Network SIS epidemic, individual-based mean field

Susceptible-Infected-Susceptible spreading on a fixed network, the individual-based form of Pastor-Satorras & Vespignani eq. (7), d rho_i / dt = -gamma rho_i + beta (1 - rho_i) sum_j A_ij rho_j, the recovered node returning directly to the susceptible pool. With k_i = sum_j A_ij and Theta_i = sum_j A_ij rho_j / k_i the creation term is beta k_i (1 - rho_i) Theta_i, which is eq. (7) verbatim; PSV set gamma = 1 so that only lambda = beta / gamma matters.

Usage

Sietrt(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 back to susceptible.

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 prevalence), prevalence, rho_final, rho_initial, theta, mean_degree, second_moment, lambda, lambda_c, endemic, n, beta, gamma, t_max, dt, method.

Details

lambda_c is the heterogeneous mean-field epidemic threshold <k> / <k^2> obtained by linearising eqs. (8)-(9) about Theta = 0; the endemic phase is lambda > lambda_c. For a k-regular graph this is 1 / k, and PSV's homogeneous result <k> lambda_c = 1 follows.

References

Pastor-Satorras, R. & Vespignani, A. (2001). Epidemic dynamics and endemic states in complex networks. Physical Review E 63, 066117, eqs. (7)-(10). doi:10.1103/PhysRevE.63.066117

Examples

G <- matrix(c(0, 1, 0, 1, 0, 1, 0, 1, 0), 3, 3, byrow = TRUE)
Sietrt(G, beta = 0.3, gamma = 0.1, initial = c(1, 0, 0))
#> $estimate
#> [1] 0.7420632
#> 
#> $prevalence
#> [1] 0.7083331 0.8095237 0.7083330
#> 
#> $rho_final
#> [1] 0.7420632
#> 
#> $rho_initial
#> [1] 0.3333333
#> 
#> $theta
#> [1] 0.7589283
#> 
#> $mean_degree
#> [1] 1.333333
#> 
#> $second_moment
#> [1] 2
#> 
#> $lambda
#> [1] 3
#> 
#> $lambda_c
#> [1] 0.6666667
#> 
#> $endemic
#> [1] 1
#> 
#> $n
#> [1] 3
#> 
#> $beta
#> [1] 0.3
#> 
#> $gamma
#> [1] 0.1
#> 
#> $t_max
#> [1] 50
#> 
#> $dt
#> [1] 0.01
#> 
#> $method
#> [1] "Network SIS epidemic, individual-based mean field (Pastor-Satorras & Vespignani 2001, eq. 7)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"