## INTRODUCTION

Although epidemics were already documented by Hippocrates (458–377 b.c.), it was not until 1760 that mathematical modelling of infectious diseases was first documented in a publication by Daniel Bernoulli [Reference Bernoulli1], who used a mathematical method to study smallpox infections. These and subsequent models at the very start of the 20th century [Reference Hamer2–Reference Kermack and McKendrick5] focused on determining the infectious disease spread and the associated basic and effective reproduction number (see Appendix for terminology used). Hugo Muench first proposed the concept of estimating the force of infection (at that time called the ‘effective contact rate’) as a key parameter in such mathematical models, in a publication entitled ‘Derivation of rates from summation data by the catalytic curve’ in the March 1934 issue of the *Journal of the American Statistical Association* [Reference Muench6]. His work became widely known only 25 years later with the publication of his book *Catalytic Models in Epidemiology* [Reference Muench7].

Before Muench's seminal work, series of physicians' case-reports, i.e. incidence data, which are often affected by underreporting and misdiagnosis, were used to study the ‘effective contact rate’ of a given disease in a given population [Reference Bernoulli1, Reference Collins8]. Muench suggested using a catalytic model on summation data to obtain a measure of the rate at which a susceptible acquires infection (and not necessarily disease). The name ‘catalytic’ was inspired by the similarity to the equations used to study the processes that drive chemical reactions. In his 1959 monograph [Reference Muench7], Muench referred to this effective contact rate as the force of infection (FOI). The importance of Muench's catalytic model is the capacity to use test results on, for example, serological or saliva samples, in addition to reported incidence data, in order to estimate the FOI, especially for infections that leave permanent markers of immunity in their surviving hosts.

More explicitly, Muench stated that the simplest approach is to assume a constant effective exposure rate λ per unit of time *t* and that this rate applies to the entire population at all times. Nonetheless, the immune proportion π increases only to the extent that previously uninfected individuals can incur new infections. In addition, the model allowed that some fraction of the population cannot be infected at all. Muench's catalytic curve can be written as

where *k* is the proportion of the population that can be infected, π is the proportion of all previously infected (and immune) individuals prior to age *t*, and *l* is the proportion of the population which may show evidence of exposure. Figure 1 illustrates the behaviour of Muench's model for different choices for *k* and *l*. Note that the model gives negative estimates of the proportion immune for *l*<1 and thus interpretation is not straightforward in this case.

Using the method of moments to estimate these parameters, Muench illustrated his approach on several datasets. These included intraperitoneal mouse protection test results for yellow fever in South America, a test which remains positive once an individual has been infected by yellow fever. He considered two regions; one in which the population was originally assumed entirely susceptible with frequent epidemics so that λ actually corresponded to a steady effective exposure rate. He compared these results with those of a second region where an outbreak of yellow fever occurred. He illustrated that these test results cannot show when individuals had been infected, only that it must have occurred at some time before taking the test. Indeed, by analysing summation data one estimates the probability of past infection at a certain time-point and derives the FOI using the analytical expression in equation (1). The latter is based on the untestable assumption of time homogeneity.

Testifying its importance, numerous later publications echoed this inherent limitation of not being able to use direct data on person-time incidence but rather summation data to derive the FOI under the assumption of time homogeneity. Muench already discussed other complications such as test reversion rates, differential mortality, and passive immunity through transfer of maternal antibodies.

Although applicable to cumulative incidence data too, we illustrate Muench's method on cross-sectionally collected seroprevalence data because of the above limitations of reported incidence data. We use two serological surveys, the first on rubella in the UK, collected in 1986–1987 for males only [Reference Farrington, Kanaan and Gay9]; and the second on parvovirus B19 in Belgium, collected in 2001–2003 [Reference Hens10]. We make a number of typical common assumptions, i.e. that no portion of the population is free from exposure, a perfect test is used, the time homogeneity assumption holds, disease-related mortality is negligible compared to all-cause mortality, infection confers lifelong immunity [in equation (1): *l*=*k*=1] and, specifically for rubella in the UK, that the schoolgirl programme that was implemented had negligible effect on the seroprevalence of rubella in males. Whereas Muench originally used the method of moments to estimate the FOI, we cast his model in the maximum-likelihood framework. More specifically, we use a binomial likelihood to relate the age-specific prevalence (or cumulative incidence) to the age-specific observed proportion immune:

where *a*=(*a* _{1}, …, *a* _{N}) denotes the age vector of length *N*; *y*=(*y* _{1}, …, *y* _{N}) and *n*=(*n* _{1}, …, *n* _{N}) denote the corresponding vectors of positive and total counts per age value. Assuming Muench's model, π(*a* _{i}) in equation (2) is given by equation (1). Maximizing the likelihood in equation (2) with respect to λ yields the maximum-likelihood estimate .

