Model Mathematics

tprime = (time 3600 1) mod day_length

M_c = \{\begin{matrix} M_night if \frac{tprime}{3600} \geq t_night \land \frac{tprime}{3600} < t_day \\ M_day otherwise \end{matrix}

\frac{d}{d time} (M) = (- km) (M - M_c)

\frac{d}{d time} (T) = c^{-1} (M - (k (T - T_a)))

k = kb + kinc (T - (T_b (1 + pEtot BR))) + f2_drug

\frac{d}{d time} (BR) = f_prime (E_slow + E_fast) (1 - BR) - (kR BR)

E_slow = AMT_dose pEs2 (gNsTs1 + gNsTs2 + gNsTs3)

E_fast = pEf2 (gNfTf1 + gNfTf2 + gNfTf3)

f2_drug = 0

T_day = T_b + \frac{delta_T}{2}

T_night = T_b - (\frac{delta_T}{2})

kb = \frac{M_b}{T_b - T_a}

M_day = (kb + kinc (T_day - T_b)) (T_day - T_a)

f_prime = delta_high_dose {(1 + ⅇ^{(- alpha) (time - (tdose1 + t_prime))})}^{-1}

M_night = (1 - f_prime) M_night_baseline + f_prime M_day

M_night_baseline = (kb + kinc (T_night - T_b)) (T_night - T_a)

gNsTs1 = \{\begin{matrix} \frac{{Ks}^{Ns}}{6} ⅇ^{(- Ks) X1} {X1}^{Ns - 1} if X1 > 0 \\ 0 otherwise \end{matrix}

gNsTs2 = \{\begin{matrix} \frac{{Ks}^{Ns}}{6} ⅇ^{(- Ks) X2} {X2}^{Ns - 1} if X2 > 0 \\ 0 otherwise \end{matrix}

gNsTs3 = \{\begin{matrix} \frac{{Ks}^{Ns}}{6} ⅇ^{(- Ks) X3} {X3}^{Ns - 1} if X3 > 0 \\ 0 otherwise \end{matrix}

gNfTf1 = \{\begin{matrix} \frac{{Kf}^{Nf}}{6} ⅇ^{(- Kf) X1} {X1}^{Nf - 1} if X1 > 0 \\ 0 otherwise \end{matrix}

gNfTf2 = \{\begin{matrix} \frac{{Kf}^{Nf}}{6} ⅇ^{(- Kf) X2} {X2}^{Nf - 1} if X2 > 0 \\ 0 otherwise \end{matrix}

gNfTf3 = \{\begin{matrix} \frac{{Kf}^{Nf}}{6} ⅇ^{(- Kf) X3} {X3}^{Nf - 1} if X3 > 0 \\ 0 otherwise \end{matrix}

X1 = \frac{time - tdose1}{24}

X2 = \frac{time - tdose2}{24}

X3 = \frac{time - tdose3}{24}

Kf = \frac{Nf}{Tf}

Ks = \frac{Ns}{Ts}