Supplementary Information for “ Parameter Scaling for Epidemic Size in a Spatial Epidemic Model with Mobile Individuals ”

1. Variability of the final epidemic size near the epidemic threshold To characterize the sample-to-sample variability near the epidemic threshold, we computed the susceptibility measure defined as χ = N(〈r∞ 2〉 − 〈r∞〉 ) 〈r∞〉 ⁄ and the variance measure defined as Δ = √〈r∞〉 − 〈r∞〉/〈r∞〉, where N is the system size, r∞ represents the final epidemic size, and the brackets indicate the sample average over the 100 trials.

For the susceptibility measure, we set = 1 for simplicity because the system size is fixed at = 100 and = 10 5 . These two measures are plotted against the proposed index as shown in Fig A. The results show that the range of (between approximately 1 and 100) for large values of the susceptibility measure (Fig A, Left) is in better agreement with that for the large standard deviation of the final size in Fig 6 than that for the large values of the variance measure (Fig A, Right). Therefore, it is possible that the epidemic threshold exists within the range of and the susceptibility measure captures the large fluctuations near the threshold.

Model
In the main text, the destination of the hopping of each individual is limited to the 8 sites surrounding the current site. However, in a more realistic case, the hopping to more distant sites would be possible. To investigate how our results change in such a case, here we consider the hopping to an extended area.
As illustrated in Fig B, the destination of the hopping includes the 16 more distant sites (dashed arrows) in addition to the original 8 sites (solid arrows). When susceptible, exposed, and recovered individuals are located in a site (i, j), each individual can hop to one of the 8 sites at (i ± 1, j ± 1), (i ± 1, j), and (i, j ± 1), with hopping rate or one of the 16 sites at (i ± 2, j ± 2), (i ± 2, j ± 1), (i ± 2, j), (i ± 1, j ± 2), and (i, j ± 2) with hopping rate /2. For infectious individuals, the hopping rates are multiplied by , where 1 − represents the mobility reduction rate. Each of the susceptible (S), exposed (E), and recovered (R) individuals hops to one of the dashed arrows with rate λ/2. For infectious individuals, the hopping rate is multiplied by the factor where 1 − represents the mobility reduction rate.

Correlation between the characteristic length and the transport distance
For the hopping to the extended area, the typical distance that an individual moves with a hopping rate λ during time τ is given by 2√14 which is derived in the same manner as in Appendix in the main text. Therefore, the characteristic length ext * , which represents the effective range that the pathogens are transported during the latent and infectious periods, and , is given as follows: ext * = 2√14 ( + (1 + ) ).
(S1) We can confirm the positive correlation between ext * and the transport distance d

Relationship between the index and the final size ∞
In the main text, the final size of epidemics ∞ is approximated by a function of the index as shown in Fig 6. In the cases with hopping to the extended area, the index can be rewritten using the characteristic length ext * as follow: ext ≡ ext * 2 0 .

Discussion
As an example of extended mobility, we have considered the cases of hopping to the extended area. The maximum distance that each individual can move at a unit time is doubled, compared with that for the hopping to neighbouring sites as studied in the main text. We have derived the characteristic length and shown that the transport distance that pathogens are carried by an infected individual is proportional to the characteristic length. Therefore, the index associated with the final epidemic size can be similarly formulated as in the case of the main text, which can predict the effect of parameter scaling on the final size. It is an interesting future issue to consider an index for other types of spatial mobility.