The probability of contracting a disease is closely linked to exposure to infectious organisms. A susceptible individual can only become infected if she/he is exposed to infectious organisms. Thereby, the transmission probability increases with an increase in the number of infectious organisms to which a susceptible individual is exposed. Subsequently, we refer to exposure as the cumulative, average amount of infectious medium ingested by a susceptible individual within a time period of interest due to close contact with an infectious person.

We base our proposed transmission model on the exponential relationship between the ingested dose and the infection risk as derived in Haas et al. 18 and used in several other publications 222324. Details describing how the following assumptions 1, 3, and 5 translate into an exponential dose-response model can be found in Haas et al. 18. As an extension to this general formulation of an exponential relation between exposure and the risk of infection, we extrapolate the actual exposure from information about the duration and intensity of a contact between an infector and a susceptible individual. The proposed mechanistic model is based on the following underlying assumptions:

1. In principle, one infectious organism is sufficient to cause infectious disease. This hypothesis has been repeatedly supported by various studies against the alternative hypothesis assuming a threshold dose of infectious organisms must be passed to cause infection 192526.

2. Every ingested infectious organism has a certain probability to survive until it reaches its target tissue and can initiate infection 1827.

3. We assume that this survival probability is a constant, i.e., factors like the susceptible hosts' immune responses are assumed to be equally effective for all individuals. This assumption is a simplification of reality since susceptibility is known to differ between individual hosts 28. However, for the purpose of this paper, such a simplification that keeps the model and the interpretation of its results manageable is justified.

4. The average dose of infectious material that is ingested by an individual is a linear function of the duration and intensity of the contact with an infectious individual. Research has shown that these measures are good predictors for individual attack rates of SARS 29. In theory, we recognize that contact can be any kind of interaction between two individuals that is sufficient to exchange body fluids that can carry infectious particles. However, for reasons of manageability and measurability, we concentrate on conversational and physical contacts.

5. The actual amount of infectious organisms ingested by an individual follows a Poisson probability distribution with the average dose (defined in assumption 4) as parameter 1819. Thereby, we model the total (i.e., cumulative) average dose ingested during an entire simulation time step. This can lead to biased results in extreme cases 30, but given the fact that this assumption has proven to work well in the past and considering other uncertainties, utilizing this simplification is justified.

Based on this, the probability that individual n becomes infected during simulation time step t_x can be derived as

where I is the total number of infectors; [s^-1] is the shedding rate (~microbial load) of infector m at simulation time step t_x; [1] is the contact intensity between the infector and the susceptible individual, which corresponds with the proportion of infectious material spread by infector m that is actually ingested by n; and [s] is the time individuals n and m actually interact during time step t_x. Finally, Θ is a calibration parameter that accounts for all relevant factors that are not explicitly represented, such as survival probability of the infectious agent. Simulation models can be fitted to measured epidemiological data, such as epidemic curves, or to targeted reproduction rates by means of Θ. We used Θ to achieve predefined reproduction rates for the contact structure introduced in the following section.

In the subsequently described test setting, empirical contact data is needed to compare the mechanistic transmission model with the classical one. We rely on contact data reported in a contact diary study that was conducted in Switzerland. A convenience sample of 54 participants was asked to report their potentially contagious contacts (as defined below) for 14 different days. Although a convenience sample is not representative for the whole population, the sample used here represents a very diverse cross-section of the population as can be seen in Table 1.

The design of the diary is similar to that used by Mikolajczyk et al. 31. A potentially contagious contact is defined as (1) a mutual conversation of more than 10 words within a short distance (<2m), (2) physical contact in general, or (3) contact involving kisses. The participants were asked to categorize their contacts according to these three categories and to estimate how long they interacted with each reported contact person during an entire day based on six provided categories. However, for the analysis, we need concrete values instead of categories to calculate transmission probabilities as defined in Equation 1. Therefore, we assume a concrete duration, the arithmetic mean of the upper and lower bounds, for each category as given in Table 2.

One diary had to be revoked due to deficient data quality; three of the remaining 53 participants provided only information for 5, 7, or 8 days, resulting in a total of 720 different person days with 7145 reported contact partners. In 36 of 7145 records, the information about contact duration was missing. These missing values were imputed based on probability distributions observed for the complete records. The processed data is provided in Additional File 1.

In the results section, we compare how the proposed mechanistic transmission model differs from the classical model assuming an equal transmission probability for all contacts. Thereby, both the contact structure and the basic reproduction number R₀ are fixed for both transmission models. We use the classical definition of R₀ as the average number of secondary cases generated by an infected individual being introduced into a fully susceptible population 4.

We first analyse the effect of the observed patterns of contact duration and assume the intensities to be equal and constant for all contacts. Then, we analyse the impact of contact intensity in a qualitative way. Information on shedding rates and inter-individual differences is available for many diseases (e.g., influenza cf. 14). However, as we are more interested in exposure differences due to contact structure than in the impact of shedding rate differences, we also assume to be equal for all infectors m. We further concentrate on hypothetical diseases with an infectious period of one day and basic reproduction numbers R₀ = 1.5, 3.0, 4.5, and 6.0. With these assumptions, the contact intensity and the shedding rate can be included in a new calibration parameter , and Equation 1 can be simplified to

The expected number of secondary cases SC generated by a specific infector m if introduced into a completely susceptible population can then be calculated as follows:

with S represents the total number of susceptible individuals infector m has contact with during the day m is infectious. Finally, the equation

reveals the basic reproduction number as defined previously when X includes the total population of interest.

