1. Abrupt Epidemic Spreading
Hans J. Herrmann
Computational Physics
IfB, ETH Zürich, Switzerland
&
Dept. de Física, Univ. Fed. do Ceará
Fortaleza, Brazil
2016 Conference on Complex Systems
Amsterdam, September 19-22, 2016

2. olive trees → square lattice
Epidemics
humans
fish
cows
olive trees → square lattice

3. SIS model for epidemics
= infection rate

4. SIS model for epidemics
directed percolation
infection rate λ
Susceptible
Infected
Susceptible
Δtinf = (MD time-steps)Dt

5. Power-law distribution of Infection Time
Agent model
soft-disks in 2d
interaction potential
collision time
Marta
González
Phys.Rev.Lett. 96, (2006)
Power-law distribution of Infection Time

6. Evolution of epidemics with healing

7. Spreading beyond percolation threshold

8. Epidemy with Global Budget
Lucas Böttcher
Dirk Helbing
Olivia Wooley-Meza
Nuno Araújo
Endogenous resource constraints trigger explosive pandemics
L. Böttcher, O. Wooley-Meza, N.A.M. Araújo, H.J.H., D. Helbing Scientific Reports 5, (2015)

9. Epidemy with Global Budget
Budget-constrained Susceptible-Infected-Susceptible (bSIS) model

10. Epidemy with Global Budget
Evolution of budget b(t):
budget function
( f(b) := Θ(b))
susceptible
infected
i(t) = 1 – s(t)
Mean-field:
Effective infection rate: τ = kp/q

11. Data from schools
Survey interviewing student from 84 schools in US
(Add Health Program)
visualization using „pajec“

12. Epidemy with Global Budget
Time evolution in the epidemic regime:
c = 2

13. Epidemy with Global Budget
Mean-field:
q = 0.8; p = 0.285

14. Epidemy with Global Budget
discontinuous transition
c = 0.833
q = 0.8

15. Epidemy with Global Budget

16. Epidemy with Global Budget
Square lattice
c = 0.833
q = 0.8

17. Imposing the connecting path to a supply center
Lucas Böttcher
Dirk Helbing
Olivia Wooley-Meza
Eric Goles
The infection rate τ is the average
number of secondary cases expected
from a single infected individual.
L. Böttcher, O. Wooley-Meza, E. Goles, D. Helbing, H.J.H., Phys. Rev. E 93, (2016)

18. Requiring a connecting path to the supply center
L. Böttcher, O. Wooley-Meza, E. Goles, D. Helbing, H.J.H., Phys. Rev. E 93, (2016)

19. Imposing the connecting path to a supply center
128 × 128 square lattice; q = 0.4
τc = (1)
p = 0.165
p = 0.3

20. Jump size distribution
finite size scaling
not a clear power-law

21. Transition time to fully infected state
square lattice; q = 0.4

22. Order Parameter
square lattice; q = 0.4

23. Finite Size Scaling of Order Parameter
square lattice; q = 0.4

24. Hysteresis square lattice for different waiting times tf
with long range connections
r =0.5; <k> = 4.99

25. School network
2539 nodes
<k> = 8.24

26. Apollonian network
1096 nodes
<k> = 5.99

27. Metastability in Recovery
Lucas Böttcher
Jan Nagler
Mirko Luković
Shlomo Havlin

28. Model definition
(i) a node spontaneously fails in a time interval dt with probability pdt (internal failure)
(ii) if fewer than or equal to m nearest neighbors of a certain node are active, this node fails due to external causes with probability rdt (external failure)
(iii) spontaneous recovery with probability qdt (internal recovery) or probability q’dt (external recovery)

29. Mean Field Calculation
i(t) = uint(t) + uext(t)
𝐸 𝑘 = 𝑗=0 𝑚 𝑘 𝑘−𝑗 𝑖(𝑡) 𝑘−𝑗 1−𝑖(𝑡) 𝑗

30. Order Parameter in Mean Field
q = q’ = 1
k=1; m = 0

31. Time evolution to a steady state
square lattice
1024 × 1024
p = 0.9
r = 0.95
q = 1.0
q’ = 0.1
m = 1

32. Order parameter
square lattice: 1024 × 1024
p = 0; q’ = 1.0

33. square lattice: 1024 × 1024 with m = 3
Critical Exponents
square lattice: 1024 × 1024 with m = 3
rc = 0.47(1)
p = 0; q’ = 1.0
→ classical exponents of contact process on square lattice

34. Phase Diagram
m = 1
2048 × 2048
q = 1.0, q’ = 0.1

35. Phase switching square lattice 50 × 50 p = 0.1065 r = 0.95 q = 1.0
m = 1
ζ = 0.1

36. Phase switching up
down

37. Absence of Spontaneous Infection

38. Nucleation square lattice 128 × 128 p = 0.05 r = 10 q = 1.0 q’ = 0.1
m = 1
no spontaneous infection

39. Going from a square lattice to a random one
ζ = 0.1
ζ = 1
ζ = 10
q = 1.0
q’ = 0.1
m = 1

40. Phase Diagram regular random graph with k = 10
q = 1.0; q’ = 0.1; m = 4

41. 512 × 512 square lattice ; q = 1.0; q’ = 0.1; m = 1
Hysteresis
512 × 512 square lattice ; q = 1.0; q’ = 0.1; m = 1
r = 0.7
r = 1.0

42. Oscillatory behavior
q’ > q

43. Oscillatory behavior mean field calculation for
p/q = 19/81; q = 0.01; r/q’ = 3125/1296; q’ = 1.0; m=1

44. Oscillatory State regular random graph with k = 10 N = 1000 p = 0.007
q = 0.01
q’ = 1.0
m = 4

45. Summary and Outlook
Global budgets produce total infection (pandemics).
Requiring contact to a supply center produces sudden infection jumps (first order transition) and subsequent pandemics.
If also spontaneous infection occurs metastability is observed (m < k-1).
If recovery from spontaneous infection is slower oscillations are found (q’ > q).

46. Thank you !