II definition and it now includes all the questions reflecting the Rome III definition of FD except for onset of dyspepsia which we can define to 3 months rather than 6 months. [START_REF] Agréus | Reproducibility and validity of a postal questionnaire. The abdominal symptom study[END_REF] The extended ASQ included the frequency of each symptom during the prior 3 months (monthly, weekly or daily symptoms) and has been revalidated. [START_REF] Aro | Validation of the translation and crosscultural adaptation into Finnish of the Abdominal Symptom Questionnaire, the Hospital Anxiety and Depression Scale and the Complaint Score Questionnaire[END_REF] The SF-36 is a generic measure of HRQoL that has been widely used in patients having dyspepsia. [START_REF] Welen | Functional dyspepsia affects women more than men in daily life: a case-control study in primary care[END_REF][START_REF] Haag | Impairment of health-related quality of life in functional dyspepsia and chronic liver disease: the influence of depression and anxiety[END_REF][START_REF] El-Serag | Health-related quality of life in functional dyspepsia[END_REF] The validated acute version of the SF-36, which has a 1 week recall period, was used to measure quality of life with the 8 standard domains of: The SF-36 scale is graded 0-100 (the higher the better) in all 8 domains It has been shown previously that a difference in SF-36 score of 5 points or greater in comparison with symptom free individuals is clinically significant [START_REF] Ware | SF-36 Health Survey: Manual and Interpretation Guide[END_REF] as long as such a difference is also statistically significant. [START_REF]Guideline on the Choice of the Non-inferiority Margin[END_REF] A medical history questionnaire, completed during the clinic visit, asked whether subjects had consulted a doctor regarding gastrointestinal and other symptoms during the past year, and asked also wether they had used any gastrointestinal medications over the past 3 months.

Definitions of symptom groups

Uninvestigated dyspepsia (UID) was defined based on the Rome III definition: weekly bothersome postprandial fullness or early satiation, or epigastric pain and/or epigastric burning (symptom onset 6 months prior the survey was not asked about in the ASQ). Pain and burning were not allowed to be relieved by defecation. [START_REF] Tack | Functional gastroduodenal disorders[END_REF] Functional dyspepsia (FD) was defined as uninvestigated dyspepsia without findings of oesophagitis, peptic ulcer, celiac disease or cancer, and no evidence of other structural disease at endoscopy that was likely to explain the symptoms. Further FD, according to the Rome III definition, was divided into: . Epigastric Pain Syndrome (EPS) consisting of pain or burning localized to the epigastric area and not generalized or localized to other abdominal or chest regions, and not relieved by defecation. Overlap between PDS and EPS was allowed according to the Rome III definition. [START_REF] Tack | Functional gastroduodenal disorders[END_REF] FD only was defined as FD without any overlap with irritable bowel syndrome (IBS) and/or gastro-oesophageal reflux symptoms (GORS).

No dyspepsia was defined as individuals not reporting any type of dyspeptic symptoms.

Organic dyspepsia was defined as individuals reporting any type of dyspepsia with concomitant finding of erosive oesophagitis, peptic ulcer disease (PUD), celiac disease, cancer or other organic disease as the likely reason for dyspepsia.

Gastro-oesophageal reflux symptoms (GORS) were defined as the presence of troublesome heartburn and/or acid regurgitation over the past three months. [START_REF] Ronkainen | High prevalence of gastroesophageal reflux symptoms and esophagitis with or without symptoms in the general adult Swedish population: A Kalixanda study report[END_REF][START_REF] Vakil | The Montreal Definition and Classification of Gastroesophageal Reflux Disease: A Global Evidence-based Consensus[END_REF] Irritable bowel syndrome (IBS) was defined as troublesome abdominal pain or discomfort located at any site plus concomitant bowel habit disturbances.This simple definition has been used previously and shown to produce results reasonably concordant with the validated Rome I criteria for IBS. [START_REF] Agréus | Identifying dyspepsia and irritable bowel syndrome: the value of pain or discomfort, and bowel habit descriptors[END_REF][START_REF] Brandt | An Evidence-Based Position Statement on the Management of Irritable Bowel Syndrome[END_REF] The validated Hospital Anxiety and Depression scale (HADs) was used define anxiety and depression. A score from 11 or more was used as cut off level for both clinically relevant anxiety and depression. [START_REF] Zigmond | The hospital anxiety and depression scale[END_REF]

