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OK, the tweet below was using a very simplistic heuristic; now let me write a thread about what the (hardly less simplistic!) SIR epidemiological model says about the attack rate of an epidemic. ⤵️ •1/23 ^{}

The SIR model is a first-order nonlinear ordinary differential equation in two variables written s,i,r. (Yes, that makes three variables, but s+i+r=1 so one is redundant.) They represent proportions of the population that is susceptible (s), infected (i) and recovered (r) •2/23
^{}

(here, “susceptible” means “not yet infected”). If I write x′ for the time derivative of x, the SIR equations are s′ = −β·i·s and i′ = β·i·s − γ·i (so r′ = γ·i), for two constants β and γ describing the epidemic (and having dimensions of inverse time): … •3/23
^{}

… here, β·i·s represents new infections, and γ·i represents recoveries (either neglecting deaths or counting them as “recoveries”). I.e., we model recoveries as a first order kinetic process (essentially, the infected population, left alone, recovers at a rate γ), … •4/23
^{}

… and infections as a kinetic process of first order in each of i and s (so, second order overall kinetic, where “order” here means degree, it's still a first-order ODE, terminology sucks 🙄). Initially, r=0, i is very small but nonzero, and s=1−i is very close to 1. •5/23
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Of particular importance is the ratio β/γ, which is the “basic reproduction number” κ (or R₀) of the infection. Note that this is the only free parameter in the equation, because we can rescale time to multiply β and γ by a common constant. •6/23
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This “basic reproduction number” is usually denoted R₀, which is absolutely awful notation 😒 when doing a model called SIR where R stands for Recovered. So I'm writing r for “recovered” and I'll write κ := β/γ for this basic reproduction number. •7/23
^{}

Initially, of course, s remains close to 1, so i grows exponentially like exp(βt). This is what we're currently seeing with #Covid19. We can measure β as the logarithmic derivative of number of cases and γ from individual patient recovery (1/γ = expected time), … •8/23
^{}

… so we can compute the basic reproduction number κ = β/γ which is essentially the number of people each infected person infects in turn. Current estimates for #Covid19 are around β ~ 0.2/d, 1/γ ~ 15d, so κ ~ 3. Very roughly. Anyway. •9/23
^{}

Typical behavior is shown in the following graphs taken from Wikimedia Commons (commons.wikimedia.org/wiki/File:Sirs…), where s is in blue, i in green and r in red, and s+i+r here is 500 because some people don't know how to divide to normalize to 1, but no matter. •10/23
^{}

What interests me is what s and r converge to as t→∞. One might think from the above graphs (and I did) that s→0, i.e., the epidemic stops when there are no susceptible individuals left, but this isn't true! Actually, s approaches a nonzero limit which I'll call s[∞]. •11/23
^{}

In other words, not everyone gets infected, only a proportion r[∞] = 1−s[∞] do (the “final attack rate”). (Of course, i[∞]=0 as is obvious from the γ·i term in the equations.) Now, how do we compute s[∞]? Well, from s′ = −β·i·s and r′ = γ·i we get, by dividing: … •12/23
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… the relation s′/r′ = −κ·s; but s′/r′ is the derivative of s with respect to r, so this can be solved as s = s₀·exp(−κ·r). In particular, s[∞] = s₀·exp(−κ·r[∞]), but since also r[∞] = 1−s[∞], we have the equation s[∞] = s₀·exp(−κ·(1−s[∞])). •13/23
^{}

Assuming we start the epidemic with an infinitesimal number of cases, s₀ is essentially 1, so we're trying to solve the equation s[∞] = exp(−κ·(1−s[∞])) in the indeterminate s[∞] (proportion never infected) with parameter κ (reproduction number). •14/23
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For κ<1, the solution is 1 (i.e., the epidemic gets nowhere). So let's take κ>1. The equation s = exp(−κ·(1−s)) is a transcendental equation, it doesn't have a solution in closed form using standard functions but it has one using a special function W: … •15/23
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… namely s = −W(−κ·exp(−κ))/κ where W is Lambert's transcendental W function (=product logarithm), see en.wikipedia.org/wiki/Lambert_W… for more about this function but basically w := W(z) satisfies w·exp(w) = z. •16/23
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(This solution comes from rewriting s = exp(−κ·(1−s)) as −κ·s·exp(−κ·s) = −κ·exp(−κ), hence −κ·s = W(−κ·exp(−κ)). Of course you need to check that we have the right branches of W, but I don't want to get into details.) •17/23
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To summarize, the SIR model predicts that the proportion of the population that remains uninfected to the end is s = −W(−κ·exp(−κ))/κ where W is Lambert's function and κ>1 is the basic reproduction number. How does this function behave? •18/23
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And how does it compare with the estimate s = 1/κ given by the simplistic reasoning given in the thread quoted in the first tweet of this thread? Well, in short, it's worse. 😕 (Essentially because SIR tracks the epidemic passed its “turning point” κ·s~1.) •19/23
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For κ = 1 + h a basic reproduction number ever so slightly above 1, we have s = −W(−κ·exp(−κ))/κ = 1 − 2·h + (8/3)·h² + O(h³): compared to 1/κ which is 1 − h + h² + O(h³), we're basically predicting twice as many cases. •20/23
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And for κ large, using the expansion W(x) = x − x² + O(x³) we find s = −W(−κ·exp(−κ))/κ = exp(−κ) + κ·exp(−2κ) + O(κ²·exp(−3κ)), so the fraction s of uninfected drops exponentially with the basic reproduction number κ. •21/23
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For κ~3, SIR predicts s~6% remain uninfected, i.e., r~94% attack rate. I'm a mathematician, not an epidemiologist, so lest anyone claim I wrote #Covid19 will infect 94% of the Earth's population, let me emphasize that this is a highly simplistic model showing its limits! •22/23
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✱ Correction: in tweet 8/23, I should have written “i grows exponentially like exp((β−γ)t) (so long as s~1)” since i′=(β−γ)i, not like exp(βt). My estimates for β should be correspondingly updated, but they are just orders of magnitude anyway. •24/(23+1)
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#Thread - 1/13

