One might wonder why dI/dt is proportional to I squared and not just I. It could be that the number of potentially infected individuals in Q around the infected individuals is proportional to I as well.
If the red circles in the figure above represent a number of infected individuals and the blue circles those around them then as each red circle grows the number of uninfected individuals remaining in the group or clique will decrease as the virus spreads. That might explain the devreasind exponential term. The negative constant term is likely to be a removal rate of infected individuals as in the SIR model. In the SIR model dI/dt equals rS-a. In the QI model rS is replaced by BI times the exponential term and -a by AI. A zero for (1/I)dI/dt may still be a good indicator of having reached a peak for I. The proportionality factor for I in Q might be thought of as a transfer or transport factor.
It seems unlikely that a removal rate is proportional to I squared so replacing the -a in the SIR model by just a constant A may work better conceptually. This might be the modified SIR model we were looking for.
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