2.1 Modeling infection within a region

To model the infections within a region, we use the renewal-equation based framework [27,28,32], where new infections at time t are described as,

(1)

where, represents the susceptible population at time t, and N denotes the total population. Eq 1 illustrates that the number of new infections at any given time depends on the prior infections, with their contributions weighted by the generation time distribution, g, which captures time interval between primary and secondary infections. Here g is derived by discretizing the pathogen-specific density function g(t) as for and . The instantaneous reproduction number R_t quantifies the average number of secondary infections caused by a single primary infection at time t. Eq 1 provides a robust foundation for understanding the transmission dynamics within a single region, which we refer to as the national model.

2.2 Extension of the framework with a spatial component

Central to this study is the advancement of the existing epidemiological framework through the integration of a spatially explicit component, which systematically incorporates human mobility data to elucidate the mechanistic role of population movement in shaping the spatiotemporally heterogeneous disease transmission dynamics. A schematic representation of the extended framework is shown in Fig 1. For simplicity, we consider two regions: region 1 and region 2 represented by two different colors. The population in each region comprises susceptible and infected individuals, with a portion of the population commuting between regions. Additionally, we assume that recovery from the disease confers lifetime immunity, as recovered individuals cannot be reinfected. The mobility between two regions , represents the proportion of the population of region j that commutes to region i. Hence, in Fig 1, m₁₂ denotes the fraction of the population of region 2 that commutes to region 1, and similarly, m₂₁ represents the reverse flow. The diagonal elements of the mobility matrix m_ii indicate the fraction of the population that does not commute, staying within their respective regions. The presence of mobile population implies that the infection in any region are influenced not only by local transmission but also by interactions with infected individuals from other connected regions. To illustrate the transmission dynamics in more detail, we focus on infection pathways affecting the population of region 1. As illustrated in Fig 1, the red ovals indicate three different routes through which individuals in region 1 can acquire infection. Susceptibles in region 1 can be exposed to infected individuals within their own region or can acquire infection by visiting region 2 and contract infection there. In summary, infection within any region can occur through three primary pathways: (i) local transmission within the region by its own infected population, (ii) infection through incoming infected commuters from other regions, and (iii) acquisition of infection during visit to another regions. These mobility-driven transmission mechanisms underscore the complexity of disease spread in interconnected populations.

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Fig 1. Schematic representation of disease transmission dynamics with mobility.

The diagram illustrates two interconnected regions (Region 1 and Region 2), where individuals are classified as susceptible (S) and infected (I) can move between regions. The colors indicate the region of origin for each individual. The arrows represent mobility flows: m₂₁ (movement from Region 1 to Region 2) and m₁₂ (movement from Region 2 to Region 1). The diagram describes the disease transmission pathways for region 1, highlighted by red ovals: (i) local transmission due to interactions between susceptible and infected individuals of region 1 within region 1, (ii) infected individuals of region 2 commute to region 1, can transmit infection to the susceptible population of region 1, (iii) susceptible individuals from Region 1 who travel to Region 2, acquire the infection in region 2, and return as infected individuals. This framework captures the role of human mobility in shaping spatial disease transmission patterns.

https://doi.org/10.1371/journal.pcbi.1013642.g001

Drawing from these observations, we develop a modeling framework that explicitly incorporates human mobility into disease transmission dynamics. Using Eq 1, as the foundational structure, we extend the model by integrating a spatial component to account for mobility flows between regions. For the modeling framework, we consider a total of M inter-connected regions, and we model the infection dynamics in a region denoted as i, with the number of infections at time t represented as I_i,t, while allowing for population movement across M regions. At time t, the susceptible individuals from region i commuting to another region j are represented as . These susceptible individuals are exposed to all the infected population in region j at that time, given by . This interaction between susceptible individuals from region i and infected population in region j results in new infection of region i, albeit occurring in region j. Since individuals from region i can commute to any regions, summing over all regions j yields the total infections in region i that is given by,

(2)

Here, the denominator serves as a normalization term, describing the effective population in region j at that time. R_t,j is the reproduction number of region j, indicates the average number of secondary infections occurring from a primary infection in that region. g is the generation time distribution, and N_i is the total population of region i, while S_t,i represents the number of susceptible individuals left in region i at time t. In the special case of completely isolated regions, the mobility matrix becomes an identity matrix (unity on the diagonal and with zero elsewhere). With this simplification, in the absence of inter-regional mobility, Eq 2 reduces to Eq 1 for each region. In the context of the framework’s time scale, if individuals do not spend an entire day in another region, the framework is flexible enough to adopt that fact. A detailed simulation is provided in S12 Fig. However, in this work, we consider a daily time scale throughout the study, based on the available data.

