How the individual human mobility spatio-temporally shapes the disease transmission dynamics

Human mobility plays a crucial role in the temporal and spatial spreading of infectious diseases. During the past few decades, researchers have been extensively investigating how human mobility affects the propagation of diseases. However, the mechanism of human mobility shaping the spread of epidemics is still elusive. Here we examined the impact of human mobility on the infectious disease spread by developing the individual-based SEIR model that incorporates a model of human mobility. We considered the spread of human influenza in two contrasting countries, namely, Belgium and Martinique, as case studies, to assess the specific roles of human mobility on infection propagation. We found that our model can provide a geo-temporal spreading pattern of the epidemics that cannot be captured by a traditional homogenous epidemic model. The disease has a tendency to jump to high populated urban areas before spreading to more rural areas and then subsequently spread to all neighboring locations. This heterogeneous spread of the infection can be captured by the time of the first arrival of the infection \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(T_{fi} )$$\end{document}(Tfi), which relates to the landscape of the human mobility characterized by the relative attractiveness. These findings can provide insights to better understand and forecast the disease spreading.

(S1) where , , " , $ , are the number of susceptible, exposed, symptomatic infectious, asymptomatic infectious, and recovered individuals, respectively. is a disease transmission rate, and is a scaling parameter taking into account the reduced infectiousness of asymptomatic infectious individuals. ( is the probability of an exposed individual to become an asymptomatic infectious individual. is an incubation rate, which is inversely proportional to the incubation period. is a recovery rate, which is inversely proportional to the infectious period, and is the total population size, = + + " + $ + . We found that the dynamics of human mobility can affect epidemic profiles. As compared to the homogeneous model, spatial heterogeneity arising due to human mobility can delay the epidemic peak of up to 15 days with a flatter and wider epidemic curve ( Figure S1a). This slower speed of the epidemic spread is partly due to an effect of the waiting time, which retards the movement of infectious individuals. Moreover, the epidemic model that integrates human mobility predicts 8.7% less the total number of recovered individuals at the equilibrium as compared to the homogeneous model ( Figure S1b). used to simulate the model stochastically.

Homogeneous epidemic model: Deterministic versus stochastic approaches
We found that the stochasticity in disease transmission is dominated when the number of infectious individuals is very low, especially at the early time of the epidemics. However, the results of the deterministic simulations will be identical to the results of the stochastic simulations when the number of initial infectious individuals is sufficiently high. As shown in Figure S2, for the deterministic simulations, using lower numbers of initial infectious individuals only shift the epidemic peak to the right but does not significantly affect the shape of the curve. In contrast, using lower numbers of infectious individuals in the stochastic simulations does not only shift the epidemic peak to the right but also decreases the epidemic peak and widens the epidemic curve.
This is because when the number of initial infectious individuals is very low, there is a very high chance for the disease to become extinct before a sustained chain of transmission can be

Estimating the waiting time parameters
In our human mobility model, we assumed that the waiting time distribution P + (∆ ) (equation 1 in the main text) is identical for all locations i. Thus, an individual will stay at a particular location for a period of time of ∆ drawn from the following truncated power law The truncated power law distribution is governed by two parameters, namely, an exponent ( ), approximately 3 km 2 , which is smaller than the grid size of 25 km 2 used in our study, we, therefore, rescaled the cutoff time by assuming that the cutoff time is proportional to the area of a location.
In our study, we, therefore, used = 10 4 hours. In order to estimate , we considered an average value of the waiting time (< ∆ >) given by By numerically integrating equation S3, the numerical relationship between < ∆ > and was obtained, as shown in Figure S3. We found that < ∆ > = 1, 3, 5, 10, and 30 days are associated to = 0.7524, 0.5392, 0.4461, and 0.1059, respectively.

Redial speed of disease spreading
To quantify the disease spreading, we computed the radial-averaged time of the first arrival of infection (< 0 >) in a location at radius centered at the first infected location and the corresponding radial speed of spread ( 0 ) as shown in Figure S4 for Belgium (above) and Martinique (bellow). We also examined the effect of average waiting time, < ∆ >, varied from 1 day, 3 days, 5 days, 10 days, and 30 days shown in different colors. The curves of < 0 > are shifted up vertically, and the slope of 0 is higher for both Belgium and Martinique when < ∆ > increases. However, in Martinique, < ∆ > does not significantly affect the disease spreading dynamics, which might be due to the fact that the distribution of population in Martinique is quite homogeneous as compared to that in Belgium.

Mobility of infectious individuals
To understand the mobility of infectious individuals, we measured the visitation frequency of infectious individuals by tracking their movement trajectories both in Belgium and Martinique with < ∆ >= 5 days (Figure S5). We found that the number of times that infectious individuals visit each location is consistent with the RA map shown in Figure 7a and 7b for Belgium and Martinique, respectively. The locations with larger RA are likely to be visited more frequently by infectious individuals than those with a smaller RA.   Figure S7a and S7b shows the relationship between the population size and the RA in each location for Belgium and Martinique, respectively. We found that the RA is proportional to the population size both in Belgium and Martinique. The populated locations have more attractiveness to attract individuals than locations with a low population density. In addition, the distributions of RA ( Figure S7c) show that the RA in Martinique is more homogeneous than in Belgium.

Characteristics of relative attractiveness
Interestingly, there are approximately six locations in which their RA is greater than the others both in Belgium and Martinique (Figure S7d). The average RA of these locations is 3 times greater in Belgium than in Martinique. It means that individuals in Belgium are likely to be trapped in these locations more than in Martinique, causing individuals in Martinique to have a tendency to visit more locations than in Belgium.

Effects of the population density
In addition, we explored how the population density affects the disease spreading by varying the population ratios; 0.01N, 0.1N, and 1N; = 1.1 × 10 * , for Belgium and 0.1N, 1N, and 10N; = 4 × 10 1 , for Martinique with < ∆ >= 5 days, as shown in Figures S8 and S9, respectively. We found that the curve of incidence cases is shifted to the right with a higher peak and a narrower width when the population ratio increases, which can be seen in both Belgium