Definition of anxiety and depression

Oesophagogastroduodenoscopy

The upper endoscopies were undertaken by three experienced endoscopists, in the two clinics (Kalix and Haparanda), which gave sole medical cover to the area. The endoscopists were unaware of the symptoms of the subjects before endoscopy. [START_REF] Aro | Valid symptom reporting at upper endoscopy in a random sample of the Swedish adult general population. The Kalixanda study[END_REF]

Statistical analysis

Mean SF-36 scores and standard deviations were determined for non-dyspeptics, uninvestigated dyspepsia, FD and subgroups of FD and for peptic ulcer disease and organic dyspepsia, and tabled (or graphed) to compare with the Swedish mean SF-36 scores. The mean values were compared with mean SF-36 domain scores of nondyspeptics to identify clinically meaningful (≥5-point) differences. Statistical significance was analyzed using a two sample T-test and Satterthwaite's approximation was used when ever variances in SF-36 scores were tested unequal. The association of UID, FD, EPS and PDS on dichotomized SF-36 domains was also analyzed using multivariate logistic regression model adjusting for anxiety, depression, use of proton pump inhibitors, age and gender. Fisher's exact tests were applied in appropriate analyses. A two sided pvalue of < 0.05 was regarded as statistically significant. The STATA 8 program was used for the analyses. [START_REF] Statacorp | The Intercoold STATA 8 program[END_REF] F o r P e e r R e v i e w

Results

Endoscopy with biopsy was performed to 1,000 responders and the data reported here are from the 999 subjects who also completed the SF-36 questionnaire. The study population (51% female; mean age of 54 years) was representative of the source population as a whole. Overall 202 (20.2%) out of a study population of 999 individuals reported UID.

FD was reported by 157 (15.7%) and of these 52 (5.2%) had epigastric pain syndrome (EPS) and 122 (12.2%) had PDS. EPS and PDS overlapped in 17 (1.7%) cases. FD only (FD without any overlap with IBS or GORS) was reported by 31 individuals (3.1%). In total, 797 (79.7%) reported no dyspeptic symptoms at all. Demographic data are presented in Table 1. Mean SF-36 scores in this study population ranged from 69.3 (general health) to 89.9 (social functioning). The differences of Kalixanda means were clinically meaningful (≥5 point difference) compared with Swedish means only in Role Physical and General Health (Table 2, Figure 1). Of the individuals having UID 5.4% had consulted a doctor due to dyspeptic symptoms the year prior to the study and of the individuals having FD 6.0% had consulted a doctor due to dyspepsia. Respective figures for EPS, PDS and organic dyspepsia were 0%, 7.7% and 3.6%.

Impact of different dyspepsia groups on SF-36 domains

Subjects with UID and FD reported a clinically meaningful (≥5 point) and statistically significant reduction in HRQoL in all SF-36 domains, except for Role Emotional, compared with non-dyspeptics (Table 2, Figure 1). UID scored lower than FD in every domain of SF-36 aside from Physical Functioning but the differences were small and not clinically meaningful or statistically significant. Women reporting FD had clinically significantly lower mean scores than men in Bodily Pain and Role Emotional and men reporting FD in Role Physical and Role Emotianal, but these differences were not statistically significant (Table 2).

All SF-36 domain scores in PDS, except for Role Emotional, were both clinically and statistically significantly lower compared with non-dyspeptic individuals. Subjects with EPS had a clinically and statistically significant impairment of HRQoL in Bodily Pain and Vitality compared with non-dyspeptics (Table 2).

The results of significance of FD, EPS and PDS on HRQoL stand also when analyzing with multivariate logistic regression adjusting for anxiety, depression and use of proton pump inhibitors, with the exception of Social Functioning in FD and Vitality in EPS (Table 3). 4).

Impact of overlap of FD with IBS and overlap of FD with GORS on HRQoL

FD overlap with GORS has meaningful clinical impact on SF-36 domain Role Physical and both clinically and statistically significant impact on Bodily Pain (p=0.02) compared with FD only (FD without any overlap with IBS or GORS) (Table 4). FD overlap with GORS and IBS is presented in Figure 2.

