Accuracy of a dynamic model of the number of cases of a disease that is spreading

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I'm trying to dynamically model the scenario of a disease spreading across the population, modelling the number of cases, and I would love to hear your feedback.

It's kinda different from the S-I-R model, and its also a crude model.

First, we consider two numbers, NT and NA.
NT is the total number of infected cases, and NA is the number of active infectious cases, those individuals who are actively spreading the disease.

I'm assuming that after a certain fixed length of time, a case becomes no longer infectious, and this length of time is labelled tS.

Making the reasonable assumption that the rate of infection, aka the rate of new cases, or the rate of change of the total number of cases, is directly proportional to the number of active cases, we get the differential equation: $$\frac{dN_{T}(t)}{dt}=k(t)\cdot N_{A}(t)$$


And for the rate of change of the number of active cases: $$\frac{dN_{A}(t)}{dt}=\frac{dN_{T}(t)}{dt}-\frac{dN_{T}(t-t_{S})}{dt}$$


Assuming that k remains constant over time, its possible for this model to have a solution in which the rate of new cases is constant as well.
The function of NT would be just a straight line. And so would the function of NA, a straight and completely horizontal line.

Its possible to have a straight line no matter how large k is, meaning no matter how infectious the disease is.
This seems counter-intuitive when we think about a scenario where each infected person infects something like 3 other people on average.

So what do you think of this counter-intuitive scenario and what do you think of the overall model?
 

Answers and Replies

  • #2
phyzguy
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Making the reasonable assumption that the rate of infection, aka the rate of new cases, or the rate of change of the total number of cases, is directly proportional to the number of active cases, we get the differential equation: $$\frac{dN_{T}(t)}{dt}=k(t)\cdot N_{A}(t)$$
So what do you think of this counter-intuitive scenario and what do you think of the overall model?
I don't find this a reasonable assumption. It assumes an infinite pool of people who can be infected. In reality, eventually the whole population has been infected, and there are no new hosts to which the disease can spread.
 
  • #3
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I don't find this a reasonable assumption. It assumes an infinite pool of people who can be infected. In reality, eventually the whole population has been infected, and there are no new hosts to which the disease can spread.
yes, of course, that's absolutely correct.

however, i'm not trying to assume an infinite pool, i'm just trying to model the early stages of spread

apart from that, may i ask what do you think of the counter-intuitive scenario?
 
  • #4
phyzguy
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I don't find it counter-intuitive. What you have found is that your equations have steady-state solutions, where at any time, the number of people getting sick per unit time equals the number of people getting well(or dying) per unit time. Note that in these steady-state solutions, for a given dNT/dt, k and NA are inversely related. This means a large k with a small number of active cases gives the same dNT/dt as a small k and a large number of active cases.

However, I wouldn't call these steady-state solutions the "early stages of the spread". These are more late time solutions. The early stages would be times before ts, where people are gettng infected, but nobody has gotten well yet. Then dNT(t-ts)/dt = 0. What do you find then?
 
  • #5
715
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I don't find it counter-intuitive. What you have found is that your equations have steady-state solutions, where at any time, the number of people getting sick per unit time equals the number of people getting well(or dying) per unit time. Note that in these steady-state solutions, for a given dNT/dt, k and NA are inversely related. This means a large k with a small number of active cases gives the same dNT/dt as a small k and a large number of active cases.

However, I wouldn't call these steady-state solutions the "early stages of the spread". These are more late time solutions. The early stages would be times before ts, where people are gettng infected, but nobody has gotten well yet. Then dNT(t-ts)/dt = 0. What do you find then?
yup, I had thought about that, when no one is out of the infectious stage yet, the disease can rapidly boom, with the rate of recovered/rip'd people increasingly unable to stem the growing tide

however, imagine if government intervention tried to force a steady-state outcome.

its difficult imagining letting coronavirus infectious people spread freely to the population, and saying, don't worry, its at a steady state.

there would be a huge public outcry, and mass panic that it would ignite the whole pandemic all over again
i have a nagging feeling that something is missing
 
  • #6
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however, imagine if government intervention tried to force a steady-state outcome.

its difficult imagining letting coronavirus infectious people spread freely to the population, and saying, don't worry, its at a steady state.

there would be a huge public outcry, and mass panic that it would ignite the whole pandemic all over again
i have a nagging feeling that something is missing
@phyzguy after considering it more carefully, i've realized that the key is integrating it with the reproduction number/ratio
 
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  • #7
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I'm not clear on whether you want to play with constructing your own model using a particular set of skills/templates, or whether you are really interested in accurate modelling of epidemics. Both are valid activities.

However, if you want an accurate model, this is the time to go out to the web and search for how the epidemiologists do it; no need to reinvent this yourself. The simple model they use is the SIR model, which you can learn about here (https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology#The_SIR_model), or on youtube, where there are lots of videos like this ().

Unfortunately, all of the good models (even the simple ones) are non-linear and can really only be solved numerically. Further more, they are heavily dependent on the data you input, which is typically unknown, of poor quality, and changes throughout the real epidemic. This is one reason why, if you want real answers, you ask an epidemiologist, that's their job and it is complicated.
 
  • #8
715
17
I'm not clear on whether you want to play with constructing your own model using a particular set of skills/templates, or whether you are really interested in accurate modelling of epidemics. Both are valid activities.

However, if you want an accurate model, this is the time to go out to the web and search for how the epidemiologists do it; no need to reinvent this yourself. The simple model they use is the SIR model,
I'm trying to see how things work if we add the specific infectious duration to the SIR model, how do you find mine?

also I think I should integrate it with the reproduction number
 

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