Is there any possibility that the next generation matrix method fails in finding R0
I am afraid I don't understand your question at all. What do you mean by next generation matrix method, and what, in this context, is R0
R0 is the basic reproduction number and it represents the number of secondary infections that occur when an infected individual is introduced into a completely susceptible population, and it is used in mathematical epidemiology to determine the possibility of the invasion of a disease.
The next generation matrix method is a method proposed by: O. Diekman, J.A.P. Heesterbeek, J.A.J. Metz.
Yes, the next generation matrix method is based on Lyapunov indirect method, which is only reliable when you are dealing with Nonlinear autonomous systems. When you are dealing with Nonautonomous systems (e.g. when the transmission rate explicitly depends on time), the Lyapunov indirect method is not available, thus the next generation matrix fails. At this moment, you may refer to Lyapunov direct method, which is way more complicated.
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