Metapopulation models

We consider a SEIR-SEI compartmental model including birth-death processes [53]. Subpopulation of hosts and vectors interact locally within nodes that are discrete and spatially distributed units. In each node i, hosts are divided into four compartments, i.e., susceptible (), exposed (), infected () or recovered () while vectors are divided into three compartments, i.e., susceptible (), exposed (), infected (). are the total host and vector population in node i, respectively. Only infected vectors can infect susceptible hosts and vice versa. We suppose that the total number of hosts and vectors remain constant (). Nodes are coupled through a mobility matrix where represents the conditional probability that an individual departing from node i travels to node j such that . In addition, air travel passengers from endemic countries are modeled as an infection reservoir which follows a homogeneous SEIR-SEI dynamics. Infected air travel passengers are imported into nodes containing airports at a rate [54] (see Fig 1). The list of parameters can be found in Table 1. Based on this metapopulation structure, the host mobility within the country is integrated according to two alternative formulations:

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Table 1. Parameters of the models.

https://doi.org/10.1371/journal.pcbi.1014646.t001

Force of infection (FoI) model.

In the FoI model, the various mobility routes are encapsulated within a spatial infection kernel that defines the force of infection between nodes. Since the mechanistic differences between the two modeling formulations are primarily reflected in the infection rates of susceptible entities, we focus our description on the equations for susceptible hosts () and susceptible vectors (); the complete SEIR-SEI systems are detailed in Text A in S1 Appendix:

(1)(2)

where is the Kronecker delta. Let us denote , and , the infection kernel from node j to node i associated with the infection rate of susceptible hosts and susceptible vectors, respectively. The exponential term captures the autonomous vector dispersal in the host infection rate in node i [28]. It shows that an infected vector in j can move in node i and infect a susceptible host ().

is the host mobility rate, i.e., the fraction of host moving per unit of time while is the transition rate between node i and node j. In this regard, provides the time scale separation between epidemic dynamics (i.e., reaction) and host mobility (i.e., diffusion). Host-mediated vector transport is modeled by , where represents the average number of vectors transported per moving host. The infection kernel is normalized so that . Finally, airline travel contributes an additional term, representing the fraction of infected individuals imported daily from the infection reservoir into airport-associated nodes , without altering the local number of hosts.

Diffusion model.

In the diffusion model, host movement is explicitly represented as a reaction–diffusion process. Note that the diffusion term is present for all host compartments (see Text A in S1 Appendix for the complete SEIR-SEI model):

(3)(4)

where is the Kronecker delta. Hosts move between nodes at the transition rates proportional to , where the scaling parameter controls the relative timescale of mobility versus epidemic dynamics [18,55]. When , the epidemics dynamics is faster than the mobility; when , mobility dominates. We denote , and , the infection kernel from node j to node i associated with the infection rate of susceptible hosts and susceptible vectors, respectively. Since the host mobility is explicitly accounted by the diffusion term, is the unit matrix, i.e., susceptible vectors are only infected locally by infected hosts. The host-mediated vector transport, the autonomous vector dispersal and the infection reservoir are modeled identically as in the FoI model.

Construction of the Italian mobility networks

We construct two real-world networks for Italy: one at the regional level with 20 regions (NUTS2) and one at the provincial level with 107 provinces (NUTS3).

Population data and distance

Population data were obtained from the Comprehensive World Cities Database (Simplemaps, version 1.77, updated in 2024). City-level data were aggregated to the appropriate administrative level (NUTS2 or NUTS3). This database reports a total Italian population of around 55.6 million. The geographical position of each node was defined as the average coordinates of the cities it contains. Distances between nodes were computed using geodesic distances, accounting for the curvature of the Earth.

