Analysis and Monte Carlo simulations of a model for the spread of infectious diseases in heterogeneous metapopulations

We present a study of the continuous-time equations governing the dynamics of a susceptible-infected-susceptible model on heterogeneous metapopulations. These equations have been recently proposed as an alternative formulation for the spread of infectious diseases in metapopulations in a continuous-time framework. Individual-based Monte Carlo simulations of epidemic spread in uncorrelated networks are also performed revealing a good agreement with analytical predictions under the assumption of simultaneous transmission or recovery and migration processes.

Figure 1

(Color online) Eigenvalues of the lower main diagonal block of the Jacobian matrix, labeled by the patch connectivity, for both types of equilibria in uncorrelated networks and nonlimited transmission. The upper half of the picture shows the eigenvalues of this matrix around the (unstable) disease-free equilibrium. These are all real and increase linearly (and becomes positive) with the degree $k$ (see main text for an explanation). The lower half shows the eigenvalues of the same matrix around the (stable) endemic equilibrium. These are all negative and decrease with $k$. The dashed line corresponds to Eq. (21) as a function of $k$. We have taken a metapopulation with scale-free degree distribution $p\left(k\right)\sim k^{-3}$ with $⟨k⟩=6$ and $k_{\text{min}}=3$. The parameter values are ${\rho }^0=16$, $\beta =0.1$, $\mu =20$, and $D_I=D_S=1$, fulfilling condition (10).

Figure 2

(Color online) Prevalence of the infection in nodes of degree $k$ of an uncorrelated scale-free network with nonlimited transmission. The continuous line corresponds to the prevalence ${\rho }_{I,k}^{\ast }/{\rho }_k^{\ast }$ predicted by the model whereas dots are the time average of the Monte Carlo simulations ${\mathcal{P}}_k\left(t\right)$ with its standard error. The mean prevalence per patch $mpp=\sum p\left(k\right){\rho }_{I,k}^{\ast }/{\rho }_k^{\ast }$ is also shown. Parameter values: the average number of individuals per patch ${\rho }^0=16,106$ (top/bottom), the migration rate $D_I=0.1,10$ (left/right), and $\beta =0.1$, $\mu =5$, and $D_S=1$.

Figure 3

(Color online) Prevalence of the infection in nodes of degree $k$ of an uncorrelated scale-free network with limited transmission. See caption of Fig. 2 for details. Here the predicted prevalence is constant and equals $1-\mu /\beta$ across the network. Parameter values: ${\rho }^0=106$, $D_I=0.1$, and $\beta =5$, $\mu =2$, $D_S=1$.

Figure 4

(Color online) Prevalence of the infection in nodes of degree $k$ of an uncorrelated exponential network with nonlimited transmission. See caption of Fig. 2 for details. Parameter values: the average number of individuals per patch ${\rho }^0=16,106$ (top/bottom), the migration rate $D_I=0.1,10$ (left/right), and $\beta =0.6$, $\mu =5$, $D_S=1$.