Scaling of the propagation of epidemics in a system of mobile agents

1 Introduction

• The study models infection spreading in a mobile population using an SIS contact process on particles following Newtonian dynamics.Agents are healthy or infected, can be reinfected after healing, and interact through motion rather than a fixed lattice.Source: §1 · p. 1
• Continuous variation of density produces a continuous range of epidemic critical exponents.The observed behavior is identified as the SIS analogue of stirred percolation models used for epidemics with immunization.Source: §1 · p. 2

2 Model

• Agents are soft disks moving continuously in a square cell with periodic boundary conditions, interacting through a truncated Lennard-Jones potential.The system uses particle number and density to specify the cell and agent population.Source: §2 · p. 2
• Collisions transmit infection from infected to susceptible agents, and infection lasts for a fixed number of time steps before healing.After healing, an agent becomes susceptible again; the infection duration Δtinf is a model parameter.Source: §2 · p. 2
• The simulation updates particle positions, velocities, and infection states using molecular dynamics with cell subdivision and leapfrog integration.Initial conditions use regularly distributed agents with equal speed, random directions, and one central infected agent.Source: §2 · p. 2

Scaling

• FIM fluctuates around a mean after the transient, and that quasi-equilibrium mean increases with infection rate λ.The infection-rate dynamics is expressed through the fraction of agents that heal and become infected.Source: Figure 1 · p. 3, §3 · pp. 3–4
• The dynamics collapse when time is scaled by the collision time τcoll and FIM by its mean-field approximation.The collision time is the inverse collision frequency and depends on agents’ mean velocity.Source: §3 · p. 3, Figure 2 · p. 4
• The critical infection rate is λc = 1, and λ contains the system’s free parameters in the mean-field description.After the transient, the mean infected fraction is obtained by setting its time derivative approximately to zero.Source: §3 · p. 4
• Mean-field predictions agree with FIM data only below density 0.1, whereas increasing density changes the transition curve.The transition is analyzed by averaging post-transient FIM over 500 initial conditions for systems of 1254 agents.Source: Figure 3 · p. 5
• Near criticality, FIM follows FIM ∼(λ−λc)^β, with β decreasing as density increases from 0.05 to 0.46.The reported slopes are β = 1.0, 0.92, 0.738, 0.66, and 0.599 for the listed increasing densities.Source: Figure 3 · p. 5

4 Transition to spreading

• At ρ = 0.2 and T = 1, the transition occurs at λc = 0.935 with β = 0.773.The fraction of infected agents follows FIM ∼(λ − λc)^β near criticality.Source: §4 · p. 5, Figure 4 · p. 6
• For densities below 0.1, the transition agrees with the mean-field approximation, whereas increasing density changes the transition-curve shape.Even in the low-density range ρ ∈ [0.1, 0.2], the critical exponent varies with density.Source: §4 · p. 5
• The analysis estimates additional critical exponents from the time dependence of survival probability, infected-agent number, and spreading distance.These quantities are averaged over approximately 10^3 realizations in systems of approximately 10^4 agents at T = 1 and ρ = 0.2, while avoiding boundary effects at long times.Source: §4 · pp. 6–7, Figure 5 · p. 6
• Critical exponents change continuously with density, spanning mean-field behavior to two-dimensional contact-process exponents.The hyperscaling relation 4δ + 2η = dz is recovered within numerical error.Source: §4 · p. 7
• The density dependence parallels the dependence on flammable-space fraction in stirred-percolation descriptions of epidemic and forest-fire dynamics.Stirred percolation models use random walkers on a changing lattice.Source: §4 · p. 8

5 Power-law distribution of infection times

• For 2 < γ ≤ 3, the epidemic threshold tends to zero because highly connected agents dominate the tail of the spreading curve.The infection spreads only when the mean spreading rate exceeds one.Source: §5 · p. 8
• For γ = 2.4, spreading has an inflection at ρc = 0.06(5) with β = 2.65.Above ρc, the infected fraction follows FIM ∼(ρ −ρc)^β.Source: §5 · p. 8
• For γ = 2.4, the predicted inflection density ρc = 0.0718 agrees with the numerical estimate ρc = 0.06(5).The prediction uses λc = 1, r0 = 2^1/6, and T = 1; both values are obtained from approximately 10^3 realizations with approximately 10^3 agents.Source: §5 · p. 8

• A continuous range of critical exponents emerges with agent density, recovering mean-field behavior at lower densities and two-dimensional contact-process exponents at higher densities.Source: §6 · p. 8
• ρc = 0.065 and β = 2.6(5) characterize the spreading curve for γ = 2.4, with the subcritical tail displayed on a log−log scale.Source: Figure 7 · p. 9
• Power-law infection times make the epidemic threshold zero, but a critical spreading rate remains dependent on the distribution exponent and mean interaction time.Source: §6 · p. 9