Talk:Compartmental models in epidemiology/Archive 1

Equation for Final Recovered Number

In the section for the "SIR model without vital dynamics" it seems like

$R_{\infty }=1-S(0)e^{-R_{0}(R_{\infty }-R(0))}$

should be

$R_{\infty }=N-S(0)e^{-R_{0}(R_{\infty }-R(0))}$

since earlier in the section it was stated that

$S(t)+I(t)+R(t)=Constant=N$

opposed to when $N=1$ where

$S(t),I(t),R(t)$

are the proportion of the population that are in that state. KNesHere (talk) 02:10, 10 February 2010 (UTC)

Add labels to diagram.

Adding labels to the red, green, and blue plots would help explain what that plot is showing. —The preceding unsigned comment was added by 74.118.13.168 (talk) 03:22, 1 May 2007 (UTC).

The label colors do not match those of the plots. Yellow for Blue is annoying, but reversing the roles of Red and Green is conducive to misunderstanding.

done GregoryKaiser (talk) 17:37, 1 April 2020 (UTC)

Needs attention of expert

I'm just a layman here, and this may sound an odd question, but how are these formulas used? Just for kicks I tried to make sense of how they might be used in a measuring an outbreak of flu, cholera, or zombies, but all I get is "For the full specification of the model, the arrows should be labeled with the transition rates between compartments." but what does that even mean? An example case or two would be nice. —Preceding unsigned comment added by 205.206.107.136 (talk) 06:29, 21 October 2010 (UTC)

An example of use would be, for instance, the H1N1 outbreak and the time it took for Governments to get emergency vaccination programs in place. If you wait too long to vaccinate,a vaccination program may be futile in terms of reducing the number of people in the population that get infected. Other uses may include knowing what demands to expect in Hospital Emergency rooms so that hospitals can staff doctors and nurses accordingly. Back to the H1N1 example in some cities in Canada there was a 2 month delay in the vaccination program that was rolled out, this resulted in a 60% infection of the populations instead what would have been closer to 25%. These models can also tell us what proportion of the population must be vaccinated in order to completely eliminate a disease from the population. Some diseases are not virulent enough to exist in populations that are 60% vaccinated against it, however there are other diseases that survive in populations with 99% vaccination. These models tell us these things.

The transition rates are how many people at each "time step" move from being susceptible to infected or from infected to recovered. for the ds/dt equation the negative term is the amount of susceptible moving to infected at each time step. You'll notice that the same amount is added to teh di/dt equation. —Preceding unsigned comment added by 99.225.12.152 (talk) 11:44, 24 November 2010 (UTC)

Needs attention of an expert whose native language is english. The article has lots of grammatical errors - to the point where I'm unable to make sense of a number of statements. --66.41.154.0 (talk) 03:48, 23 September 2014 (UTC)

I've tagged the article. It isn't just that it needs to be rewritten in plain English, some of these changes need to be verified. SW3 5DL (talk) 12:58, 23 October 2014 (UTC)

Parts of this article definitely seem really wrong, particularly some of the equations and stuff about the SIR model. Just visited it in preparation for an epidemiology exam, and I have a textbook in front of me contradicting a lot of this page. (Modeling infectious disease in humans and animals, Matt J. Keeling and Pejman Rohani - may help others). I'm not enough of an expert to fix it, although perhaps in a few weeks I'll give it a shot. Whole page needs a comprehensive review rhodesj971 (talk) 10:57, 3 May 2015 (UTC)

Thanks for pointing that out. It's probably down to vandalism. Science, math, biology, etc usually suffer the most because editors on page patrol are not familiar with the topics and can't recognize that the charges are detrimental. Good luck with your exam. SW3 5DL (talk) 14:24, 3 May 2015 (UTC)

Need to redo the Bibliography

To make this article accessible, we need to reduce the size of the bibliography. I see a lot of very specific articles that are place there but not used to validate any claims in the main body of the article. I think therefore we should remove those which are not used, and cite some more broad text book resources, or cornerstone research articles in the field.