As the next step, for each region i, the model is fitted to available observational data, such as infection data, case counts, or death data, to estimate the region-specific instantaneous reproduction number R_t. The model can be adjusted based on the nature of the available datasets and the specific context. We demonstrate the utility of our model through two distinct applications. First, we employ simulated daily incidence data as observations to validate the model in a controlled setting. Second, we apply the model to real-world data by analyzing the early spread of SARS-CoV-2 in England, using weekly mortality data as observations. These examples illustrate the flexibility of the framework in accommodating both different types of data and their aggregated nature. This spatially extended framework offers a comprehensive understanding of the impact of human mobility on disease transmission dynamics across interconnected regions. By explicitly incorporating mobility flows into the infection process, the model captures both local and mobility-driven transmission, providing a more realistic representation of pathogen spread in spatially structured populations. Here, we provide a detailed description of three sources of infection.

2.3 Sources of infections

Infections within a region can originate from three different sources. Using Eq 2, we segregate the infected population based on the three different sources of transmission:

1. Infections driven by own infected population within the region: These infections occur within the region when susceptible individuals contract the disease from the infected population of their own region. This category captures purely local transmission dynamics and can be expressed as:(3)
2. Mobility induced infections within the region: These infections are caused by infected individuals from other regions visiting the region and transmitting the disease to the local susceptible population. This category is defined as infections due to mobility inside the region and can be expressed as,(4)
3. Mobility induced infections outside the region: These infections occur when individuals from a region commute to other regions, become exposed there, and get infected. This category is defined as infections due to mobility outside the region and can be expressed as :(5)

2.4 Simulated data generation

We use a simulated infection dataset to validate the framework. The simulated dataset assumes three regions with total populations of 20,000,000, 10,000,000, and 15,000,000, respectively. These regions are interconnected through a mobility matrix (see Fig 2b), where each column corresponds to the outflow of individuals from a specific region (i.e., the jth column represents outflows from region j), while each row represents the inflows into a region (i.e., the ith row represents inflows into region i). Consequently, the sum of each column equals unity. According to the intrinsic characteristic of the framework, infections are assumed to occur homogeneously within each region governed by the corresponding reproduction numbers, R_t,i,i = 1,2,3 (see Fig 2c). We randomly choose three different countries (IND, GRB, ITA) from the repository [40], and take the time series of R_ts for a time period of 350 days. Based on the reproduction numbers, mobility matrix, and using Eq 2, simulated data are generated as shown in Fig 2d. The colors of the reproduction number and incidence data correspond to their respective regions. The generation time distribution is obtained by discretizing the density function [28,41], a SARS-CoV-2 specific generation time distribution. Here, we consider the gamma distribution with mean 6.5 and CV 0.62. However, in real-world scenarios, human mobility, total population, and observational datasets are typically the only available data sources. Using these, the goal is to estimate the instantaneous reproduction numbers R_t, for all regions to understand the epidemic dynamics in the presence of human mobility.

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Fig 2. Simulated dataset.

(a) Three regions with the associated mobility flow. (b) The mobility matrix where each element m_ij denotes the fraction of population of region i commutes to region j. Therefore, the columns show the outflows from the respective region and each row represents the inflow to that specific region. (c) Instantaneous reproduction numbers R_t for the three regions. (d) With the mobility matrix and reproduction numbers, we simulate daily infection data using Eq 2.

https://doi.org/10.1371/journal.pcbi.1013642.g002

2.5 Model adjustments and assumptions for estimating R_t with the simulated data

We define R_t,i as the effective reproduction number for the ith region at time t and is modeled as geometric random walk, where log(R_t,i) is updated according to a normal distribution with the mean log(R_t−1,i) and a standard deviation as follows:

(6)

Here, R_0,i is the reproduction number at t = 0 defined as the average number of secondary cases coming from an infected individual at the start of the outbreak. It is assigned a weak prior to consider a wide range of values. follows a prior of truncated normal distribution (constrained to positive values). For connected R_t estimation, we use the mobility matrix as defined in Fig 2b. We use CmdStan [42] for model fitting, with NUTS sampler. 4 chains are generated in parallel for 500 iterations after 500 warm-up. There are no divergent transitions.