Mobility data

Mobility matrices were derived from Cuebiq data (publicly available [17]). We focused on mobility flows on 18th January 2020 before any COVID-19-related lockdowns. The conditional probability that an individual departing from node i travels to node j was computed as , is the number of time a user from node i move to node j, is the population of node i and the sample size of users recorded in node i. Finally, we impose . Airline traffic data were obtained from the Italian civil aviation authority (ENAC) [56] to determine the size and position of airports. We computed the total number of arrivals from international destinations per airports and integrate at the level of NUTS2 and NUTS3. We then normalized by the total number of arrivals (more than 63 million passengers in 2019) to estimate the proportion of each unit in capturing the total Italian airline traffic. Then we estimated the proportion of arrivals and departures from and to countries currently at risk of vector-borne diseases over the total international traffic (approximatively 6%). Countries at risk are defined as countries where the risk of dengue is frequent or sporadic as assessed by the CDC [57]. Assuming that the arrivals are similar to the departures it allowed us to estimate the average daily arrivals in Italy to daily passengers from countries at risk.

Estimating the time-scale separation parameter

The mobility parameter is estimated from time-resolved mobility matrices provided by the Meta Data 4 Good Program [58]. These matrices are used exclusively for the estimation of ; all other analyses in the Results section are based on the pre-lockdown mobility matrix derived from Cuebiq data, as described in the previous section. The Meta Data 4 Good Program is a collaborative initiative that provides aggregated GPS-based datasets of Facebook users that opted-in for the collection of their data. The Meta Data 4 Good Program provides estimates of the mobility flow between local administrative areas at the NUTS3 level for European countries [59]. These flows are built by aggregating weekly movements of Facebook users that opted-in to the location-based services. The flows are reported by sequence of 8 hours (0–8, 8–16, and 16–24) along with the userbase in each unit. The precise methodology is detailed in [59]. For Italy, we constructed weekly mobility matrices covering 12 weeks between 16th June 2020 and 7th September 2020 corresponding to the absence of lockdown and the seasonal presence of mosquitos. We averaged the 8 hours userbase for each week to obtain the weekly active users in each node. The database provides the weekly mean co-locations , i.e., the average fraction of time in which pairs of users from province i and j were observed together. The conditional probability that an individual departing from node i travels to node j is upscaled by computing , where is the population of node i. Finally, we impose . Here, we assume that the population of Facebook users in node i active a time t is a good proxy of the global population of node i, i.e., .

For any epidemiological state, the evolution of the number of host in node i read:

(5)

We assume that within the time span days, the evolution of population is small. This choice balances the need to reduce short-term fluctuations (e.g., weekends) while retaining enough samples to resolve dynamics. In full generality, can be node dependent . Rewriting the above equation under these assumptions yields:

(6)

where and is the population of active Facebook user in node i in the week t and in week t + 1. Since the total population of active users changes from one week to another, the population are normalized () to account only for the change in the population due to mobility.

Estimating the local importation risk for chikungunya

To estimate the local importation risk, we perform stochastic simulations of the FoI and diffusion model using the Gillespie algorithm on the Italian provinces (NUTS3). In particular, we use the -leaping approximation scheme with days to have a reasonable computational time while ensuring that the average of the stochastic trajectories are consistent with the deterministic dynamics [60]. We define the local importation risk as the ratio of simulations where a province (NUTS3) is infected over the total number of simulations, i.e., the probability that a province is infected. Probabilities are estimated over 1000 simulations of the stochastic version of the FoI and diffusion model, for each scenario. Serological evidence from a historical chikungunya outbreak in Italy shows an estimated 45% detection rate [61]. We therefore assume that infection in a province can be detected only when at least two infections occur, since single cases are likely to remain unnoticed. The ISS (the Italian institute of health) reports that Italy experienced 41 imported chikungunya cases from January to September 2025 [62] corresponding to approximatively 1 per week nationwide. Dividing per the number of provinces, we fix the rate of injection from the airline mobility to 0.01 people per week, i.e., .For assessing the importation risk, we parameterize the models for chikungunya (CHIKV) carried by Aedes albopictus (see Table A in S1 Appendix).

Analysis of the basic reproduction number R₀ and the attack rate

To asses the impact of our mobility parameters on the vector-borne disease dynamics, we compute the basic reproduction number R₀ using the next-generation matrix method [64].