Impact of organic dyspepsia and PUD on HRQoL

Individuals having organic dyspepsia scored lower in every SF-36 domain except for Physical Functioning compared with individuals having FD; the differences were clinically meaningful in Role Physical and General Health but statistically significant in none of the SF-36 domains (Table 2). Individuals having organic dyspepsia had clinically meaningfully lower scores in all SF-36 domains compared with non-dyspeptics and the difference was statistically significant in all domains aside from Role Emotional (Table 2). The associations of organic dyspepsia with impared HRQoL were confirmed are also shown in multivariate logistic regression aside from Social Functioning (Table 3). Role Physical and Role Emotional but these differences were not statistically significant (Table 2).

PUD had a clinically meaningful and statistically significant impact on HRQoL in all other domains of SF-36 compared with non-dyspeptics except on Mental Health (Table 2). The associations of PUD with clinically meanigfully worsened HRQoL are also shown in multivariate logistic regression (Table 3). and their mean values are very near the respective values found in the Finnish general population, [START_REF] Hagman | SF-36-terveyskysely koetun terveyden ja toimintakyvyn mittarina[END_REF] noting approximately 20% of the participants were of Finnish origin. The impact of UID and FD on HRQoL appears to be significant both clinically and statistically in all domains with the exception of Role Emotional.

Our findings are consistent with the published literature. A literature review in 2003 found six studies that used a validated questionnaire measuring HRQoL, although none applied the Rome III criteria. [START_REF] El-Serag | Health-related quality of life in functional dyspepsia[END_REF] There was evidence for impairment of HRQoL in these studies, but none of these examined HRQoL among samples of the general population. [START_REF] El-Serag | Health-related quality of life in functional dyspepsia[END_REF] A small case control study applied the Rome III definition for dyspepsia and found significantly decreased HRQoL in FD and in both FD subgroups (PDS and EPS) as well as an association of symptoms with abnormal sensory and motor function; [START_REF] Haag | Is there any association between disturbed gastrointestinal visceromotor and sensory function and impaired quality of life in functional dyspepsia?[END_REF] however, this was not a population-based sample and may have been subject to serious referral bias. Our results are also in line with the results of Japanese and Korean studies showing a significant impact on HRQoL in physical and pain related SF-36 domains. Both these studies did apply the Rome III definition of FD and the Korean study did also exclude organic reasons of FD by endoscopies but it was a case-control study and the Japanese study was not based on the general population, and gastrointestinal endoscopy was not included in the study protocol. [START_REF] Kaji | Prevalence of overlaps between GERD, FD and IBS and impact on health-related quality of life[END_REF][START_REF] Lee | Depressive mood and quality of life in functional gastrointestinal disorders: differences between functional dyspepsia, irritable bowel syndrome and overlap syndrome[END_REF] Our study shows also that FD overlaps with both IBS and GORS, and the overlap impacts HRQoL. According to expert opinion in persisting dyspepsia the precence of heartburn does not exclude the diagnoses of PDS or EPS and the precence of IBS does not exclude the diagnoses of any of functional gastroduodenal disorders. [START_REF] Tack | Functional gastroduodenal disorders[END_REF] We therefore believe the approach taken here is suitable.

Our results do not confirm the results from a Swedish case control study where it was shown that women scored lower than men in domains describing physical activity. [START_REF] Welen | Functional dyspepsia affects women more than men in daily life: a case-control study in primary care[END_REF] Women scored only clinically meaningfully lower in our population-based study in the domain describing Bodily Pain, and men had clinically meaningfully lower mean scores in domains describing Physical and Emotional Roles. Further exploration of gender differences in FD would be valuable. Haag et al. suggested that co-excisting anxiety and depression have significant impact on HRQoL in FD. [START_REF] Haag | Impairment of health-related quality of life in functional dyspepsia and chronic liver disease: the influence of depression and anxiety[END_REF] Our results show that FD in a general adult population independently impacts HRQoL regardless of the presence of anxiety or depression.

