There is more to life than ODEs but my question is why these reports do not discuss these problems as dynamical systems ("ODEs++") so that other questions can be answered beyond numerical ODEs. I wonder why no mention is made of the Lyapunov, Poincare-Bendixson theory etc. What happens to the ODEs when [$]t \rightarrow \infty[$] (sounds like a logical question). A Sapir-Whorfe problem ("give hammer then all is a nail")
They do. Many epidemiologists use network models of contagion / directed percolations
. The critical behaviour is at the threshold between the active and the absorbing regime (cf R0).
Funny, in the other forum I saw some Jan Dash posting papers on Regge field theory (from high energy physics, any CERN people here to give more insight?) as "generalisation" of a Brownian motion. RFT, Brownian motions and the network model of epidemic/directed percolation mentioned above are in the same universality class - every model with an underlying stochastic Markov process qualifies to this class (every reaction-diffusion system: contagion, reacting chemical substances, 2nd order phase transitions like ferromagnetic - note percolation threshold vs critical, and so on. Scientists are copycats...
Anyway, let's assume a naive model the contagion as an isotropic percolation, i.e. Bethe lattice
A classic result is that large clusters and long-range connectivity arise when the probability of "jumping" to the next node is Pc = 1/(z-1) (*), where z is the coordination number. In our model the latter corresponds to the number of regular contacts made by an infectious (it's a static model - contacts don't change).
Assume that for Covid the probability of "jumping", i.e. infecting the contact, is 15% per each meeting. This probability is due to the Poisson process with intensity given by the equation 0.15 = 1-exp(-lambda*1), which yields lambda = -ln(0.85).
Say the disease period is 21 days and you meet all your contacts once a day. Then the probability that you infect each contact accumulates to P(21 days) = 1-exp(ln(0.85)*21) = 1 - 0.85^21. For this probability to be the percolation/epidemic outbreak threshold, it takes z susceptible contacts given by Eq (*), which is 2.0340679919043606448571830240222 persons.
Don't tell Boris!