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Michael PC Watts, Retired 

Austin TX 




Infections in the  vaccinated  and unvaccinated are now similar in the UK and Israel, two countries with the highest level of vaccination, and the US must be close behind.  Up to the fall of 2021,  infections have been mostly limited to the unvaccinated, and have been controlled by social distancing. Now, society has largely reopened and  enough of the population is protected by  vaccinations and prior infection (symptomatic or asymptomatic) that "breakthrough" infections in the majority vaccinated have become as common as in the unvaccinated. 

Vaccinations continue to provide significant protection against hospitalization and death, with the elderly remaining at much higher risk of bad outcomes. Vaccinated the remaining population and taking precautions to avoid catching a "breakthrough" infection remain essential to protect the vulnerable. 

Variants are going to spread quickly through our re-opened society. Fortunately it looks like the latest Omicron variant seems  infectious even to vaccinated people, but fortunately fairly benign. 

Evidence that the entire population has some degree of protection 

The UK has one of the most complete data sets  on infections broken down by vaccination status.  The vaccinated now represent about half of those hospitalized, the cases per day per million vaccinated are similar to the cases per day per million unvaccinated, and 98% of blood donated has Covid antibodies. Effectively, nearly everyone in the UK is protected to at least some degree.

Download full paper here 


                        Figure 1a) Daily cases rates                                                                                                                 Figure   1b) Case outcomes 

Data from the UK for  November 2021 showing (a) that more daily cases are occurring in the vaccinated, except for very old and very young. The case outcomes (b) showing the fraction of cases that result in serious illness "Case Seriousness Rate"  (CSR), and the fraction of cases that result in death "Case Fatality Rate" (CFR). The outcomes are much more serious in the elderly. Vaccination provides 7x reduction in both serious illness and death. 

As shown in Figure 1a, the case rates for vaccinated are actually higher than unvaccinated, except for the under 18's who are just starting vaccinate and the very old. The outcomes once you are infected are shown in Figure 1b,  as the fraction of cases that result in serious illness  "Case Serious Rate (CSR)", and death "Case Fatality Rate (CFR)". The risk of serious illness and death is exponentially higher with increasing age.  The solid line for serious illness and dashed line for death, in both vaccinated and unvaccinated, converge with age confirming the dangers of Covid in the elderly. 


Vaccination is still very effective, because there is  a 7x reduction in the risk of serious illness as can be seen by comparing the 2 solid lines,  and 7x reduction in risk of death shown by  the 2 dashed lines.   

The data from Israel is less detailed, but in Nov 2021 daily cases per million for vaccinated are 75% of the unvaccinated, and Case Seriousness Ratio is 4.5x.


This seems to confirm that Covid is spreading through the entire population, both as breakthroughs in the vaccinated and nominally unvaccinated, as society reopens.  Case rates / million in the  the nominally unvaccinated, relative to the vaccinated are lower because the unvaccinated are in the minority AND  due to prior infection. Vaccination provides 7x greater protection from serious illness and death compared to the nominally unvaccinated.  The elderly still need to be careful because of the increased risk of bad outcomes.

Cycles of the pandemic 

The core question at every stage of the pandemic seems to be "where are we" and "what is going to happen next".  Our model suggests that the  of waves of the pandemic can be best understood through the number of infections that are passed on by each infected individual known as the  "Reproduction Number" (Rn). At the start of the pandemic, the  vast majority were totally unprotected against infection just going about their usual activities, being exposed to a few infected people. The Rn for the virus with pre-pandemic social activity was around 5, which caused the daily cases to rise 10x in only 7 days. This triggered a world wide shut down to try and contain the pandemic. 


A wave in cases is created when there is some combination of  sufficient; community protection, barriers to infection such as masking, or reduction in  social contacts, so that  each infected person transmits the virus to less than 1 other person and  daily cases fall.  Otherwise, there is transmission to more than 1 person (Reproduction Number greater than 1 and cases rise.  The changing balance between protection, barriers, and social contacts  creates the waves in infection. 


Eventually, protection from  vaccinations and prior infection (symptomatic or asymptomatic) means that  the unprotected are a very small fraction of the population which causes the daily cases to drop even as society reopens. Because vaccinations are not 100% effective, there are people who are vaccinated but not protected. At some point, "breakthrough" infections in the vaccinated become larger than cases in the unvaccinated. The domination of  "breakthrough" infections  is an inevitable outcome of the cycle of the pandemic, when some protection has spread to the vast majority of the population.  Unfortunately, because breakthroughs occur across the entire population, daily cases can be significant even with a low Reproduction Number in the protected. 

Our model suggests that in late Oct 21,  the US may  have reached the point where some protection has spread across virtually the entire population, and a new wave has started across the entire population. The cases data for Nov 21, broken down by vaccination status will become available in mid Dec 21. It looks like the UK reached the same point in Sept 21, and they have collected data based on vaccination status for a longer time. The case rate in the vaccinated compared to unvaccinated changed in July , and dominate by Sept 21. In the 4 weeks  ending Dec 8 21, 

The good news is that vaccines still offer excellent (7x) protection against serious illness and death compared to the unvaccinated, equivalent to 85% vaccine efficiency. 

