### Derivation of VRTP formula

For mathematical consideration of VRTP, we derive the VRTP formula with three theoretical assumption model. Those are the fully-mixed model (mean field model), the pairwise approximation model and the degree-based model.

For fully-mixed model based on ‘mass action principle’[16], we solved the nonlinear differential Eq. (1) algebraically remaining *r* at TP. The exact form of *r* at TP is like following,

$$ {r}_{TP}=\frac{ \ln \left({s}_0{R}_0\right)}{R_0} $$

(2)

where *r*
_{
TP
} is VRTP and *R*
_{
0
} is reproduction number, *s*
_{
0
} is the initial value of the susceptible population.

Likewise, we solve the differential equations in the pairwise approximations model or moment closure method [17] (configuration model),

$$ <{r}_i\left( t= TP\right)>=\frac{ \ln \left(<{s}_i(0)>{R}_0{k}_i\right)}{R_0{k}_i} $$

(3)

where *k*
_{
i
} is the degree of the i^{th} node.

And in the degree-based model [18],

$$ {\displaystyle \sum_{k=0}^{\infty }}{q}_k{r}_k(t)=\frac{ \ln \left({R}_0 k{s}_k(0)\right)}{R_0 k} $$

(4)

where *q*
_{
k
} is the probability that a vertex with degree *k* is present.

As we can see, the whole form of the functions is like *ln(x)/x* for *s*
_{
0
} ~ 1 and *x* is the reproduction number. The following figure (Fig. 7) shows the change of VRTP values by reproduction number.

From the result of theoretical consideration, finally, we can guess the range of the VRTP values in the SIR spreading model. In observation of the function form of VRTP, we can see that VRTP is low than *1/e* ~ 37% at *R*
_{
0
}
*= e*.

So, the range of VRTP is [0, *1/e*] and epidemic characteristics is divided into two regions, lower (*R*
_{
0
} < *e*) and upper (*R*
_{
0
} > *e*) region. The VRTP increases by *R*
_{
0
} till the *R*
_{
0
}
*= e* and after that point, *R*
_{
0
} decreases by *R*
_{
0
}. From the value of recovered population has the upper bound of 30% of the whole population when the infected population is maximum, we concluded that before the recovered being under 37%, and infected population would be decreased.

### VRTP surface and curves

As far as we know, the reproduction number *R*
_{
0
} consists of two parameters, contagiousness *β* and recovery rate γ and specifically *R*
_{
0
}
*= β/γ*. We did the epidemic simulation by the parameters of two epidemic parameters, contagiousness *β* and recovery rate *γ* and of one network structure parameter *k*. From the simulation result, we constructed the distribution of VRTP and calculated the representative mean value of VRTP. Then we constructed the surface of VRTP and observed the change of VRTP varying those epidemic parameters.

With both *β* and *γ*, VRTP increased rapidly from 0 to maximum value and decreased. The surface of VRTP shows some fluctuations while k increases, and the location of the peak of VRTP moves toward *β* = 0, *γ* = 0. Also, VRTP always has its maximum value below 30% of the whole population (Fig. 8).

If we magnify the surface of *k* = 2 and *k* = 10, we can observe the area of the low beta area, we can see the smooth change between two curves of low *β* with sustaining same function form (Figs. 9 and 10).

With network parameter *k*, VRTP surface changes drastically in the region of low γ_{d}. If we magnify the surface and observe the curve of low γ_{d} by *k*, we do not miss that the change between VRTP curves of low γ_{d} with changing the function form (Figs. 11 and 12).

We can observe same findings again in the VRTP surfaces with varying k (Fig. 13), which we find out in the previous curves.