We report new data on the impact of PDS and EPS on QoL. The impact of PDS is clinically and statistically significant compared with controls except for Role Emotional and the results are consistently the same when analyzing for possible confounders such as anxiety, depression and use of proton pump inhibitors. Subjects with EPS had a clinically and statistically relevant impairment of HRQoL only in two domains (Bodily Pain and Vitality) compared with non-dyspeptics. Organic dyspepsia had a more clinically meaningfully impact on quality of life only in the domains Role Physical and General Health compared with FD. Compared with FD, PUD subjects reported lower Role Physical and Role Emotional scores, while the mean level of Bodily Pain, Physical Functioning and Vitality were on the same level as individuals having FD. PUD seems to impact HRQoL significantly even though a remarkable proportion of individuals with PUD in this study were asymptomatic. [START_REF] Aro | Peptic ulcer disease in a general adult population. The Kalixanda study: A random population-based study[END_REF] Our study shows that 5-6% of individuals reporting UID or FD did consult a doctor for dyspepsia the yer prior to the study. This result is in line with earlier studies. (1,[START_REF] Koloski | Epidemiology and health care seeking in the functional GI disorders: a population-based study[END_REF] The strength of our study is its population-based design with a randomly selected adult population undergoing symptom surveillance and then an endoscopic sub-study of this adult population. We obtained a very high response rate and thus the study is most probably representative of the general adult population in Sweden and in other western countries with similar socioeconomic circumstances. [START_REF] Aro | Valid symptom reporting at upper endoscopy in a random sample of the Swedish adult general population. The Kalixanda study[END_REF] FD was defined by carefully collecting individuals' medical history and by performing an oesophagogastroduodenoscopy with biopsies in all the participants. One potential weakness is that due to the study design before the era of the Rome III definition for dyspepsia, we could define the symptom onset 3 months prior to the study but not 6 months prior as it should be according to the Rome III definition. Some groups in the study are relatively small and type II errors cannot always be excluded. shows that FD has a significant impact on HRQoL in all domains describing physical activity and physical role, and in all other psychological domains but not the domain describing emotional role. PDS seems to impair HRQoL in all SF-36 domains more than EPS, while FD overlap both with IBS and GORS has a significant impact on Bodily Pain.

  that captures non-presenters where structural disease has been excluded by upper endoscopy; no such data have been published.Our aim was to explore if HRQoL in a general population sample, using the validated Short Form 36 questionnaire (SF-36), is impaired in FD and among FD subgroups defined according to the Rome III definition.

1 . 3 . 5 .

 135 Physical Functioning describing the individual's ability to perform physical activities 2. Role Physical describing the individual's performances of work and other daily activities Bodily Pain describing limitations in life due to physical pain 4. General Health evaluating personal judgment of general health Vitality measuring personal energy in daily life 6. Social Functioning measuring impact of physical or emotional problems on social life 7. Mental Health showing impact of nervousness and depression on daily life and 8. Role Emotional describes problems with work or other daily activities as a result of emotional problems.[START_REF] Sullivan | The Swedish SF-36 Health Survey-I. Evaluation of data quality, scaling assumptions, rerliability and construct valitity across generall populations in Sweden[END_REF]

1 .

 1 Postprandial Distress Syndrome (PDS) consisting of bothersome postprandial fullness and/or early satiation.

  had a clinically meaningful impact on the SF-36 domain Role Physical and both a clinically and statistically significant impact on Bodily Pain (p=0.002) and General Health (p=0.02) compared with FD only (FD without any overlap with IBS or GORS) (Table

  clinically meaningful impact on HRQol compared with FD in the domains

Discussion

  This is the first population-based cross-sectional endoscopic study to our knowledge applying the Rome III definition for FD to study the impact of dyspepsia on HRQoL in a general adult population. The SF-36 domain scores in the Kalixanda endoscopic study population differ from Swedish mean values only for Role Physical and General Health,

  our population-based study applying the Rome III definition for dyspepsia

Figure 1 .

 1 Figure 1. Means of SF-36 domains in Sweden and in different dyspepsia groups. The

Figure 2 .

