Least Infection Probability Path (LIPP)

• Least Infection Probability Path (LIPP) is a pandemic-aware routing concept that minimizes infection risk by aggregating mode-specific probabilities using a multiplicative survival model.
• It employs a probabilistic framework that models segment risk via exposure durations, hazard rates, and environmental factors across various transportation microenvironments.
• The approach balances infection risk with travel time and convenience, offering actionable insights for optimizing safe multimodal transit during pandemics.

Least Infection Probability Path (LIPP) denotes the problem of selecting an origin–destination route that minimizes the probability of infection during travel. In the formulation that explicitly names the problem, LIPP is a pandemic-aware routing objective over a multimodal transportation network in which infection risk is accumulated across route segments rather than approximated by distance or travel time alone. Related work does not generally define LIPP by name, but contributes upstream components that are directly relevant to it: estimation of edge infection probabilities from diffusion observations, temporal directed interaction graphs for path-based contagion inference, and contact-level transmission factors derived from Bluetooth proximity and exposure duration (Mirgalooyebayat et al., 1 Oct 2025, Bota et al., 2017, Liu et al., 2020, Mahawar et al., 2021).

1. Definition and conceptual scope

LIPP is introduced as a routing problem in which the objective is not shortest distance or fastest travel time, but minimum transmission probability. The setting is explicitly multimodal and segment-based: a complete route is decomposed into segments corresponding to walking, subway, BRT, city bus, or car, and the preferred route is the one with the least infection probability between source and destination. In the paper that names the problem, the optimization target is written as

$$\min_{\pi \in \Pi(s,t)} P_n \quad\text{where}\quad P_n = 1 - \prod_{i=1}^{n-1}(1-P_i),$$

with $P_i$ the infection probability on segment $i$ (Mirgalooyebayat et al., 1 Oct 2025).

This formulation differs from conventional shortest-path routing in two stated ways. First, edge or segment costs are probabilistic and mode-dependent rather than purely geometric or temporal. Second, path composition is multiplicative in survival terms, since infection on any segment contributes to route-level infection. The same source also notes that the optimization “balances infection risk against travel time and convenience” at a conceptual level, while the actual mathematical development and case study are centered on minimizing infection probability.

A central implication is that LIPP is not merely a reweighting of an existing route planner by an ad hoc safety score. It is a probabilistic path problem whose semantics depend on how segment risk is modeled, how exposures are aggregated, and what independence assumptions are adopted. This makes LIPP a bridge topic between transportation routing, epidemic risk modeling, and graph-based inference.

2. Probabilistic formulation

The direct LIPP formulation uses a multiplicative survival model across route segments. If a route contains $n-1$ segments, the route-level infection probability is

$P_n = 1 - \prod_{i=1}^{n-1}(1-P_i).$

This treats consecutive route segments as independent in the sense made explicit by the paper: survive each segment with probability $(1-P_i)$, then subtract the total survival probability from 1 (Mirgalooyebayat et al., 1 Oct 2025).

The same work develops an interval-based infection probability function $$\hat{f}_{\rho,p}: \mathrm{Int}(\mathbb{R}^+) \to [0,1]$$, where $\rho$ denotes environmental density and $p$ denotes the probability that an individual is a carrier. Its consistency condition over intervals is

$$\hat{f}_{\rho,p}([0,a+b]) = \hat{f}_{\rho,p}([0,a])+\hat{f}_{\rho,p}([0,b]) -\hat{f}_{\rho,p}([0,a])\hat{f}_{\rho,p}([0,b]),$$

which is then used to motivate an exponential segment model. The resulting segment-level infection probability takes the form

$P_i$0

where $P_i$1 is a risk coefficient or hazard parameter determined by density and carrier prevalence, and $P_i$2 is exposure duration.

For routing, the operational segment formula is

$P_i$3

with hazard rate

$P_i$4

Here $P_i$5 and $P_i$6 are environment-specific coefficients, $P_i$7 is air inhalation rate used as activity intensity, $P_i$8 is the expected number of infected individuals in the segment environment, and $P_i$9 is the mean interpersonal distance. The expected infected count is modeled as

$i$0

The paper further assumes an inverse-square distance law in the derivation of infection pressure, and it provides an exact geometric expression for the mean distance between two random points in a rectangular environment. The model therefore combines prevalence, occupancy, geometry, activity, and duration in a single hazard-based segment risk.

