Scope and Limits

“All models are wrong, but some are useful” (George Box)

The SEIR model, like many models of mathematical biology, ecology, or medicine, has never been designed to serve as “epidemiological weather forecast”. There are many questions it does not answer or even address, notably all issues that refer to smaller scales than the sizes of the compartments that it is built on. For our simulation tool, this excludes spatial patterns as well as practical questions concerning the relative roles of schools and work places (or public transport, care homes, sports events, etc.) for transmission. Although the scheme is flexible and can readily be extended to account for heterogeneities within compartments, added detail does not help much when the values of key parameters for fatality and transmission are unknown, or if they depend on human behavior, which is hard to predict.

What makes deliberately simple models like SEIR useful, is that they can help build intuition and coin basic concepts like reproduction number or herd immunity. They also provide a rough overview of possible or plausible scenarios for the epidemiological dynamics across wide parameter regions. They can help to assess the relative importance of model parameters and the feasibility of certain coping strategies (for example, a herd immunity strategy). We provide examples for some of these uses on our Scenario page.

While many extensions and additional compartments do little to change the qualitative model predictions, several assumptions of our SEIR simulation scheme play a more important role. Below, we briefly comment on some of these assumptions, with a focus on factors that have been discussed in the context of COVID-19.

Partial immunity, vaccines, and medication

Our implementation of the SEIR model builds on the double assumption that immunity is absent in the starting population, but complete and permanent for recovered individuals. Both assumptions are likely wrong. On the one hand, there is emerging evidence that immunity against other human coronaviruses could provide partial protection against COVID-19. On the other hand, acquired immunity may be weak or short-lived (e.g., after an asymptomatic infection). Depending on their prevalence, both factors can have far-reaching consequences for the course of the epidemic, which are not accounted for in our model.

Similarly, the model does not account for the prospect that the course of the epidemic may be fundamentally altered once vaccination becomes broadly available. Efficient vaccination would simply transfer susceptibles directly to the “recovered” class, without ever getting infected. If vaccines only provide partial protection, people may still get infected (and even be infectious), but may only experience mild symptoms. The latter also holds if medication is discovered. Indeed, any improvement in medical practice can lead to an IFR that declines with time. To some (unknown) degree, this has probably been already the case in the first months of the worldwide pandemic. It may significantly reduce the rate of future deaths, e.g., during a potential second wave.
Extensions of the SEIR model to include effects of medication, vaccination, or partial and temporary immunity are possible. However, currently robust empirical evidence for all these factors is lacking and there are no reliable data for a parametrization of such a model. If this changes, it may be included in a future update.

Age-dependent infection risk and detection ratio

In Austria and many other countries, the age distribution of detected infected individuals has seen a clear shift towards younger age classes between the early phase of the epidemic in spring and later phases in summer. There two potential reasons for this effect:

It appears likely that both factors have contributed to the observed effect, but it is not easy to disentangle them on the basis of the available data. Unfortunately, Austria currently does not provide any information on the age distribution of severe (hospitalized) cases to research, which would serve as an important piece of information in this case. There is, however, clear independent evidence for the second factor (increased detection ratio) from the proportion of positive PCR tests among all tests. This proportion has dropped from values > 15% in the early phase of the epidemic to < 1% in late spring and summer, indicating an increase in the detection ratio δ(t).

We have therefore included a time-dependent detection ratio δ(t) into the model, but currently we have not included a time-dependent age-specific infection risk. Note that an alternative model with constant detection ratio, but variable, age-dependent risk would lead to higher estimates of the effective reproduction number Reff, because increased detection can no longer serve as an explanation for an increase in observed case numbers.