The black curves in Figure 2 show the fitted seroprevalence and FOI curves together with the observed seroprevalence for both infections. The constant FOI was estimated as 0·104 and 0·053 for rubella and parvovirus B19, respectively. The assumption of a constant FOI seems visually appropriate for the rubella data but inappropriate for the parvovirus B19 data (cf. Fig. 2). Such observations led other researchers to develop new methods allowing for an age-dependent FOI.

### Standing on the shoulders of Muench

Muench's work has led to the development of many models to estimate the FOI. These models can broadly be divided into *parametric* and *non-parametric methods*. Whereas parametric methods rely on specific functional relationships (e.g. Muench's model is parametric since it assumes a constant FOI), non-parametric models relax such assumptions. In this section we present an overview of these different approaches. Starting with Muench's work, Table 1 summarizes the key contributions with respect to the shape of the FOI, the focus of the analysis and the data used.

#### Parametric methods

Griffiths [Reference Griffiths11] was the first author to propose the use of a linear rather than a constant FOI to model measles incidence data from England and Wales for the period 1956–1969. With the above notation, his model could be formulated as

where λ(*t*)=γ_{0}(*t*+γ_{1})*I* _{{t>τ}}, with γ_{0}, γ_{1} parameters to be estimated and τ the threshold for inherited immunity. *I* _{{t>τ}} denotes the indicator function taking value 1 if *t*>τ and 0 otherwise. In general, equation (3) is the solution of the differential equation

(with initial condition *x*(0)=1 meaning that everyone is assumed susceptible at birth) that describes the changes in the proportion of susceptibles *x*(*t*)=1−π(*t*) with respect to time. Note that denotes the cumulative FOI up to age *t* acting on susceptibles. Whereas Griffiths [Reference Griffiths11] described the estimation procedure outlined above as simple and straightforward to apply, he actually used an alternative method to model the measles incidence (rather than cumulative incidence) using a multinomial model and thereof estimated the attack rate.

Note that the catalytic model as presented in equation (3) is actually a survival model and that the probability of past infection is the cumulative distribution function of the time to infection or alternatively 1 – survival function. The (cumulative) FOI is the (cumulative) infection hazard.

In general the change in the susceptible portion could be both age- and time-dependent. It is under the assumption of time homogeneity that age can be used to determine the time of infection and thus age and time are identified and equation (3) is typically denoted in terms of age rather than time. Time homogeneity implies that neither the pathogen's transmissibility, nor susceptible people's receptiveness to infection (irrespective of their age), nor the frequency and intensity of interactions necessary for transmission to occur (e.g. social contact patterns for air- and saliva-borne infections, sexual intercourse for sexually transmitted infections) have changed substantially with time. In particular feco-orally transmitted infections (e.g. hepatitis A virus, cholera) are documented to be time heterogenous, due to the drastic improvements in sanitary conditions in many settings over the last century [Reference Gay12, Reference Schenzle, Dietz and Frösner13]. Note that Schenzle *et al*. [Reference Schenzle, Dietz and Frösner13] argued that, for the case of hepatitis A, it is more appropriate to assume age homogeneity and time heterogeneity, and thus modelled time effects. The limitation of having to choose for either time or age may be overcome by use of data which is both age and time structured (see below).

Griffiths used maximum-likelihood theory to estimate the parameters in the catalytic linear model and for the first time addressed the goodness-of-fit to the data using Pearson's χ^{2} test. Interestingly, Griffiths justified his choice of a linear FOI by using a non-parameteric estimate for the FOI which was plotted against age and showed a linear trend. Note that Griffiths applied his model only up to age 10 years as by then most children had been infected with measles. Griffiths' work was later used by several other authors to model measles and other common childhood infections [Reference Anderson and May14]. Griffiths' 1974 contribution should be seen as the completion of the basic building block for the estimation of the FOI.

