<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class="">Hi William,<div class=""><br class=""></div><div class="">Thanks very much for this. You make great points, and I completely agree - it is very important to explore transmission by coupling between scales in the way that you describe. This is, in fact, something that we have been looking at too (<a href="https://royalsocietypublishing.org/doi/10.1098/rsif.2020.0230" class="">https://royalsocietypublishing.org/doi/10.1098/rsif.2020.0230</a>), although not in the context of COVID-19. I have heard of the beta proportional to log-VL being assumed for other pathogens, but I was not aware of whether or not this is reasonable for SARS-CoV-2, so thanks for this info :-)</div><div class=""><br class=""></div><div class="">One thing that immediately comes to mind is that, by comparing the two different but complementary approaches, it might be possible to say something about the magnitude of behavioural effects (e.g. isolation later in infection reducing transmission, even if the viral load remains high). This could be interesting to explore.</div><div class=""><br class=""></div><div class="">Best wishes,</div><div class="">Robin</div><div class=""><br class=""><div class="">
<meta charset="UTF-8" class=""><div dir="auto" style="caret-color: rgb(0, 0, 0); color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><div dir="auto" style="caret-color: rgb(0, 0, 0); color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><div>----------------------------------------------------------------------<br class="">Dr Robin Thompson<br class="">Assistant Professor of Mathematical Epidemiology<br class="">Mathematics Institute<br class="">University of Warwick, UK<br class=""><a href="http://www.robin-thompson.co.uk" class="">www.robin-thompson.co.uk</a><br class="">----------------------------------------------------------------------</div></div></div>
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<div><br class=""><blockquote type="cite" class=""><div class="">On 29 Apr 2021, at 10:23, William Waites <<a href="mailto:wwaites@ieee.org" class="">wwaites@ieee.org</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class="">Hi Robin!<div class=""><br class=""><blockquote type="cite" class="">Thanks for this. All the generation time distributions (i.e. expected infectiousness as a function of the time since infection) that we have been considering are normalised so that they integrate to 1. </blockquote><div class=""><br class=""></div>This is very interesting. The shape of all of the distributions for generation that you find (particularly striking in Fig 4a of your paper) have the characteristic shape of the curves for log viral load, and if we suppose that chance of developing symptoms is proportional to this quantity we indeed get a cumulative fraction of infectious individuals that is lagged, as would be expected. See, for example,</div><div class=""><br class=""></div><div class=""> <span id="cid:5E02B9E4-1AE2-4A8D-9B21-C20474288320"><Screenshot 2021-04-29 at 09.45.37.png></span></div><div class=""><br class=""></div><div class="">modulo perhaps some stretching. The vertical axis for n is arbitrary units where n is allowed to vary between 0 and 2 proportional to log virus population within the host.</div><div class=""><br class=""></div><div class="">Coupling this to a transmission model at the population level is straightforward. If we also suppose that infectiousness is also proportional to log viral load (traditionally the quantity denoted β) then we can get to generation time distributed as you find it. The point is, we can get at this mechanism from an underlying model of adaptive immune response, as opposed to the more phenomenological approach of estimating the chance that someone is infectious or has symptoms as a function of time. It is good that this latter approach agrees, but this same result seems to follow from some more basic (not to say simpler, just more fundamental) considerations. </div><div class=""><br class=""></div><div class="">The implications for combining models with these two approaches are important. It is possible (though a bit subtle for various reasons) to derive generation time distributions from immune models like this. It is easier and more “correct” in a certain sense to couple them directly to transmission by giving an account of the actual mechanism of transmission (giving a bit of virus to another person). It seems to me that something important might get lost by going via a population-level distribution derived from a process that happens within individuals and exhibits quite a lot of heterogeneity between them. This particular jumping between scales needs to be looked at closely and properly understood — it’s not going to be as simple as just agreeing to normalise in a certain way.</div><div class=""><br class=""></div><div class="">Cheers,</div><div class="">-w</div><div class=""><br class=""></div></div></div></blockquote></div><br class=""></div></body></html>