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

Abstract

Early 2020, catastrophic consequences of COVID-19 was predicted in the do-nothing scenario, based on mathematical models for epidemiology. As data began to emerge, several scientists noted that growth did not seem exponential, as the models predicted, leading to speculations of pre-existing immunity or immunological dark matter to explain this pattern. On the other hand, reports of choir-rehearsals infecting most members seemed to refute this, and the topic remained inconclusive. We provide a mathematical theory in which both observations are true; on a population level, pre-immunity exists, on an individual level, it doesn't. This theory demonstrates that established formulas relating e.g. R0 and the herdimmunity threshold are wrong. We derive new mathematical formulas, which applies to any virus whose transmission dynamics is associated with large individual variability in susceptibility to the infection. Contrary to great variability in infectivity, which we show has no bearing on the mathematical modeling, variability in susceptibility actually manifests itself as pre-immunity on a macroscopic scale, thus making pre-immunity a necessity for accurate mathematical modeling.

COVID-19 Modeling Outcome versus Reality in Sweden

“…In a separate study [6], we demonstrate that, from a modeling perspective, a number of population heterogeneities, such as variable susceptibility, variable activity patterns, as well as self-isolatory measures, seem to have an effect identical to assuming that a certain proportion of the population had a sterilizing pre-immunity. This "pre-immunity", if it indeed existed, is thus likely a manifestation of a much more complex phenomenon than the name suggests, and to underline this we will refer to it as "artificial pre-immunity".…”

“…Pre-immunity, as well as immunity developed after an infection, is not a binary variable that either gives a 100% protection (so-called "sterilizing immunity") or none at all. In the separate study [6], we show that there is practically no difference mathematically between an advanced model taking variable susceptibility into account, and a more naive model which simply stipulates that a certain proportion of the population has sterilizing immunity. Hence, the idea of a binary pre-immunity works well on a macro-level, as long as one is aware that this is a simplification that does not apply on a micro-level.…”

“…We now update the model so that a(t), the human contact rate, is constant throughout the modeling time-span, but, instead, we include a certain level of artificial pre-immunity, which we choose manually in order to obtain a good fit with measured data. As discussed earlier, if variable susceptibility on an individual level exists, it will manifest itself as (artificial) pre-immunity on a macroscopic level [6]. Thus, if a certain level of pre-immunity, say 50%, gives a good fit between model and reality, it does not mean that 50% are immune to SARS-CoV-2.…”

“…To complicate things further, a fraction of the population went into various degrees of self-isolation, and clearly it is impossible to distinguish such individuals from individuals with high-levels of natural defense against contracting SARS-CoV-2. However, it is shown numerically in [6] that such variations are also manifested as an artificial level of sterilizing immunity. In practice, this means, for example, that the black curve in Figure 3, produced by a model with 72 compartments (Age-Act-SEIR), could be obtained easily from the simpler 24-compartment Age-SEIR model, upon including artificial sterilizing immunity.…”

“…This is a mathematical modeling simplification that comprises a number of complex factors, and it is impossible to know how much of this immunity comes from actual variations in susceptibility compared with variations in social interaction patterns. To underscore this, we have coined the term "Artificial Sterilizing Immunity" [6], but since this paper concerns the period when SARS-CoV-2 was relatively new, artificial sterilizing pre-immunity, or artificial pre-immunity for short, seems more accurate.…”