Talk:Mathematical modelling of infectious diseases/Archive 1

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Mutations

What about the probability of a mutation that circumvents the vaccine? The more the disease is exposed to the vaccine, the greater the probability that it will suffer a mutation that permits it to survive the vaccine. This is over-simplified, but on point.

Over a sufficiently long period of time, this probability becomes greater if not inevitable. The factors are, simplistically:

  • the mutability of the virus,
  • the capacity for the vaccine to handle variations.

In reality, there are a large but discrete number of viable mutations, M={M0, M1, ... Mn}, and a discrete set of mutations that the vaccine protects against V={V0, V1, ... Vm}. The difference M-V={E1, E2, ... Ek}=E, is the set of mutations to which the vaccine is ineffective to a significant degree.

The probability of a circumvention is the sum of probabilities all of Ei, for i=1..k, occuring, which is a function of time (E(t)). It is however, not singly a function of time. It is also a function of opportunity. If the virus can spread without mutation, as M0, the probability of E(t) spreading broadly is decreased because M0 is predominant in the hosts. Unless Ei has a selective advantage (faster transmission, greater infectious rate, etc), it will likely remain curtailed by the presence of the dominant strain.

The situation is different in the presence of a vaccine. If the vaccine reduces transmission of the original strain, M0, then the opportunity arises for a circumventing mutation, Ei to spread. Presupposing that the vaccine is ineffective for Ei at X percent of the time, the probability of Ei spreading is the probability of Ei occuring (over time, asymptotically approaches 1) Ei(t), times the ineffectiveness (X): Ei(t) * X. As t approaches infinity, the probability of spreading to a given host goes to X.

Thus, on the one hand, a vaccine can increase the probability of an immune mutation occuring. This is in line with expectations.

In the contrapositive, a vaccine-less disease such as AIDS will decrease in mortality over time, as longer lifespans translate into greater transmission rates. In effect, diseases tend toward an equilibrium with their hosts. Vaccines (and cures, such as antibiotics) increase the probability of mutations that circumvent the treatment or prophylactic. On the other hand, prolonged exposure to a disease weakens it; variants of the disease with lower short-term mortality increase transmission rates.

All of these elements directly affect the mathematics of epidemology, and are relevant to this article, I believe.

Referencing

Hi there. Nice article, but it desperately needs some referencing in there. Not least for the great gods of epidemiological modelling, Bob May and Roy Anderson. Otherwise, it's a good intro to the subject. --Plumbago 17:22, 9 March 2006 (UTC)

Following up on that 2006 comment and seconding the header in the main article, there is still a need for the references in this article to be included in the text. I also worry that this article is a bit too mathsy for the average reader. More work needed on both these fronts. Emble64 (talk) 18:09, 26 December 2012 (UTC)

Partial immunity

How is the maths changed when instead of a proportion q of the population being 100% immune, 100% of the population is q immune? Cyberia. anjurtupil.k

What about the variation in levels of antibody responses.Themaths should change depending on the different levels as they depend on a large number of Biological ,Environmental,agent factors. Kannan

Rename?

Might I suggest that this article would more accurately be named "Mathematical modelling of infectious disease"? Quite a bit of mathematical modelling is done in non-infectious disease epidemiology too. Please let me know what you think by replying here. Qwfp (talk) 18:07, 21 February 2008 (UTC)

Small populations

It would be nice to have some material here about the effects of small populations on infectious diseases - specifically, stochastic fade out. Unfortunately I don't know the topic well enough to tackle it myself. --GenericBob (talk) 07:45, 23 April 2009 (UTC)

TeX code

I found a bunch of stuff that looked like this:

{A_q} = \frac {L} {({L}/{A})({1}-{q})} = \frac {AL} {{L}({1}-{q})} = \frac {A} {{1}-{q}}

This looks like a parody. A feeble attempt to incite laughter, where it's inappropriate. I changed this to this:

A_q = \frac {L} {(L/A)(1-q)} = \frac {AL} {L(1-q)} = \frac {A} {1-q}

There was a huge amount of this stuff in this article. Notice how this gets rendered:

Either way, that's how it gets rendered, but the first way, with things like {{1}-{q}}, just clutters and complicates things for those who edit.