Grenfell & Anderson [Reference Grenfell and Anderson15] extended Griffiths' approach to encompass a general polynomial description of changes in the FOI with age and derived a stepwise maximum-likelihood method for parameter estimation from datasets consisting of either case notifications or serological information. This encompasses both Muench's and Griffiths' catalytic model. The appropriate polynomial degree can be found by minimizing the relative deviance. Grenfell & Anderson discussed the advantages and disadvantages of using case-notification data and serological data and stressed that availability is the key criterion for which type of data to use [Reference Grenfell and Anderson15]. For case- notification data, the quality largely depends on the biases and inaccuracies of the case-notification system. Serological databases too can suffer from the representativeness of the collected samples, as well as implicit assumptions for analysis, such as time homogeneity, lifelong immunity and the other complications listed above, duly noted by Muench [Reference Muench6]. Several of these issues gave rise to further research.

A first complication is that under the assumptions stated above, the model for the prevalence should be monotonically increasing with age. The monotonicity issue was not relevant for Muench and Griffiths since a model with constant or linear FOI always estimates a monotone prevalence. However, this is not necessarily the case with high-order polynomials. In 1990, Farrington [Reference Farrington16] placed the issue of monotonicity in the heart of the estimation problem. He noted that for measles, mumps and rubella, the FOI typically rises linearly from birth up to about 10 years of age, after which it drops off again for older ages. This qualitative form was also observed by Griffiths. Farrington considered a nonlinear model, i.e. an exponentially damped linear model, in age

where γ_{0}, γ_{1} and γ_{2} are positive parameters that can be estimated using maximum-likelihood and constrained optimization. The FOI in equation (5) is 0 for age 0, then shows a linear increase and ends in an exponential decay. Following Griffiths, Farrington [Reference Farrington16] assessed the goodness-of-fit of his model using a non-parametric estimate of the FOI, i.e. a moving average, indicating again that a more formal non-parametric approach for the estimation of the FOI is needed, at least in the exploratory stage.

Note that for specific choices of λ(*a*), such as those proposed by Muench and by Grenfell & Anderson (see also [Reference Massad, Raimundo and Silveira17]), equation (3) corresponds to a generalized linear model (GLM [Reference McCullagh and Nelder18]) with binomial response and log-link. Other parametric models fitted within the framework of GLMs have been proposed since then using different link-functions such as the logit and cloglog link [Reference Keiding19–Reference Beutels29]. Of those, the most popular parametric model employed is the piecewise constant FOI where for predetermined intervals a constant FOI is assumed. The choice of these intervals is usually inspired by the intuitive relevance of the ages of mixing groups in the population (e.g. pre-school, school, high school, etc.). A drawback for the model of Grenfell & Anderson and many other parametric models is the probable occurrence of a negative FOI for some age values or an unrealistic steep increase for higher age values because of the chosen functional relationship. Farrington's model is not a member of the GLM family, and deals with these issues by constraining the model based on prior knowledge. However, if that knowledge is unavailable or questionable, non-parametric options could be explored.

#### Non-parametric methods

Although non-parametric techniques were used before to assess the fit of a parametric, possibly nonlinear, function [Reference Griffiths11], Niels Keiding [Reference Keiding30] was the first to explicitly use a non-parametric technique to estimate the FOI from serological data, based on the isotonic Kaplan–Meier estimator of 1−π(*a*) which finds its origin in survival analysis. Keiding also addressed the issues of time homogeneity, monotonicity, and censoring.

Most of the various non-parametric methods in the GLM framework developed since Keiding, involved estimating the (sero)prevalence by a non-parametric technique and subsequently deriving the FOI by . Note that this latter expression for the FOI is a discretized version of equation (4) with *x*(*a*)=1−π(*a*).

Of these non-parametric applications, spline-based methods have become very popular [Reference Hens10, Reference Nagelkerke31–Reference Shiboski34]. A spline can be seen as a concatenation of a rich number of local polynomials which are glued together in a ‘smooth’/continuous way. In 1996, Keiding and colleagues proposed to replace the kernel smoother in his earlier work with a smoothing spline [Reference Keiding19]. Subsequent work involved semi-parametric models [Reference Hastie and Tibshirani35–Reference Namata37], in which the age-specific prevalence is modelled non-parametrically and possible covariate effects such as gender are included in the parametric component of the model. The green (blue) curves in Figure 2 represent the (monotonized) estimated prevalence and FOI based on the spline methodology [Reference Hens10, Reference Keiding19, Reference Namata37].

