The usual situation is that, if R0 1, there is a locally asymptotically stable EE and the infection persists.

As R0 increases through 1, there will be an exchange of stability between DFE and EE. The change in the equilibrium behaviour at R0=1 is called forward bifurcation, it is similar to transcritical bifurcation (just exchange in stability).

The case when there are multiple EE when R0

I don't think these terms are clearly defined, but most bifurcations have a standard way in which the 1-parameter diagram is presented. For instance if somebody says (supercritical) Hopf bifurcation, you normally think of a stationary state losing stability while a stable limit cycle emerges. If you see it in the opposite way (a stable limit cycle converging onto an unstable state) you see the bifurcation backwards so to speak. In the transcritical bifurcation forwards and backwards, look qualitatively the same (two stationary states in two stationary states out) So it looks the same forwards or backwards.

Forward and backward are just a matter of one's perspective and have no unique definition. In general, the parameter space has many dimensions, and it is often useful to examine the dynamics along some one-dimensional path in that space which need not be a straight line. Traversing the line in both directions without reinitializing the system allows one to observe any regions of hysteresis which is one signature of multistability.

Thank you all for your contributions. They're all very helpful

Thanks for your support.

It is an interesting question. From the view of mathematics, I think there should be no difference, but from the perspective of the evolution of the dynamic system, there may be some differences, which can be verified by simulation experiments.

Forward bifurcation and backward bifurcation are both types of bifurcations that can occur in a dynamical system. The main difference between the two is the direction of the bifurcation.

In a forward bifurcation, a stable equilibrium point loses stability as a parameter is varied, and two new equilibrium points, one stable and one unstable, are created. The stable equilibrium point moves forward in parameter space as the parameter is increased. This means that a small change in the parameter can cause the system to transition from one equilibrium point to the other.

In a backward bifurcation, a stable equilibrium point loses stability as a parameter is varied, but instead of creating two new equilibrium points, the system moves back to a region where the original stable equilibrium point is still stable. This means that the system can exhibit hysteresis, where a small change in the parameter can cause the system to switch between two stable equilibrium points.

Transcritical bifurcation is another type of bifurcation that can occur in a dynamical system. In this type of bifurcation, a stable equilibrium point of the system collides with an unstable equilibrium point and they exchange stability. In other words, the system undergoes a qualitative change in its behavior as a parameter is varied.

There is a relationship between forward bifurcation and transcritical bifurcation in that a forward bifurcation can lead to a transcritical bifurcation if the stable and unstable equilibrium points created in the forward bifurcation collide and exchange stability. However, there is no direct relationship between backward bifurcation and transcritical bifurcation.