Compute the time-varying effective reproduction number Rt

Description

Computes R_t for the heterogeneous-network SEIR-MFG equilibrium using the next-generation spectral radius formula (Theorem 3.6 of Wang, 2026):

R_t = \frac{\beta}{\gamma \langle k \rangle} \sum_k k^2 P(k) n_k^{S*}(t) S_k(t).

This scalar formula holds for uncorrelated networks.

Usage

compute_Rt(result, network, gamma = 1)

Arguments

Value

A numeric vector of length M+1 giving R_t at each time step.

References

Wang, W. (2026). Learning contact policies for SEIR epidemics on networks: A mean-field game approach. doi:10.5281/zenodo.19381052.

Examples

# Simple network setup (fast)
net <- make_poisson_network()


# Full solve then compute Rt (slow)
res <- seir_mfg_het(network = net, sigma = 2, verbose = FALSE)
Rt  <- compute_Rt(res, net)
plot(res$t_grid, Rt, type = "l", xlab = "Time", ylab = "Rt")

Build a truncated Poisson degree distribution

Description

Constructs the degree distribution P(k) for an uncorrelated Markovian network with a truncated Poisson degree distribution, and computes the transmission rate \beta required to achieve a target basic reproduction number R_0 = \beta \langle k^2 \rangle / (\gamma \langle k \rangle).

Usage

make_poisson_network(mu = 8, k_max = 25, R0 = 4, gamma = 1)

Arguments

Value

A list with components:

k_vals

Integer vector of degree values 1, \ldots, k\_max.

PK

Numeric vector of probabilities P(k), summing to 1.

K_bar

Mean degree \langle k \rangle.

K_bar2

Second moment \langle k^2 \rangle.

beta

Transmission rate achieving the target R_0.

R0

The target R_0.

References

Wang, W. (2026). Learning contact policies for SEIR epidemics on networks: A mean-field game approach. doi:10.5281/zenodo.19381052.

Examples

net <- make_poisson_network(mu = 8, k_max = 25, R0 = 4, gamma = 1)
net$beta   # approximately 0.4444
net$K_bar  # approximately 8

Solve the heterogeneous-network SEIR mean-field game

Description

Computes the Nash equilibrium contact policy and epidemic trajectory for the SEIR mean-field game on an uncorrelated heterogeneous network using the iterative forward-backward sweep (FBS) algorithm (Section 6.1 of Wang, 2026).

Usage

seir_mfg_het(network, sigma = 2, gamma = 1, r_I = 50, n_min = 0.1,
             eps = 1, C_E = 0, C_I = 0, S0 = 0.98, E0 = 0.01,
             I0 = 0.01, T_end = 30, dt = 0.05, omega = 0.3,
             tol = 1e-7, max_iter = 2000, verbose = TRUE)

Arguments

Value

A list with components:

S, E, I, R

Matrices of shape K \times (M+1): compartment fractions.

nS

Matrix of shape K \times (M+1): equilibrium effort n_k^{S*}(t).

I_agg

Numeric vector: aggregate infectious fraction \sum_k P(k)I_k(t).

t_grid

Numeric vector of time points.

converged

Logical. Whether FBS converged.

n_iter

Integer. FBS iterations used.

References

Wang, W. (2026). Learning contact policies for SEIR epidemics on networks: A mean-field game approach. doi:10.5281/zenodo.19381052.

Examples

# Simple network setup (fast)
net <- make_poisson_network(mu = 8, k_max = 25, R0 = 4)


# Full FBS solve (slow: ~40 iterations x 25 degree classes)
res <- seir_mfg_het(network = net, sigma = 2, verbose = FALSE)
plot(res$t_grid, res$I_agg, type = "l", col = "red",
     xlab = "Time", ylab = "Aggregate I(t)")

Solve the homogeneous-network SEIR mean-field game

Description

Computes the Nash equilibrium contact policy and epidemic trajectory for the SEIR mean-field game on a homogeneous network where all nodes have degree k and the effective per-contact transmission rate beta_k is fixed.

Usage

seir_mfg_hom(k = 8, sigma = 2, beta_k = 4, gamma = 1, r_I = 50,
             n_min = 0.1, eps = 1, C_E = 0, C_I = 0, S0 = 0.98,
             E0 = 0.01, I0 = 0.01, T_end = 30, dt = 0.05,
             omega = 0.3, tol = 1e-7, max_iter = 2000, verbose = TRUE)

Arguments

Value

A list with components:

S, E, I, R

Numeric vectors of length M+1: compartment fractions.

nS

Numeric vector: equilibrium effort n^{S*}(t).

t_grid

Numeric vector of time points.

converged

Logical.

n_iter

Integer. Iterations used.

References

Wang, W. (2026). Learning contact policies for SEIR epidemics on networks: A mean-field game approach. doi:10.5281/zenodo.19381052.

Examples

# Full FBS solve (slow: ~40 iterations)
res <- seir_mfg_hom(k = 8, sigma = 2, beta_k = 4, verbose = FALSE)
plot(res$t_grid, res$I, type = "l", xlab = "Time", ylab = "I(t)")
lines(res$t_grid, res$nS, col = "red", lty = 2)
legend("topright", c("I(t)", "nS*(t)"),
       col = c("black", "red"), lty = 1:2)

Solve the heterogeneous-network SIR mean-field game

Description

Computes the SIR-MFG Nash equilibrium on an uncorrelated heterogeneous network. This corresponds to the limit \sigma \to \infty of seir_mfg_het (Lemma 5.7 of Wang, 2026). The initial infectious seed absorbs the exposed compartment: I_0^{\mathrm{SIR}} = I_0 + E_0.

Usage

sir_mfg_het(network, gamma = 1, r_I = 50, n_min = 0.1, eps = 1,
            C_I = 0, S0 = 0.98, I0_sir = 0.02, T_end = 30,
            dt = 0.05, omega = 0.3, tol = 1e-7,
            max_iter = 2000, verbose = TRUE)

Arguments

Value

A list with components S, I, R, nS, I_agg, t_grid, converged, n_iter.

References

Wang, W. (2026). Learning contact policies for SEIR epidemics on networks: A mean-field game approach. doi:10.5281/zenodo.19381052.

Examples

# Simple network setup (fast)
net <- make_poisson_network()


# Full FBS solve (slow)
res <- sir_mfg_het(network = net, verbose = FALSE)
plot(res$t_grid, res$I_agg, type = "l", col = "blue",
     xlab = "Time", ylab = "Aggregate I(t)")

Solve the homogeneous-network SIR mean-field game

Description

Computes the SIR-MFG Nash equilibrium on a homogeneous network. The initial infectious seed absorbs the exposed compartment: I_0^{\mathrm{SIR}} = I_0 + E_0.

Usage

sir_mfg_hom(k = 8, beta_k = 4, gamma = 1, r_I = 50, n_min = 0.1,
            eps = 1, C_I = 0, S0 = 0.98, I0_sir = 0.02, T_end = 30,
            dt = 0.05, omega = 0.3, tol = 1e-7,
            max_iter = 2000, verbose = TRUE)

Arguments

Value

A list with components S, I, R, nS, t_grid, converged, n_iter.

References

Wang, W. (2026). Learning contact policies for SEIR epidemics on networks: A mean-field game approach. doi:10.5281/zenodo.19381052.

Examples

# Full FBS solve (slow)
res <- sir_mfg_hom(k = 8, beta_k = 4, verbose = FALSE)
plot(res$t_grid, res$I, type = "l", col = "blue",
     xlab = "Time", ylab = "I(t)")