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Disease-induced resource constraints can trigger explosive epidemics.

Böttcher L, Woolley-Meza O, Araújo NA, Herrmann HJ, Helbing D - Sci Rep (2015)

Bottom Line: The onset of explosive epidemics is very sudden, exhibiting a discontinuous transition under very general assumptions.We find analytical expressions for the critical cost and the size of the explosive jump in infection levels in terms of the parameters that characterize the spreading process.Our model and results apply beyond epidemics to contagion dynamics that self-induce constraints on recovery, thereby amplifying the spreading process.

View Article: PubMed Central - PubMed

Affiliation: ETH Zurich, Computational Physics for Engineering Materials, CH-8093 Zurich, Switzerland.

ABSTRACT
Advances in mathematical epidemiology have led to a better understanding of the risks posed by epidemic spreading and informed strategies to contain disease spread. However, a challenge that has been overlooked is that, as a disease becomes more prevalent, it can limit the availability of the capital needed to effectively treat those who have fallen ill. Here we use a simple mathematical model to gain insight into the dynamics of an epidemic when the recovery of sick individuals depends on the availability of healing resources that are generated by the healthy population. We find that epidemics spiral out of control into "explosive" spread if the cost of recovery is above a critical cost. This can occur even when the disease would die out without the resource constraint. The onset of explosive epidemics is very sudden, exhibiting a discontinuous transition under very general assumptions. We find analytical expressions for the critical cost and the size of the explosive jump in infection levels in terms of the parameters that characterize the spreading process. Our model and results apply beyond epidemics to contagion dynamics that self-induce constraints on recovery, thereby amplifying the spreading process.

No MeSH data available.


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Discontinuous transition from the low to the high epidemic regime.(a) Asymptotic fraction of infected individuals i(∞) on a square lattice with 10242 sites (we consider a von Neumann neighborhood, which has k = 4 neighbors), for fixed costs c = 0.833, and recovery rate qb = 0.8, as a function of τ (we vary p, and hold k and qb fixed). A discontinuous transition from the low (grey) to the high epidemic regime (red) is observed. The inset shows the approach of i(∞) to τ* = 1.6488 ± 0.0001, confirming the predicted power-law scaling. The black solid line is a guide to the eye, corresponding to (c − c*)β, where β = 0.586 ± 0.014 is the critical exponent of the order parameter for the SIS model on the square lattice. The values of τ* and β where obtained from ref. 39. (b) Same as in (a) but on the friendship network, where the average number of neighbors is 〈k〉 = 8.2355. The inset shows that in the epidemic regime i(∞) ~ (τ − τ*), just as in the mean-field limit of the SIS model32. (Results are averages over 3600 samples.)
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f4: Discontinuous transition from the low to the high epidemic regime.(a) Asymptotic fraction of infected individuals i(∞) on a square lattice with 10242 sites (we consider a von Neumann neighborhood, which has k = 4 neighbors), for fixed costs c = 0.833, and recovery rate qb = 0.8, as a function of τ (we vary p, and hold k and qb fixed). A discontinuous transition from the low (grey) to the high epidemic regime (red) is observed. The inset shows the approach of i(∞) to τ* = 1.6488 ± 0.0001, confirming the predicted power-law scaling. The black solid line is a guide to the eye, corresponding to (c − c*)β, where β = 0.586 ± 0.014 is the critical exponent of the order parameter for the SIS model on the square lattice. The values of τ* and β where obtained from ref. 39. (b) Same as in (a) but on the friendship network, where the average number of neighbors is 〈k〉 = 8.2355. The inset shows that in the epidemic regime i(∞) ~ (τ − τ*), just as in the mean-field limit of the SIS model32. (Results are averages over 3600 samples.)

Mentions: The dynamic change at t*, and the consequent drastic increase in the fraction of infected individuals, is a continuous process in time (see Fig. 2b). However, the transition between the regimes is discontinuous in the model parameters (the healing costs as well as the recovery and infection rates). For example, as illustrated in Fig. 4, the transition is always discontinuous in the basic reproduction number τ. The transition is also discontinuous in the sense that there is a jump, Δi∞, in the infection level (see Fig. 2b). We discuss below how the jump size scales with the model parameters. The discontinuity of this jump in the final infected fraction i(∞) has important implications for the resilience of the health system and the control of disease, since it implies that a minute change in the properties of the disease or average contacts of an individual can abruptly shift the infection levels from a low equilibrium to one that is higher and harder to control.


Disease-induced resource constraints can trigger explosive epidemics.

