### Are We Closer to Herd Immunity than Most Experts Say?

A recently published article suggests that we are much closer to herd immunity than the experts suggest. In a nutshell, the authors suggest that variation in connectivity (mobility / contact rates) and susceptibility (variation in viral resistance) will significantly lower the herd immunity threshold relative to standard theory. Their hypothesis is this: the greater the variation in the population, the lower the threshold for herd immunity. It is an interesting concept, and if true, it means we may be closer to getting back to normal than experts say. The paper presented by Gabriela M. Gomes and colleagues is here:

Individual Variation in Susceptibility or Exposure to SARS-CoV-2 Lowers the Herd Immunity Threshold

Below we present discussion and analysis of 4 cases. We conclude that this affect will drive the herd immunity threshold down, and we demonstrate that even conservative assumptions will push it well below 50%. If actual distributions are wider, as evidenced by demographic data, herd immunity thresholds will be even lower.

Simulations for a Population of 1 Million in an Unmitigated High Density Outbreak

**Standard Herd Immunity Calculations for COVID-19**

Herd immunity calculations predict that the percentage immune population (P) needed is in the range of 60% to 72% for COVID-19, based on a reproduction factor, R0, of 2.5 to 3.5. R0 is number of people an infected individual infects over infectious period at the outbreak start.

The equation for proportion the population needed to establish herd immunity is simple:

- P = 1 – 1/R0

The calculation of R0 is defined as:

- R0=B*T
- B = effective infectious contact rate for transmission: infected people per day.
- T = period of infectiousness: days.

**Reproduction Factor Varies by Region**

We know that B and T can vary depending on demographics, regions, healthcare systems, behaviors. Higher density populations will have higher infectious contact rates (B). Poor access to healthcare would drive longer period of practical infectiousness (T) as sick individuals would remain in the general population. R0 varies by regional COVID-19 outbreaks. This is shown in the detailed analysis and assessments of the outbreak by economists Jesús Fernández-Villaverde and Charles I. Jones:

Estimating and Simulating a SIRD Model of COVID-19 for Many Countries, States, and Cities

**Populations are Diverse**

If we see variation in factors affecting B and T across geography are these factors homogeneous (constant) or heterogeneous (varied) over the population of a region? Common sense would suggest that within a given population there will be variation in connectivity (some individuals stay at home while others are out and about, some are hands on and others are not). Common sense would also suggest that there would be a variation in susceptibility (for the same virus exposure, some individuals will get sick and others will be resistant).

**How a Diverse Population Affects an Outbreak Dynamically**

If the characteristics of a population are all the same, the infection plays out according to standard equations. This article below, based on the standard model for heard immunity, has a visualization tool for an outbreak using the standard homogeneous population model:

Without A Vaccine, Herd Immunity Won’t Save Us

If a population is diverse in connectivity and susceptibility, as common sense would suggest, the dynamics of the outbreak change over time. Those who are highly connected and more susceptible will be infected early on. As those individuals are removed from the susceptible population, B (infectious contact rate per day) will change. As the remaining population is less mobile and more susceptible, B will drop. There will be less exposure and the rate of infectious transmissions per day will drop.

**As an Infection Progresses the Remaining Population is Less Likely to Be Infected**

As the effective R0 drops, the required herd immunity threshold percentage also drops. For example, if R0 drops to 1.5 from 3, the required percentage of the population required for herd immunity goes from 66% to 33%. We call the time variable reproduction factor R (t) or R effective. R (t) is dynamic and reduces over time as those who are most susceptible and those who and most connected are more likely to be infected. The remaining population becomes dominated by those who are resistant and less connected.

*The greater the variation in the population with respect to connectivity and susceptibility factors, the lower the ultimate threshold for herd immunity.*

**What We See From High Density Outbreaks**

In areas such NYC, Italy, Spain, and Cruise Ships where the virus spread rapidly and unabated, we are finding about penetration rates in the range of 20% to 25%. In these high density areas, the exposure to the virus was very high, and yet we see a limit to the penetration. Why is that? Social distancing and other measures likely occurred after the peak of the infection spread in dense areas.

In areas where the disease penetration was high, most people who had higher contact rates and lower susceptibility likely got sick. Those who were less likely to contact people and had a lower exposure rate would have reduced infection rates. Those who were less susceptible (virus resistant) were exposed, but likely did not contract the disease. The remaining population is not immune, but due to their characteristics, they are resistant to infection and less likely to get sick.

**What the Data Tells Us: Young People**

A modeling approach that accounts for the susceptibility variation for age will give insight into how this affects herd immunity. Data for infectious distributions over age show that for the 20 and under population, the infection rates are very low compared to the rest of the population.

The CDC shows the spread of confirmed infections versus age: https://www.cdc.gov/coronavirus/2019-ncov/cases-updates/cases-in-us.html. Children account for less than 2.5% of the positive tests, but represent approximately 25% of the overall population. This would indicate that they are less susceptible to contracting the virus by an order of magnitude. Note that a more scientific assessment of actual infection ratios would need to remove sample set age bias factors to establish a uniform representation.

