**Appendix A. Review of Delay Models**

Time delay in epidemic models accounts for the delay for an individual to become infectious after being infected. Time delay for vector-born diseases characterizes the time needed for the pathogen to reach a certain threshold sufficient for infection transmission. Under the assumption that the infection transmission rate at time *t* depends on the number of infected at time *t* − *τ* and the number of susceptible at time *t*, the delay SIR model for vector-born diseases is given by the following system of equations (see [41] and the references therein):

$$\begin{array}{rcl}\frac{dS(t)}{dt} &=& -f(S(t), I(t-\tau)),\\\frac{dI(t)}{dt} &=& f\left(S(t), I(t-\tau)\right) - \delta I(t).\end{array}$$

where *S*(*t*) represents susceptible and *I*(*t*) infectious compartments, the function *f*(*S*(*t*), *I*(*t* − *τ*)) characterizes the disease transmission rate, *δ* is the clearance rate, and *τ* is the time delay from the moment of infection to disease transmission. Another type of delay model describes temporary immunity [28]:

$$\begin{array}{rcl}\frac{dS(t)}{dt} &=& -f\left(S(t), I(t)\right) + \delta I(t-\omega),\\ \frac{dI(t)}{dt} &=& f\left(S(t), I(t)\right) - \delta I(t),\end{array}$$

where *ω* is the time period after which an infected individual becomes susceptible again.

For long-lasting epidemics such as HIV, new generations of susceptible individuals influence epidemic dynamics. In this case, time delay corresponds to the maturation period after which young adults become susceptible to infection (see [42–44] and the references therein):

$$\begin{aligned} \frac{dS(t)}{dt} &= \mathcal{J}(N(t-\tau)) - h(\{S(t), I(t)\}),\\ \frac{dI(t)}{dt} &= \mathcal{J}(S(t), I(t)) - \delta I(t). \end{aligned}$$

Here, *N*(*t*) = *S*(*t*) + *I*(*t*), *g*(*N*(*t* − *τ*)) is the susceptible recruitment function, which incorporates the maturation delay *τ* and *h*(*S*, *I*) is the disease transmission rate.
