A note on variable susceptibility, the herd-immunity threshold and modeling of infectious diseases.

The unfolding of the COVID-19 pandemic has been very difficult to predict using mathematical models for infectious diseases. While it has been demonstrated that variations in susceptibility have a damping effect on key quantities such as the incidence peak, the herd-immunity threshold and the final size of the pandemic, this complex phenomenon is almost impossible to measure or quantify, and it remains unclear how to incorporate it for modeling and prediction. In this work we show that, from a modeling perspective, variability in susceptibility on an individual level is equivalent with a fraction θ of the population having an "artificial" sterilizing immunity. We also derive novel formulas for the herd-immunity threshold and the final size of the pandemic, and show that these values are substantially lower than predicted by the classical formulas, in the presence of variable susceptibility. In the particular case of SARS-CoV-2, there is by now undoubtedly variable susceptibility due to waning immunity from both vaccines and previous infections, and our findings may be used to greatly simplify models. If such variations were also present prior to the first wave, as indicated by a number of studies, these findings can help explain why the magnitude of the initial waves of SARS-CoV-2 was relatively low, compared to what one may have expected based on standard models.

The role of super-spreaders in modeling of SARS-CoV-2.

In stochastic modeling of infectious diseases, it has been established that variations in infectivity affect the probability of a major outbreak, but not the shape of the curves during a major outbreak, which is predicted by deterministic models (Diekmann et al., 2012). However, such conclusions are derived under idealized assumptions such as the population size tending to infinity, and the individual degree of infectivity only depending on variations in the infectiousness period. In this paper we show that the same conclusions hold true in a finite population representing a medium size city, where the degree of infectivity is determined by the offspring distribution, which we try to make as realistic as possible for SARS-CoV-2. In particular, we consider distributions with fat tails, to incorporate the existence of super-spreaders. We also provide new theoretical results on convergence of stochastic models which allows to incorporate any offspring distribution with a finite variance.

COVID-19 Modeling Outcome versus Reality in Sweden.

It has been very difficult to predict the development of the COVID-19 pandemic based on mathematical models for the spread of infectious diseases, and due to major non-pharmacological interventions (NPIs), it is still unclear to what extent the models would have fit reality in a "do nothing" scenario. To shed light on this question, the case of Sweden during the time frame from autumn 2020 to spring 2021 is particularly interesting, since the NPIs were relatively minor and only marginally updated. We found that state of the art models are significantly overestimating the spread, unless we assume that social interactions significantly decrease continuously throughout the time frame, in a way that does not correlate well with Google-mobility data nor updates to the NPIs or public holidays. This leads to the question of whether modern SEIR-type mathematical models are unsuitable for modeling the spread of SARS-CoV-2 in the human population, or whether some particular feature of SARS-CoV-2 dampened the spread. We show that, by assuming a certain level of pre-immunity to SARS-CoV-2, we obtain an almost perfect data-fit, and discuss what factors could cause pre-immunity in the mathematical models. In this scenario, a form of herd-immunity under the given restrictions was reached twice (first against the Wuhan-strain and then against the alpha-strain), and the ultimate decline in cases was due to depletion of susceptibles rather than the vaccination campaign.