 2 Figure 2. Overlap of functional dyspepsia (FD) with irritable bowel syndrome (IBS) and

  Foot notes: FD=functional dyspepsia (n=157) FD only=functional dyspepsia without any overlap with gastro-oesophageal reflux symptoms or irritable bowel syndrome (n=31) Overlap FD-IBS=overlap of functional dyspepsia with irritable bowel syndrome (n=29) Overlap FD-GORS=overlap of functional dyspepsia with gastro-oesophageal reflux symptoms (n=39) Overlap FD-GORS and IBS= overlap of functional dyspepsia both with gastrooesophageal reflux symptoms and irritable bowel syndrome (n=58)

Table 1 .

 1 Demographic data of the Kalixanda study population (n=999). Kalixanda population-based endoscopic study December 1998-June 2001

	F o
	r
	P
	e e r
	R e v i e w

Table 2 .

 2 Mean values of Short Form-36 domains in Sweden, Kalixanda study population and different dyspepsia groups and PUD. Kalixanda population-based endoscopic study December 1998-June 2001

		PF	RP	BP	GH	VT	SF	RE	MH
		Mean	Mean	Mean	Mean	Mean	Mean	Mean	Mean
		95% CI	95% CI	95% CI	95% CI	95% CI	95% CI	95% CI	95% CI
	Swedish mean values	87.9	83.2	74.8	74.8	68.8	88.6	85.7	80.9
	Kalixanda means n=999	84.4
		83.1-85.8

Table 3 .

 3 Associations of UID, FD, EPS, PDS organic dyspepsia and PUD with Short Form-36 domains in multivariate logistic regression. Kalixanda population-based endoscopic study December 1998-June 2001 RP=role physical, BP=bodily pain, GH=general health, SF=social functioning, RE=role emotional, MH=mental health, UID=uninvestigated dyspepsia, FD=functional dyspepsia, EPS=epigastric pain syndrome, PDS=postprandial distress syndrome, PUD=peptic ulcer disease

		PF	RP	BP	GH	VT	SF	RE	MH
		OR	OR	OR	OR	OR	OR	OR	OR
		95% CI	95% CI	95% CI	95% CI	95% CI	95% CI	95% CI	95% CI
	UID n=202	2.46	2.80	2.63	2.56	2.11	1.53	1.50	1.87
		1.68-3.61	1.84-4.25	1.85-3.73	1.79-3.65	1.49-2.99	1.03-2.27	1.03-2.20	1.31-2.67
	FD n=157	2.34	2.56	2.36	2.25	1.98	1.47	1.57	1.98
		1.53-3.58	1.62-4.06	1.60-3.48	1.52-3.34	1.34-2.92	0.94-2.29	1.03-2.40	1.33-2.95
	EPS n=52	1.50	1.99	2.06	1.53	1.69	0.99	1.20	1.59
		0.74-3.04	0.93-4.28	1.09-3.89	0.80-2.91	0.90-3.20	0.46-2.13	0.58-2.49	0.83-3.03
	PDS n=122	2.73	2.68	2.62	2.53	2.04	1.75	1.55	2.31
		1.71-4.37	1.62-4.43	1.69-4.05	1.63-3.92	1.33-3.14	1.09-2.80	0.98-2.47	1.49-3.58
	Organic dyspepsia n=45	2.74	3.64	3.88	3.72	2.61	1.81	1.43	1.51
		1.35-5.59	1.61-8.26	1.97-7.65	1.88-7.33	1.32-5.14	0.89-3.70	0.70-2.93	0.78-2.96
	PUD n=41	2.84	3.41	3.26	2.29	1.93	3.00	2.20	1.10
		1.36-5.94	1.53-7.58	1.67-6.38	1,15-4.57	1.00-3.76	1.50-6.00	1.10-4.42	0.52-2.29
	PF=physical functioning,

Table 4 .

 4 Mean values of Short Form 36 domains in Functional dyspepsia and overlap groups FD-IBS, FD-GORS and FD-both GORS and IBS. Kalixanda population-based endoscopic study December 1998-June 2001

	PF	RP	BP	GH	VT	SF	RE	MH
	Mean	Mean	Mean	Mean	Mean	Mean	Mean	Mean
	95% CI	95% CI	95% CI	95% CI	95% CI	95% CI	95% CI	95% CI

Acknowledgements

Declaration of personal interests: Elisabeth Bolling-Sternevald is an employee of AstraZeneca R&D Mölndal, Sweden, others have nothing to disclose.