If nothing changes in virus virulence or social distancing, it seems likely that the current increase in cases will continue until the spring, at which point the cases should naturally fall as the remaining vaccinated but unprotected are infected.  However, it seems likely that society will continue to move slowly to pre-pandemic levels of interaction which will increase Rn and daily cases. Variants in the virus are the biggest wild card. The latest data on Omicron is that both vaccination and pre-infection protection from earlier variants is weaker. The good news is that to date South Africa has seen in no noticeable increase in hospitalization and  deaths even though cases have gone up over 10x, suggesting that Omicron bypasses the vaccine but infection is relatively benign.

The situation today is much more complicated, there are multiple variants, life has not returned completely to "normal", around 70% of the population  have been vaccinated and are probably 80-90% protected. Most of the unvaccinated have probably been infected either symptomatically or asymptomatically, but we have little idea how much protection they have acquired. A further complication is that the level of protection also depends on the variant of the virus.  

Increasing  infections are caused by a combination of the number of contacts with unprotected individuals, and the probability of getting infected. 


The  number contacts compared to pre-pandemic is determined by social distancing. Commute traffic  is back,  restaurants seem busy, everyone is complaining that they cannot find enough workers, sports events are well attended, indoor concerts are starting. The Google Mobility metrics  for Recreation & Retail, Grocery and Pharmacy, Parks are all close to pre-pandemic, Transit and Workplace are still down.  We know that mobility is a pretty good proxy for the effect of  social distancing  on the number of new infections transmitted  by each infected  individual, usually called the "Reproduction Number".


The probability of getting infected depends on mask wearing, the level of resistance that each person has acquired by vaccination or previous infection,  and the virulence of the virus. There is much less masking than the spring of 2020, although a few organizations are still requiring masks. 

The different variants have complicated the question of virulence. The Delta variant seems to be significantly more infectious than the original Alpha. The Omicron variant has just appeared, at the time of writing the best analysis is that it is no more virulent, but vaccines and prior infection my not provide as much protection. 

 The cycle of the pandemic 

As a highly infectious disease, the waves of Covid have been managed by a combination of cycles of social distancing and vaccination. As the pandemic progresses, a larger fraction of the population is effectively protected by prior infection or vaccination. The standard model for infectious diseases is the “SIR” model that identifies 3 populations in a pandemic: the Susceptible who have not been infected; the Infected; and the Recovered who are now effectively immune.  We have modeled the dynamics of the pandemic with an extended SIR model that includes the effects of vaccination and asymptomatic infections. The SIR model uses the number of people infected by each infected person or “Reproduction Number (Rn)” to drive the calculation of cases per day. In this model we allowed Rn to vary over time to reflect changes in social isolation and virus variants. The simplest models of the daily case data in TX and NY were obtained when the asymptomatic infection rate was approximately 2.3x the symptomatic infection rate.  

The observed changes in Reproduction Number in TX are consistent with Google Mobility as a proxy for distancing and the spread of the Delta variant. We fit the daily death data with a time varying Case Fatality Rate and lag between cases and deaths that appears to be consistent heath care improvements.

A more detailed write up is available at download button at the top of the page. 

TX case WriteUP.jpg

Figure 1 Case and Deaths per day per Million 7 day rolling average for TX. The Asymptomatic infections = 2.3x the Symptomatic infections. Population vaccinated at least once, lagged by 10 days are used to correct Susceptible population. The model uses a time varying Reproduction Number (Rn(t)). By Nov 2021, the fraction of previously infected individuals combined with the fraction of vaccinated individuals is so large that the progress of the infection is now dominated by breakthrough infections with a much smaller Rn(t) = 1.2. The deaths in TX were modelled by a time and lag varying Case Fatality Rate.   

As shown in Figure 1 for TX, the Reproduction Number  starts high while there is pre-pandemic social behavior, and dropped sharply due to social isolation. In fall of 2020, there was a cycle of re-opening followed by re-isolation. By 2021, protection from infection and vaccination took over, and Rn increased again as social isolation relaxed and infectious variants took over. The model illustrates that when social distancing controls the Reproduction Number, there will be a point (Christmas 2020) at which the level of protection is high enough so  the infection will naturally decay.  Then when either society starts to reopen or a new, more infectious variant arrives (Sept 2021), the infection rate will increase until a new natural decay point is achieved consistent with the new Reproduction Number and level of protection. Eventually cases in unvaccinated individuals drop and breakthrough cases constitute the majority of cases (Nov 2021). Breakthrough infections probably occur throughout the both the vaccinated and the unvaccinated ex-infected populations. We model the breakthrough infection rate as if it were a new infection distributed across the entire protected population with an appropriately low Reproduction Number.

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