Since R₀ is defined at equilibrium, it is not possible to consider the contribution of airline mobility as the number of individuals from the infection reservoir entering the system at equilibrium is zero. The airline mobility contribution is visible only out-of-equilibrium. The parameter represents the time-scale separation between mobility and epidemic dynamics. When , host mobility is much faster than the infection process (fast mobility regime). Conversely, when , host mobility is slow relative to disease spread (slow mobility regime), and local epidemiological dynamics dominate over spatial displacement. For the sake of readability, we first present the analysis of the FoI model, then the diffusion model before comparing them.

Force of infection model.

We can show that the basic reproduction number associated with the FoI model read (see Text C in S1 Appendix for derivation):

(7)

with , the spectral radius of the matrix and matrix respectively. The latter are defined as:

(8)(9)

with , , and ⊙ the element-wise multiplication. Next, we compute the phase diagram for in the () space for synthetic networks and Italian networks (see Figs 2B, 3B and 4B). We divide the phase diagram into three regimes.

Case and . It corresponds to the case of no mobility interactions. Using perturbation analysis, we derive at the first order (see Text D in S1 Appendix for derivation):

(10)

By setting , we recover the classical SEIR case [65]. Numerical simulations confirm the absence of outbreak across synthetic and Italian networks in our case study (see Figs 2B, 3B and 4B).

Case and . Here hosts move extremely fast between nodes but vectors remain localized. For our case study, the spectral radius of and are both proportional to the one of the mobility matrix (see Text D in S1 Appendix). Since , remains unchanged compared to the no-mobility case, we observe an absence of outbreak. This analytical prediction matches our numerical observations (see Figs 2B, 3B and 4B).

Case and . In this regime, both vectors and hosts interact across all nodes according to the mobility matrix. In our case study, we show that the spectral radius of and is proportional to the one of . Since , R₀ < 1. Again, this leads to the absence of outbreak which is again consistent with our numerical observations (see Figs 2B, 3B and 4B). By deriving a lower bound for in this regime, we estimate the critical point separating slow and fast mobility (see Text D in S1 Appendix):

(11)

with the maximum of the diagonal of . Here denotes the conditional probability for an individual departing from node i to travel to node j after traveling to any node k and its diagonal elements capture the contribution of return paths to the same node. In particular, its maximum diagonal element reflects the largest weights of such return paths. Higher values of this quantity widen the slow-mobility region of the phase space allowing importation-driven outbreaks to occur under a broader range of conditions. Our numerical simulations are in good agreement with our analytical estimates (see dashed lines in Figs 2F, 3F and 4F).

Across synthetic and Italian networks, the attack rate matches the prediction of the basic reproduction number (see igs 2D, 3D and 4D). The standard deviation of the attack rate for the synthetic networks ensemble is negligible since we observe that an absence of dynamics happens in this case (see Fig A in S1 Appendix). Incorporating airline mobility reveals 2 zones in the mobility phase space (see Figs 2F, 3F and 4F). For fast internal host mobility (), we observe an absence of outbreaks whereas for slow internal mobility, the injection of infected air travelers sustains outbreaks. Mechanistically, in the FoI model, fast internal mobility prevents outbreaks in visited nodes because the time spent by hosts in each node become too short for local vectors to effectively acquire infection from transient infected travelers (see Eq. 2).

Diffusion model.

The basic reproduction number associated with the diffusion model read (see Text C in S1 Appendix for derivation):

(12)

with , the spectral radius of the matrix and matrix , respectively. The latter are defined as:

(13)(14)

with , the transpose of the Laplacian matrix of the mobility network. Next, we compute the phase diagram for in the () space for synthetic networks and Italian networks (see Figs 2C, 3C and 4C). We remark that in the diffusion model.