This structure places LIPP closer to reliability-style path modeling than to conventional additive routing. A common inferred reformulation is to transform $i$1 by logarithms so that shortest-path solvers can be used, but that reformulation is not explicit in the paper. The paper itself minimizes the route infection probability in its multiplicative form rather than through a derived additive shortest-path objective.

3. Transportation microenvironments and mode-specific risk

The direct LIPP model is mode-specific through coefficients, capacities, dimensions, and activity assumptions. The environment-specific coefficients given are pedestrian $i$2, subway $i$3, BRT $i$4, and city bus $i$5; the car specification is exceptional, because the table omits $i$6 and gives only $i$7, and the routing implementation uses a car formula with no $i$8-term (Mirgalooyebayat et al., 1 Oct 2025).

The model also assigns concrete microenvironment dimensions, capacities, and mean interpersonal distances. The pedestrian environment is $i$9 meters with capacity $n-1$0 and $n-1$1 m; subway is $n-1$2 with capacity $n-1$3 and $n-1$4 m; BRT is $n-1$5 with capacity $n-1$6 and $n-1$7 m; city bus is $n-1$8 with capacity $n-1$9 and $P_n = 1 - \prod_{i=1}^{n-1}(1-P_i).$0 m; car is $P_n = 1 - \prod_{i=1}^{n-1}(1-P_i).$1 with capacity $P_n = 1 - \prod_{i=1}^{n-1}(1-P_i).$2 and $P_n = 1 - \prod_{i=1}^{n-1}(1-P_i).$3 m. Activity levels are parameterized by inhalation rate $P_n = 1 - \prod_{i=1}^{n-1}(1-P_i).$4, with sedentary $P_n = 1 - \prod_{i=1}^{n-1}(1-P_i).$5, light $P_n = 1 - \prod_{i=1}^{n-1}(1-P_i).$6, moderate $P_n = 1 - \prod_{i=1}^{n-1}(1-P_i).$7, and vigorous $P_n = 1 - \prod_{i=1}^{n-1}(1-P_i).$8 L/hr; the case study uses moderate activity for walking and light activity for transit modes.

In the Tehran case study, the paper uses active cases $P_n = 1 - \prod_{i=1}^{n-1}(1-P_i).$9, residents $(1-P_i)$0, and therefore $(1-P_i)$1, corresponding to about $(1-P_i)$2 prevalence. Candidate routes were taken from the routing applications Neshan and Balad for travel from Sadeghiyeh Square to Amirkabir University during the evening peak period $(1-P_i)$3 in September 2021. The study analyzes $(1-P_i)$4 total routes, $(1-P_i)$5 from Neshan and $(1-P_i)$6 from Balad, spanning $(1-P_i)$7 km to $(1-P_i)$8 km and $(1-P_i)$9 minutes to $$\hat{f}_{\rho,p}: \mathrm{Int}(\mathbb{R}^+) \to [0,1]$$0 minutes.

The reported per-hour hazard rates are $$\hat{f}_{\rho,p}: \mathrm{Int}(\mathbb{R}^+) \to [0,1]$$1 for car, $$\hat{f}_{\rho,p}: \mathrm{Int}(\mathbb{R}^+) \to [0,1]$$2 for city bus, $$\hat{f}_{\rho,p}: \mathrm{Int}(\mathbb{R}^+) \to [0,1]$$3 for BRT, $$\hat{f}_{\rho,p}: \mathrm{Int}(\mathbb{R}^+) \to [0,1]$$4 for subway, and $$\hat{f}_{\rho,p}: \mathrm{Int}(\mathbb{R}^+) \to [0,1]$$5 for pedestrian travel. The model-based explanation is likewise explicit: cars have very small $$\hat{f}_{\rho,p}: \mathrm{Int}(\mathbb{R}^+) \to [0,1]$$6 m and poor ventilation; buses have smaller interior size and poorer ventilation than BRT or subway; subways and BRT benefit from larger geometry and better airflow; pedestrian routes are open-air with large spacing. These parameter choices produce the route-level conclusion that mixed pedestrian–subway/BRT routes are safer than car or city-bus routes.

