**Problem:** Given the equation: $$\beta S_0 \sigma \phi_e \phi_i + \beta S_0 \eta_1 p \delta h \sigma \phi_e \phi_i \phi_h + \beta S_0 \eta_2 \kappa \delta t \sigma \phi_e \phi_i \phi_t + \beta S_0 \eta_3 \sigma(p \delta h m h d \phi t + \kappa \delta t m t d \phi h + \eta \phi_h \phi_t) \phi_e \phi_i \phi_h \phi_t \phi_c$$ where $$\kappa=1-p$$ $$\phi_e = \sigma + \mu$$ $$\phi_i = \alpha + p \delta_h + \kappa \delta_t + \mu + \mu_i$$ $$\phi_h = \gamma + \mu_h + \mu + m h d$$ $$\phi_t = m t d + \mu + \mu_t + \psi$$ $$\phi_c = \xi + \mu_c + \mu$$ Make $\beta$, $\sigma$, $\eta_1$, and $\mu$ the subject of the formula. --- **Step 1: State the expression clearly and define variables** Let $$A = \beta S_0 \sigma \phi_e \phi_i$$ $$B = \beta S_0 \eta_1 p \delta_h \sigma \phi_e \phi_i \phi_h$$ $$C = \beta S_0 \eta_2 \kappa \delta_t \sigma \phi_e \phi_i \phi_t$$ $$D = \beta S_0 \eta_3 \sigma (p \delta_h m h d \phi_t + \kappa \delta_t m t d \phi_h + \eta \phi_h \phi_t) \phi_e \phi_i \phi_h \phi_t \phi_c$$ So the equation can be written as $$A + B + C + D$$ Our goal is to isolate $\beta$, $\sigma$, $\eta_1$, and $\mu$. --- **Step 2: Isolate $\beta$** Note that each term contains a factor $\beta S_0$. Factorize this: $$\beta S_0 \sigma \phi_e \phi_i [1 + \eta_1 p \delta_h \phi_h + \eta_2 \kappa \delta_t \phi_t + \eta_3 (p \delta_h m h d \phi_t + \kappa \delta_t m t d \phi_h + \eta \phi_h \phi_t) \phi_h \phi_t \phi_c]$$ Let the bracket be $K$: $$K = 1 + \eta_1 p \delta_h \phi_h + \eta_2 \kappa \delta_t \phi_t + \eta_3 (p \delta_h m h d \phi_t + \kappa \delta_t m t d \phi_h + \eta \phi_h \phi_t) \phi_h \phi_t \phi_c$$ So the whole original expression is: $$\beta S_0 \sigma \phi_e \phi_i K$$ Then for a given value on the right hand side (say $R$), we have: $$R = \beta S_0 \sigma \phi_e \phi_i K$$ Hence: $$\beta = \frac{R}{S_0 \sigma \phi_e \phi_i K}$$ --- **Step 3: Make $\sigma$ the subject** Substitute $\phi_e = \sigma + \mu$ Write $\beta S_0 \sigma \phi_i K (\sigma + \mu) = R$ Rearranging: $$R = \beta S_0 \phi_i K \sigma (\sigma + \mu)$$ Rewrite as: $$\beta S_0 \phi_i K (\sigma^2 + \mu \sigma) = R$$ Which gives quadratic in $\sigma$: $$\sigma^2 + \mu \sigma - \frac{R}{\beta S_0 \phi_i K} = 0$$ Solve for $\sigma$ using quadratic formula: $$\sigma = \frac{-\mu \pm \sqrt{\mu^2 + 4 \frac{R}{\beta S_0 \phi_i K}}}{2}$$ --- **Step 4: Make $\eta_1$ the subject** Recall from $K$: $$K = 1 + \eta_1 p \delta_h \phi_h + \eta_2 \kappa \delta_t \phi_t + \eta_3 (...)$$ Isolate $\eta_1$: $$K - 1 - \eta_2 \kappa \delta_t \phi_t - \eta_3 (...) = \eta_1 p \delta_h \phi_h$$ So: $$\eta_1 = \frac{K - 1 - \eta_2 \kappa \delta_t \phi_t - \eta_3 (p \delta_h m h d \phi_t + \kappa \delta_t m t d \phi_h + \eta \phi_h \phi_t) \phi_h \phi_t \phi_c}{p \delta_h \phi_h}$$ --- **Step 5: Make $\mu$ the subject** Recall $\phi_e = \sigma + \mu$ Also, $\phi_i = \alpha + p \delta_h + \kappa \delta_t + \mu + \mu_i$ $\phi_h$, $\phi_t$, $\phi_c$ also depend on $\mu$ but only $\phi_h = \gamma + \mu_h + \mu + m h d$, $\phi_t = mtd + \mu + \mu_t + \psi$, and $\phi_c = \xi + \mu_c + \mu$ depend on $\mu$ If we want to solve $\mu$ explicitly, the equation becomes nonlinear and more complex requiring implicit or numerical methods. Rewrite the expression for $R$ in terms of $\mu$ explicitly as: $$R = \beta S_0 \sigma (\sigma + \mu) (\alpha + p \delta_h + (1-p) \delta_t + \mu + \mu_i) K$$ where $K$ also depends on $\mu$ via $\phi_h, \phi_t, \phi_c$. The explicit formula for $\mu$ involves solving a nonlinear equation; further data or numerical approach is needed. --- **Final answer summary:** $$\beta = \frac{R}{S_0 \sigma \phi_e \phi_i K}$$ $$\sigma = \frac{-\mu \pm \sqrt{\mu^2 + 4 \frac{R}{\beta S_0 \phi_i K}}}{2}$$ $$\eta_1 = \frac{K - 1 - \eta_2 \kappa \delta_t \phi_t - \eta_3 (p \delta_h m h d \phi_t + \kappa \delta_t m t d \phi_h + \eta \phi_h \phi_t) \phi_h \phi_t \phi_c}{p \delta_h \phi_h}$$ $$\text{where } K = 1 + \eta_1 p \delta_h \phi_h + \eta_2 \kappa \delta_t \phi_t + \eta_3 (p \delta_h m h d \phi_t + \kappa \delta_t m t d \phi_h + \eta \phi_h \phi_t) \phi_h \phi_t \phi_c$$ $\mu$ requires solving an implicit nonlinear equation due to its presence in multiple $\phi$ terms.