Looking further at the different methods as applied to the rubella and parvovirus B19 data in Figure 2, several observations can be made. First, Muench's model seems to fit the serological profile of rubella well, whereas it does not follow the pattern of the parvovirus B19 seroprofile. Farrington's exponentially damped linear model shows an improved performance for rubella whereas the fit to the parvovirus B19 data seems reasonable except that it is not able to capture the decay in seroprevalence at about 25 years of age. The spline model shows a similar fit to the seroprevalence data as the exponentially damped linear model for the rubella data, but some quantitative differences appear on the scale of the FOI.

Moreover, the spline model is able to capture the decaying seroprevalence at around the age of 25 years whereas its monotone version is a regularization to ensure a positive FOI. Indeed, when looking at parvovirus B19 in Figure 2*a*, the spline fit reveals a non-monotone pattern (green curve) whereas its monotonized version, applying a smooth-then-constrain approach, shows a monotone trend (green-blue-green curve) for the prevalence and a positive FOI.

Based on the non-parametric fits, one might question which of the underlying assumptions is violated for parvovirus B19 in Belgium. This could for instance be due to antibody titres declining with time post-infection, and eventually falling below the cut-off for positivity; or a time dependence in the FOI resulting in a cohort effect, or the use of a manifestly imperfect test. It is only by contrasting different methods and the graphical exploration of the goodness-of-fit that such distortions appear [Reference Hens21].

When estimating the FOI from summation data, phenomena like these are often not taken into account because of the lack of information in the data and thus result in what is often referred to as an average profile. The methods that have been used to investigate the potential mechanisms behind such features mostly rely on contrasting mathematical models with incidence data. It seems hardly possible and beyond the scope of the current paper to list all papers that may have used Muench's catalytic model as a starting point (without necessarily referencing it) to make extensions for such mathematical modelling studies. Nonetheless we further illustrate the vast influence of Muench's pioneering work by listing some of the main examples of such extensions: (1) models assuming infection confer no or no lasting immunity. Examples of such analyses include *Bordetella* pertussis [Reference van Boven38] and *Haemophilus influenzae* type b [Reference Coen39], transmission models in which waning immunity was taken into account; (2) analyses describing the possible occurrence of seasonality [Reference Fine and Clarkson40–Reference Zaaijer, Koppelman and Farrington42] or regular epidemics [Reference Whitaker and Farrington43]; (3) analyses on chronic infections (see e.g. [Reference Mathei44–Reference Becker48]); (4) analyses taking into account vaccination programmes (see e.g. [Reference Becker48–Reference Amaku50]); (5) further extensions of the more basic catalytic models have been used (see e.g. [Reference Zhang51]).

### A practical guide to estimate the FOI

The diversity of methods to estimate the FOI raises the question, given a particular dataset, which method should be used? Figure 3, presents a guiding flow chart, which starts from an exploratory stage in which a non-parametric model is fitted to the data. A graphical representation of such a fit is given in Figure 2. Given the epidemiology of the infectious disease under consideration and specific complications, such as diagnostic uncertainty, the shape of the non-parametric estimate and the goodness-of-fit to the data could show distortions with respect to monotonicity, maternal immunity and time homogeneity. For each of these distortions, different remedial techniques are available, which are listed in the flow chart (Fig. 3). These techniques enable obtaining a ‘regularized’ estimate of the FOI, which can be studied in detail, parametrized [Reference Beutels52–Reference Thiry56] and in turn can be used for the estimation of related parameters, such as the basic reproduction number [Reference Farrington, Kanaan and Gay9, Reference Farrington16].

### Muench today: a road map to another 75 years?

Although, Muench's model was overly simplistic in assuming a constant FOI, his 1934 paper already raised concerns with respect to model assumptions and data constraints, which are still today important research topics. There are a number of areas where recent research has produced interesting results, and in this section we highlight four of these.

(1) Time homogeneity is often assumed, as a necessary condition, if one can only use a single cross-sectional serological survey for estimating the FOI. In some cases (e.g. parvovirus B19 in Fig. 2), this seems questionable and might be untenable. If so, one needs longitudinal type of data (see e.g. [Reference Whitaker and Farrington57]) or alternatively several prevalence surveys at different points in time [Reference Whitaker and Farrington57, Reference Ades and Nokes58]. Since similar distortions were observed for parvovirus B19 in four other countries [Reference Mossong59], other hypotheses such as waning of naturally acquired antibody levels with time post- infection need further investigation.

