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. 2020 Oct:139:110077.
doi: 10.1016/j.chaos.2020.110077. Epub 2020 Jul 3.

Spreading of infections on random graphs: A percolation-type model for COVID-19

Fabrizio Croccolo  1 , H Eduardo Roman  2
Affiliations

Spreading of infections on random graphs: A percolation-type model for COVID-19

Fabrizio Croccolo et al. Chaos Solitons Fractals. 2020 Oct.

Abstract

We introduce an epidemic spreading model on a network using concepts from percolation theory. The model is motivated by discussing the standard SIR model, with extensions to describe effects of lockdowns within a population. The underlying ideas and behaviour of the lattice model, implemented using the same lockdown scheme as for the SIR scheme, are discussed in detail and illustrated with extensive simulations. A comparison between both models is presented for the case of COVID-19 data from the USA. Both fits to the empirical data are very good, but some differences emerge between the two approaches which indicate the usefulness of having an alternative approach to the widespread SIR model.

Keywords: Critical percolation; Monte Carlo simulations; Random graphs; SIR Model.

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Conflict of interest statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Figures

Fig. 1
Fig. 1
Structure of a population of N individuals before the outbreak of the disease: Susceptible, N−N0, versus non-susceptible or dormant, N0. The latter are assumed to be inaccessible to the infection and being disseminated uniformly within the population. Although this classification is apparently superfluous within a SIR approach, it becomes useful for spreading phenomena on networks.
Fig. 2
Fig. 2
(Upper panel) SIR model for β=1.1,γ=1/30,R0=β/γ=33 and f0=N0/N=0.5, yielding β¯=0.55 and R¯0=16.5. The case f0=N0=0 (dashed lines) is shown for comparison. (Lower panel) SIR model with distancing effects for β¯(t)=β¯(τ0/t)q, with τ0=18 and q=2. The dashed lines represent the solution without lockdowns, and are shown for comparison.
Fig. 3
Fig. 3
The graph of connected individuals used in the simulation. Each site of the (100x100) square lattice represents an individual belonging to one of the four categories: (S) Susceptible (green), (I) Infected (red), (R) Recovered (blue), (D) Dormant (yellow). Panels: (Upper left) Starting configuration (t=0) for f0=0.5 with S=5042,I=1,R=0 and D=4957; (Upper right) t=50; (Lower left) t=60; (Lower right) t=70. The model parameters are: τI=2 (transmission time) and β=1/τI=1/2,τH=30 (healing time) and γ=1/30, yielding R0=β/γ=15, and τL=10 (long range transmission time) yielding βL=1/10. Times are expressed in days. The average node degree for the starting configuration is 〈k〉=2, while additional links are added dynamically as the network evolves in time. The newly created links yield an additional mean degree 〈Δk〉=1110/5043≅0.22, corresponding to an effective mean node degree 〈keff〉=2.22. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 4
Fig. 4
Same as in Fig. 3 in the case of lockdowns: Panels: (Upper left) Starting lockdowns (t=20); (Upper right) t=50; (Lower left) t=60; (Lower right) t=70. The model parameters are: τ0=20 (start of lockdowns) and τL=∞ (no long range transmissions) yielding βL=0. The average node degree for the starting configuration is 〈k〉=2, while additional links are added dynamically until t=τ0. The newly created links yield an additional mean degree 〈Δk〉=156/5043≅0.03, corresponding to an effective mean node degree 〈keff〉=2.03.
Fig. 5
Fig. 5
Time evolution of the normalized SIR functions for the lattice network-spreading model. (Upper panel) No lockdowns (Fig. 3). (Lower panel) Lockdowns: t≥τ0=20 (Fig. 4).
Fig. 6
Fig. 6
Time evolution of COVID-19 in the USA. SIR model (upper panel): N=3106,f0=1/2,β=0.65,γ=1/30,R¯0=9.75; Lockdowns: τ0=36,βD=0.45,q(t)=2.5−t/100,d(t)=0.17[0.3+1/(1+t/60)]. Time lag tLag=29. Network model (lower panel): L=400,N=90106,β=0.65,γ=1/30,R¯0=9.75,βL=1/8,〈k〉=1.994,〈Δk〉=661/79900≅0.008 and 〈keff〉=2.002; Lockdowns: τ0=20,βD=0.55,βLD=1/32,q=2.5,d=0.06. Time lag tLag=45 for cases and tLag=21 for deaths. Data up to May 25, 2020.
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