The model
The idea was to develop a version of the standard SIR
model that allows for differences in isolation, and a
transition from open  to isolation.  

The basic SIR model is the base for most epidemic model.  
It splits the population into 3 buckets; Susceptible, Infected
and Recovered. There are 3 differential equations, the
change in infections over time dI/dt depends on the rate of
new cases minus the rate of recovery. The rate of new
cases is given by the constant beta, multiplied by the
concentration of Infections and the concentration of
Susceptible. The rate of recovery dR/dt is given by the
constant gamma and the concentration of Infections.

The ratio of beta to 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. An Ipp of 2
means that every infected person infects 2 more, and so
on. Over 8 cycles of infection, the total new infections will
be 2^8  which is of course 256. It is this chain reaction that
creates an epidemic.

There are 2 common additions, a rate of Death which is
formulated just like Recovery with a constant that is the
probability that an infected person dies per day. The
second addition, are Infected Asymptomatic individuals who
are treated just like the Infected Symptomatic (I).  

The simple model can be used to extract metrics of the
process. At the beginning of the infection, the susceptibles
can be assumed close to 1. The daily infections dI/dt, new
cases dC/dt, deaths dD/dt, all have the form = K*I, so they
will track each other.

There is a solution for I(t) in the early stage of the infection,
where I(0)  is patient zero and usually set to 1.


The infections per person (Ipp) are given by the log slope
of  Infections. The exponential function of infections as a
function of time (I(t)) means that conveniently, the daily
infections (differential) has the same  log slope, as does
daily new cases, recoveries and deaths. Hence the
universal use of log plots to analysis pandemics.
Now we have a way to quantify the stages of the infection.
We can use the log slope to quantify the differences
between countries approaches. The days for a 10x change
can be converted to Infections per person using the table
below.

A model for the time line that includes changes in
isolation

In the standard formulation, Ipp(0) is fixed for the entire
infection. The reality of life, in the middle of a pandemic, is
that isolation is used to try and stall the infection by
reducing the Infections per person below 1.  The slope of
the log plot while the infection is falling varies from country
to country presumably depending on their isolation.

To make this work in the model, beta will be allowed to vary
with time, and hence Ipp(t) varies as well. The equations
were numerically integrated in an Excel spread sheet. The
resulting Daily New Cases and Deaths are compared to the
7 day rolling average of Daily data for the US. When Ipp(t)  
just has 2 fixed levels open  and isolation, the resulting
profile is obviously unrealistic.


The first step was to manually adjust Ipp(t) to match the US
data between 2 constant values, open  and  isolation.  
The manual Ipp(t)/Ipp(0)  profile looks like an exponential,
so that was the function I used and show to the right. The
the time of start of the transition is ti, and the exponential
time constant (torr) is set to  7.5 days based on best fit.  
The same process can be easily generalized to multiple
stages of re-open and re-isolation each with an Infections
per person and a transition between stages.  The Ipp(t)
profiles and resulting fit to TX, NY and WA data are shown
to the right. Small changes in the time of onset of isolation
relative to the start of the infection have a huge impact on
the infection.

The idea that the transition between 2 different levels of
exponential growth 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
2). 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.

Measuring social distancing

As soon as people found that isolation was working but also
causing serious economic damage there was interest in
ways of measuring isolation. The Google data group have
proposed using cell phone data as a measure of social
distancing. These have been published by IHME, an
example is shown on the right.  They measure the %
reduction in mobility (m) compared to a pre-Covid baseline,
for each state.












There is a simple model for the effect of  social distancing
on Ipp, where “f” is a fraction of the population, and “a” is
the reduction in number of social contacts (Ref 4). The idea
is that when it takes 2 people to meet to infect, if you halve
the number of contacts you would expect to 1/4  the
number of infections.  When all the population responds
and f = 1, there is  a simple connection from the change in
mobilty to Ipp.







As shown, the shape of the transition in Ipp(t)/Ipp(0) and
mobility are in broad agreement. There should be a time
lag between a change in isolation squared and the effect
being seen in the infections, which can be seen below as
about 14 days. The fit is to 7 day rolling average, so the lag
between mobility change and impact on infections is
roughly 9 days and is an estimate of the incubation period
of Covid.   