Böttcher L, Woolley-Meza O, Araújo NA, Herrmann HJ, Helbing D - Sci Rep (2015)

Discontinuous transition from the low to the high epidemic regime.(a) Asymptotic fraction of infected individuals i(∞) on a square lattice with 10242 sites (we consider a von Neumann neighborhood, which has k = 4 neighbors), for fixed costs c = 0.833, and recovery rate qb = 0.8, as a function of τ (we vary p, and hold k and qb fixed). A discontinuous transition from the low (grey) to the high epidemic regime (red) is observed. The inset shows the approach of i(∞) to τ* = 1.6488 ± 0.0001, confirming the predicted power-law scaling. The black solid line is a guide to the eye, corresponding to (c − c*)β, where β = 0.586 ± 0.014 is the critical exponent of the order parameter for the SIS model on the square lattice. The values of τ* and β where obtained from ref. 39. (b) Same as in (a) but on the friendship network, where the average number of neighbors is 〈k〉 = 8.2355. The inset shows that in the epidemic regime i(∞) ~ (τ − τ*), just as in the mean-field limit of the SIS model32. (Results are averages over 3600 samples.)
© Copyright Policy - open-access
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4644972&req=5

f4: Discontinuous transition from the low to the high epidemic regime.(a) Asymptotic fraction of infected individuals i(∞) on a square lattice with 10242 sites (we consider a von Neumann neighborhood, which has k = 4 neighbors), for fixed costs c = 0.833, and recovery rate qb = 0.8, as a function of τ (we vary p, and hold k and qb fixed). A discontinuous transition from the low (grey) to the high epidemic regime (red) is observed. The inset shows the approach of i(∞) to τ* = 1.6488 ± 0.0001, confirming the predicted power-law scaling. The black solid line is a guide to the eye, corresponding to (c − c*)β, where β = 0.586 ± 0.014 is the critical exponent of the order parameter for the SIS model on the square lattice. The values of τ* and β where obtained from ref. 39. (b) Same as in (a) but on the friendship network, where the average number of neighbors is 〈k〉 = 8.2355. The inset shows that in the epidemic regime i(∞) ~ (τ − τ*), just as in the mean-field limit of the SIS model32. (Results are averages over 3600 samples.)
Mentions: The dynamic change at t*, and the consequent drastic increase in the fraction of infected individuals, is a continuous process in time (see Fig. 2b). However, the transition between the regimes is discontinuous in the model parameters (the healing costs as well as the recovery and infection rates). For example, as illustrated in Fig. 4, the transition is always discontinuous in the basic reproduction number τ. The transition is also discontinuous in the sense that there is a jump, Δi∞, in the infection level (see Fig. 2b). We discuss below how the jump size scales with the model parameters. The discontinuity of this jump in the final infected fraction i(∞) has important implications for the resilience of the health system and the control of disease, since it implies that a minute change in the properties of the disease or average contacts of an individual can abruptly shift the infection levels from a low equilibrium to one that is higher and harder to control.

Bottom Line: The onset of explosive epidemics is very sudden, exhibiting a discontinuous transition under very general assumptions.We find analytical expressions for the critical cost and the size of the explosive jump in infection levels in terms of the parameters that characterize the spreading process.Our model and results apply beyond epidemics to contagion dynamics that self-induce constraints on recovery, thereby amplifying the spreading process.

View Article: PubMed Central - PubMed

Affiliation: ETH Zurich, Computational Physics for Engineering Materials, CH-8093 Zurich, Switzerland.

ABSTRACT
Advances in mathematical epidemiology have led to a better understanding of the risks posed by epidemic spreading and informed strategies to contain disease spread. However, a challenge that has been overlooked is that, as a disease becomes more prevalent, it can limit the availability of the capital needed to effectively treat those who have fallen ill. Here we use a simple mathematical model to gain insight into the dynamics of an epidemic when the recovery of sick individuals depends on the availability of healing resources that are generated by the healthy population. We find that epidemics spiral out of control into "explosive" spread if the cost of recovery is above a critical cost. This can occur even when the disease would die out without the resource constraint. The onset of explosive epidemics is very sudden, exhibiting a discontinuous transition under very general assumptions. We find analytical expressions for the critical cost and the size of the explosive jump in infection levels in terms of the parameters that characterize the spreading process. Our model and results apply beyond epidemics to contagion dynamics that self-induce constraints on recovery, thereby amplifying the spreading process.

No MeSH data available.


Related in: MedlinePlus