The key case-study findings are numerically specific. The lowest-risk route has $$\hat{f}_{\rho,p}: \mathrm{Int}(\mathbb{R}^+) \to [0,1]$$7 infection probability and is described as a pedestrian–subway combination matching route R10. Car routes have $$\hat{f}_{\rho,p}: \mathrm{Int}(\mathbb{R}^+) \to [0,1]$$8–$$\hat{f}_{\rho,p}: \mathrm{Int}(\mathbb{R}^+) \to [0,1]$$9, city bus routes $\rho$0–$\rho$1, and pedestrian-only routes $\rho$2–$\rho$3. The results also state that R4 and R10, which combine subway/BRT/pedestrian segments, have the lowest transmission risks. One common misconception addressed by these results is that private car travel is automatically safest during a pandemic; under this model, it is not.

4. Algorithmic realization and graph-theoretic status

The paper that introduces LIPP is stronger on route-risk modeling than on formal combinatorial optimization. It does not provide a theorem proving correctness of a shortest-path algorithm, formal graph notation such as $\rho$4, or an explicit statement that the problem is solved by Dijkstra’s algorithm or any other label-setting method. Instead, it gives an implementation workflow based on route decomposition, segment parameterization, segment-risk computation, route-risk aggregation, and ranking of candidate routes supplied by routing applications (Mirgalooyebayat et al., 1 Oct 2025).

The operational procedure is: split a complete journey into segments and label each segment by environment type; assign expected infected individuals, mean distance, inhalation rate, exposure duration, and time-of-day adjustments; compute each segment’s infection probability $\rho$5; aggregate them as

$\rho$6

then rank candidate routes by total route risk and recommend the lowest-risk route. Complexity is not given asymptotically, but in the implemented setting it is described as proportional to the number of candidate routes times the number of segments per route.

This algorithmic status matters because it places LIPP, in its initial named formulation, closer to path evaluation and ranking than to a fully formalized shortest-path problem over a graph. A common inferred graph-theoretic interpretation is straightforward: if each segment has infection probability $\rho$7, then

$\rho$8

would convert the multiplicative survival objective into an additive path cost. This suggests that standard shortest-path solvers could be used if segment probabilities were known and the independence assumption were retained. That interpretation is mathematically natural, but it is not stated by the paper itself.

5. Relation to edge-probability estimation and temporal contact inference

LIPP depends on how segment or edge infection probabilities are obtained. One upstream approach is the generalized inverse infection model of Bota and Gardner, which studies the problem of estimating unknown edge infection probabilities $\rho$9 on a known graph $p$0 from observations of an infection or diffusion process. The core formulation is

$p$1

and the task is to find $p$2 minimizing $p$3. The framework supports IC, SI, SIR, and SEIR, allows an arbitrary number of observations, and uses Fully Informed Particle Swarm Optimization with a von Neumann neighborhood to search the hypercube $p$4. The same work emphasizes that the inverse problem is often underdetermined: many distinct edge-weight assignments can explain the same observations. This makes it directly relevant to LIPP as an upstream estimator of edge risks, while also showing that those risks may not be uniquely identifiable (Bota et al., 2017).

A second adjacent line of work is temporal contact-graph inference. CLIIP constructs an individual directed graph whose nodes are individuals and whose edges connect people who stayed at the same location for overlapping time intervals; direction runs from the person who left earlier to the person who left later. It then performs incubation-constrained backward tracing and forward tracking, uses multi-layer path information and neighborhood features, and outputs a node-level infection probability or ranking score via LightGBM. The framework is explicitly path-informed, but it does not define a Least Infection Probability Path; instead it ranks individuals, especially asymptomatic carriers, according to graph-derived risk. This suggests a temporal LIPP variant in which time-respecting paths in a directed interaction graph are scored by inferred edge risks, although that path optimization is not part of the original method (Liu et al., 2020).

