Fig. 4.

(left) Plot of the asymptotic total affected population fraction, $\overline{R}/N,$ as a function of the reproductive number $R_0$. The parameters used are ${\beta }_a=0.33,{\beta }_p=0.5,\sigma =0.33,{\gamma }_a=0.125,{\gamma }_p=0.083,r=0.0\text{and}u=1.0$. We also plot the quantity $(I^{\left(m\right)}/N)(\sigma +{\gamma }_e)/\sigma ,$ obtained numerically from many different parameter sets, and compare it with the theoretical predicted curve $1-(1+\ln R_0)/R_0$ (green line). (right) Verification of the $\ln \left(N\right)$ dependence of $t^{\left(m\right)}$ in Eq.  (40) for different choices of $R_0$. The slopes of the straight lines compares well with ${\mu }^{-1}$ as stated in Eq.  (40). The parameters used which correspond to Eqs.  (34)–(37) are $\sigma =1/3,{\gamma }_e=0.1$ for all values of $R_0$ and $\beta =R_0{\gamma }_e$. The dominant eigenvalue $\mu$ are computed using the linearised version of Eqs.  (34)–(37) and slopes are obtained from the data. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)