Spatial structure and heterogeneous transmission

Data show that virus outbreaks occur in local or regional clusters, but our compartment model ignores any spatial structure. Any representation of the “true” spatio-temporal dynamic of COVID-19 in Austria would require a huge number of parameters to capture the exchange between regions. This is outside the scope of our model, but raises the question which effects spatial structure can have for the global dynamics. Generally, spatial structure can lead to a slower spread of the epidemic, in particular if contacts between regions are strongly reduced. However, the total number of infections by the end of the epidemic (so-called final size) is not much affected by spatial structure alone. For Austria, this can be seen, for example, in the spatially explicit model by N. Popper, where 77% to 95% of the population gets infected in a baseline scenario, depending on the level of compliance to hygiene standards and social distancing.

Population structure has much larger effects if it also implies heterogeneity in transmission. As a thought experiment, imagine that the disease does not transmit at all in half of the country, obviously limiting the final size to below 50%. But also less extreme heterogeneities (e.g., between cities and the countryside) can have profound effects, generally reducing both the level of herd immunity and the final size. While it is very plausible to assume that there is at least some heterogeneity in transmission, e.g. due to differences in population density, patterns for the COVID-19 epidemic are as yet hard to discern. For example, while New York City has been hardest hit in the US, the epicenters in Lombardy have been villages and small towns rather than the biggest city Milan. In Austria, as of end of June, total attack rates in sparsely populated Tyrol are higher than in Vienna and the region around Salzburg is harder hit than the city of Salzburg itself. Matters are further complicated by superspreading transmission, as discussed further below.

Heterogeneity effects can be discussed in an extended SEIR model, but for the reasons given a reliable parametrization for the Corona outbreak in Austria is not possible. We therefore have not included this factor in the current model.

Superspreading and overdispersion

Accumulating evidence of the past months points to the importance of “superspreading” for COVID-19. This means that a large proportion of all infections may go back to a small fraction of infectious individuals. For example, one recent study (Endo et al., 2020) estimates that 10% of all infectious individuals may be the cause of 80% of all infections. This phenomenon is also called “overdispersion” and is indicated by small values of the so-called dispersion parameter k.

Standard SEIR models (like our current one) effectively assume a dispersion parameter k = 1. This implies some dispersion in the number of secondary infections, but no strong overdispersion (e.g., as indicated by k=0.1, Endo et al., 2020). We can ask how this affects the model outcome.

The primary effect of superspreading is that it increases the stochasticity of the outbreak. With high variance in transmission, there is a higher chance that the epidemic simply dies out locally, but there is also a higher risk of sudden large outbreaks. This alone does not bias the model estimates, but it increases the uncertainty of these estimates, in particular if case numbers are low.

Superspreading can also have further consequences, but this is where things get complicated. The effects depend more subtly on the cause for the observed overdispersion. Superspreading can result from

Most likely, several factors contribute to overdispersion of SARS-Cov-2 transmission. Cluster analyses (e.g. by AGES) suggests a large role of the transmission context, but the relative importance of various contexts is not yet well understood.

There are two main ways how superspreading can affect the results beyond creating stochasticity. First, if it also leads to a positive correlation of susceptibility and transmission, i.e., superspreaders are also individuals, who are more likely infected themselves. This is plausible for some transmission contexts (i.e., regular bar or church goers), but less so for others (funerals, conferences) and unknown for factors such as individual disposition or genetics. The second scenario where overdispersion leads to deviating results are factors connected to the virus.

Superspreading with correlated effect on susceptibility and transmission results in a stronger reduction of Reff due to emerging immunity than assumed in the model. This is because transmission is more strongly reduced once most bar goers and choir members already had it, or once the virus has "run out of meat-packing plants". This effect is similar to spatial heterogeneity, but augmented by temporal aspects. Interestingly, sporadic superspreading can also work to reduce effects of spatial heterogeneity. For example, imagine a rural region with low transmission that has largely been spared from the virus until everybody got infected at the annual beer fest.

Reduction of Reff (and therefore, effectively, a lower herd immunity level) is thus not a necessary consequence of superspreading, but depends on context and additional factors that are hard to assess. Indeed, also the opposite effect of an increase of Reff is possible. This occurs if overdispersion is caused by differences in transmission rates among viral strains, because a high-transmission strain naturally outcompetes the other strains.