doi: 10.1098/rsos.191504. eCollection 2020 Feb.
Improved susceptible-infectious-susceptible epidemic equations based on uncertainties and autocorrelation functions
Affiliations
- PMID:32257317
- PMCID: PMC7062106
- DOI: 10.1098/rsos.191504
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Improved susceptible-infectious-susceptible epidemic equations based on uncertainties and autocorrelation functions
Gilberto M Nakamura et al. R Soc Open Sci..
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Abstract
Compartmental equations are primary tools in the study of disease spreading processes. They provide accurate predictions for large populations but poor results whenever the integer nature of the number of agents is evident. In the latter instance, uncertainties are relevant factors for pathogen transmission. Starting from the agent-based approach, we investigate the role of uncertainties and autocorrelation functions in the susceptible-infectious-susceptible (SIS) epidemic model, including their relationship with epidemiological variables. We find new differential equations that take uncertainties into account. The findings provide improved equations, offering new insights on disease spreading processes.
Keywords: Monte Carlo; epidemic models; fluctuations; stochastic process.
© 2020 The Authors.
Conflict of interest statement
The authors declare no competing interest.
Figures
Deviations from compartmental predictions. Predicted values ofρeq −ρ versus observedρ with (full circles) and without (cross) corrections. Corrections are related toσ2/ρ, whereσ2 is the variance ofρ. Monte Carlo simulations are performed with 106 samples in the complete graph withN = 50 agents,γ = 1/2 andα = 1. Linear fit (solid line) producesγdata = 0.50(3) andαdata = 1.00(0).
Linear chain. Simulated data for SIS agent-based model withN = 50 agents, in a linear chain with periodic boundary condition. The system features translation symmetry but its low connectivity violates the random-mixing hypothesis. (a) Simulated data agree with predictions obtained from compartmental equations as recovery events dominate the dynamics. (b) Diffusion of the disease in the linear chain creates correlations between agents. The operator formalism and translation symmetry (dashed line) provide an improved prediction for variation rate of the average density of infected,.
Rate of change for the variance in agent-based simulations in finite populations. Simulations are performed over 106 Monte Carlo samples andN = 50 agents. Forward time derivative ofσ2(t) using simulated data (circles), withγ/α = 0.1 and 0.5. The solid line represents equation (3.5).
Finite size effects in the complete graph withγ/α = 1/2. (a) ForN = 20 (dotted lines), the influence of absorbing state drives 〈ρ〉 below the expectedρeq = 1/2, whileσ2(t) increases over time (inset). ForN = 50 (solid line), 〈ρ〉 lies slightly belowρeq, with constantσ2 for larget. (b) Both cases are in agreement with equation (3.4). Simulated data with 106 samples under the same initial condition.
Deviations from Gaussian behaviour. Simulations are performed in the complete graph withN = 50 agents, and 106 samples. The quantity Δ3 − 3〈ρ〉σ2 measures the deviation of the system compared to Gaussian fluctuations. Curves forγ/α = 0.1 and 0.5 imply Δ − 3〈ρ〉σ2 ∼o(σ2/N). This behaviour is not observed forγ/α = 0.9, suggesting that the variance vanishes more rapidly than Δ3 − 3〈ρ〉σ2, in disagreement with Gaussian behaviour. Error bars omitted.
Contributions for |Dρρ(t)/〈ρ〉|2. Simulation results comprehend 106 simulation samples in the complete graph withN = 50. Gaussian fluctuations occur forγ/α = 0.5 (green circles). An exponential decay is observed during the transient. The divergence appears as 〈ρ〉 approachesρeq. Finite size corrections drive 〈ρ(∞)〉 to slightly lower values thanρeq in the steady state. Non-Gaussian fluctuations create an exponential growth during the transient regime forγ/α = 0.9 (black asterisk).
for various ratiosγ/α. Data extracted from numerical simulations withN = 20 agents (106 samples). After a sharp divergence, |Dρρ(t)/〈ρ〉|2 either moves towards a constant value (two lowermost curves,γ/α = 0.3 and 0.4) or increases exponentially.
Change rate ofσ2(t). Simulations withN = 20 agents (106 samples). (a)γ/α = 0.9. Forward derivative data are consistent with predictions using equation (4.4) (solid line). The validity of the approximationDρρ = −D1/〈ρ〉 is restricted to non-Gaussian regime (red dashed line). Inset: 〈ρ(t)〉 quickly deviates from classical predictions of compartmental equation (dashed line). (b)γ/α = 0.5. Non-Gaussian regimes takes a lot longer to start.
Evolution ofDρρ with 〈ρ〉−1. Data are colour coded with time. The region inside the rectangle demarks the time interval corresponding to the transition between distinct fluctuation regimes. The linear relationship betweenDρρ and 〈ρ〉−1 dictates the system evolution, in the non-Gaussian regime. The dotted line depicts the corresponding line equation that crosses the origin.
Second-order differential equation. Numerical simulations withN = 20 agents,γ/α = 0.9, and 106 samples. The exact formula in equation (4.5) agrees with simulated data for arbitraryt. The approximationDρρ = −D1/〈ρ〉 fails to replicate the data during initial times (thick dashed line). The dotted line represents the compartmental prediction d2ρ/dt2 =α2 (ρeq − 2ρ)(ρeq −ρ)ρ, obtained by taking the derivative of equation (2.1).
Direction field and critical points. The critical points (red circles) in the phase plane (ρ,σ2) are (0, 0), (ρeq, 0), and. The first two critical points are equilibrium points for the usual compartmental equation, while the remaining one lies at the separatrixσ2 =ρ2 (dashed line). Above the separatrix, equations (5.1a) and (5.1b) fail to converge.
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