Case and . In this regime, diffusion is infinitely slow (), so the system effectively behaves as if nodes do not interact. Once equilibrium is reached, hosts remains localized and the basic reproduction number of each node can be approximated by the classical SEIR formula for the basic reproduction number with updated equilibrium host population i, . In our case study, read:

(15)

The number of vectors in node i is fixed in our case study. As a result, the vector-to-host ratio increases in nodes where the equilibrium host population decreases, which can drive the system above the epidemic threshold (). Comparing theoretical predictions with numerical simulations confirm the epidemics onset: and the observed basic reproduction number are in good agreement for a synthetic network of the considered ensemble (2.74,2.74), the Italian regions (4.83,4.83) and the Italian provinces (3.10,3.24) (see Figs 2C, 3C and 4C).

Case and . In the cases where , the inverse matrix of Eqs.13, 14 is ill-defined. Nonetheless, by expressing the matrix in terms of its (left-)right eigenvectors, it is possible to show that and reduce to (see Text E in S1 Appendix for derivation):

(16)(17)

where is matrix where each column is the same and equal to , being the strength of node i and m the total strength of the mobility network.

In our case study, and reduce to the no mobility case. This result matches the numerical simulations since we observe no outbreak for and in our numerical simulations (see Figs 2C, 3C and 4C).

Case and . Using the exact same canvas as before, we show that and reduce again to the no mobility case in our case study (see Text E in S1 Appendix for derivation). This is again consistent with our numerical simulations (see Figs 2C, 3C and 4C).

For (fast mobility), the attack rate matches the prediction of the reproduction number consistent with the absence of epidemics. However, for (slow mobility), the diffusion model predicts outbreaks at equilibrium () while the simulated attack rate remains negligible (see Figs 2E, 3E and 4E). This discrepancy arises because the system fails to reach equilibrium within the two-year horizon, highlighting that R₀ is not always a valid predictor over short- to medium-term time horizons. We therefore quantify the validity domain of R₀-based predictions with a blue dotted line in Figs 2B, 2E, 2G, 3B, 3E, 3G, 4B, 4E and 4G. Only to the left of this line do outbreaks reach quasi-equilibrium within two years. In the vicinity of this boundary, the dynamics remain only partially developed within the analyzed time span (see Fig B in S1 Appendix). In contrast with the FoI, the addition of airline mobility and local vector interaction (see Eq. 4) leads to outbreaks even in the fast mobility regime, as imported infections can sustain local transmission, despite rapid host movement. Its effect is instead marginal where internal mobility alone already sustains transmission. Finally, for slow mobility, the diffusion model behaves similarly to the FoI although its predicts significantly larger outbreak sizes, consistent with previous analyses (see Figs 2G, 3G and 4G).The standard deviation of the attack rate for the synthetic networks ensemble is maximal at the transition since each network undergoes it for slightly different parameters (see Fig A in S1 Appendix).

Comparison between mobility formulations with fixed host population

While the previous analysis highlights important differences between the FoI and diffusion formulations, the use of the diffusion formulation may primarily apply to scenarios in which the human population physically relocates, such as seasonal or long-term migration. To further investigate the effects of population redistribution -and to address whether model differences persist when the host distribution is stable- we compare the two models while keeping the human population fixed. Specifically, we set the host population in each node equals to the equilibrium distribution obtained from the diffusion model, while leaving all other parameters unchanged. Under this setup, the distribution of local reproduction numbers becomes heterogeneous, reflecting conditions typical of the early phase of the peak season in non-endemic areas [2].

Case . In this regime, the basic reproduction numbers for the FoI and the diffusion model are identical with values consistent those obtained in the previous analysis ( for Italian provinces). These results are in agreement with numerical simulations which show similar attack rates for both model, regardless of whether airline mobility is included (see Fig 5). This convergence is expected as, in the absence of host mobility, the models are mathematically equivalent.

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Fig 5. Phase diagrams in the () space for Italian provinces (NUTS3) where host population is at equilibrium.