Respected: @narendramodi @AmitShah @PMOIndia @NSAGov

I'm might be wrong but I'd like to share my views as this is a very serious matter now!

I'm fed up, angry & feeling helpless now about one-way hate for Hindus.

Cc: @BesuraTaansane

@desimojito

@theskindoctor13

Respected: @narendramodi @AmitShah @PMOIndia @NSAGov

I'm might be wrong but I'd like to share my views as this is a very serious matter now!

I'm fed up, angry & feeling helpless now about one-way hate for Hindus.

Cc: @BesuraTaansane

@desimojito

@theskindoctor13

2/13

It's not unprecedented but the Hinduphobia agenda is now rising up in gallop, it's not just about India but the #LutyensMedia, #UrbanNaxals, #Congress peddled Hindu hate to a level that the entire world is abusing Hindus now, just bcoz some set of ppl narrated that way..

It's not unprecedented but the Hinduphobia agenda is now rising up in gallop, it's not just about India but the #LutyensMedia, #UrbanNaxals, #Congress peddled Hindu hate to a level that the entire world is abusing Hindus now, just bcoz some set of ppl narrated that way..

3/13

I'm being a Hindu, it really hurts & is disturbing at the same time. Just imagine if someone is being harassed & abused in his own house, how does he'll feel? Likewise, Hindus are going thru similar kinda situations here. It's appalling to see Wikipedia published an..

I'm being a Hindu, it really hurts & is disturbing at the same time. Just imagine if someone is being harassed & abused in his own house, how does he'll feel? Likewise, Hindus are going thru similar kinda situations here. It's appalling to see Wikipedia published an..

#thread

#MangalSutra

Why women wear Mangal Sutra.

(Scientific Reason)

Today I want to talk on a controversial topic that is why women wear manga sutra or you can say science behind it.

@meenakshisharan @LevinaNeythiri @jkd18 @Atheist_vashali @Lisha_Cryptic @3Enigmatic @sanj9

#MangalSutra

Why women wear Mangal Sutra.

(Scientific Reason)

Today I want to talk on a controversial topic that is why women wear manga sutra or you can say science behind it.

@meenakshisharan @LevinaNeythiri @jkd18 @Atheist_vashali @Lisha_Cryptic @3Enigmatic @sanj9

I am saying controversial because as a male I should not talk on such sensitive topics but as a human I have something to share it with you that’s why I am writing this thread I don’t want that knowledge that I have go waste & got finished with me that’s why

@Praveen84822453

@Praveen84822453

I am sharing with you so let’s start . There are many stories behind it. Let’s discuss it. Mangalsutra, literally meaning the sacred-thread, is an important part of the Hindu culture .There are many types of mangalsutra available &

@BeenaPP1 @Dnyaneshwari04 @almightykarthik

@BeenaPP1 @Dnyaneshwari04 @almightykarthik

#THREAD The #LobbyDay militia rally tomorrow in Richmond already had one planned terrorist attack foiled and is awash in disinfo.

It's a little late to do so, but here's my preview thread for what's going on.

It's a little late to do so, but here's my preview thread for what's going on.

What the hell *is* this event that might pop off tomorrow? Here's the tl;dr from @IwriteOK's newest article for @bellingcat, which I'll be discussing and linking to later in this thread.

The broad right wing militia/patriot movement, particularly in the United States, is the conjoined twin brother of the white nationalist movement. That doesn't mean they have the same ideas about race. This article provides some important background. itsgoingdown.org/ammon-bundy-th…

[#Thread on #Algeria] 1/ @Redasbox got in touch to ask if I could have a look at some unusual tweeting in Algeria. I analysed around 20,000 tweets from around 5,769 unique accounts. What I found, was alarming, and clear evidence of a disinformation campaign. Read on #Algerie

@Redasbox 2/ First some very brief context. Algeria has seen the 32nd weekend of pro-democracy demonstrations. Peaceful protests successfully pressured Abdelaziz Bouteflika, the ruler of 20 years, to step down in April. An election has been called for on December 12th to help try and

3/ resolve the political stalemate in Algeria. However, many Algerians are not happy as they believe the elections will be rigged by a military/elite junta. This is not uncommon in MENA. Thus the elections are seen as giving democratic legitimacy to continued authoritarian rule

Ok. It’s time to go into Joshua’s memory box and tell his story #thread

Joshua was very much a rainbow baby 🌈 and we were so excited and over the moon when I finally made it into the second trimester. (Alex and I are both big rugby fans!)

I was working as a 1st year Registrar in A&E when I suddenly broke my waters at 24 weeks. I was transferred to Southampton @UHSFT who have a specialist level 3 Neonatal Intensive Care Unit. They were wonderful, and did everything they could to stop my labour over Father’s Day.