Fig. 3.

(a) Regions in the phase plan $(J,S)$ induced by Eqs. (9), (10) and example of trajectory accounting for epidemic waves induced by ARI (dashed line). (b) Proportion of individuals infected by influenza (solid line) and by ARI (dashed line) over time. A “very small” epidemic wave, not shown here, arises at $t\approx 3300$. Parameters employed: $\beta =0.27$, $\gamma =0.25$, $\alpha =0.104$, $\delta =0.1$, $\rho =100$, $t_0=t_0^{\mathrm{ari}}=0$. (c) Effective reproduction number (see Eq. (8)) over time for simulation in (b). (d) Solid line represents the possible solutions of the equation $R_e(J,S)=1$ (Eq. (9)) and filled circles represent actual pairs of values $(J,S)$ at the beginning of each wave in (b) (but for the first one, which is not a solution of $R_e(J,S)=1$). Diamond refers to the 6th (not shown) wave. Dashed line represents the trajectory of $(J(t),S(t))$.