Network SIR epidemic, individual-based mean field

Susceptible-Infected-Recovered spreading on a fixed network. The individual-based mean-field system, obtained from Pastor-Satorras & Vespignani's eq. (7) by adding an absorbing recovered compartment instead of returning recovered nodes to the susceptible pool: dS_i/dt = -beta S_i sum_j A_ij I_j, dI_i/dt = beta S_i sum_j A_ij I_j - gamma I_i, dR_i/dt = gamma I_i.

Usage

Siaepi(G, beta, gamma, initial, t_max = 50, dt = 0.01)

Arguments

G

Square adjacency matrix, one row/column per node.

beta

Per-edge infection rate.

gamma

Recovery rate into the absorbing R compartment.

initial

Initial infection probability of each node, in [0, 1].

t_max

Integration horizon. Default 50.

dt

Runge-Kutta step. Default 0.01.

Value

List with estimate (final attack rate), S, I, R, attack_rate, prevalence, susceptible, peak_prevalence, peak_time, mean_degree, conservation_error, n, beta, gamma, t_max, dt, method.

Details

With gamma = 0 the R compartment never fills and the system is exactly the SI model of Siepid. The per-node total S_i + I_i + R_i is conserved by construction and is reported as conservation_error so the integration can be checked.

References

Pastor-Satorras, R. & Vespignani, A. (2001). Epidemic dynamics and endemic states in complex networks. Physical Review E 63, 066117, eq. (7). doi:10.1103/PhysRevE.63.066117

Examples

G <- matrix(c(0, 1, 0, 1, 0, 1, 0, 1, 0), 3, 3, byrow = TRUE)
Siaepi(G, beta = 0.3, gamma = 0.1, initial = c(1, 0, 0))
#> $estimate
#> [1] 0.9696585
#> 
#> $S
#> [1] 0.000000000 0.003122758 0.051928496
#> 
#> $I
#> [1] 0.006737947 0.010914712 0.018320633
#> 
#> $R
#> [1] 0.9932621 0.9859625 0.9297509
#> 
#> $attack_rate
#> [1] 0.9696585
#> 
#> $prevalence
#> [1] 0.0119911
#> 
#> $susceptible
#> [1] 0.01835042
#> 
#> $peak_prevalence
#> [1] 0.5113556
#> 
#> $peak_time
#> [1] 5.3
#> 
#> $mean_degree
#> [1] 1.333333
#> 
#> $conservation_error
#> [1] 2.664535e-15
#> 
#> $n
#> [1] 3
#> 
#> $beta
#> [1] 0.3
#> 
#> $gamma
#> [1] 0.1
#> 
#> $t_max
#> [1] 50
#> 
#> $dt
#> [1] 0.01
#> 
#> $method
#> [1] "Network SIR epidemic, individual-based mean field (Pastor-Satorras & Vespignani 2001, eq. 7)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"