2.6 Model adjustments and assumptions for case study: SARS-CoV-2 in England and North East region of England (2020)

We apply the framework 2 to analyze the initial phases of the COVID-19 pandemic in England during 2020 as a case study. To address the challenge of low reporting rates in the early stages, we use weekly death data to estimate R_t for the nine regions of England: North East, North West, Yorkshire and the Humber, East Midlands, West Midlands, East, London, South East and South West. The mobility matrix (see S3a Fig) is constructed over the nine regions using data from [43], which reports 2011 census estimates for individuals aged 16 years and over in England and Wales with their place of residence and workplace. We calculate the proportions of outflows to obtain the mobility matrix that is compatible with the model 2. During the lockdown periods, we set the mobility matrix to identity , ignoring inter-regional mobility. The population data are obtained from [44], and weekly death data [45] for each region are shown in S4 Fig. We outline below the framework used to estimate reproduction numbers.

We use the same generation time distribution, [28,41], as in the previous section to simulate infections. For death modeling, we consider the infection-to-death delay distribution , which combines the infection-to-onset and onset-to-death [28,39,46] delays. Here, the gamma distributions are parametrized as Gamma(mean, CV).

The weekly death data D_w,i for region i in week w is modeled using a negative binomial distribution:

(7)

Here, [39]. Here, d_w,i is the expected number of weekly deaths, calculated as the sum of daily deaths over seven days. Daily deaths d_t,i are derived from daily infections I_t,i using the discrete sum,

(8)

where represent the infection fatality ratio [28] of England with a noise around it corresponding to region i.

To estimate the instantaneous reproduction number R_t,i for each region, we account for temporal changes in human behavior utilizing activity trends derived from the google mobility report [9]. We use five different types of information from Google mobility data: retail and recreation, groceries and pharmacies, parks, transit stations, and workplaces. Here, we define R_t as follows:

(9)

where, , is twice the inverse logit function. with for all regions [28,39]. Here, R_0,i is the initial reproduction number for region i. In this context, it is the value of the reproduction number before fitting starts. is the weekly effect that follows a random walk centered at 0, with the week index , where t₀ is the first day of seeding.

X_i is the region-specific effect for covariates , which are the averages of five different dimensions observed from Google mobility data. The implementation of NPIs typically leads to a significant decline in both population mobility and disease transmissibility. However, in many countries, the adoption of social distancing behaviours weakened the correlation between mobility and transmissibility following the relaxation of control measures [47–49]. To account for this effect, we introduce the parameter P_t, which captures behavioural changes associated with different lockdown phases, with distinct values x, y, and z representing the periods up to the end of the first lockdown, from the end of the first lockdown to the end of the second lockdown, and the post-second lockdown period, respectively.

We assume that initial seeding begins six weeks prior to the date on which each region in England reaches 10 cumulative reported deaths. From that date, we seed equal initial infections for six consecutive days at each region, modeled as [39]:

(10)

By tailoring the model to account for spatial and temporal variations in epidemic trends, the framework provides a robust tool for understanding and managing regional dynamics of SARS-CoV-2 in England in the year 2020. The model fitting is shown in S5 Fig for nine regions of England.

We then apply the framework to a finer spatial scale, focusing to the North East region, which is selected for detailed analysis because nearly 99% of its population does not commute to other regions. This characteristic makes it an ideal area for studying the relationship between human mobility and disease transmission on a more localized scale. The mobility matrix (see S3b Fig) is constructed for the twelve lower-tier local authorities (LTLAs) within the North East region based on the same mobility data [43], ignoring mobility to regions outside of it. We use the same model described in Eq 2 and Sect 2.6, with death data specific to the North East region (see S9 Fig). The model fitting across the 12 LTLAs are illustrated in S10 Fig.

We use CmdStan [42] for model fitting, with NUTS sampler. 4 chains are generated in parallel for 500 iterations after 1000 warm-up. There are no divergent transitions.