R₀ for the force of infection model a) and the diffusion model b). Attack rate without infection reservoir for the force of infection model c) and for the diffusion model d). Attack rate with infection reservoir for the force of infection model e) and for the diffusion model f). is the variable associated with human mobility and p₀ is the one associated with human-mediated vector mobility. Horizontal and vertical plain lines are the estimation of the uncertainty range for empirical and p₀. The blue dashed line is the estimated threshold between slow and fast mobility. The blue dotted line corresponds to , the limit after which the host mobility diffusion does not reach equilibrium. The host population of the FoI and diffusion model are constant equal to its equilibrium value for the diffusion model. for the infection reservoir.

https://doi.org/10.1371/journal.pcbi.1014646.g005

Case . In this regime, the basic reproduction numbers for the FoI and the diffusion model diverge. While the FoI model remains above threshold ( for the Italian provinces), the diffusion model falls below threshold (see Fig 5). This behavior is consistent with analytical predictions for this regime, as the reproduction number of the diffusion model does not explicitly depend on the equilibrium human population (see Text E in S1 Appendix for derivation). The corresponding attack rates confirm the difference between the two models. Mechanistically, the effective redistribution of the human population at equilibrium in the diffusion model causes the system to behave like a single, well-mixed patch, which in this case lies below the epidemic threshold.

Case .

In this intermediate regime, the basic reproduction number for the diffusion model is above threshold but consistently lower than that of the FoI model (see Fig 5). Despite this, the attack rate is substantially higher in the diffusion model. This discrepancy highlights a key mechanistic difference: the effective mobility of susceptible hosts, which is absent in the FoI model. In the diffusion model, susceptible individuals continuously move into nodes where transmission potential is high, sustaining the epidemic. At the same time, infected hosts also diffuse across the system maintaining epidemic growth in high-risk areas. In contrast, in the FoI model the pool of susceptible individuals is not replenished (aside from demographic processes) and progressively declines. As a result, although the FoI model exhibit a higher basic reproduction number, it leads to a lower overall attack rate. Similar results are observed at the regional scale (see Fig C in S1 Appendix).

Overall, while the two formulations yield comparable results under slow human mobility, their divergence at intermediate and high mobility underscores the importance of the mobility formulation in shaping large-scale epidemic dynamics, even when the host population is kept fixed.

Importation risk of vector-borne diseases in real-world conditions

To further assess the comparative impact of mobility parameters in the two formulations, we estimate the local importation risk of vector-borne diseases under real-world conditions. In particular, we evaluate how the models’ assumptions shape the interplay between different mobility pathways using data-driven parameter values.

First, we estimate the time-scale separation parameter, and the human-mediated vector mobility parameter for the Italian networks. We interpret the latter as the average number of Aedes albopictus mosquitoes transported per human trip between nodes. Based on empirical studies including observations from Barcelona reporting values between 0.002 and 0.01, we fix this parameter to p₀ = 0.005 [49] (see Supplementary Material). While the two models differ in how mobility affects epidemic dynamics, they rely on the same underlying host mobility representation, and thus share a consistent definition of the time-scale separation parameter. Using time-resolved mobility data, we estimate this parameter across nodes and time intervals (see Methods). Negative values reflect flows from smaller to larger cities, such as return mobility after summer holidays. To ensure consistency, we consider both absolute and positive-only distributions (see Fig D in S1 Appendix). For Italian regions, the average of the medians yields a representative value of 390, with an uncertainty interval from 151 to 1514. For Italian provinces, the corresponding value is 285, with an uncertainty interval from 74 to 812. These relatively high values indicate a slow diffusion regime, consistent with observed summer mobility patterns in Italy.

Next, we focus on the Italian provinces to investigate how the mobility formulation shapes the estimation of the local importation risk in a realistic setting. We define the local importation risk as the probability for each province to be infected at a given time knowing a particular initial condition. We consider that an infection in a province can be detected only when at least two infections occur (see Methods for details). We assume the local R₀ = 2 for each province which is consistent to the observed basic reproduction number during the peak season on a historical chikungunya outbreak in Italy [7]. We fixed the total rate of infected international travelers injected in Italy to 0.01 and we set the mobility parameters to and p₀ = 0.005. Epidemic parameters are calibrated for chikungunya (see Table A in S1 Appendix) (see Methods for details).

