{"id":22921,"date":"2020-06-19T15:56:15","date_gmt":"2020-06-19T19:56:15","guid":{"rendered":"http:\/\/chcollins.com\/100Billion\/?p=22921"},"modified":"2022-08-01T07:42:47","modified_gmt":"2022-08-01T11:42:47","slug":"coronavirus-what-are-my-chances","status":"publish","type":"post","link":"https:\/\/chcollins.com\/100Billion\/2020\/06\/coronavirus-what-are-my-chances\/","title":{"rendered":"Coronavirus: What Are My Chances?"},"content":{"rendered":"

Asked & Answered 9.0<\/strong><\/em><\/span><\/h3>\n

\"Coronavirus<\/a>We have been in isolation mode for the coronavirus for so long that maintaining distance in public has pretty much become an automated, fear-induced behavior.\u00a0 This is sad.\u00a0 Like everyone else, I want to get out and go to restaurants and live life normally, but the case numbers here have not been encouraging.\u00a0 I have been keeping a chart (at right<\/a>) which shows we are averaging 7 new positive tests a day.<\/p>\n

Is there some “magic number” of new cases per day that would make me feel comfortable increasing my degree of exposure to the world?\u00a0 There must be some number, otherwise it wouldn’t be worth my while tracking this data.\u00a0 So, how do I translate the rate of positive test results in my area to my personal risk level of going out and about?<\/p>\n

Having found no clear answer to this on the internet, I thought I would play Dr. Fauci and guesstimate it myself, given the data available to me, plus some assumptions.\u00a0 Here goes.<\/p>\n

\u2022 \u2022 \u2022<\/p>\n

At its most basic level, the formula may be expressed as chance of infection per contact<\/em> = chance the other person is infected<\/em> times chance of transmission upon contact.<\/em><\/p>\n

Let’s start with the chance that the contacted person is infected.\u00a0 Here I define contact<\/em> as passing through a person’s airspace for some length of time.\u00a0 I will ignore transmission via objects for now, as I use gloves, hand-washing and object-cleaning protocols to minimize that risk factor — for me<\/em>.<\/p>\n

My chance of contacting an infected person on a given outing depends on the number of contacts I make and the percentage of infected people among them.\u00a0 This is a good place to introduce my framework and assumptions, starting with the chart below:<\/p>\n

\"Populations<\/a>My first assumption is that I only come in contact with the local population (P<\/em>), defined as those who live in my county.\u00a0 This population consists of two groups, those infected with the virus (I<\/em>) and those not infected (N<\/em>).\u00a0 The infected group is further divided into three sub-groups: asymptomatic (I<\/em>A<\/sub>); symptomatic but untested (I<\/em>S<\/sub>); and those who have tested positive (I<\/em>T<\/sub>) and are considered contagious.\u00a0 Those who test negative and those who have recovered from the virus are included in group N<\/em>.<\/p>\n

For our purposes, I assume that infected people are evenly distributed around the county and may travel anywhere within its borders.\u00a0 Whether I meet an infected person depends on their propensity to travel, which may differ among the various sub-groups.\u00a0 If we define the baseline (i.e., non-infected group) propensity to travel as f<\/em>N<\/sub> = 1.0, then we can assign (by guesswork) travel factors f<\/em>A<\/sub> , f<\/em>S<\/sub>\u00a0 and f<\/em>T<\/sub> to the infected sub-groups, where each\u00a0f<\/em>-factor is between 0 and 1.\u00a0 For instance, we might suppose that those who have tested positive and are still contagious would have a low propensity to travel — so f<\/em>T<\/sub> might be 0.1, say.<\/p>\n

This framework leads to Equation 1, the chance C<\/em>I<\/sub> that a random contact is infected:<\/p>\n

(1)\u00a0     C<\/em>I<\/sub> = ( f<\/em>A <\/sub>I<\/em>A<\/sub> + f<\/em>S <\/sub>I<\/em>S<\/sub> + f<\/em>T <\/sub>I<\/em>T<\/sub> ) \/ P<\/em><\/pre>\n
Now, we know P,<\/em> and county health officials provide a weekly update on the number of positive tests, but that’s about all we know.\u00a0 We have to deduce the sizes of the sub-groups I<\/em>A<\/sub>, I<\/em>S<\/sub> and I<\/em>T<\/sub> and the travel factors f<\/em>A<\/sub> , f<\/em>S<\/sub>\u00a0 and f<\/em>T<\/sub> using official estimates and semi-informed guesswork.\u00a0 Let’s consider each of these in turn.<\/p>\n
First, we estimate the number of people in the I<\/em>T<\/sub> (tested positive) group as I<\/em>T<\/sub> = d<\/em> Q<\/em> , where d<\/em> is the duration of a typical infection (also the quarantine time following a positive test) and Q<\/em> is the rate of new positive test results per day.\u00a0 We assume that the positive test rate is at steady-state, i.e., Q<\/em> is constant.\u00a0 For simplicity, we also assume that those who test positive are tested on the first day of their infections.\u00a0 So, if Q<\/em> = 10 new positive test results per day and d<\/em> = 21 days, then the number of people currently in the I<\/em>T<\/sub> sub-group is 210.<\/p>\n
But there are plenty of stories about symptomatic people who do not bother to get tested and who only slightly modify their behaviors (the I<\/em>S\u00a0 <\/sub>sub-group).\u00a0 I have found few clues as to the size of this group, and none of the testing statistics are relevant.\u00a0 The best I can do is take a wild guess at the fraction of infected people who develop symptoms but go untested. Based on human nature (see hurricanes<\/em>), I would bet that at least a third of symptomatic people “ride it out” without ever seeking care or getting tested, at least in this county.<\/p>\n
So the size of the symptomatic group I<\/em>S<\/sub> = k<\/em>S\u00a0 <\/sub>( I<\/em>S\u00a0<\/sub> +<\/em> I<\/em>T<\/sub> ), where k<\/em>S\u00a0 <\/sub>is the fraction of those symptomatic and infected who go untested.\u00a0 This may be rearranged as I<\/em>S<\/sub> = I<\/em>T<\/sub> k<\/em>S\u00a0 <\/sub>\/ (1 – k<\/em>S <\/sub>).<\/p>\n
Finally, the infected-but-asymptomatic group I<\/em>A <\/sub>.\u00a0 There has been much debate about how many infected people are asymptomatic and how infectious they are.\u00a0 The CDC estimates<\/a> that 35 percent of infected people are asymptomatic — I have seen figures as high as 80%. For our purposes, we will assume that asymptomatic people also never get tested, and that their propensity to travel is the same as the non-infected group.<\/p>\n
The relevant formula here is I<\/em>A<\/sub> = ( I<\/em>S<\/sub> + IT<\/sub><\/em>\u00a0 ) k<\/em>A\u00a0 <\/sub>\/ (1 – k<\/em>A <\/sub>), where k<\/em>A\u00a0 <\/sub>is the fraction of infected people who are asymptomatic.<\/p>\n
Given all the above, we are now ready to plug some numbers, guesses and estimates into our equations:<\/p>\n