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## Main Question or Discussion Point

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,

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

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

The function of

Its possible to have a straight line no matter how large

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?

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

First, we consider two numbers,

**N**and_{T}**N**._{A}**N**is the total number of infected cases, and_{T}**N**is the number of active infectious cases, those individuals who are actively spreading the disease._{A}I'm assuming that after a certain fixed length of time, a case becomes no longer infectious, and this length of time is labelled

**t**._{S}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

**N**would be just a straight line. And so would the function of_{T}**N**, a straight and completely horizontal line._{A}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?