First, we analyze a monocentric scenario where an infected individual is introduced in a province hosting an airport at time t = 0 and where the total rate of infected travelers is concentrated there. We apply the monocentric scenario on the four biggest international airports of Italy (see Methods): Rome, Milan, Bergamo and Venice. Results show that 30 days after the introduction of an infected individual in the Rome province, the probability of spillover to another province is below 1% for FoI and diffusion models (see Fig 6). In contrast, 4 provinces have a probability to be infected superior to 5% after 90 days in the FoI model, but only after 60 days in the diffusion model. In the same vein, results show that 13 provinces have a probability to be infected greater than 1% after 90 days in the FoI model while the very same provinces have probability to be infected greater than 5% in the diffusion model after 90 days.

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Fig 6. Local importation risk for Italian provinces (NUTS3) in function of time knowing that the infection is seeded in Rome province.

Probability for a province to be infected 30, 60 and 90 days after the arrival of an infected individual in Rome for the a) FoI model and the b) diffusion model. R₀ = 2 for each province. Fixed airline injection rate in Rome is 0.01 individual per week. Province is considered infected if at least 2 individuals are present. The red points indicate where the infection is seeded. Probabilities are computed over 1000 simulations. Basemap administrative boundaries are realized by ISTAT data, via the Openpolis geojson-italy repository under the Creative Commons Attribution 4.0 (CC BY 4.0) license.

https://doi.org/10.1371/journal.pcbi.1014646.g006

While the FoI and the diffusion model follow very similar spatio-temporal patterns of infection spreading at short time (before 30 days), the diffusion model leads to significantly shorter arrival times than the FoI model at longer time (after 30 days). Note that the diffusion model always conserves similar patterns as the FoI model. Similar results are found when the infection is seeded in Milan (Fig E in S1 Appendix), Bergamo (Fig F in S1 Appendix) or Venice (Fig G in S1 Appendix). Then, we analyze a polycentric scenario where the infected individual is introduced in a province i according to , the proportion it represents in the Italian inflow of international passengers. The fixed rate 0.01 infected passengers per week is distributed accordingly. Results show that there are 7 provinces with probabilities of being infected superior to 5% after 60 days in the FoI model corresponding to the biggest airports cumulating around 80% of the international passengers inflow (see Fig 7). In contrast, 10 provinces have corresponding probabilities in the diffusion model, among which three do not host international airports but are direct neighbors of provinces hosting big international airports. At 90 days, there are 10 provinces in the FoI model and 45 provinces in the diffusion model with probabilities of being infected superior to 5%.

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Fig 7. Local importation risk for Italian provinces (NUTS3) in function of time knowing that the infection is seeded proportionally to the incoming international airline flow.

Probability for a province to be infected 30,60 and 90 days after the arrival of an infected individual for the a) FoI model and the b) diffusion model. Fixed airline injection rate is 0.01 people per week distributed across the international airports according to their weight. A province is considered infected if at least 2 individuals are present. Red points indicate the presence of an international airport and their size is proportional to their weight in the incoming international airline flow. Probabilities are computed over 1000 simulations. Basemap administrative boundaries are realized by ISTAT data, via the Openpolis geojson-italy repository under the Creative Commons Attribution 4.0 (CC BY 4.0) license.

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Both models exhibit similar spatial patterns where the most important risk of spillover is concentrated in provinces neighboring the ones hosting an international airport. Strong mobility interaction between neighboring provinces can explain this effect (see Fig 4A). By computing the odds ratio between the diffusion and FoI models (see Fig H in S1 Appendix), we show again that the estimated importation risk for a given province is much more important in the diffusion model than in the FoI model at longer time horizons.

We hypothesize that the effective movement of infected hosts, combined with the replenishment of susceptible individuals described in the previous analysis, explains the faster spread observed in the diffusion model compared with the Fol one. Overall, these results confirm, using data-driven mobility parameters, that the choice of mobility formulation has a crucial impact on how the interplay between mobility pathways translates into epidemic outcomes in realistic settings.