A recent paper published in the New England Journal of Medicine presents a detailed analysis of COVID-19 test data from Iceland. The analysis of age and susceptibility shows that the rate of infection in young people is very low. Spread of SARS-CoV-2 in the Icelandic Population

Standard models do not include this age bias factor, which will drive down the herd immunity threshold.

**Reasonable Assumptions about Connectivity**

A modeling methodology that accounts for variation in mobility and contact due to travel habits, personal habits and other factors would give insight into how this factor affects the dynamic spread of a virus and the selection (infection) of agents who had had factors more likely to result in infection. These factors can be simple to account for with a probability distribution of contacts per day or a contact multiplier for health habits (handshaking, hand washing, face touching and other factors). Standard models do not include these factors. These factors can be derived from common sense or from more sophisticated studies.

**Reasonable Assumptions about Susceptibility**

Variations in susceptibility by for a given population are missing from the standard models. These factors can be estimated by generating a range of susceptibility based on general health status. For example, we can assume that some individuals are twice as susceptible as the average and others are half. This is a coarse assumption, but first order, is reasonable. These and other factors can be modeled in a dynamic simulation to create a reasonable susceptibility factor. More data is needed to accurately estimate these factors, but the first order effect based on common sense can give a reasonable starting point.

**Modeling a Typical COVID-19 Outbreak for Herd Immunity Factors**

A baseline analysis using our COVID Decision Model (CDM) has been constructed to study the effects of connectivity and selectivity on the herd immunity threshold. The threshold is defined in the dynamic simulation as the % population who has been infected when the reproduction factor over time, R (t), crosses unity during the infection decay.

We examine three scenarios below to gauge these effects on the herd immunity threshold. In all cases we are using a standard set of disease parameter timers with a simulation population of 1 Million.

The integrated probability distribution below is used in all three simulation cases. Uniform distributions across all demographics are used.

The outcome table based on age that includes a distribution of co-morbidity factors and an assumption that 90% of deaths involve a preexisting co-morbidity factor. We also assume that 50% of the infections are asymptomatic. The comorbidity factors are assigned by age group randomly according to the distribution below. When infected, the victim’s outcome is determined by the appropriate probably distribution outcome table.

All simulations are calibrated to an initial R0 of 3.0.

- Case 1 is homogeneous: there is no variation in the population.
- Case 2 has the connectivity adjustment for age selectivity to better align with actual infection distributions.
- Case 3 has a nominal heterogeneous population for both connectivity and susceptibility.
- Case 4 has a wider heterogeneous population for both connectivity and susceptibility.

We compare the herd immunity threshold in each case. The range of variation is meant to demonstrate the effect of a heterogeneous population. The analysis clearly demonstrates that there is a significant shift in herd immunity with allowance for a distribution in connectivity and susceptibility. The assumptions relative to distributions here can further be refined by data.

**Simulation Case #1: Homogeneous Population**

In this simulation all factors are normalized to represent a homogeneous population. In this case R0 is in the range or 3.5 and percentage of people who have been infected is just under 63% when R (t) crosses unity around the end of April. As the infection plays out, ultimately approaching the 80% of the population is infected. The output summary table for the simulation shows a uniform infection rate as a function of age (this does not match real data trends, this is addressed in Case #2). This is consistent with typical models for herd immunity predictions. All parameters affecting connectivity and susceptibility are normalized to create a homogeneous population.

**Simulation Case #2: Homogeneous Population with Age Connectivity Adjustment**

In this simulation all factors are normalized to represent a homogeneous population with the exception of an age connectivity adjustment to better match the actual infection rates we see in the 20 and under demographic. In this case R0 tracks to 3.0 and the percentage of people who have been infected approximately 56% when R (t) crosses unity in mid May. As the infection plays out, ultimately approaching the 65% of the population is infected. The output summary table for the simulation shows a distributed infection rate as a function of age. All parameters affecting connectivity and susceptibility are normalized to create a homogeneous population with the exception of age. In this case the connectivity is only degraded in a subset of the population so the net effect is to slow the infection. The overall deaths are reduced by over 10% as we have established herd immunity at a lower threshold.

**Simulation Case #3: Heterogeneous Population with Nominal Distributed Connectivity and Susceptibility**

In this simulation all factors are spread to represent a nominal homogeneous population. In this case R0 tracks to 3.0 and percentage of people who have been infected is approximately 43% when R (t) crosses unity in mid April. As the infection plays out, ultimately approaching the 54% of the population is infected. The output summary table for the simulation shows a distributed infection rate as a function of age. This is inconsistent with typical models for herd immunity predictions. All parameters affecting connectivity and susceptibility are spread to create a heterogeneous population. In this case deaths are 20% less than the baseline.

**Simulation Case #4: Heterogeneous Population with Wider Distributed Connectivity and Susceptibility**

In this simulation all factors are spread to represent a wider homogeneous population. In this case R0 tracks to 3.0 and percentage of people who have been infected is approximately 34% when R (t) crosses unity in mid April. As the infection plays out, ultimately approaching the 44% of the population is infected. The output summary table for the simulation shows a distributed infection rate as a function of age. This is inconsistent with typical models for herd immunity predictions. All parameters affecting connectivity and susceptibility are spread to create a heterogeneous population. In this case deaths are 32% less than the baseline.