I have been working with a compartmental model and I am now trying to compute the basic reproduction number. To do this I must find the spectral radius (dominant eigenvalue) of the following matrix.
$$K=\begin{bmatrix}\frac{\beta_{HH}( \pi_H/ \mu_H)}{(\alpha + \gamma + \mu_H)(\mu_H)}&\frac{\beta_{TH}(\pi_H/\mu_H)}{\mu_T(\mu_T * IT)}&\frac{\beta_{CH}(\pi_H/\mu_H)}{\mu_C(\mu_C * IC)}\\0&\frac{(\beta_{TTV} + \beta_{TTH})(\pi_T/\mu_T)}{\mu_T(\mu_T * IT)}&\frac{\beta_{CT}(\pi_T/\mu_T)}{\mu_C(\mu_C*IC)}\\0&\frac{\beta_{TC}(\pi_C/\mu_C)}{\mu_T(\mu_T * IT)}&0\end{bmatrix}$$
I understand this is done by taking the determinant of $K - \lambda I$. I have found the determinant but cannot correctly compute the eigenvalues. I know when using a $2 \times 2$ Next Generation Matrix there is an easy formula to find the basic reproduction number but I am unsure for a $3 \times 3$. If anyone can give their insight that would be great. Thank you.
The characteristic polynomial of $\pmatrix{a & b & c\cr 0 & d & e\cr 0 & f & 0\cr}$ is $(\lambda - a)(\lambda^2 - d \lambda - e f)$ so the eigenvalues are $a$ and $\dfrac{d \pm \sqrt{d^2+4ef}}{2}$. Assuming the parameters are all positive, the greatest eigenvalue is either $a$ or $(d + \sqrt{d^2+4ef})/2$, whichever is greater.