A third relevant component comes from Bluetooth-based contact tracing. The fog-based infection probability evaluation model defines a per-contact transfer fraction

$p$5

where $p$6 is a binary RSSI gate and $p$7 is a duration-based fraction that saturates at $p$8. For a user $p$9, the contact-based infection probability is accumulated as

$$\hat{f}_{\rho,p}([0,a+b]) = \hat{f}_{\rho,p}([0,a])+\hat{f}_{\rho,p}([0,b]) -\hat{f}_{\rho,p}([0,a])\hat{f}_{\rho,p}([0,b]),$$0

and the total probability adds a symptom term. This is not a path formulation, but it supplies an explicit contact-level transfer mechanism that can be mapped into LIPP-style edge risks under additional assumptions. In that sense it functions as a partial foundation for LIPP in dynamic contact networks rather than as a direct solver of the problem (Mahawar et al., 2021).

6. Technical limitations, misconceptions, and open issues

Several limitations recur across the LIPP literature and its adjacent frameworks. In the direct transportation formulation, the route objective depends on independence across consecutive route segments, static prevalence during the trip, constant or representative interpersonal distance within each microenvironment, inverse-square dependence on distance, and expected infected count based on capacity rather than real-time occupancy. The paper also states that variability in ventilation-related factors such as window usage is omitted through simplifying assumptions, and that additions such as waiting times and line changes remain future extensions even though the implementation says it automatically adds $$\hat{f}_{\rho,p}([0,a+b]) = \hat{f}_{\rho,p}([0,a])+\hat{f}_{\rho,p}([0,b]) -\hat{f}_{\rho,p}([0,a])\hat{f}_{\rho,p}([0,b]),$$1-minute transfer risks at interchange points (Mirgalooyebayat et al., 1 Oct 2025).

A second issue is identifiability. If LIPP uses learned edge infection probabilities rather than directly parameterized mode hazards, then the upstream estimation step can itself be underdetermined. The generalized edge-estimation framework shows exactly this: even simple multi-parent infection equations admit many edge-weight assignments with the same observed node probabilities. The same study also shows that binary observation tasks can achieve ROC AUC $$\hat{f}_{\rho,p}([0,a+b]) = \hat{f}_{\rho,p}([0,a])+\hat{f}_{\rho,p}([0,b]) -\hat{f}_{\rho,p}([0,a])\hat{f}_{\rho,p}([0,b]),$$2 and RMSE $$\hat{f}_{\rho,p}([0,a+b]) = \hat{f}_{\rho,p}([0,a])+\hat{f}_{\rho,p}([0,b]) -\hat{f}_{\rho,p}([0,a])\hat{f}_{\rho,p}([0,b]),$$3 while remaining highly underdetermined. For LIPP, this means that a route computed from one fitted edge-weight assignment need not be stable under another equally good assignment (Bota et al., 2017).

A third issue concerns path semantics in temporal contact graphs. CLIIP is deliberately designed as a path-aware ranking framework rather than as an explicit path-probability calculus. It does not define edge transmission probabilities, a closed-form path aggregation rule, or a source–destination optimization problem. Similarly, the fog-based model uses HMM terminology without a full HMM specification, and it aggregates multiple contact contributions additively, capped at $$\hat{f}_{\rho,p}([0,a+b]) = \hat{f}_{\rho,p}([0,a])+\hat{f}_{\rho,p}([0,b]) -\hat{f}_{\rho,p}([0,a])\hat{f}_{\rho,p}([0,b]),$$4, rather than through the independent-event complement form $$\hat{f}_{\rho,p}([0,a+b]) = \hat{f}_{\rho,p}([0,a])+\hat{f}_{\rho,p}([0,b]) -\hat{f}_{\rho,p}([0,a])\hat{f}_{\rho,p}([0,b]),$$5. These choices are reasonable for ranking or alerting, but they complicate direct translation into an exact LIPP optimization (Liu et al., 2020, Mahawar et al., 2021).

A final misconception is that LIPP is already a mature shortest-path theory. The available work suggests otherwise. The named LIPP formulation gives a clear route-level infection objective and a practical route-ranking method, but not a full graph-optimization treatment. The estimation and contact-tracing papers provide strong building blocks—edge-probability estimation, temporal path tracing, and per-contact transfer factors—but not a single unified framework with uncertainty quantification, formally justified path transforms, and real-time dynamic routing under changing prevalence and occupancy. This suggests that future LIPP research will likely need to combine probabilistic calibration, temporal feasibility, and robust optimization rather than rely on any one existing component alone.