The Bidirectional Mobility Model, when implemented as an Agent-Based Model (ABM), is designed to capture the dynamic spatial spread of infectious diseases driven by the movement of individuals between discrete, geographically distinct locations, such as neighborhoods, cities, or regions. In this framework, the population is represented as a collection of agents, each possessing both a health state and a location attribute that evolves over time according to predefined, probabilistic movement patterns, including commuting, routine travel, or migration.
By explicitly modeling movement between locations, bidirectional mobility ABMs account for how human travel connects otherwise separated populations, enabling long-range disease transmission and repeated seeding events. This approach moves beyond static networks and uniform diffusion assumptions, providing a mechanistic representation of spatial mixing and epidemic propagation.
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π 1. Compartmental Structure and Flow
The bidirectional mobility ABM is typically built upon a classical compartmental disease structure, most commonly the SusceptibleβExposedβInfectiousβRecovered (SEIR) framework. Each agent belongs to a specific location i at time t and occupies exactly one epidemiological state.
A defining structural feature of this model is the dual stratification of the population:
β’ by disease state (S, E, I, R), and
β’ by geographic location i.
Transmission occurs locally within each location, while spatial spread emerges through the explicit movement of agents between locations.
Flow Dynamics:
- Disease Progression:
Within a given location i, agents transition from Susceptible to Exposed, from Exposed to Infectious, and from Infectious to Recovered according to biological parameters governing transmission, latency, and recovery. These transitions depend on local mixing among individuals present in the same location. - Mobility:
Agents move bidirectionally between locations i and j according to specified movement rates or probabilities. When an agent moves, it carries its current disease state, potentially introducing infection into a new location or altering local prevalence elsewhere.
The interaction between local disease dynamics and inter-location movement determines the speed, direction, and spatial pattern of epidemic spread.
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π 2. Mathematical Formulation
Although the ABM itself is stochastic and agent-based, its structure is informed by an underlying deterministic multi-location compartmental framework. Let Sα΅’, Eα΅’, Iα΅’, and Rα΅’ denote the numbers of susceptible, exposed, infectious, and recovered individuals in location i, and let Nα΅’ denote the total population in that location.
The temporal evolution of the susceptible population in location i is governed by local transmission and net mobility flows:
dSα΅’/dt = β (Ξ²α΅’ Β· Sα΅’ Β· Iα΅’ / Nα΅’)
+ Ξ£ over j β i of ( Mα΅’β±Ό Β· Sβ±Ό β Mβ±Όα΅’ Β· Sα΅’ )
The first term represents local disease incidence driven by mass-action mixing within location i. The summation represents the net gain or loss of susceptible individuals due to bidirectional movement between locations.
This mobility structure applies analogously to all epidemiological compartments. For a general compartment J in {S, E, I, R}, the dynamics are:
dJα΅’/dt = Reaction term for compartment J in location i
+ Ξ£ over j β i of ( Mα΅’β±Ό Β· Jβ±Ό β Mβ±Όα΅’ Β· Jα΅’ )
The reaction terms encode biological progression:
β’ Exposed individuals become infectious at rate Ο.
β’ Infectious individuals recover at rate Ξ³.
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π Parameter Definitions
| Parameter | Definition | Role in the Model |
|---|---|---|
| Ξ²α΅’ | Transmission rate in location i | Governs the local rate of contact and infection within location i |
| Mα΅’β±Ό | Movement rate from location j to location i | Quantifies the intensity of bidirectional travel or migration between locations |
| Ο | Latent period rate (1 / T_inc) | Governs the transition from Exposed to Infectious |
| Ξ³ | Recovery rate (1 / T_inf) | Governs the transition from Infectious to Recovered |
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π 3. Parameter Ranges (General Viral Disease Context)
Parameter values depend on both disease biology and the spatial scale of mobility being modeled. Typical ranges for acute viral respiratory diseases are as follows:
| Parameter | Typical Range | Unit | Context |
|---|---|---|---|
| Recovery rate (Ξ³) | 0.07 β 0.14 | dayβ»ΒΉ | Corresponds to an infectious period of approximately 7 to 14 days |
| Latency rate (Ο) | 0.14 β 0.25 | dayβ»ΒΉ | Corresponds to an incubation period of approximately 4 to 7 days |
| Local transmission rate (Ξ²α΅’) | 0.2 β 0.5 | dayβ»ΒΉ | Calibrated according to local population density and contact intensity |
| Mobility rate (Mα΅’β±Ό) | 10β»β΄ β 10β»Β² | dayβ»ΒΉ | Reflects daily probability of movement between locations, from rare travel to frequent commuting |
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π§ 4. Applicability and Limitations
Applicability β When to Use Bidirectional Mobility ABMs:
- Evaluation of Travel Restrictions:
These models are well suited for assessing the impact of reducing movement rates between locations, such as travel bans, commuting reductions, or regional lockdowns. - Spatial Resource Allocation:
By anticipating where infected individuals are likely to arrive next, bidirectional mobility ABMs support planning for healthcare capacity, vaccination deployment, and testing resources. - Analysis of Metastable Spread:
The framework is effective for studying how localized outbreaks seed secondary epidemics in connected regions and how highly connected locations accelerate large-scale spread.
Key Assumptions and Weaknesses:
- Dependence on Mobility Data:
Model accuracy relies heavily on detailed, time-resolved originβdestination data describing movement flows. Such data are often incomplete or inferred, introducing uncertainty. - Homogeneous Mixing Within Locations:
Transmission within each location is commonly modeled using mass-action assumptions, neglecting internal heterogeneity such as household structure or contact networks. - Limited Behavioral Adaptation:
Many implementations assume fixed mobility patterns, making it difficult to capture adaptive responses such as voluntary travel reduction or behavioral changes in response to rising infection risk.
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π References
Anderson, R. M., & May, R. M. (1991). Infectious Diseases of Humans: Dynamics and Control.
Arino, J., & van den Driessche, P. (2003). The basic reproduction number in a multi-city compartment model.
Lai, S., et al. (2021). Assessing the effect of global travel and contact restrictions on mitigating the COVID-19 pandemic.
Murray, J. D., Stanley, E. A., & Brown, D. L. (1986). On the spatial spread of rabies among foxes.
Zhou, Y., Wang, L., Zhang, L., et al. (2020). A spatiotemporal epidemiological prediction model to inform county-level COVID-19 risk in the USA.