Modeling epidemics by the lattice Boltzmann method

In this paper, we demonstrate that the lattice Boltzmann method can be successfully adopted to investigate the dynamics of epidemics. Numerical simulations prove the excellent accuracy properties of the approach, which recovers the solution of the popular SIR model. Because spatial effects are naturally accounted for in the lattice Boltzmann formulation, the present scheme appears to be more competitive than traditional solution procedures. Interestingly, it allows us to simulate scenarios characterized by selective lockdown configurations.

Figure 1

Linear stability analysis: Map of the maximum complex modulus of the eigenvalues of $\mathbf{G}$.

Figure 2

Time evolution of the fraction of infected people for different values of $R_0$, i.e., 1.5 (top), 3 (center), and 5 (bottom), and $\mathcal{I}(t=0)$, i.e., $0.1\%$ (solid line and squares), $1\%$ (dash-dotted line and circles), and $10\%$ (dotted line and triangles). Lines and symbols correspond to results obtained by the SIR model [see Eq. (1)] and the present LB one, respectively.

Figure 3

Time evolution of the fraction of infected people for different values of $R_0$, i.e., 1.5 (top), 3 (center), and 5 (bottom), and $d$, i.e., 0.0005 (blue dash-dotted line), 0.001 (red dotted), and 0.01 (black solid line).

Figure 4

Time evolution of the fraction of infected people at $d^{\mathcal{I}}=0.001$ and $d^{\mathcal{S}}=d^{\mathcal{R}}=0.0005$ (blue dash-dotted line), 0.001 (red dotted line), and 0.1 (black solid line).

Figure 5

Time evolution of the fraction of infected people at $d^{\mathcal{S}}=d^{\mathcal{R}}=0.001$ and $d^{\mathcal{I}}=0.0005$ (blue dash-dotted line), 0.001 (red dotted line), and 0.01 (black solid line).

Figure 6

Uniform diffusivity: Map of the percentage fraction of infected people in Midtown Manhattan on different days.

Figure 7

Nonuniform diffusivity: Map of the percentage fraction of infected people in Midtown Manhattan on different days.

Figure 8

Uniform vs nonuniform diffusivities: Map of the percentage fraction of infected people in Midtown Manhattan on day 12.

Figure 9

Time evolution of the fraction of infected people by uniform (black solid line) and nonuniform (red dotted line) diffusivities.