3.1 Model validation with simulated dataset

To validate the proposed framework (described in Sect 2), we compare the estimated R_ts with the ground truth (see Fig 2c) used in the simulation. This validation ensures that the framework reliably captures the transmission dynamics under controlled conditions before applying it to real-world scenarios. Fig 3a–3c presents the results for the simulated dataset, showing estimated reproduction numbers for three regions. The disconnected and connected R_t estimates are shown in red and green, representing scenarios without and with inter-regional mobility, respectively. In the disconnected model, mobility-driven infections are ignored, assuming individuals remain within their respective regions and infections occur exclusively within regional boundaries. This simplification produces biased R_t estimates. Table 1 shows the correlation between one-step increments in R_t for the connected (green) and disconnected (red) estimates against the ground truth (blue). The consistently higher correlation between the connected R_t and ground truth supports the observation that disconnected R_t can lead to a severe misleading understanding of transmission patterns.

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Fig 3. Estimation of time-varying reproduction number (R_t) using simulated data and contribution of mobility-driven infections.

Top row (a–c): Blue curves represent the true R_t trajectories for three regions, with horizontal dotted lines indicating the epidemic threshold (R_t = 1). Green and red curves show the estimated R_t values with and without incorporating human mobility, respectively. Shaded areas around the curves denote the 95% credible intervals. The deviations between the two estimates highlights the importance of accounting for human mobility in transmission modeling. Middle row (d–f): Stacked plots illustrate the three sources of daily infections. Orange shows the infections caused by the infected population of that same region. Blue shows the infections that generated within the region by infected incoming population form the other regions. Purple reflects mobility-induced infections acquired outside the region by the outgoing susceptibles. Bottom row (g–i): The proportion of infections from three different sources. The colors are considered same as defined for the Fig 3(d)–3(f). These figures represent the considerable number of mobility driven infections in the disease transmission dynamics.

https://doi.org/10.1371/journal.pcbi.1013642.g003

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Table 1. Correlation between one-step increments of true and estimated R_ts.

https://doi.org/10.1371/journal.pcbi.1013642.t001

To further investigate the relationship between mobility-driven infections and overall transmission dynamics, Fig 3d–3i shows the absolute number and proportion of infections arising from three distinct sources. These figures indicate that a considerable proportion of infections is influenced by mobility. This effect becomes particularly important when R_t varies across regions and the epidemic behaves differently from one region to another, an effect overlooked by the disconnected R_t.

Table 2 produces a quantitative support that shows a positive correlation between the residuals of disconnected R_t with respect to the ground truth and the proportion of mobility-driven new infections. When the mobile population gets fewer infections in other regions than they would without commuting, disconnected R_t underestimates (residual negative), and if the number of such infections is higher, it overestimates the epidemic (residual positive). For example, around day 100, the disconnected R_t underestimates the true R_t in Region 2. Around that time, as 15% population of region 2 commutes to regions 1 and 3, where reproduction numbers are lower than region 2, it reduces mobility driven infection outside the region, and as well as the total number of infections in region 2. However, disconnected R_t fails to capture this mobility-driven lowering effect outside the region and leads to an underestimation of the true epidemic situation. On the other hand, around day = 220 in region 2, by failing to account for that mobility-driven increasing infection outside the region, disconnected R_t overestimates the epidemic situation. Therefore, disconnected R_t underestimates or overestimates an epidemic when mobility reduces or increases infections compared to a no-mobility scenario. Further discussion of the effects of mobility and regional epidemic conditions on transmission dynamics is provided in S1 and S2 Figs.

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Table 2. Correlation between the residuals (disconnected R_t− ground truth) and the proportion of mobility-driven infections.

https://doi.org/10.1371/journal.pcbi.1013642.t002

These results validate the effectiveness and necessity of incorporating human mobility into the model to accurately estimate reproduction numbers and better understand the epidemic situation. This approach is essentially critical in scenarios involving spatial heterogeneity in transmission dynamics.