Michael Hardy (talk) 06:34, 10 June 2013 (UTC)

Probabilistic Models

I wonder why nobody is presenting probabilistic models, i.e. something that has noise terms. Are there any in epidemiology? Wikipedia should be the place where these things are discovered. Limit-theorem (talk) 16:01, 19 October 2014 (UTC)

Very much so. It's a matter of updating the article. Adpete (talk) 00:35, 28 June 2017 (UTC)

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Reed and Hampton (1928) preceded by Kermack and McKendrick (1927)?

The article contains the sentence: "Lowell Reed and Wade Hampton Frost developed the Reed–Frost epidemic model to describe the relationship between susceptible, infected and immune individuals in a population'". Go to Reed-Frost model and you will see it was developed in 1928. But surely we should instead mention Kermack–McKendrick theory which has the SIR compartments and was first published in 1927? Adpete (talk) 04:16, 5 September 2016 (UTC)

@Adpete: Why not mention both? :D Mvolz (talk) 17:02, 26 June 2017 (UTC)
I'm fine with both if they were independent, though we should mention the year of each. The two look very similar - I once understood the difference, but now I've got to go and re-read :) Adpete (talk) 23:24, 26 June 2017 (UTC)
And do you know anything about William Hamer? The cite given to him is 1928, which is 18 years later than Ross, and the same year as Reed and Frost. It seems strange then to mention Hamer first, unless he published earlier. Adpete (talk) 00:17, 27 June 2017 (UTC)

6.2 Steady State Solutions Undefined symbols

There are undefined and not explained symbols, mu, delta, and the integration variable x, for example. Could someone please add text to make it more understandable to the non-initiated. Thank you — Preceding unsigned comment added by 67.86.77.196 (talk) 15:02, 17 March 2020 (UTC)

I agee the whole 6.2 section is a load of nonsense without having a copy of the referenced book. It shouls either be rewriten or removed. It looks like a paper using similar parameters is available here https://www.jstage.jst.go.jp/article/sss/2017/0/2017_146/_pdf Please note this is a completely different setup as in 6.1.

Semi-faulty logic

   However, it is important to consider this effect when vaccinating against diseases that are more severe in older people. A vaccination programme against such a disease that does not exceed qc may cause more deaths and complications than there were before the programme was brought into force as individuals will be catching the disease later in life. These unforeseen outcomes of a vaccination programme are called perverse effects.[citation needed]

I see what they're trying to say "because the disease lingers around longer now, people will catch it decades later and die" - but the "more deaths" part is wrong: more deaths will always result from more virulent outbreaks, because more old people will die now, than will the young people now who will later grow old and die in the future.

A different example needs to be found that's true and clear.

Multiple uses of S

S is sometimes used as the proportion of the population that is susceptible. In the SIR model, it is used as the number of people in the population that are susceptible. Could we differentiate so that there is no confusion? Perhaps a lowercase s for proportion? Or a subscript? — Preceding unsigned comment added by Patman Gaughan (talk • contribs) 03:04, 3 April 2020 (UTC)

Variable p in the "Mathematics of mass vaccination" section

Since p is defined by the equation S(∞) = 1 - p, shouldn't p be the fraction of the population that's been infected, and not "the fraction of the final size of the population that is never infected"? That is, as S approaches zero (almost no one is still susceptible), p will approach 1 (i.e. almost everyone has been infected). ErikH17 (talk) 14:19, 19 April 2020 (UTC)

Merge

looks very similar to me. maybe a better distinction would help too. Biggerj1 (talk) 23:42, 2 September 2020 (UTC)

I disagree. Compartmental modeling is a huge topic on its own, particularly in light of the COVID-19 epidemic. If anything, this article on mathematical modeling needs to be expanded to cover all the types of modeling. And just a quick summary of compartmental modeling, and a link to the main article on compartmental modeling.Jaredroach (talk) 23:47, 19 September 2020 (UTC)

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