Thoughts? 149.171.172.161 (talk) 04:46, 21 October 2014 (UTC)

Maybe you could list here the ones you'd remove and add. SW3 5DL (talk) 12:48, 23 October 2014 (UTC)

Have already started, although only have done one for now. 149.171.172.161 (talk) 00:04, 24 October 2014 (UTC)

Delete the lot, I say. The only ones I think are of general relevance are Anderson + May's book, and the Kermack-McKendrick article, and they're both in the references anyway. Most (all?) of the bibliography was added by a single editor in 2008, who I suspect was just dropping in a paper or thesis they wrote. Adpete (talk) 03:09, 2 March 2018 (UTC) Done. Adpete (talk) 03:19, 2 March 2018 (UTC)

Merger proposal

This article covers much of the same ground as Epidemic model. I think that they should be merged, and have stated so in the other article's talk page.

I have added the tags for this. I agree, although Mathematical_modelling_of_infectious_disease is also very similar and "epidemic model" is more general than just compartmental models, although that is largely what it covers. I suggest maybe some of the historical/other non-compartmental content in epidemic model actually go into mathematical modelling of infectious disease instead of here. Mvolz (talk) 17:29, 26 June 2017 (UTC)

looks very similar to me. maybe a better distinction would help too. Biggerj1 (talk) 23:43, 2 September 2020 (UTC)

I think Compartmental modeling deserves it's own page, particularly in light of the COVID-19 epidemic. Compartmental modeling is a huge topic in its own right. It is a sufficiently large sub-topic of Disease modeling that it deserves its own page. Jaredroach (talk) 23:44, 19 September 2020 (UTC)

On a related note, I will merge Susceptible individual into this article (Compartmental models in epidemiology), as Susceptible individual does not merit a separate article.

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Major cull

I deleted 2 entire sections, and most of the bibliography. This was added by a single user in 2008, and reads suspiciously like a paper or project they wrote. More importantly, it was off topic. The section "Modelling mass vaccination programmes" is not about compartmental models, but about one (of many) applications of compartmental models, and there's no reason given why that application in particular should be in the article. The section "The influence of age: age-structured models" is going a step beyond compartmental models, but again no reason is offered why to take this extension in particular. Then most of the bibliography was deleted because it was very specific to those two topics. I won't protest if someone sees fit to restore these sections (last version is here [1]) but I think it's inappropriate for a general article, and was cluttering it. Adpete (talk) 01:51, 3 March 2018 (UTC)

@Adpete: I mean, the reason to take that extension in particular is that modelling vaccination strategies is practically the entire function of epidemiological models to begin with :). But I agree the section as written is not very good; are these text book models? Where do they come from? How do they compare to other published models, of which there are many? But I think rather than deleting it should be fixed. At any rate, it's not an application of a model, so this is why I reverted. It IS a model. Applying a model involves showing how predictive a model is with real data. So theoretically speaking, it belongs in the article. Mvolz (talk) 09:35, 1 July 2018 (UTC)

Yes but there are probably thousands of papers which apply compartmental models in different ways. Why should this application make it into the article, and not all the others? That is my concern. Especially the very detailed "The influence of age: age-structured models" section. Adpete (talk) 13:03, 1 July 2018 (UTC)

Inconsistent use of γ

Here γ is used as inverse of the infectious period (which leads to $R_{0}={\frac {\beta }{\gamma }}$), but in Basic reproduction number it is used as that period (which leads to $R_{0}=\beta \gamma$). Is one of them wrong? Or do different authors use different definitions? Anyway, this is a confusing situation. --mfb (talk) 01:24, 11 March 2020 (UTC)

RfC: consistency of definition and use or symbol R

In the article, at some points the symbol R is used for the recovered, at other points it is used for the recovered and the deceased. I suspect there may be conditions or assumptions under which no essential mathematical difference arises from this inconsistency. But I prefer to have the definitions clear, and their use consistent. So, could we agree to sticking to defining and using R for recoveries only?Redav (talk) 16:00, 19 March 2020 (UTC)

Birth Rate

Both in the SIR section with vital dynamics and, since it was changed by some IP to be that way, and in the SEIR section, the birth rate $\Lambda$ is a constant term, independent of the population size. To me it is unclear why that should be the case. Can anyone comment on this? Cheers --Sensorpixel (talk) 22:31, 21 March 2020 (UTC)

The model assumes a constant total population size $N$. So if the per capita birth rate, $\lambda$, is the same for each compartment (S, E, I and R) we get $\lambda (S+E+I+R)=\lambda N=\Lambda$. OpenScience (talk) 15:56, 22 March 2020 (UTC)

If $\Lambda =\mu$ as stated in the beginning of the SEIR model, then the model should be ${\frac {dS}{dt}}=\mu N-\mu S-\ldots$. The version before 11 march seems better. Or we should precise $\lambda =\mu$ and $\Lambda =\lambda N$ .

