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Most classical epidemic models such as SIR and variants are formulated as differential equations. However, to me discrete-time models feel more natural to measure the evolution of a disease on a day-by-day basis:

  • human activity come in daily cycles, so that most people are active by day and sleep at night, hence infections have at minimum a highly daily seasonality.
  • all available data such as the number of new positive tests have a time resolution of one day.
  • since there is a long delay between infection and infectivity period, the behavior on a very short time-scale is unlikely to matter.
  • in the end, differential equations have to be discretized to simulate them. Even if one can do more sophisticated things than simple time-stepping (Runge-Kutta methods, for instance), the seasonality of the data and the fact that we rarely need to simulate for longer than approx. 100-day period suggest that the time accuracy obtained in solving a differential equation exactly is unlikely to matter.
  • checking if a continuous-time model converges to zero is no simpler than checking if a discrete-time one does.

So why do people use SIRs and variants rather than equivalent discrete-time models? What is the advantage I am missing?

Federico Poloni
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    Using *differential equations* to model disease sounds more sophisticated than using *difference* equations - the first sounds like rocket science, the second like grade school... – Stephan Kolassa Feb 28 '21 at 14:43
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    Differential equations may be easier to solve than difference equations. This motivates (at least part of) the use of continuous time models in finance, so perhaps it might be relevant here? – Richard Hardy Feb 28 '21 at 15:08

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This is an interesting question.

  • I mean, I guess you could ask the same thing about derivatives. What does it really mean to be going 60 kms per hour instantaneously if speed is distance travelled per unit time? It seems inherently discrete (measuring distance over a specific interval of time, even if that interval is very small) and yet the derivative continues to be used.

  • I largely suspect that the reason differential equations are used over difference equations is not a matter them being "better" but because they were developed to further understand the dynamics of the simple epidemics that motivated their development. If difference equations operate on the unit of days, how can one ask about the concept of an $\mathcal{R}_0$? Under the difference equation model, a single person might infect multiple people in a single day, who may then go on to infect several more people. How can we parse out how many new infections an index case creates without examining the dynamics as the time step we take becomes infinitesimally small?

Because the SIR model and other models like it have been studied for nigh a century, I highly suspect an answer exists "out there". It might be beneficial to start here in which the author begins with the difference equations for an epidemic and derives the SIR model from them.

Demetri Pananos
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  • R_0 is originally defined as the average number of people an infected individual passes the illness to (over all the time they are infected). This definition works perfectly fine also in discrete time, doesn't it? – Federico Poloni Feb 28 '21 at 16:20
  • @FedericoPoloni I think that is sufficiently close to the definition I use. What if the time the individual is infectious is not measured in days, but fractions of a day (say, 2.5)? My point is that since the model is measured in days, it might be awkward to interperet measures like R0, especially if the length of infectiousness is something like 2.38 days. These concerns likely necessitated smaller time steps, after which it is natural to ask about the dynamics in the limit as the time steps go to 0 from above, which yield ODEs. – Demetri Pananos Feb 28 '21 at 22:04
  • I don't think this needs a continuous-time model at all. Most likely you are modelling the infectious time by introducing in your model a certain probability of recovering for each infected individual, and this can be done equally well in both the continuous and the discrete-time model, giving a non-integer average infectious time. – Federico Poloni Mar 01 '21 at 13:26