Fig. 1.

A visualisation of the phase plane generated by Eq. (6) (with $k=0.3$ and $\gamma =0.08$) where black solid and dashed lines correspond to the two branches of equilibrium solution numerically obtained by satisfying Eq. (9). In both panels the heatmap corresponds to the strength and direction of the first derivatives M′(t) (and second, up to a negative constant $-(\mu +{\mu }_1)M^{\prime }\left(t\right)$ — see Eq. (8)) computed through Eq. (6) in the vertical direction (lines of constant R₀), where lighter colours correspond to strongly positive values (strong forces upwards) and darker colours correspond to strongly negative values (strong forces downwards) — consequently, intermediate colours have the weakest forces in either direction. The right panel is a zoomed version of the left panel with enhanced colour contrast for illustration purposes. The increasing timescales to travel between distances of, e.g., $M=[1.0\to 1.1,0.9\to 1.0,0.83\to 0.9]$ for a fixed value of $R_0=2.12$ are $t-t_0\simeq [9.6\text{years},12.9\text{years},23.4\text{years}]$.