$\Lambda =\mu$ is rather imprecise, $\Lambda =\mu N$ at least has consistent units of measurement. I would probably leave out vital dynamics to make things simpler. But I don't know what's customary. OpenScience (talk) 01:26, 25 March 2020 (UTC)

Reproductive rate

Why does the basic reproductive rate/number have a negative sign in front of it? R_0 is supposed to be positive. Lambda and delta are both positive constants.....

/////////// Someone has reverted hours of work and even removed the picture which was needed. Please discuss before reverting. Other people have edited the article since and have not reverted the improvements. If you want to substitute your opinion for mine and for other editors please have very good reasons. Other people have said the like the improvements. I am an inexperienced editor and I don't know much about people who do this or how to fix it but I am very upset by you. Kelly222 (talk) 03:08, 29 April 2020 (UTC)

Constant population

In the section about the SEIR model, it says: "We have S + E + I + R = N, but this is only constant because of the (degenerate) assumption that birth and death rates are equal; in general N is a variable." Isn't a better justification for this lack of demography a separation of timescales: disease dynamics being much faster than population dynamics so that a quasi-steady state is an accurate approximation? --OpenScience (talk) 11:29, 22 March 2020 (UTC)

Yes, probably that is a better justification. Also because deaths can eliminate either susceptible or immune individuals, whereas births tend to create mostly susceptible individuals. Although neonatal immunology can be very complicated and dynamic. I don't strongly feel the need to alter the text here, as maybe if one is that interested in the details, it might be time to grab a text book or start reading the primary sources. Jaredroach (talk) 20:12, 3 November 2020 (UTC)

Also, the section says: "vital dynamics with birth rate Λ equal to death rate μ". This cannot be correct because form the equations it is clear that μ is a per capita rate, measured in units of 1/time, whereas Λ is measured as individuals per unit of time. OpenScience (talk) 12:43, 22 March 2020 (UTC)

I believe the problem is that different primary sources use different units. Some use 1/time and some use individuals per unit time. And this Wikipedia article has been cobbled together by different Wikipedians using different schools of usage. You can see this by comparing these two equations which appear in the article:

${\begin{aligned}{\frac {dS}{dt}}&=\Lambda -\mu S-{\frac {\beta IS}{N}}\\[8pt]{\frac {dS}{dt}}&=\Lambda N-\mu S-{\frac {\beta IS}{N}}\\[8pt]\end{aligned}}$

So someone could consider taking on the task of going through and check the entire article for consistency and also add a note that there are different conventions in different textbooks. Jaredroach (talk) 20:20, 3 November 2020 (UTC)

SIR without vital dynamics example for Mathematica using the RK4 method

   Clear[IC, i, f, NN, β, γ];
   IC = {499, 1, 0} (*Initial Conditions: 499 Susceptible, 1 Infected, 0 Recovered*);
   NN = Sum[i,{i,IC}] (*Total Population: 499+1+0, NN used because N is a reserved word in Mathematica*);

   β = 0.5; γ = 0.1;
   (*dS/dt*) f[1, t_, S_, I_, R_] := -((β I S)/NN);
   (*dI/dt*) f[2, t_, S_, I_, R_] := (β I S)/NN - γ I;
   (*dR/dt*) f[3, t_, S_, I_, R_] := γ I;

