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Michael PC Watts, Retired 

Austin TX 



We have developed an extended SIR model  suggesting that the primary Covid 19 pandemic may have run its course with almost all  the population protected to some degree; by  vaccination, and by either symptomatic or asymptomatic  infection of the unvaccinated.  Because the protection is not  100%,  It seems inevitable that there will be a secondary "breakthrough" infection that will spread through the entire protected population. The secondary infection will have  a much lower virulence that reflects the level of protection. Unfortunately, because the population is large, the daily cases may continue to challenge the health care system. 

Case rates per million in the unvaccinated, compared to vaccinated,  will continue fall because the unvaccinated are in the minority and make fewer contacts with other at risk unvaccinated individuals, AND they  have been exposed to  prior infection.  Vaccine efficiency against infection appears to be dropping, but in reality it is the unvaccinated case rate that is lower NOT  the vaccine case rate rising.  Vaccine efficiency based on case outcomes, such as hospitalization or death is the only reliable test of the vaccine. 

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.  

Download full paper here 

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.

Data from the UK shows trends consistent with some  protection in virtually all the unvaccinated by November of 2021. 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.



                        Figure 2a) Daily cases rates                                                                                                                 Figure   2b) 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 2a, 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 2b,  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.   


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.

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. 

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