# A mannequin of vaccine-derived herd immunity to COVID-19 in heterogeneous populations

A recently published theoretical paper currently available on the medRxiv * preprint server reveals a novel vaccination model that shows how a herd immunity threshold (HIT), which takes heterogeneity into account, versus a HIT value assuming a homogeneous population is much lower – with many effects on further vaccination planning.

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Since the beginning of the ongoing and devastating coronavirus disease (COVID-19) pandemic in 2019, one of the major issues raised by the global research and public health community has been the proportion of the susceptible population that needs immunization to get rid of the Successfully stopping the pandemic. This intensified after several vaccines became available for different countries.

One of the commonly used metrics is the HIT vaccination rate, which is dependent on the base reproduction number (i.e. the highly cited R0), the latter representing an average number of secondary infections caused by an infected person when confronted with a fully susceptible population while she / he you is contagious.

For example, if the R0 is 2.5, we need to immunize more than 60% of people to get HIT and remove the disease from the population. However, this derivation depends on several assumptions about population characteristics and vaccine properties.

In particular, it predisposes that vaccine protection will be perfect and last a lifetime without diminishing the declining effectiveness observed with other vaccines. Can we then apply such clean calculations to somewhat unpredictable real-world scenarios under ideal circumstances?

Without careful mathematical analysis, some have suggested that (assuming a homogeneous population) the classical vaccine-induced HIT can be significantly higher than the minimum HIT obtained considering population heterogeneity.

Dr. Elamin H. Elbasha from Merck & Co., Inc. in Kenilworth (USA) and Dr. Abba B. Gumel of Arizona State University in Arizona (USA) and the University of Pretoria (South Africa) decided to investigate the above claim by developing and rigorously testing a vaccination model that includes various forms of heterogeneity, and is then compared with a model of a homogeneous population.

Vaccine coverage above which herd immunity is achieved through group and base reproduction numbers.

## Modeling the optimal vaccination approaches

By using a two-group vaccination model in heterogeneous populations, these researchers have theoretically recognized the conditions under which heterogeneity can lead to many different HIT scores – depending on the relative contact rates for groups, their size, and the nature of the mixing between them as well as relative effectiveness of the vaccine.

For example, in cases of biased random mixing and in cases where vaccination of a particular group results in disproportionate prevention of higher transmission rates, an optimal approach would be to vaccinate that very group before beginning vaccination of other groups.

In addition, assuming biased assortative mixing, the researchers have also found situations where it is optimal to vaccinate more than one group for optimal results. All of this is evidenced by many reproducible calculations.

## Heterogeneity and lower HIT score

In short, the theoretical results for a two-group model developed by these two researchers suggest that deriving the vaccine-induced herd immunity threshold using the heterogeneous population model is much more cumbersome than for the homogeneous population model.

In addition, they have shown that population heterogeneities tend to result in much lower HIT scores compared to the homogeneous population case. This has been found to be true of both proportional and other types of intermingling between heterogeneous populations.

Using realistic numerical examples and the assumption of biased assortative mixing with a vaccine efficacy of 95% and a base reproduction number of 2.5, they have shown that the HIT value considering heterogeneity compared to a HIT value assuming a homogeneous population can be significantly lower (40%) (63%).

## A blueprint for future studies

Of course, when discussing their results, we must take into account that variables such as the base reproduction number, and in particular the effectiveness of the vaccine, cannot be taken strictly as fixed. Therefore, the results must be interpreted with caution.

Although a two-group model was used for this analysis, the results can be extended to multi-group models. Likewise, rigorous analysis of models with many heterogeneous groups (which are more realistic) can be performed using the methods included in this article.

In any case, modeling studies similar to this prove to be indispensable, especially since vaccinations are not advancing at the desired pace worldwide. Achieving vaccine-induced herd immunity is one of the holy grails in our fight against COVID-19.

## * Important NOTE

medRxiv publishes preliminary scientific reports that are not peer-reviewed and therefore should not be considered conclusive, guide clinical practice / health-related behavior, or be treated as established information.

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