   Clear[a, b, h, n, m, t, k1, k2, k3, k4, w];
   (*Graph boundaries: t goes from 0 to 60 at steps of 0.5*)0;
   a = 0; b = 60; h = 0.5;
   (*Graph data points: n x m = Number of steps x Number of equations*)0;
   n = (b - a)/h; m = Length[IC];

   t[i_] := a + i*h;
   For[i = 1, i <= m, i++, w[i, 0] = IC[[i]]];
   k1[i_, j_] := h*(f @@ Flatten[{i, t[j - 1], Table[w[k, j - 1], {k, 1, m}]}]);
   k2[i_, j_] := h*(f @@ Flatten[{i, t[j - 1] + h/2, Table[w[k, j - 1] + k1[k, j]/2, {k, 1, m}]}]);
   k3[i_, j_] := h*(f @@ Flatten[{i, t[j - 1] + h/2, Table[w[k, j - 1] + k2[k, j]/2, {k, 1, m}]}]);
   k4[i_, j_] := h*(f @@ Flatten[{i, t[j - 1] + h, Table[w[k, j - 1] + k3[k, j], {k, 1, m}]}]);
   w[i_, j_] :=  w[i, j] = w[i, j - 1] + (k1[i, j] + 2*k2[i, j] + 2*k3[i, j] + k4[i, j])/6

   Clear[dataSet1];
   dataSet1 = Table[Table[{t[j], w[i, j]}, {j, 0, n}], {i, 1, m}](*//TableForm*);
   ListPlot[dataSet1, PlotStyle -> {Blue, Green, Red}]

47.13.58.75 (talk) 15:55, 31 March 2020 (UTC)

Does a epidemic really end because the number of infected people falls?

The claim is made in this article that, "this means that the end of an epidemic is caused by the decline in the number of infected individuals rather than an absolute lack of susceptible subjects." I put it to you that this is simply false, leaving the math to one side, it is clear that it has to be false since most epidemics start with a small number of infected individuals, probably only one in the case of a new strain. Where the number of infected the relevant issue, then such epidemic would never be able to get started.

The writer of the claim is trying to convey the key point that when an epidemic ends, there are typically still susceptible people in the population. A common misconception in didactic settings is that all epidemics end because everyone in the population was infected. And right before the epidemic ends, there will have been a "decline in the number of infected individuals". So the sentence is not "simply false". Because I knew what the writer intended to say, the sentence was clear to me, and true. But the sentence is clearly poorly worded, as you point out. I will try to revise it. Jaredroach (talk) 00:29, 4 November 2020 (UTC)

I argue that the real key to the end of an epidemic is actually a fall in the proportion of susceptible individuals in the population. In most developments of SIR theory R0 is defined as: β/γ. I have a seen a number of papers in which there is considerable confusion or even downright error arising form this. In this particularly paper, this confusion is avoided by defining R0 as, "the expected number of new infections (these new infections are sometimes called secondary infections) from a single infection in a population where all subjects are susceptible." The qualification, "... where all subjects are susceptible." makes this definition work, but I suggest that there is a better option, that is to define R0 as:

R0 = β.S(0)/γ Where s(t) = the proportion of susceptible individuals within the population at time t.

This has two advantages:

Firstly, it allows us to address cases where s(0)<1.

Secondly it enables us to generalize from the Basic Reproduction Number (that at the start of the epidemic) to a Current Reproduction Number R(t). The Current Reproduction number will fall off as s(t) decreases, and I would suggest that the point at which the epidemic begins to end will the point at which the Current Reproduction Number falls below 1. In other words, the point at which an epidemic ends depends not on I(t) but on the interaction of s(t) with Β and γ.

The "effective reproduction number" is the term of art (for what you are suggesting as "current reproduction number"). Check out the basic reproduction number article for more on this concept. Jaredroach (talk) 00:29, 4 November 2020 (UTC)

I really do think this important, because some politicians are trying to use declining death or infection rates as the basis for ending measures such as the lock-down, when in fact they have little direct relevance to when it will be safe to do so. This seems a mistake to me, but as my background is in social sciences, psychology and philosophy rather than epidemiology I am only putting this up as a suggestion for discussion.Michael Gowland (talk) 14:46, 19 April 2020 (UTC)

Any real pandemic is a particular instance of an infinite number of possible pandemics, each of which are different due to different initial conditions and parameters, which may be time-dependent. The basic reproduction number R0 is intended to be a fundamental parameter of the pandemic, independent of the particular proportions of susceptible, infected, recovered, etc. and independent of the initial conditions which give rise to the pandemic. Your definition makes it a function of the fraction of susceptibles at a particular time, which makes it no longer a fundamental descriptor of a pandemic, but rather a variable, depending on the particular initial conditions of that particular instance of the pandemic. If your definition is intended to make R0 less prone to misinterpretation, this is not the way to do it. Errors in interpretation should simply be corrected. Suppose you have a population in which 1/3 have been vaccinated and are immune, the rest susceptible, and zero infected, then you have no pandemic, but a pandemic will still occur if one infected individual is introduced, it just won't be as bad. R0 never comes directly into play here, the infected individual is not introduced to a completely susceptible population. Nevertheless, the resulting pandemic is certainly described in terms of R0, as are all other possible pandemics of that class. The fundamental nature of R0 is not lost simply because there never was a completely susceptible population.

