Our understanding of the dynamics of Covid is mostly based on models of the infection process. The basic SIR model is the base for most epidemic model.  It splits the population into 3 buckets; Susceptible (S), Symptomatic Infected (Is) and Recovered (R).


I have made a number of additions to the model. 


 1) The infections per person (Ipp), often called "Reproduction Rate (R(t)), changes over time as social distancing, and masking effects the transmission of the virus. There is a exponential time constant for the effect of behavior changes on Ipp .

 2)  There is a significant population who are Asymptomatic Infected (Ia) and develop some immunity without knowing it. The model shows 75% of the infected are in fact asymptomatic.  With around 10% of the population having been symptomatic infected, the total infected is around 40%.

 3) The vaccinated population decreases the Susceptible population but most people who been infected will still chose to get vaccinated. Therefore the additional protected  population from vaccination is reduced by the fraction of already protected. With 60% vaccinated and 40% infected, the total protected is 76% which means the US is very close to  herd immunity assuming that asymptomatic infection gives significant protection.  

4) The treatments for Covid have also improved enormously, the doctors have learnt how keep really sick  people alive and eventually cure them. The deaths per day are calculated using a Case Fatality Rate and a time delay from infection to death. In the first wave in Texas, the CFR was 3% after 5 days, and has improved to 1.7% after 20 days. 

The dynamics of Covid infections can be seen in the log graph of daily new cases for different countries, early in the pandemic. In the first days, there was a very rapid rise  of 10x in only 10 days. Almost immediately countries instituted lockdowns which stalled the infections and then saw them fall at different rates.  The straight line rise and fall in the log graph is characteristic of the exponential growth of an infection. 



The idea was to develop a version of the standard SIR
model of infections that accounts for  changes in social isolation behavior  and health care.

In the classic SIR model, the change in number of  infections over time dI/dt depends on the rate of new cases minus the rate of recovery. The new cases per day is given by the infection rate constant beta, multiplied by the number of Infected people I(t) and the fraction of people available to be infected S (Susceptibles). The rate of recovery dR/dt is given by the recovery time constant gamma multiplied by the concentration of Infections I(t).

The ratio of infection to recovery rate constant (beta/gamma) has useful meaning as the initial Infections per person or Ipp(0), often called R0. It is the gain in the chain reaction of infections. For Covid, the Ipp of 5 means that every infected person infects 5 more for the roughly 10 days they are infectious.  In thirty days there will be 5^3 =125 x increase in  infections !! That is why Covid is so dangerous.  


In reality, peoples social behavior changes to avoid infection so the rate of infection  changes over time beta(t) and so does Infections per person Ipp(t). 

It has become clear that there is a large number of "hidden" infected individuals who are  asymptomatic and are not caught by current US testing that is mostly limited to asymptomatic. At current testing levels, the number of asymptomatic people caught by random chance is very low. Estimates vary, but using a ratio of protected = 4 x symptomatic (Asymp : Symp = 3:1) results in a best fit with the case rate rolling off in early Jan. When the asmp ratio is 3:1 and vaccinations are included, the roll off in cases occurs without any change in distancing (constant Ipp). 

These differential equations were numerically integrated in an Excel spreadsheet, to model an infection. 
The model can be used to extract metrics of the
process. At the beginning of the infection, the susceptibles
can be assumed close to 1, so the infection grows exponentially and can visualized as a straight line in a log graph, with a slope of Ipp.


The daily infections d log(I)/dt, new cases d log(C)/dt, deaths d log(D)/dt, all have the form = f(I), so they will track each other. Hence the universal use of log plots to analysis pandemics.

Now we have a way to quantify the stages of the infection illustrated in the graph of daily new cases for different countries. In very the early stages, the daily cases have a steep log slope consistent with a initial Ipp of over 5. In response, various levels of lockdown were instituted, and the infection stalled and then started to drop with countries showing different declining log slopes with a Ipp of 0.7-0.9. The effect of the lockdown was to change behavior, which in turn changed in the infections per person. It appears that the evolution of Ipp over time can be generalized as a series of constant levels.

Obviously human behavior does not change as step function. Manually adjusting Ipp to fit gave a curve that looked like an exponential.  Using a exponential function, gave a reasonable fit to US, NY, TX  and WA data with a 1/e time constant of 7  days, and around 20 days for the change in infection per person to stabilize.  This leads to a generalized expression for Ipp(t) with a series of constant levels linked by exponential approach, that was applied to TX data. 

The idea that the transition between 2 different levels of social
behavior is an exponential seems inherently   reasonable, and  has been used to describe transitions in isolation for  other infections (Ref 1). A possible physical interpretation has been  proposed, borrowed from finance (Ref 2), which suggested that the transition is an "Ornstein-Uhlenbeck" process.