(2) Antibody levels are commonly used to classify an individual's sample as positive, negative or inconclusive based on a given test-specific threshold. This allows estimation of the proportion of susceptible people at each age, and to derive from these proportions the FOI by age. Mixture models are a natural alternative for this type of data. Gay [Reference Gay60] was the first to model age-stratified serological data using two-component mixture models with age-dependent mixing probabilities and age-dependent mixture components. Using this mixture approach, Bollaerts

*et al*. [Reference Bollaerts61] proposed a direct estimator for an age-dependent FOI (i.e. without using a predefined threshold to distinguish susceptible from immune people).(3) As sera are often tested for more than one antigen, joint analysis for diseases with similar transmission routes can lead to new insights. Hens

*et al*. [Reference Hens10] introduced the age-dependent joint and conditional FOI, a framework allowing formal statistical tests such as testing the assumption of separable mixing. Earlier work on bivariate models [Reference Farrington, Kanaan and Gay9, Reference Kanaan and Farrington62] focused on the estimation of the basic reproduction number from serological survey data incorporating the effect of individual heterogeneity (see also [Reference Coutinho63]) and testing for separable mixing [Reference Farrington, Kanaan and Gay9].(4) A fundamental concept for the estimation of the basic reproduction number is the mass action principle, relating the FOI with the transmission rate, i.e. the

*per capita*rate at which an individual of a particular age makes an effective contact with a person of a specific age, per year. The transmission rates populate the so-called WAIFW matrix [Reference Anderson and May64], which is traditionally imposed to be of a particular structure (rather*ad hoc*, albeit inspired by, e.g. social and schooling systems). Wallinga*et al*. [Reference Wallinga, Teunis and Kretzschmar65] were the first to conceptually link seroprevalence data with data on conversational contacts per person, whilst assuming that transmission rates are proportional to rates of conversational contact. Using data from a social contact survey, Ogunjimi*et al*. [Reference Ogunjimi66] and Goeyvaerts*et al*. [Reference Goeyvaerts67] proposed the disentanglement of the WAIFW matrix into two components: the surface of conversational contacts and an age-dependent proportionality factor.

## DISCUSSION

Although the basic reproduction number is a very powerful and elegant summarizing parameter at the heart of infectious disease transmission dynamics, the paucity of opportunities, and difficulties to obtain direct estimates of *R* _{0}, make the FOI extremely relevant. Indeed, it may often be the parameter, which through its estimation allows all other parameters to be estimated.

In disease burden estimates, it determines estimates of the occurrence of infections, of clinical cases and all consequences arising from these cases (e.g. hospitalizations, deaths, life-years lost, disability adjusted life years). Moreover, in studies estimating the impact of infectious disease interventions, such as vaccination, in terms of effectiveness and cost-effectiveness, the FOI will time and again rank high if not first of the most influential parameters determining the outcomes, and hence the public health policy measures based on these outcomes. Therefore estimating the FOI as accurately as possible, is essential.

Muench's seminal works published 75 and 50 years ago, are still inspirational today for the conceptual basis he provided for the FOI, and the pitfalls and problems he identified. We have described not only the various extensions those standing on the shoulders of Muench have proposed, but also noted that we are still trying to deal with the same pitfalls and problems already identified by Muench.

There appears to have been an increased interest in developing approaches for the estimation of the FOI since the mid-1980s [Reference Hens68]. We hope the guidance we have provided will be useful for researchers to decide, after exploration with a non-parametric method, which of the parametric methods is best suited for the dataset under consideration.

## ACKNOWLEDGEMENTS

We thank the editor and both referees for their valuable suggestions that have led to an improved version of the manuscript. This work was supported by research project (MSM 0021620839), funded by ‘SIMID’, a strategic basic research project funded by the institute for the Promotion of Innovation by Science and Technology in Flanders (IWT) (project number 06008); by the Fund of Scientific Research (FWO, Research Grant G039304) in Flanders, Belgium; and by the IAP research network (no. P6/03) of the Belgian Government (Belgian Science Policy). The R-code used to analyse the datasets in this manuscript is available from the authors.

## APPENDIX Glossary of words in alphabetical order

## DECLARATION OF INTEREST

None.