(As for the politicians, they are totally focused on the political impact of a particular interpretation, not on its validity) PAR (talk) 19:05, 26 June 2020 (UTC)

Declining rates indicate that the effective reproduction number is likely less than 1. This is one of many reasonable statistics to take into account with other data (e.g., hospitalizations, test positivity, availability of treatments, economic externalities, etc.) in deciding to modify public health measures. As for whether the basic reproduction number is truly "a fundamental descriptor", I am not so sure. It is a bit more slippery than similar fundamental parameters from fields such as physics. See the discussion at basic reproduction number. Jaredroach (talk) 00:29, 4 November 2020 (UTC)

Work on lede

@Kelly222: I reduced the number of compartments listed in the lead - having that many made it a bit word-salady and harder to read. I'd like to keep this to under 4 compartments, if possible. Sorry for the revert before - that was rather lazy of me. Cheers Mvolz (talk) 10:51, 3 May 2020 (UTC) Thank you. It still needs a picture. Kelly222 (talk) 21:02, 23 July 2020 (UTC)

Issues with Compartmental_models_in_epidemiology#The_SIS_model

The text for this model mentions flu, but the image here is of the SI model - where there's no recovery at all. This is kind of a rare thing, but it does happen in parasitology - Stiven 1967 for example - but not much in human infectious diseases. Cold and flu epidemiological models look nothing like this, so it's a bit misleading to have this text with this model.

• Stiven, Alan E. (1967-12). "The influence of host population space in experimental epizootics caused by Hydramoeba hydroxena". Journal of Invertebrate Pathology. 9 (4): 536–545. doi:10.1016/0022-2011(67)90135-8. ISSN 0022-2011. {{cite journal}}: Check date values in: |date= (help)

Mvolz (talk) 11:10, 3 May 2020 (UTC)

Unification Suggestion

I have noticed that beta*SI and beta*IS are both used randomly throughout the article without clear reasons. I suggest this to be unified, and since I couldn't choose which one should be used, so I request for someone to do this job! Thank you so much~ Luke Kern Choi 5 (talk) 13:39, 28 July 2020 (UTC)

+I'm meaning the equations such as dS/dt= ...Luke Kern Choi 5 (talk) 13:43, 28 July 2020 (UTC)

Counter-measures

I am searching for any sources, books or something, that the measurments were a part of the epidemiology science or practice. What I found is this site. Andh ere: "Counter-measures such as masks, social distancing and lockdown will alter the contact rate in a way to reduce the speed of the pandemic." - but, is there any reference for this? --Struppi (talk) 15:58, 14 August 2020 (UTC)

This is discussed in Basic reproduction number. So maybe just put in a Wikilink rather than a reference. Jaredroach (talk) 20:08, 3 November 2020 (UTC)

Need mathematical consistency in the "Transition rates" section

Two types of differentials appear in the "Transition rates" section. First we have "d(S/N)/dt = -βSI/N2" which has units of 1/time, i.e. the rate of change of the fraction of individuals in S. Next we have "between I and R, the transition rate is assumed to be proportional to the number of infectious individuals which is γI.", which has units of individuals/time. To make this section comprehensible we should use rates that share the same units. Since the SIR state diagram, which I just fixed, has units of individuals/time for both rates, I recommend that we standardize on that. A good paper we can follow and refer to is Allen, L.J., 2017. A primer on stochastic epidemic models: Formulation, numerical simulation, and analysis. Infectious Disease Modelling, 2(2), pp.128-142 — Preceding unsigned comment added by Arthur.Goldberg (talk • contribs) 19:39, 25 October 2020 (UTC)