Rabies in raccoon contact networks

Robert M. Sholl · Kansas State University

Question

How do the infection rate β and the removal rate δ shape a rabies outbreak on a contact network? I simulated stochastic SIR epidemics on a proximity network of 23 raccoons and on a 113-node network rescaled from it with the Orbis 2K method, which keeps the joint degree distribution.

Results

0.1 0.25 0.4 0.55 0.7 0.85 Infection rate β 0.1 0.25 0.4 0.55 0.7 0.85 Removal rate δ 80 98 96 99 100 99 53 82 89 93 99 97 45 74 83 84 87 92 36 61 78 90 70 91 30 52 60 76 82 90 32 37 63 78 73 83 Original network 0.1 0.25 0.4 0.55 0.7 0.85 Infection rate β 0.1 0.25 0.4 0.55 0.7 0.85 85 95 97 98 98 97 58 88 94 93 95 96 41 78 88 91 92 94 28 65 79 88 90 91 21 55 76 79 88 89 20 52 66 80 83 88 Rescaled network 0.2 0.5 1 2 5 Ratio β/δ (log scale) 0% 20% 40% 60% 80% 100% Share of animals infected Final size against β/δ original rescaled
Figure 1. Final epidemic size (% of animals ever infected) on a 6 × 6 grid of infection rate β and removal rate δ, 10 runs per cell, for each network (left, centre), and the same runs against β/δ (right).

Final outbreak size depends on β and δ only through their ratio, and both networks fall on nearly the same curve. Raising removal works as well as cutting transmission, which supports removing or vaccinating infected animals.

0 2 4 6 8 10 Effective spreading rate τ relative to the threshold τc 0% 20% 40% 60% 80% 100% epidemic threshold rescaled (τc = 0.151) original (τc = 0.187) Mean-field SIS steady-state prevalence Prevalence is zero below the threshold and climbs fast above it
Figure 2. Mean-field SIS steady state for both networks. The threshold τc = 1/λmax comes from the largest eigenvalue of each adjacency matrix.

The mean-field epidemic threshold is τc = 1/λmax, the reciprocal of the largest eigenvalue of the adjacency matrix: 0.187 for the observed network and 0.151 for the rescaled one. Below it an SIS epidemic dies out. Above it prevalence climbs quickly, a little more slowly on the rescaled network.

1 2 3 4 5 6 7 8 Season 1 10 100 1,000 population infected Animals (log scale) Twenty runs with a birth pulse each season 2 3 4 5 6 7 8 still active Season the outbreak ended 0 1 2 3 4 5 Runs When outbreaks end
Figure 3. Each line is one run; a line stops when its outbreak ends. With a birth pulse each season and no natural deaths, the population roughly doubles every season, yet most outbreaks die out by the seventh.

With a birth pulse each season and no natural mortality, the population roughly doubles every season. The new susceptibles don’t sustain transmission for long: 15 of the 20 outbreaks ended between the third and seventh season, and 4 were still active after the eighth.

Methods

Epidemics run in continuous time with the Gillespie algorithm, using the GEMF model specification: a susceptible node is infected at rate β for each infected neighbour, and an infected node is removed at rate δ.

In the population model each season adds litters to the network, infection status carries over from one season to the next, and removed animals leave the network. Runs stop after eight seasons, when the population is too large to simulate quickly.

The network metrics, the simulator and the threshold are all written in base R, with no network packages.

The two networks

Observed: 23 animals, 50 contacts Rescaled: 113 animals, 274 contacts
Figure 4. The observed proximity network (left) and the network rescaled from it with the Orbis 2K method (right), which keeps the joint degree distribution. Node size shows the number of contacts.
Network metrics
OriginalRescaled
Nodes23113
Links50274
Density0.19760.0433
Mean degree4.3484.85
Largest degree717
Smallest degree11
Degree assortativity−0.07901−0.4884
Mean clustering0.48990.03974
Mean shortest path2.7433.068
Diameter56
Algebraic connectivity0.23030.5781
Largest eigenvalue, λmax5.3366.627