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 In the standard formulation, Ipp(0) is fixed for the entire isolation 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 distancingAs 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 reopeningThe 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 rateThe 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) |

model. Ipp is often designated R0.

start of the infection when S is close to 1, leading

to an expression for Ipp(0) from the log slope.

days for 10x change in cases or deaths.

for the impact of isolation.

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.

contacts on Infections per person.

based on cell phone mobility data.