The following two analyses are used to contrast the effect of the mechanistic model (Equation 2) against the classical model:

1) We illustrate the relationship between the expected number of secondary cases SC and the number of contacts S by calculating SC for the 720 person days as separate units of observation. SC is calculated according to Equations 2 and 3 and based on the contact durations measured with the contact diaries. We group the SC-values by S and show the so grouped SC-values in box plots. We do this for different values of Θ'; Θ' is determined such that R₀ = 1.5, 3.0, 4.5, and 6.0 for the given test population according to Equation 4. The contact intensity i and the shedding rate q are assumed to be constants. We then compare the number of secondary cases SC of the simplified mechanistic model with the analogue SC value when the assumption of a constant per-contact transmission probability is used (also grouped by S).

2) In the second analysis, we calculate how the contact intensity is related to the duration of a contact in the empirical data set of potentially contagious contacts. This allows a qualitative discussion related to how the inclusion of variable contact intensities instead of a constant might affect the results found in analysis 1.

The presented results elucidate the implications of accounting for contact duration and intensity in simulations of disease spread. Figure 2 suggests that the importance of highly connected individuals, so-called super-spreaders, is strongly overestimated when all contacts are assumed to be equally likely to transmit infectious disease. This finding is particularly important because some well cited publications have concluded that highly connected individuals are major drivers of disease spread without accounting for the heterogeneity of the inter-individual transmission probabilities 2021. The results suggest that in the case of a disease with a low reproduction number, the expected secondary cases induced by individuals with many contacts are in the same range as those induced by individuals with medium numbers of contacts. Only when R₀ is close to the mean number of contacts is the expected number of secondary cases approximately linearly related to the number of contacts (see Figure 1 and the linear regressions shown in the results section).

This finding can be easily explained by the fact that the marginal total contact time decreases with every further contact person. In other words, most people have only a small set of persons (usually at home or at work) with whom they meet and interact for long periods during a day. Those individuals who meet with far more people than the average spend on average less time with every single contact person than those persons who have some or only a few contact partners per day. This can be illustrated with the example of train conductors, flight attendants, or supermarket cashiers; indeed, all of them have contact with hundreds of people a day, but they interact with each single contact only for a very short time.

As a consequence, highly connected individuals have more potentially contagious contacts than others, but these contacts are simultaneously on average less likely to transmit disease. Highly connected persons can reach their full "super-spreading" potential only if a disease is so contagious that almost every contact with a susceptible person leads to infection. Similar findings have been reported for sexually transmitted diseases: Research has shown that individuals with many different sexual partners per year are less important for disease propagation than often assumed because they have less sex acts per partner and in total than individuals with a smaller number of partners 3233.

The conclusion that highly connected individuals are overestimated in their importance if a constant per-contact transmission probability is assumed is further supported by the analysis of the contact intensity as reported in the contact study. Theoretically, a short interaction between a susceptible and an infectious person could lead to a comparable amount of ingested infectious material as that of a long interaction assuming that the short interaction is more intensive than the long one. However, prolonged contacts tend to be more intensive than short contacts because they more often involve closer interaction, such as physical contact and kisses. This finding is plausible because those persons with whom the individuals spend much time are in most cases their loved ones, thereby indicating the higher likelihood of more intimate interactions than with casual acquaintances. As a result, the conclusions from the analysis of the pure contact durations are even further pronounced by taking contact intensities into account.

Therefore, our results suggest that sole concentration on the connectedness of individuals to explain super-spreading events is not valid. The explanation for super-spreading events might lie in a combination of many contacts and high shedding rates (cf. the notion of "super-shedder" 6). Extreme numbers of secondary cases can only be achieved when the shedding rate of an infector m is much higher than the average shedding rate and only if this infector m has many susceptible contacts S.

We see three limitations in our study. First, the empirical data set used to test the proposed exposure-based transmission model is rather small and not representative of the population. Furthermore, the person days used to calculate secondary cases are not statistically independent from each other as every person participating in the study contributed diary entries for several days. Finally, contact patterns are dependent on the cultural background and may look differently in Italy, Germany, Thailand, or Sudan 34. Thus, the generalizability of our results may be questioned. Although this limitation exists, it is not likely to bias the presented results in a relevant manner. The observed contact patterns are plausible and theoretically grounded. An increasing number of contact partners per time unit naturally results in a decrease in the time spent with each single contact partner. Additionally, most people plausibly have only a very limited set of persons with whom they interact very closely. Additionally, the attributes of our contact structure are in complete agreement with other empirical studies on potentially contagious contacts that have also addressed similar attributes 34.

Secondly, the six time categories of the diary study offer rather imprecise information on the actual time that two persons interacted, and the three intensity categories are too vague to be translatable into concrete values for use in Equation 1. Hence, the results presented have to be qualified in a quantitative rather than qualitative sense. For every time category, we defined a precise value (the arithmetic mean of the upper and lower boundaries) that was used for all calculations. However, a sensitivity analysis that alters the actual duration defined for every category within the given boundaries does not lead to qualitatively different results (see Additional File 2). Although the measured intensity indicator is not sufficiently precise to allow inclusion in a mathematical sense, the analysis clearly indicated that inclusion of contact intensity would amplify the observed phenomena rather than falsify our conclusions.

Finally, this paper makes statements about the expected number of secondary cases of infected individuals in a fully susceptible population. In a simulation model of disease spread, the importance of an individual also depends on her/his position within the contact network structure; i.e., the network position of every individual determines the likelihood of becoming infected as well as the susceptibility status of the surrounding individuals. Due to the complex nature of simulation models of disease spread, complete simulation models must be designed and tested for sensitivity to the changed transmission model proposed in order to allow precise statements on the impact of exposure-based transmission models on simulation outcomes.