Why are most epidemic models continuous-time? 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:

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*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?
 A: This is a really interesting question and I doubt that there is any one 'correct' answer to it, but here are my thoughts on the reasons, which can be split into three categories.
History
I suggest you have a look at Section 2.7 Discrete-Time Models in the book Modeling Infectious Diseases in Humans and Animals by Keeling and Rohani. This book points out that most of the literature on epidemic modelling is focussed on continuous time differential equation models, following on from the SIR differential equation model which was first analysed by Kermack and McKendrick in 1927. The authors state

This [focus on differential equation models in our book] is partly because the vast majority of models in the literature are based
on this framework. The inherent assumption has been that the processes of disease
transmission occur in real time and that variability in factors such as the infectious period
may be dynamically important.

However, they also acknowledge that some discrete time models have been developed, for example the "chain binomial" models which assume that there are successive generations of new infections.
Mathematics
Most differential equation models based on the SIR model cannot be solved explicitly. However they can usually be simulated in a straightforward manner, so the lack of easy solvability is not really a problem.
One potential issue (also discussed in Section 2.7 of the book) with discrete time models is that we have to choose a time step, and it may not be obvious what the time step should be. Perhaps the time step should be the 'generation time' i.e. the time between successive individuals being infected in a chain of infection. However, these times may vary between individuals, and may be very uncertain in real applications. So we may not want to embed an assumption regarding the generation time in the formulation of the model. Hence continuous time models can seem more attractive.
Another issue is that the process of formulating a discrete time model can 'fail', in the sense that the discrete time model does not exhibit the expected properties of the real disease system e.g. see Glass, Kathryn, Yincun Xia, and Bryan T. Grenfell. Interpreting time-series analyses for continuous-time biological models—measles as a case study.  Journal of theoretical biology 223, no. 1 (2003): 19-25.
Relating model to data
One apparent motivation for discrete time models is that much of the data are collected at discrete time intervals e.g. daily reports of new infections. However, more sophisticated techniques for model parameterisation have made this consideration less relevant. We can use Bayesian techniques to fit epidemic models with latent variables: we can formulate a continuous time process model (which is not directly observed) and use discrete time data to parameterise it. There are countless examples of this approach in the literature e.g. Ster IC, Singh BK, Ferguson NM (2009) Epidemiological inference for partially observed epidemics: The example of the 2001 foot and mouth epidemic in great britain.  Epidemics 1: 21–34.
A: 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.