Modelling reopening

The 7 day rolling average of daily new cases for Texas
shows the latest time line of the infection. The time line of
infection per person (Ipp), drives the model calculation. The
infection starts with  growth of the cases  at fixed Ipp. Next,
at the end of March, the community starts to isolate which
produces a transition to no growth with Ipp <1 through
April.  In the case of Texas, almost immediately,  isolation
was relaxed and there was slow growth in cases through to
mid May. Finally the state was allowed to re-open which
causes the cases  rise in late May.

Before cases increased in  re-opening,  there was a sharp
7 day step down in cases. It is not clear whether the step
down was data hiccup or a real change. To evaluate this, I
manually changed the Ipp profile to follow the case data in
the step down, and obtained a new transition that mimicked
the shape of the previous transition.

There is a possibility that the step down in cases that
preceded the re-opening, is a result of the incubation time
of the infection. At the start of the infection, one person
infects others, these people will only be infectious after the
virus has incubated (Ref 5).  Exponential growth can start
once there a steady state of transmission and infectious
transmitters. Self-isolation obviously is a situation with few
stable contacts, and a few external contacts such has going
to the grocery store.  Re-opening, returning to work, going
to restaurant’s etc. does involve a complete change in
people’s pattern of contacts which might be like restarting
the infection (Ref 5).

Based on these ideas, we can fill in the time line. The re-
open behavior probably started around in early May. Two
weeks later the existing network of contacts collapses,
temporarily stalling the infection. The infection count drops
as the remaining  infected people recover. Simultaneously,
a new network of contacts are being incubated.  In 7 days
a  new contact network is established and the cases start to
rise again.  

The time line of deaths shows a few more details. After the
infection stalls, the death count falls tracking the cases.
Next, the disease progresses through its new group of
victims, and then the death count starts to rise again, with a
different fatality rate discussed below.  There is evidence of
the same sequence in late April, when the isolation is
relaxed.


Modelling the time line for fatality rate

The fatality of the virus also changes as the infection
proceeds. As noted on the page on Fatality, the fatality rate
seems to rise with infection level, probably from stress on
the health care system. There was also a correlation to
poverty in the city probably through a well documented
dependence on the demographics of the infected.

The fatality rate measured by  daily deaths per infection is
used in the model. The effect of health care stress is
included using the equation;

              Fatality rate = K1 + K2*Cases/N.

For New York , Ki = 0.006, and K2 = 12.
For TX, K1 = 0.004 with K2 = 12.

The fatality rate also depends on the demographics of the
infected population, as was found in the regression
analysis of CFR with poverty for states. The same thing
happens in the time line. Many states recovering from the
first wave  have high fatality rates even as cases count
drops because of infections in closed communities such as  
nursing homes filled with old and sick individuals.  In the
case of Texas, the process of re-opening seems to have
changed the infection to mostly young people who have
ignored distancing. The fatality rate for the second wave in
Texas is half the value in the first wave
. The change in the
infected population is modeled using the same exponent as
the time line for a change in isolation.



Thanks to Andrew Zachary for pointing out the link to financial
modelling.

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

Ref 2 https://www.linkedin.com/pulse/modeling-term-structure-
pandemic-negative-interest-peter-cotton-phd/?
trackingId=K6StB6cgwAAA8dSumxqOfQ%3D%3D

Ref 3 https://en.wikipedia.org/wiki/Ornstein%E2%80%
93Uhlenbeck_process

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

Ref 5 "Repeat contact and the spread of disease" by Peter Cotton
(2020)
The differential equations for the basic SIR
model. Ipp is often designated R0.
The differential equations  can be solved at the
start of the infection when S is close to 1, leading
to an expression for Ipp(0) from the log slope.
A convenient look up table to find Ipp from the
days for 10x change in cases or deaths.
The limitation  of using a two level approximation
for the impact of isolation.
The final model applied to NY,TX and WA. Ipp(t) has a exponential
transition between the initial growth period and stable levels of isolation.
The Ipp time line drives the SIR calculation. Small differences in Ipp have
large impacts on the infection.
A screen shot of the mobility data for TX from the IHME site.
The model equation for the effect of changes in the number of social
contacts on Infections per person.
The modelled transition in Ipp and measured values of isolation for TX
based on cell phone mobility data.