# Table 1 Expressions for derivatives in all models

Ross

Macdonald

McKenzie

S h

-

-

q R h (t)-h I m (t)S h (t)

e h ,E h

-

-

h I m (t)S h (t)-k E h (t)

i h ,I h

a b m i m (t)(1-i h (t))

a b m i m (t)(1-i h (t))

k E h (t)-p I h (t)

-

-r i h (t)

-r i h (t)

r h ,R h

-

-

p I h (t)-q R h (t)

S m

-

-

f-h I h (t)S m (t)-d S m (t)

e m ,E m

-

a c i h (t)(1-e m (t)-i m (t))-μ2e m (t)

h I h (t)S m (t)-g E m (t)-d E m (t)

-

-

$-{\mathit{\text{aci}}}_{h}\left(t-{\tau }_{m}\right)\left(\right1-{e}_{m}\left(t-{\tau }_{m}\right)-{i}_{m}\left(t-{\tau }_{m}\right)\left)\right{e}^{-{\mu }_{2}{\tau }_{m}}$

i m ,I m

a c i h (t)(1-i m (t))

a c i h (t-τ m )(1-e m (t-τ m )-μ2i m (t)

g E m (t)-d I m (t)

-

$-{i}_{m}\left(t-{\tau }_{m}\right)\left)\right{e}^{-{\mu }_{2}{\tau }_{m}}$

Anderson/May

e h ,E h

$-{\mathit{\text{abmi}}}_{m}\left(t-{\tau }_{h}\right)\left(\right1-{e}_{h}\left(t-{\tau }_{h}\right)-{i}_{h}\left(t-{\tau }_{h}\right)\left)\right{e}^{-{\tau }_{h}\left(r+{\mu }_{1}\right)}\right)$

i h ,I h

a b m i m (t-τ h )(1-e h (t-τ h )

$-{i}_{h}\left(t-{\tau }_{h}\right)\left)\right{e}^{-{\tau }_{h}\left(r+{\mu }_{1}\right)}-{\mathit{\text{ri}}}_{h}\left(t\right)-{\mu }_{1}{i}_{h}\left(t\right)$

e m ,E m

a c i h (t)(1-e m (t)-i m (t))-μ2e m (t)

-

$-{\mathit{\text{aci}}}_{h}\left(t-{\tau }_{m}\right)\left(\right1-{e}_{m}\left(t-{\tau }_{m}\right)-{i}_{m}\left(t-{\tau }_{m}\right)\left)\right{e}^{-{\mu }_{2}{\tau }_{m}}$

i m ,I m

a c i h (t-τ m )(1-e m (t-τ m )-μ2i m (t)

-

$-{i}_{m}\left(t-{\tau }_{m}\right)\left)\right{e}^{-{\mu }_{2}{\tau }_{m}}$

Chitnis

S h

q R h (t)

-h I m (t)S h (t)

e h ,E h

$\left(\right\frac{{\sigma }_{m}{\sigma }_{h}{N}_{m}\left(t\right){b}_{\mathit{\text{hm}}}{i}_{m}\left(t\right)}{{\sigma }_{m}{N}_{m}\left(t\right)+{\sigma }_{h}{N}_{h}\left(t\right)}\left)\right\left(\right1-{e}_{h}\left(t\right)-{i}_{h}\left(t\right)-{r}_{h}\left(t\right)\left)\right$

$-\left(\right{\nu }_{h}+{\psi }_{h}+\frac{{\Lambda }_{h}}{{N}_{h}\left(t\right)}\left)\right{e}_{h}\left(t\right)+{\delta }_{h}{i}_{h}\left(t\right){e}_{h}\left(t\right)$

i h ,I h

${\nu }_{h}{e}_{h}\left(t\right)-\left(\right{\gamma }_{h}+{\delta }_{h}+{\psi }_{h}+\frac{{\Lambda }_{h}}{{N}_{h}\left(t\right)}\left)\right{i}_{h}\left(t\right)$

+δ h i h (t)2

r h ,R h

${\gamma }_{h}{i}_{h}\left(t\right)-\left(\right{\rho }_{h}+{\psi }_{h}+\frac{{\Lambda }_{h}}{{N}_{h}\left(t\right)}\left)\right{r}_{h}\left(t\right)+{\delta }_{h}{i}_{h}\left(t\right){r}_{h}\left(t\right)$

N h

Λ h +ψ h N h (t)-(μ1h+μ2hN h (t))N h (t)-δ h i h (t)N h (t)

e m ,E m

$\left(\right\frac{{\sigma }_{m}{\sigma }_{h}{N}_{h}\left(t\right)}{{\sigma }_{m}{N}_{m}\left(t\right)+{\sigma }_{h}{N}_{h}\left(t\right)}\left)\right\left(\right{b}_{\mathit{\text{mh}}}{i}_{h}\left(t\right)+{\stackrel{~}{b}}_{\mathit{\text{mh}}}{r}_{h}\left(t\right)\left)\right\left(\right1-{e}_{m}\left(t\right)-{i}_{m}\left(t\right)\left)\right$

i m ,I m

ν m e m (t)-ψ m i m (t)

N m

ψ m N m (t)-(μ1m+μ2mN m (t))N m (t) 