3.2 Case study: Instantaneous reproduction number estimates over the regions of England

In Fig 4, we present the estimates of the connected and disconnected instantaneous reproduction number (R_t) (mean and 95% credible interval) for the nine regions of England, comparing results with and without account for mobility. The vertical dotted lines indicate the start and end of the first and second lockdowns. We observe that the connected and disconnected estimated R_ts are nearly identical across all regions. The absolute difference between connected and disconnected R_ts are shown in S8 Fig. After an early discrepancy for a few regions (East, London, South East) maximum mean absolute difference reaches 0.25. This suggests that at regional scale, mobility does not have a significant impact on infections after the early transmission phase. Upon examining the mobility matrix (see S3a Fig), it is observed that the fraction of the population commuting between regions is very small relative to the total population of each region. This makes the regions of England treatable as disjoint areas, where the disconnected model is sufficient for R_t estimation at this spatial scale. Transmission at the regional level can be treated as homogeneous spreading within regions, with minimal influence from inter-regional mobility. This suggests that, when the disease is already established in all regions, mobility can be reasonably excluded from models at such spatial scales. The absolute number and the proportion of daily infections originating from three distinct sources across the 9 regions are shown in S6 and S7 Figs. The Rhat statistics regarding the estimated R_ts are shown in S15 Fig.

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Fig 4. Estimated R_t over different regions of England.

The red curve shows the disconnected R_t which is the reproduction number estimated separately for each region without considering mobility across them. Green curve is the connected R_t that is the estimated reproduction number with the considered mobility matrix showed in S3a Fig. Each curve is shown with its corresponding 95% credible interval. The vertical dotted lines show the first and second lockdown period, and the horizontal black dashed line is the threshold for reproduction number that is R_t = 1.

https://doi.org/10.1371/journal.pcbi.1013642.g004

3.3 Findings on higher spatial resolution: LTLAs of North East region

To assess whether the estimates provide any additional or different insights when applied at a higher spatial resolution, where the mobility is considerably greater than the previous case. The high proportion of mobile populations between the LTLAs of North East region has a notable impact on disease transmission, as evidenced by Fig 5. Significant differences in the estimated reproduction numbers are observed in LTLAs such as Hartlepool, Middlesbrough, and North and South Tyneside. The absolute difference between connected and disconnected R_ts are shown in S11 Fig. In Middlesbrough, the difference reaches 1.09(0.7-1.3) in the second week of September. Similarly, for North Tyneside it reaches 1.15(1.03-1.2). These variations highlight that a considerable number of infections arise from movement between LTLAs, as shown in Fig 6, and the estimated disconnected R_ts fail to accurately capture these transmission dynamics, often resulting in underestimation or overestimation of the disease transmission. This underscores the importance of incorporating intra-regional mobility into models to better understand localized disease transmission patterns.

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Fig 5. Estimated R_t over the LTLAs of North East region.

The green and red curve shows the estimation of real-time reproduction number with and without considering mobility respectively. The dotted vertical lines represents the lockdown period and black horizontal line shows the threshold for reproduction number at R_t = 1. Red and green bands show the 95% uncertainty interval corresponding to the disconnected and connected R_t estimates. The mobility matrix is shown in S3b Fig.

https://doi.org/10.1371/journal.pcbi.1013642.g005

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Fig 6. Daily infections from three different sources over the LTLAs of North East region.

Orange color represents the infections occur within the LTLA due to the infected population of that same LTLA. Blue and purple depict the mobility induced infections happen within and outside the LTLA. The dotted vertical lines indicate two lockdown periods while mobility driven infections go down and most of the infections are generated by the local transmission. Here, the lines represent the mean of the estimates and band shows the 95% uncertainty around it.

https://doi.org/10.1371/journal.pcbi.1013642.g006

In Figs 6 and 7, we examine the absolute number and the proportion of daily infections originating from three distinct sources across the 12 LTLAs, providing direct evidence of a substantial portion of infections arising from mobility. In particular, areas such as Hartlepool, Middlesbrough, Newcastle upon Tyne, and North Tyneside show mobility-driven infections contributing to approximately 50% of the total cases, with the proportion rising to as high as 75% in North Tyneside. During the lockdown periods, the reduction in mobility led to a marked decrease in mobility-driven infections, resulting in transmission dynamics being predominantly driven by local spread. These findings underline the crucial role of inter-LTLAs mobility in shaping transmission dynamics and emphasize the importance of accounting for such mobility in public health interventions and resource allocation strategies. Additionally, in S13 and S14 Figs, we present an analysis showing that the main findings are robust to the choice of time scale.

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Fig 7. Proportion of daily infections from three different sources across the LTLAs of the North East region.

Orange, blue, and purple indicate the proportions of daily infections attributed to three distinct sources. The dotted vertical lines represent two lockdown periods, during which nearly all infections resulted from local transmission.

https://doi.org/10.1371/journal.pcbi.1013642.g007