In mathematics, the Ornstein–Uhlenbeck process is a
stochastic process. Its original application in physics was as
a model for the velocity of a massive Brownian particle
under the influence of friction (Ref 3). More generally it
describes the long term transition between two states by  
many short term random changes, and has been used in
many application including the change in bond prices (Ref
4). The maths of this have a solution of exponential change
that is identical to the empirical equation I used. The idea
that a change in isolation, producing a change in infection
rate per person, would be accomplished by multiple small
random changes in behavior seems plausible.


Finally, published data on mobility based on cell phone data also shows an exponential response to the lockdowns in March.


Vaccinations started in late December. Vaccinations reduce the Susceptible population, and hence the new infections. However, there is a random chance that people getting vaccinated were already protected by being infected, so the contribution must be reduced by the fraction of already protected. Therefore the new vaccinations per day multiplied by (1-cume cases*asymp ratio) were added to the Susceptible every day.  The vaccinated were lagged by 10 days to account for jab to work. 


It looks like the average distancing behavior has remained unchanged from early November to April which seems consistent with observations on the ground. 


Finally in April and June, distancing and mask use effectively end, causing increases in Ipp bumps in cases through June when Ipp has returned to the levels at the beginning of the pandemic. 

The wave in August finally rolled off in Sept. Using a Asym : Sym ratio of 3 resulted in the model for the waves in April and August to roll off  with constant distancing (Ipp). This suggests that the model is on the right track. In Oct, the remaining unprotected from either vaccine or infection is down to 7%,

The slope of the roll off is 5x per month, similar to the roll off that is happening in Florida and Israel. Israel has a one of highest vaccination rates. The UK had a sharp peak in July, followed by a slow rise of 30% per month. In the UK over half the hospitalizations  were breakthroughs. This suggests that France, UK,  Texas and Israel may be at herd immunity from the combination of vaccination and earlier infection.

Other indicators of herd immunity are the fact that in Sept, 98% of blood doners in UK had Covid antibodies, and falling vaccine efficiency (modelled on the right) as the unvaccinated are nearly all ex-infected. 

After herd immunity, the vaccine will now spread through the entire protected population at a much lower Ipp. The Delta Ipp(0) is around 10. If each infection has a 10% chance of success in a vaccinated individual, the Ipp post herd should be 1+9*0.1/10 = 1.1. An infection with an Ipp =1.1 should have a max cases/d/M of around 100. This should be noticeable but manageable. The case rate will increase 10x in 6 months, roughly matching the UK trend. 



Modelling the time line for fatality rate. 


For the unlucky few who die, there is a time lag L between infection and  death, and there is evidence (shown below) that this time has lengthened as the doctors have learnt how to manage the disease. Therefore, the number of deaths per day is given by the death rate times the number of infected people at the time of infection  I(t-L). Similarly, the Case Fatality Rate can be followed over time as deaths per day / cases per day lagged by time to death L.  

The number of  deaths per day are obviously to the number of cases per day. The log plot of daily  deaths and cases shows the log distance between deaths and cases varying, which  means that the ratio of deaths to cases (CFR) is changing. Also the time lag between the peaks in March are smaller then than the lag in August, which means that the time lag from infection to death is also changing. 

TX had a relatively mild infection in March, so it avoided any effects of patient load on outcomes. A series of CFR's at different lags for TX are shown. Assuming that CFR is likely to be relatively constant, it appears that prior to June, the lag was 7 days with a CFR of 3%, and post June the  lag was 20 days with a CFR of 2%.  The changing CFR and lag presumably a result of improvements in medical care. 

Based on these observations the death rate was modelled as a series of constant steps. 


In January, vaccinations have started to take hold with about 8% of the population. This is focused on the >65 group who constitute 15% of the population and are where almost all of the deaths occur. About 40% of the at risk community have been vaccinated. There is a 20 day lag from case to death and 10 days to generate immunity. In 1 months time the death rate should be halved. The graph for TX  shows the projected death account after adjusting for vaccination. 


The modelling analysis led to a number of key characteristics of Covid infection. 

The Infections per person (Ipp) vary as behavior changes.


Under "normal life" behavior, Ipp is around 5. 

Under lockdown conditions Ipp follows an exponential approach to a new constant value that takes about 20 days. 

The death rate and the time from infection to death both change, probably due to improvements in medical care. 


Thanks to Andrew Zachary for pointing out the link to financial

Ref 1 https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5348083/

Ref 2 https://www.linkedin.com/pulse/modeling-term-structure-

Ref 3 https://en.wikipedia.org/wiki/Ornstein%E2%80%

Ref 4 "Modelling to Inform Infectious Disease Control" by Neils
Becker (2013)

Ref 5 "Repeat contact and the spread of disease" by Peter Cotton

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