Table 5.

Variables substitutions for the stochastic differential equations for the fluctuations about the endemic equilibrium (2) at time t (A.12)

Variable	Expression
H ∗	$\frac{{\mathit{N}}_{\mathit{h}}\left(\mathit{k}{\left(\mathit{t}\right)}^2{\mathit{N}}_{\mathit{m}}\right(\mathit{t})\mathit{pq}-{\mathit{N}}_{\mathit{h}}\mathit{\delta }(\mathit{t}\left)\mathit{\mu }\right(\mathit{t}\left)\right)}{\mathit{k}\left(\mathit{t}\right)\mathit{q}\left(\mathit{k}\right(\mathit{t}\left){\mathit{N}}_{\mathit{m}}\right(\mathit{t})\mathit{p}+{\mathit{N}}_{\mathit{h}}\mathit{\mu }(\mathit{t}\left)\right)}$
M ∗	$\frac{\mathit{k}{\left(\mathit{t}\right)}^2{\mathit{N}}_{\mathit{m}}\left(\mathit{t}\right)\mathit{pq}-{\mathit{N}}_{\mathit{h}}\mathit{\delta }\left(\mathit{t}\right)\mathit{\mu }\left(\mathit{t}\right)}{\mathit{k}\left(\mathit{t}\right)\mathit{p}\left(\mathit{k}\right(\mathit{t})\mathit{q}+\mathit{\delta }(\mathit{t}\left)\right)}$
a 11	−k(t)p M ∗/N h−μ(t)
a 12	k(t)p(N h−H ∗)/N h
a 21	k(t)q(N m(t)−M ∗)/N h
a 22	−k(t)q H ∗/N h−δ(t)
d	a 11 a 22−a 12 a 21
T	−a 11−a 22
α H	${\mathit{a}}_{22}^2{\mathit{D}}_{11}+{\mathit{a}}_{12}^2{\mathit{D}}_{22}-2{\mathit{D}}_{12}{\mathit{a}}_{12}{\mathit{a}}_{22}$

If the statistics were evaluated about the trajectory of the fast-slow system, then H ^∗ and M ^∗ in the table below are replaced with H(t) and M(t) respectively (e.g., Fig. 4). The expressions for the D _ij coefficients are in Table 6. For fluctuations about an endemic equilibrium, the value of control parameters k(t), N _m(t), μ(t) and δ(t) are constant